Fundamentals of Engineering Mechanics | Leon Petrou | Skillshare

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Fundamentals of Engineering Mechanics

teacher avatar Leon Petrou, Engineer

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Taught by industry leaders & working professionals
Topics include illustration, design, photography, and more

Watch this class and thousands more

Get unlimited access to every class
Taught by industry leaders & working professionals
Topics include illustration, design, photography, and more

Lessons in This Class

56 Lessons (6h 49m)
    • 1. Introduction

      2:12
    • 2. SI Units and Prefixes

      2:05
    • 3. Significant Figures

      7:12
    • 4. SI Units and Significant Figures

      5:44
    • 5. Vector Opperations

      6:13
    • 6. Geometry Recap

      2:43
    • 7. Vector Operations Example 1

      7:40
    • 8. Vector Operations Example 2

      9:29
    • 9. Algebraic Vector Addition

      4:07
    • 10. Algebraic Vector Addition Example

      15:51
    • 11. Cartesian Vectors in Three Dimensions

      4:30
    • 12. Cartesian Vectors in Three Dimensions Example

      10:40
    • 13. Position Vectors and Vectors Directed Along A Line

      3:29
    • 14. Position Vectors and Vectors Directed Along A Line Example

      9:54
    • 15. Dot Product and its Applications

      5:53
    • 16. Dot Product and its Applications Example

      12:27
    • 17. Particle Equilibrium in Two Dimensions

      2:54
    • 18. Equilibrium in Three Dimensions

      4:50
    • 19. Free Body Diagrams

      5:19
    • 20. Free Body Diagrams Example 1

      12:01
    • 21. Free Body Diagrams Example 2

      10:01
    • 22. Particle Equilibrium in Three Dimensions

      10:45
    • 23. Moment of a Force

      4:42
    • 24. Moment of a Force Example

      8:21
    • 25. Moment of a Force Using Cross Product

      5:04
    • 26. Moment of a Force Using Cross Product Example

      9:32
    • 27. How to do the Cross Product on your Calculator

      6:04
    • 28. Moment of a Force About a Point

      8:57
    • 29. Moment of a Force About an Axis or Line

      3:47
    • 30. The Moment of a Couple (Scalar)

      4:02
    • 31. The Moment of a Couple (Scalar) Example

      6:42
    • 32. The Moment of a Couple (Vector)

      3:27
    • 33. The Moment of a Couple (Vector) Example

      7:49
    • 34. Reduction of Force and Couple system to its Simplest Form

      4:05
    • 35. Reduction of Force and Couple system to its Simplest Form Example

      14:42
    • 36. Further Simplification of a Force and Couple System

      2:00
    • 37. Further Simplification of a Force and Couple System Example

      10:27
    • 38. Rigid Body Equilibrium in Two Dimensions

      4:02
    • 39. Rigid Body Equilibrium in Two Dimensions Example

      12:51
    • 40. Two Force Members

      1:20
    • 41. Method of Sections

      3:31
    • 42. Method of Sections Example

      14:27
    • 43. How to Determine if a Member is in Tension or Compression

      3:50
    • 44. Zero Force Members

      4:54
    • 45. Zero Force Members Example

      10:28
    • 46. Frames and Machines

      1:48
    • 47. How to Calculate the Centroid

      6:05
    • 48. Centroid of a Wire

      10:44
    • 49. Reduction of a Simple Distributed Loading

      8:15
    • 50. Reduction of a Simple Distributed Loading Example

      12:23
    • 51. Moments of Inertia

      5:52
    • 52. Moments of Inertia Example

      15:21
    • 53. Internal Forces in Structural Members

      3:46
    • 54. Internal Forces in Structural Members

      9:32
    • 55. Shear Force Diagram Example

      12:21
    • 56. Bending Moment Diagram Example

      12:03
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About This Class

'Fundamentals of Engineering Mechanics' makes complicated mechanics calculations easy!

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This course includes video and text explanations of everything in a first year engineering mechanics module. It includes more than 60 worked through examples with easy-to-understand explanations. 'Fundamentals of Engineering Mechanics' covers five sections:

  1. Particle Equilibrium.
  2. Rigid Body Equilibrium.
  3. Structural Analysis.
  4. Centroids and Inertia.
  5. Internal Forces in Structural Members.

These are the five fundamental chapters in the study of engineering mechanics.

We start from the beginning... First I teach the theory. Then I do an example problem. I explain the problem, the steps I take and why I take them and how to simplify the answer when you get it.

Meet Your Teacher

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Leon Petrou

Engineer

Teacher

Hello, I'm Leon. I graduated with a Bachelor's of Engineering degree with distinction from the University of Pretoria. I specialized in Industrial and Systems Engineering. I have a passion for problem solving and education. I love teaching others and simplifying the learning curve for engineering students around the world.

See full profile

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Transcripts

1. Introduction: welcome to engineering mechanics. My name is Leon, an industrial engineer and are your instructor for this course, along with my partner joining we are the creators off up engineering videos. The reason we created this course is because of the extremely high drop operates for first year engineers. Our vision is to increase the pass rate for first engineering. Students around the world, in this course are pay attention to the small details and step usted procedures to answering the questions that electric often tend to overlook. It isn't that those full details, but that are essential for understanding the core foundation and concept off the work and like off slogan says We don't lecture. We teach the major component of this course include particle equilibrium, rigid body equilibrium, structural analysis center, roids and inertia as well as internal forces. Now I'm going to ask you two questions. Are you stressed about getting the moxie once and second me? Do you want to effectively covered in time modules content in as little time as possible? If your answer to both those questions is yes, then this is the course for you. There are no requirements for prerequisites in order to take this course. We only ask that you come with an open mind and ready to go. So whether you're looking to just pass or get you A, we will provide you with the necessary tools on knowledge. In order to leave you stress free, feel free to look at the course description. You know, on at the end of this course, you will be able Teoh not only walk into that examined confidence, but you'll be able to walk out with the marks he wants. 2. SI Units and Prefixes: in this video, I'll be discussing s i units on its prefixes. This is something you should have done in high school. But I'll just to revise it for you. Nah, quickly. So basically, standard index units or the standard units. Um, well, the standard index units. So, for example, length with S I unit for length is meters for time at seconds for mass, it's kilograms on for force. It's Newton's. So mass is the only, or kilograms is the only s I unit which has a prefix, which is kilo which is this K over here, which has a kilo in front of it. So now, um, if you want to convert these s i units to, um to like exponential or scientific notation, you then use the following. So here we have Giger, Mega Kilo, Hector, Decca Desi Senti Millie Micro Nano PICO Fintor on at So it goes bigger this way and smaller this way. So this is times 10 to the one Thompson to the 2369 And this is times 10 to the minus. One minus two minus 369 12 15 18. So, just as an example, 50 million Newtons is the same as 50 times. 10 to the negative three mutants. So you can team Millie is times 10 to the negative three. So you basically literally replacing a time to 10 to the negative three with a 1,000,000. So, um, this the prefixes are basically just for the sake off writing the final answer in a simple form. So instead of having a time since the *** three year time center negative six, you could just replace it with its appropriate prefix. 3. Significant Figures: in this video, I'll be teaching you guys what you need to know for significant figures. Andi, Ill. Then do some examples, so basically there are five rules when you need to figure out how many significant figures a number has, or if you want to, just leave your final answer in it with a certain number off significant figures. So first rule is all non zero digits are significant, so non zero means like 1234 Any number that's not zero all zeros between significant digits are significance. So if a zero is between a three and a four for examples at 304 then it then the zero is significant in all cases, whether it's before or after the dismal place, all zeros to the left off, the first non zero number or insignificant. So, for example, 01 23 calmer 45 then that zero in the beginning is insignificance all they rose to the right of the decimal point and to the right off all non zero significant digits or significance. Andi, If zeros occur at the end of a whole number, use engineering or scientific notation to avoid these ambiguities, so We will do many examples now now, just to clarify all the rules, all right? And you still have to use rounding off when you're using significant figures. So when you rounding off, if the values above five year round up, if it's below five, you don't round off. If it's equal to five on the preceding digits is even then you don't round off. And then if it's equal to five in the preceding digital, odd, then you're on up. So let's do some examples just to clarify significant difference for you. All right, so basically, harmony, significant digits. Do the following numbers have all right. So in blue, I have the number of significant digits and in red, I have the rules refer t So if it's number one, it's three foot. We using rule number one. So, for example, here we have three comma 14159 This has six significant figures. 123456 Because Almanzora digits are significant. We move on here. This has 1234 five significant digits because if you look at rule number four all they rose to the right of the decimal point and to the right off. All non zero significant digits are significance. So to the right of a non zero number, which is one and to the right off the decimal point. So all these figures here are significant. Here we have 12 significant digits, 0.35 has two significant digits and that's referring to Rule number three. All zeros to the left of the first non zero number are insignificance. So in this case, three is the first non zero number. So all the zeros to the left off that are insignificant. So the 1st 4 zeros are insignificant. All right, so living on here, we have three significant digits. 35 and zero are the significant digits, referring to rules three and four. So three all there is to the left or the first non zero number are insignificant. So it's basically saying those for insignificant and then all zeros the rights off the decimal point and to the right off all non zero significant digits, our significance. So as you can see, zero is on the right of the decimal point because it decimal point is there and zeros afterwards, and it's to the right off all non zero significant digits. So three and five on non zero significant digits and zeros to the right of that. So that's why we have 123 significant digits. Basically, here we have the number 8200 you could write it with either 34 or two significant digit digits. I'm depending on how you want to, Ah, knitter it or what they ask, but basically this is referring to rules four and five, but mainly number five. So if zeros occur at the end of a whole number, use scientific notation to avoid these ambiguities. So, basically, over here, 8200 as it is, you would say it has four significant digits. But if you wanted to have three or two significant digits and can write it as eight comma 20 times 10 to the three, which is the same as 8200 if you want, you want to have to significant digits, which is eight and two. Then you say 8.2 without zero afterwards times 10 to the three, and then obviously that's the same as 8200. If you have two zeros. So it's 8.200 times into the three that has four significant digits cause 8 to 0 and zero Orel significance. So over here I have the wrist off the examples. I'm not going to go through all of them cause we just don't have time for that. But what you can do, you can just practice. You could look at these numbers on, then figure out how many significant digits they have or sorry, significant figures. This one smudged it. But this is actually a three. Over here. Iraq was is 123 significant digits, referring to rules three and four. This one's referring to Rule number one. All right, sorry about that. But so four has one significant digit figure. 0.413 has three, which is 41 and three. Those are the three significant digits. Then, if you want to know which rules it's refer to you, look here. So one and three, then I can just pours around here. Rules number one and three are the rules, which determined how many significant figures they have 4. SI Units and Significant Figures: just a useful tip that you need to know for, um, going about this module is for your intermediate steps You must use for significant digits or figures on your final answer. Must have three significant figures unless otherwise stated. All right, so here we have a few examples with the application off s I units. All right. So, like, for example, a small imm represented times 10 to the negative three Millie, for example. So that's basically what we're going to do in this example. Evaluate the falling answer in S I units. All right. So good. Like this. 16 is first of all, I rot 16 milli grams times 10 to the negative. Three grams times 58 times. 10 to the positive. Three meters. Here we have grams were meters divided by 0.412 times 10 to the positive. Three mutants multiplied by six times 10 to the positive six meters. All right, that's you know, for example, mega. You know, mega is times into the six. So all these, um, prefixes on numbers you need to memorize. All right, So if we were cheap like this oil into our our calculator, we'd get a value off. Eight, 8543689 Common 32 On the end, that's grams meters. Mutiny divided by mutant. There are two these Newton's will in council art. But, um, we want our final answer in three significant digits, right? But still in S i units. So that's with equal 88 comma, five times 10 to the positive six grand meets is all right. So I know this is the same as 88.5 mega grand meters. But if you wrote put the M for mega, you will then be in conflict with the question which states that it wants the on ST S I units. All right, so let's do the sit second example. So here we have 66 millimeters squared, so that's 66 times 10 to the negative three squid and they noticed the meters is also squid meters squared. Causes squared is outside the brackets. Then we have multiplied by 12 squared kilograms. Um, we can't say 12 grand, 12 squared times 10 to actually, we will just leave kilograms kilograms sa units. So it's just talk skip squared kilograms. Murat's just remember that that mass is the only s I unit which has a prefix, which is the kilo. All right. On 12 square, multiplied by seven squared and then the kilograms. In this case, it is also squared. All right, so 66 times 10 to the negative three squared. It's equal to four comma 356 times 10 to the negative three notice are are living in four significant digits for their intermediate steps for significant digits All right, multiplied by 12 squared which is 144 kilograms. So I forgot the meat to square here, multiplied by seven squared which is 49 kilograms squared, sir. We can then multiply all the values for common 356 times 10 to the negative three. Um multiply by 1 44 multiply by 49 equals 30 0.7 meters Squared kilograms Cute pick was meets a square is there and the kilograms multiplied by kilograms squared would give you kilograms cubed and thats three significant digits. Well, figures on that is your final answer 5. Vector Opperations: in this video, we will be disgusting. Victor, Operation on trig laws. Just a small recap on trigger vision from high school. What? So we have what you call it a parallelogram law. So if you have to forces acting from the same 0.4 say and force be, then the result in force will form a a result in force in the same direction as the two forces, but as a resultant So, um, it would then form a parallelogram shape. Um, based on the two forces you have. So the parallelogram law states that the result in force is equal to force a plus force be to the two forces that make up the resultant are then added, and then this is how you draw it. So just make a mental note off how that looks. And then that's how you will be drawing your result in forces when you have to forces acting from the same point. All right, the triangle law. It's similar, except that when you have a head to tail arrangement or forces to force, If a on it starts from this point and then if a to this point in f b the tail off the force of B is from the head or force of a and then act in this direction. And then the result in force is from the tail or force of a to the head or force of being. And then obviously force of a plus force off, be equal to the result in force. Just the thing to note. They like to test thes parallelogram laws and triangle laws in multiple choice. Hard to try to catch you is they turn these arrows around, and then and then they try. I'm test whether you know the application of the parallelogram law or the triangle. So with the triangle Lord, just it'll nice thing to remember is I'm treated as if it's on the convey about. So if you put like, let's say, if you're standing here on the conveyor belt moves this way, then you'll eventually end up at this point and then if you go along, if are you should end up at the same point. But if this if he was acting the other way around, the conveyor belt would move this way, and then you'd stop because the conveyor belts now act in the opposite direction. That's just a little way I like to remember how to do it, but it's pretty easy to remember, So just make a mental note off that as well. All right, Trig laws. If you have a right angle triangle and you have the 90 degree angle there and then you have your angle V, then this would be your Jason side. The Jason side is always the side where your angle is. If you're working with this angle as your theater, then this would be there, Jason Side and this would be the opposite. So I like to work with opposite adjacent and high Potter news instead, off X y and R mainly because if you use X and Y could be deceiving because sometimes the triangles all the other way around on the Cartesian plane and then your ex becomes rewind, why becomes your X? And then you can get very confused when using these formulas. So that's why I like to just stick to opposite the Jasons and high part in use. All right, the if you have a triangle, right, and then they have different lengths, A, B and C and different angles. Ace A, B and C. It can be the same angles that can be different. That can be the same. Length can be different, but basically you use these formulas over here to find unknowns. So you use this formula here. If you know all three lengths off the sides of a B and C on, then you want to figure out an angle. All right. But that could also be calculated using these. Or you could use this formula if you have to length and an angle. But the two links I need to have its corresponding angles. So if you have the lengths A and B, then you need you need angle A. In order to use this formula. If you have sides BNC, you need angle. See, you can't have sides A and B and then use angle A for the formula that won't work. All right. And then here we have beside a the length of a over sign a small a equals B of a sign Be equal, See over science. See All right. Notice a B and C up the numerator refers to the lengths on A B and C Inside. The sign refers to the angles notice the's formulas could also be used if was flipped around. So sign a over A is equal to sign the Overbey, which is equal to sign. See, oversee. So you had used these formulas when you know two links on one angle. Well, no. Two angles on one length. So that's basically the theory you need to know. For trigonometry, this is going to be used everywhere in this module, especially these things are very, very common. In the next video, we will be doing a recap off geometry because geometry is also highly used in this marginal . 6. Geometry Recap: In this video, we will be doing a recap on high school geometry. It is highly used in this module, so that's why I thought it would be appropriate to just revise it quickly. All right, so if we have a line perpendicular on another line, it is then 90 degrees forms a 90 degree angle, and that's represented by this little square at the corner. All right, a line that's just straight forms a 180 degrees on angle. So if this is origin here, it's 90 degrees by the origin. This is the origin. Here it's 180 degrees. All the angles within a triangle at up to 180 degrees. Offer plus Peter plus gamma equal to 1 80 over angles in a quad, which is a four sided shape. All the angles at up to 360 degrees on then, of course, this I don't know how I was taught to you, but they used to tell us maths is fun if you in. All right, so that's how you'd remember it. But basically, um, if you have, like, an EF shape and then those two lines are parallel, then offer and beater off equals. If that angle vase offer and that angle days beater, it's equal. That's called corresponding angles. Then let me just zoom in a bit for you guys. If we have these two lines that are parallel and it forms a U shape, then offer plus beater is equal to 180 degrees. That's called co interior angles on. Then if it forms, is dead shape and in those two lines off the said or the end, Ah, parallel in those two angles are equal. Offer is equal to beater, and that's called alternating angles in this module is not necessary to know what they called. Um, I just did it for the Sekoff explaining purposes, but these are very highly used in this module, so make sure you know what I just explained. There 7. Vector Operations Example 1: a crime needs to be towed by two metal cables. A result in force along the positive X access is 888 mutants. Andi. Then it costs us to calculate force a force, be on theater. All right, so just in theory, you need to know the it will be a minimum. All right. When if a And if he or perpendicular. So that would mean it creates an angle of 90 degrees between them. But if we could just stroll a redraw these forces using the parallelogram rule, it would look something like this. So here we have the force resultant at 888 Mu Tim's here. We have forced air, and here we have four speed. You know, the angle here is 40 degrees. That'll here his theater. All right, so 90 degrees, minus 40 degrees is equal to 50 degrees. All right, and then that is the value off theatre. 50 degrees over there. All right, now, to solve this, we could use Trig, so we know that that's 90 degrees. All right, so that means this is 90 degrees, and that's 90 degrees. So, basically it's a square, right? This forms of square. Um, but we could redraw this force a all right, and then force be all right. So this is this forces just fish. I'm shifted along this line here on, then. Here we have the force. Resultant. All right. And theater. We have a B here as 50 degrees. All right, so now, um, we could look here, and that line is alternating. So then that means that 40 degrees and that angle vase equals So means this is 40 degrees here, and then this is 90 degrees quits the corner off the square over there. So it's 90 degrees. So now we have all the angles on the inside of this triangle. Uh, we could literally so fis either by using this or with all this, right? So I'll just solve it using these. All right, So if our is 888 all right, so we want to calculate f b. We want to calculate if b divided by signs. So this is if b over here and in the angle opposite it is the 50 degrees said sign 50 degrees equals 888 divided by sine off 90. All right. And in that is equal to sign of 19 is equal to one. All right, so basically, that is equal to 888 sign 50. And therefore, if B is equal to 888 signed 50 which is 618 mutants. That's three significant figures. All right, let me just show you what would have happened if you calculated using the trig laws. All right. First you choose an angle feta, and then you make that side you adjacent side. So let's work. Worth, um, this side here, our adjacent this side here opposite and then decide here will be all high part in use. All right, so we want to calculate the f B, which is the Jason side. All right, so Jason's over high part in use is equal to cause. All right, Are you over age? Use the cost, Peter. And then obviously the theater would be the 40 degrees, because that's aside off the adjacent. All right, So a. The Jasons is the same as F b. Age is the same as the force resultant, which is 888 cause 14. All right, then, if b cause 40 multiplied by 888. Then give you an answer off cause 14 times 888 680 and you'll notice that it gives you the same answer by doing it. This method. All right, So if you want to cutlet if a, you know, using the opposite over the high part in use. All right, so here we'll have opposite over hypotenuse signed 40. All right, then that would be signed or 40 multiplied by 888 on the calculator would been given on so off. 571 mutants. Right on then. That is the answer to the question we have the on the very foot. If a is the on feta. Andi, that is what the question oss for. All right, So there is another way off doing this, which will learn late in the course. It's using equilibrium equations with the some of the forces in the X, equal to some of the forces in the why and all that. But that is how you go about doing it. Using the parallelogram on triangle lol 8. Vector Operations Example 2: Hey, guys. So before I do this example, I just like to stake that there isn't more than one way to do a problem on. That's probably why so many students struggle with mechanics. And that's because it is not once it way to solve a problem. All right, so the way I'm going to do this is not the anyway you can solve. This problem is other ways you can go about doing it, but this is how I would do it. The question states. Three forces act on a point, and all three forces create a result in force off 400 Newtons. Two of the cables are subject to known forces determined the direction theater off the third cable such that the magnitude off force F is added to minimum. All forces act on the X Y plane. All right, so in order for the force, if to be at its minimum, it would have to be along the same direction as the result in force. All right, so that's one thing you need to pick up from this question when it is a target minimum that this force needs to be along the same direction as this 400 newton result in force. All right, so that does not mean it is equal to 400. That just means if is a part off this 400 Newtons. All right, so let's go, Andi. So for if all right, so the best way Teoh, go barking. This would be using the triangle or the parallelogram law. So if you would read or that we have the 300 Newton force, which is that one there. All right, then. This 100 Newtons, we just shifted up along this line. All right, 100 Newtons. And then we have force if in this direction. All right, so now this makes an angle with the horizontal. She just threw a solid line. So this is the why act the X axis and this is the Y axis. And then here in between we have the angle. Feta! All right, here we have the angle 40. All right, Andi, if this rope bristol here being parallel to this line, this angle here is equal to 20 degrees. So knowing about this angle and this angle are both parallel. That means you can use alternating angles. And that means this angle here is equal to theater, plus 20 degrees. All right, this angle over here is equal to 50 minus theater. Why is it equal to 50 minus teeter? Because here we have 90 degrees. And then the angle between here and here is 50 quid. You say 90 minus this. 40 degrees. So there in the distance or sorry, the angle between here and here would be 50 minus theater. All right, so then you are left with an angle over here. It's just call it beater for now. All right? So in order to solve for beater, we would thin go on and some the angles in a triangle to equal 1 80 So beater plus theater plus twin e plus 50 minus. Tita is equal to 180 degrees. Right? Some of the angles in a triangle equals 1 80 you'll notice the theaters cancel out. All right. And then you would be left with beater equals 110 degrees when you take that over. All right, So now that we have that, let me just redraw that quickly. So you have the 300 new tunes. 100 you have. If all right, and we have 110 degrees. All right, so now we want to find if the which is the magnitude of force that this cable if is applying. All right. But if we calculate if using what we have here, that will give us the resultant off the 300 and 100 Newton forces, so that will give us a resultant off these two forces. And you still need to add the force of this to get the resultant off 400. All right, So And to calculate f over here, we can then use are formulas. Here it is and can then use this formula. All right, when you have two lengths and an angle, all right, so I'll just leave that on the screen. So want to calculate if so, f is equal to the square, root off a squared? Let's call this a and this be a squared, which is 300 squared, all right, plus 100 squid minus two times 300 times, 100 times cause of 110 degrees. And then that will give you an F value off 347. Comet two mutants Notice how it's four significant figures because it's an intermediate step. All right, so we could actually call this f prime, because that's not actually the value off if or we just calculated is a resultant off thes two cables, right. To calculate the value of this force, you then take the total result in force, which is given in the question as 400 Newtons. Right? Because it says all three forces create a resultant off 400. So you're then say force is equal to 400 minus f prime. If prime is this value, and then that will give you a value off 52.8 mutants. All right, so that's if calculated. But we still need to calculate Peter over here. All right, so in order to do that, we can use the sign rule. All right, so, looking at this again, we want to calculate this angle over here offer, All right, because if we have offer, offer is equal to deter plus 20 so we can equate offer to theater plus 20 on we can, then soul four feet. Er All right, so let's do this. You to use the sign. It'll sign off. Offer divided by 300 equals. Sign off. Now we will use the side. We know so if but there's actually f prime. All right, sign off. 110 degrees, divided by 347.2. All right. And this will give you an answer off is equal to 54.29 Well, actually, if we b 54.3 Zira all right. For significant digits, we can then equate offer to Peter plus 20 because, um, we call this value offer. All right? It's office equal to theater plus 20. Therefore, theater is equal to offer minus 20 54.30 minus 20 is equal to 34 point three degrees. And that is your final answer for theater. And that is your final answer. For if 9. Algebraic Vector Addition: Hey, guys, In this video, I'll be discussing Algebraic Victor Edition with rectangular components in a plane. So just something to note that rectangular components means the same thing as a Cartesian victor. So if you have a Cartesian plane, x and Y axes on, you have a force, all right. We could call this a result in force or just the force that could be broken up into both its X and Y components. So it will be broken up into X components, right? And your draught until the X points off the head off. The result in force on then the why force would act till the head off the resultant. All right. So you could call this if why and that if x All right, so in the positive X direction you label that positive I with 1/2 on top of it on in the negative exit would be negative. I hat here, we have positive Jay hat on Dhere. We have negative jihad in the negative y direction. So always remember, eyes refer to X on Jay's refer to why later you'll also learn that you have is a access coming out. So it's pointing out the access which is makes it three dimensions. And then that would then be K. So you'd have I for X J for y and K for is it all right? So the Cartesian victor, if if you want to express it in rectangular components in your right f x the magnitude off off this force in the X direction I kept, plus the magnitude of the force in the Y direction if why multiplied by J. Cap. So this I and J basically tells you what direction that magnitude is acting. All right, So if we do it in a little example here to find the force resultant add the i components and add the J components. So here we have three forces acting on a Cartesian plane. 4th 1 forced to and forced three. So to find the result in force, we will add if one if two and a three. So you're basically just add all these valued minus three I plus four j plus four I plus two j plus three I minus five j So are highlighted all the eyes and all the jays of in you . Then say, minus three plus four. I plus tree that gives you four I and then four J plus two J minus five j gives you one J. Then you put them in brackets. What they usually do is sorry it. So it's four I plus one j in U turns what they usually use. The squiggly brackets didn't use the normal brackets for some reason. So you put it all in the squiggly brackets and then mutants, all right. And then just the way to define a quotation, Victor is you put a hat on top of it. If you ever see a letter, for example, if on its own, then that means it's the magnitude off if. But when you see an F with a little line on top of the if it means that a Cartesian victor , which means it has thes I and J components, all right. In the next video, we will be doing a more complicated example with Cartesian victors 10. Algebraic Vector Addition Example: expressed. The four forces acting on the post in Cartesian Victor Form determine the magnitude and direction off theatre off Force One. So the magnitude of Force One and the direction which is the angle feta so that the result in force is in the direction, as indicated. If our which is 700 mutants, all right, and that 20 degrees from the horizontal X axis. So just little things, remember, you need to know positive. Why is positive Jay hat positive? It is positive, I hat and in the negatives, um, for the negative for the opposite direction. All right, so let's define it. Starts with a simple ones. Let's start with, if full so, if forced magnitude is equal to 10. Right notice There's no hats because it's a magnitude 10 mutants. But if you want to rot it as a Cartesian victor, you put a line on top of it. Wherefore would equal to 10 and it's acting in the positive Y direction. So that means it's positive Jay hat and can put the squiggly brackets and right in Newton's . So that's if full all right if three is acting downwards. If three has a magnitude off 110 Newtons, the Cartesian victor off a three is equal to minus 110 j. All right, mutinous because it's acting in the negative y direction. So it's negative, J. And then we have if two over here, which is acting in the positive X direction. So if two is equal to 400 Newtons and then the Cartesian victor, if two is equal to 400 I hat and you tense on, then if one if, if one is the trickier one, all right, because we haven't unknown. So if you have a triangle, all right, this is if one all right, it's then broken up into its X component if x one and it's why components if Why one? All right, so you know the angle here is Dita so we can venues. Trigonometry. We know this is the adjacent side over here, you know, this is the opposite side, and you know, this is the high part in use. So to find the adjacent we use sign Andi, or we use cause. So basically this angle, this is a Jason's opposite high part in use on. We want to find a I want to find a over age which is equal to cause theater. So a which is equal to if x one is equal to cause theater multiplied by hype. What news? Which is if one All right, then, to find if why now we want to find the opposite. I'm sorry. Opposite over high part in U equals sine theater party news we noticed if one the opposite side is if y one equal to sign theater if one All right. So, no, we can write if one as a Cartesian victor, if one is equal to they fixed one which is equal to cause, Peter, if one acting in the positive i direction. All right, and then plus if Why one which is sign feta if one acting in the positive Jay direction. Ah, that is all. And mutants. So now that we have that we have the values off. If 123 and four pull in Cartesian victors, so we have a four ef three ef two. And if one oil express and Cartesian vectors Now we need to find if our as a Cartesian victor. So once again same thing force resultant. And here we are, 20 degrees here we have a 90 degrees angle sa adjacent opposite High Potter news When you find the adjacent side we used to cause. So if our is the high part in use Onda we want to find the Chasen's over High Potter news. So used cause theater so that adjacent side, which is, if our in the X direction equal to cause 20 times the result in force If our which is equal to 700 Newtons given in the question Look here 700 Newtons. All right, And then if all why would equal to sign off 20 700 Newtons? So if he would write out the result in force as a Cartesian victor, that would be caused 2700 I plus sign 2700 j because it's acting the positive I and positive Jay direction. And don't forget your units Mutants. All right, so we know that when you add or to find the result in force as mentioned in the previous video, he add, the are components on add the J components. So if you could write out force resultant is equal to if one plus if to plus if three plus if for all right? No. You'll notice how we have are unknowns theater. And if one which means we have two unknowns, which means we would need two equations. We're right. So let's go about solving this force Resultant is equal to that cause Twin 27 times 700 I plus Sline 20 time 700 j That is all equal to if one which is cause Vita if one I plus sign deter if one j plus the value for if three which is actually minus 110 j on the plus if full which is plus 10 j and then if two we left out of two, which is plus 400 I all right? So now we have or how values equated. So here we have the eyes. I like to circle the eyes or just highlight them in a different color just so I can keep track of all the values. I mean, you create a with the eyes, all right. It's, uh we would then have Krauze 2700 course 20 times, 700. Hi. Which is equal to cause Peter if one plus 400 all right, that actually wouldn't solve anything. So how you actually have to go about doing this is simply fine. Both sides off the equations. Let me just think about this equals 400. All right, All right. So basically, you have two equations and two unknowns, so you can then equate them. All right, so here, we'll have caused 20. I'm just going to simplify this so I don't have to keep writing that out. 6 57.8 Minus 400. Divided by cause feta. And that equals if one that's what happens when we equate all the I values no to he creates all the J values. You know, We have a J A, J A, J and J. We have signed 20 time 700 which is to 39 0.4. That's equal to sine theta minus 110 plus 10. I lived out the before in here, So if one Santita All right, so now we can So, for if one So that's to 39.4. All right, then you take that over. So miners 110 plus 10 is equal to minus 100 you take that over. So it's plus 100 divided by sine theta equals F one. So now we have two values for F one. All right, this is where it gets a bit tricky. So you say, if one is equal to if one writes so then we have 6 57.8 to minus 400 at the top. So we have 257.8 ever cause Peter all right. Equal to 300. 259.4 It is 300 359.4 over. Sign theater, right? I just added that so equated Thies too. So this represents the why, um, this represents the eye or the X side X components. So now we no sign of a cause represents tan. So sign theater over cause Peter is equal to 339.4 over 257.8. All right, So that is equal to 10 theater off 3 39.4 Divide about 2 57.8 1.317 So then that implies that 10 to the minus 11.317 Architect off that value. Right shift 10. 1.3 17 That will give you an answer off 52 point eight degrees. And then that is your answer for feta. So I can't go back to the question and make sure you answered everything. Determine the magnitude and direction off Theatre off if one. All right. So basically, we need feta, and we need f one. We've only calculated Peter. Alright. So, to calculate F one, we were just substitute theater into one of these equations, so we would then have 6 57.8 minus 400 divided by cause off 52.8. Then that gives you an answer off. 400 26 mutants. All right. So just I noticed that the best way to avoid making the stakes with the subject is to write sneaky. If you write neatly, it'll eliminate 80% of your mistakes. Because mechanics is such a tedious subject with the numbers and all the steps you have to follow. Writing neatly will help a lot 11. Cartesian Vectors in Three Dimensions: In this video, we will be discussing Cartesian victors in three dimensions. So if you have a three dimensional Cartesian plane with the X axis, the Y axis and visit access and you have a force, um, as a Cartesian victor all right with the little hat on top, if it can then be broken up into its three components. So it's the X component off the force, the wire component of the force, and there's their components off the force, all right. And then, if it travels along the X direction, it's defined as I had ever travel along the Y direction. It's J hat, and if it travels along is their direction. It's known as K hat. All right, so that's how you would write out the Cartesian victor if the magnitude off the component of the foot force multiplied by then. If why multiplied by J? Then if Zedd looks like if two. But it's if said multiplied by K and that all measured in Newtons. So now if you want to calculate the magnitude off this force so that if with Arthur Little hat on you then use the following formula where you just say if X squared plus F Y spread. Plus, if that squid and your roots all of that, the same would work. If you work in the X Y plane in two dimensions on, you just have X and Y. You don't have said then you would just used that to find the magnitude of the force in order to find the direction off. If use the following formulas All right, the angle between the Cartesian for Victor force on the X axis is offer. That's the angle between it. The angle or the direction between the Cartesian victor and the Y axis is known as beater on the angle between the zip access and the forces known as gamma all rights. In order to calculate this use, these formulas cause offer equals the force. The X component of the force divided by the magnitude off the force you'll see are color coded it. So if X if y if said refer to if X If y if said or rights on, then the blue F over here represents the magnitude of the force calculated with this formula. All right, just another useful formula to know is that cost grade offer plus cause Quaid beater plus course squared gamma equals one. You would use this formula when you have two off the angles and you want to calculate the third, you can then use this formula. All right now, Unit victors, a unit vector is a victor off length one, all right, and it basically defines the direction off the victor that sometimes even called a direction victor. So if you have the three unit vector or the direction Victor multiplied by the magnitude of the force, you will then get the Cartesian vector off the force, right? It's the magnitude multiplied by the direction, and then you will end up with something like that, all right, But this formula here is the same as this year. It's just saying the unit Victor divided by F All right, so it's the same thing, not the unit Vector, the Cartesian victor divided by the magnitude of the force that equals the unit vector. All right, that's what we have in here. And then that is equal to effects of If I Plus if, why over if J. Plus visited over. If K, This is used quite a lot in this section of work, so it's important you understand it. But we will be doing some examples off that in the following videos 12. Cartesian Vectors in Three Dimensions Example: determine the result in force Magnitude on its direction angles if if one is equal to so much killing kittens. So here we have the magnitude of F one, which is four killer Newtons. We have the direction angles, but we don't need to calculate the the Cartesian victor if one because it's given to us. But the Cartesian victor off if two is not given to us. All right. But we have they give us this little triangle 5 to 7, which we can use to calculate the Cartesian vector. So basically it also is to calculate the results in force magnitude. All right, so you know the result in force in Cartesian Victor is equal to force one plus forced to you some all the forces in the in the plane or right? So if one we have and if to we need to calculate the Cartesian victor if to So what I usually like to do is I like to read draw this plane in two dimensions just that easier to work from. So here we have the Y axis and those aid access. All right. It's this plane here that I'm working with and then you'd have a force along those it Why plane? All right on it has little triangle like that on it with that dimensions. 52 and seven. All right, so now what we can do is we can write art the Cartesian victor. So if to is equal to the X over the heart bottom, use 2 to 5/7, All right. Multiplied by five killer mutants in the J direction. Plus, now we work in the positive direction plus 2/7 five Killer Newtons in the que direction that's up because it says it is K. Why is Jay all right? So that is all in mutants? No. You could have calculated this angle here, this angle, feta. And then, um, worked. It aren't using signing cause, but if you'll notice 5/7 5/7 is the adjacent over the high Potter news, which is equal to cause feta. So you could have calculated theater on. Then, if you do calculate, it would be equal to 21.8 degrees. So you'd be left with five Killer Newtons multiplied by cause 21.8 in the J direction plus five killing U turns multiplied by sine 21.8 could not be working with 2/7 Sign the opposite of the hard part in use in the que direction all rights And then that is basically the exact same thing that will both give you the same value. I'm just saying you, the two of you could go about doing it. You could choose which way you prefer. This weighs obviously much quicker and easier. But if you used to this rather just stick to this way, so you avoid making the stakes. So then we plug this into the calculator five times, cause 21.8 No. Four comma 642 j. All right, Plus, then you multiply, but sign 1.857 Okay, Killer mutants, right. It's mutants here because I left the 1000 inside with the five. That's 5000. That's 5000. But when I cook it on the calculator, I rode five instead of 5000. So let's just make sure your units all correct or consistent. All right, so you will notice if two has no. I component only has a J and K, but now we can go ahead and calculate the result in force so forth resultant equal to if one plus if to if one is given in the question a z this over here. So I'm just going to write that out. 2.45 i plus one point full one j minus two point 83 k That's all in Killer Newtons that if one now, we can add if two plus four. Comment. Six Full to J plus 1.857 k All right, so now we can add all the eyes together and addle the jays, so results in force would be equal. Teoh, Are we adding the eyes? Searched 2.4145? It's only one eye adding the Jays. That's 1.41 plus 4.642 That's plus 6.52 J and then adding the K, we have minus 2.83 plus 1.857 and that is equal to 4.687 K killer mutants. All right, but you'll notice the question asked for the result in force magnitude, not the result in force Cartesian victor. So to get the magnitude we would do the falling, The force is equal to you. 2.45 squared plus 6.52 squared plus four 0.687 squid. All right, we simply using this formula here. The magnitude off if is equal to that if excreted plus if y squared plus if is it squid. So that is then equal to to 0.45 squared plus 6.52 squared plus 4.687 squid. All right, The magnitude is then 8.4 Killer mutants. All right, so that's your magnitude off if. But the question also also is to calculate the direction angles. All right, so to do that, we use the following formulas these year cause offer, cause beater and cause gamma. So offer. Because Alfa is equal to the X Coordinate if x if X is this value here? 2.45 rights divided by the magnitude. All right, which we calculated now as eight point 04 All right, then Cosby tha is equal to 1.41 divided by 8.4 and cause gamma is equal to minus 2.83 divided by the magnitude, which is once again 8.4 All right, so we can now plug these into a calculator. So you say shift cause so that our cause off 2.4 5/8 commas era For now, this should give you the value of offer. Offer is 72 points, three degrees. Next one, we have 1.41 That's 70 9.9 degrees, and then we have minus 2.83 and then that's 110 point six. Or if you leave it to three significant digits, triple one degrees on that is your answer for the direction angles off their results and force. 13. Position Vectors and Vectors Directed Along A Line: position. Victor's on Victor's along a line. Now, what is a position, Victor? A position. Victor is a Cartesian victor that is measured in units of length directed between two points in space. So if you have any two points in space or within the X Y z it plain, All right, so here we have point A and point B, if you want to find the Cartesian vector between there, it's called a position vector. All right? So just another thing to note the position Victor is measured in units of length, so that will be either in meters or in millimeters or whatever the question states. All right, now, the formula to calculate a position vector is the following. So if this line this position victims called our hat, then our hats is equal to our hat B minus our hats A which is equal to our hat a B. So something you'll notice in this module. If you're working with a position victor a B, then you say B minus A. If it was our hat B A, which means traveling from B to a. You wouldn't say the courts coordinate our A minus bi, so it always all right. Minus the left. That's something you need to remember. And then he has little application of what I just stated. So the position victor are is XB, or this is our baby. So it's XB minus X a. Why B minus y a zit be mine. Is it a now, these X? Why? And said values inside the brackets refer to the coordinates at the two points in space. All right, And then, as you can see here, it's measured in units of length. In this case, it's meters now. A unit Victor, dozens have any units? All right. It doesn't have units because of formula. Has meters at the top and meters at the bottom. So they just cancel each other art. And you left with no units. So here we have the position vector in Cartesian Victor form at the top of the equation, divided by the magnitude off are. So it's the square root off that squared. Plus that squared. Plus that square. All right. And then that will give you the unit. Victor, Notice this are here. It represents the distance between A and B. If you remember from high school, this is the distance formula between two points. All right, if you want to calculate, calculate the force Victor directed along the line, you then multiply this unit, Victor by the magnitude off the force. So the magnitude of the force multiplied by the unit. Victor gives you the Cartesian victor off the force. All right. And then this would be then measured in mutants. In the next video, we will be doing an example with position vectors. 14. Position Vectors and Vectors Directed Along A Line Example: a rope causes a force off. And this is a Cartesian victor off Force 10 I plus a J minus seven K Killer Newtons on the hook. A. So here we have hook a And then if this force being applied, they're acting in this direction from a to B. All right, so this is right from a to be So then says calculate the X and Y locations off be. So here we have point B, the X And while occasions is whether also to calculate notice that zit off B is at zero, all right, and then also called osos to calculate the zero off a All right, so it's this height off a. The length of the rope is five meters. All right, so what's you would usually do is you look at what's given, and then you look at your formulas and you just solve the problem from there. So if you look at what's given with us, we given a length, were given some positions off the coordinates and you're given the force. If so, looking at this formula, we have the force Victor all right. Directed along a line which is from a teepee which we could then used to calculate the unit vector. Once we have the unit Victor, we can substitute it into this equation where we can then calculate the position Victor are . And once we've calculated the position, Victor are we can substitute are known positions for X A and B Y n B and sit and be, and then we can solve for the unknowns. All right, so let's do this. So starting with Force Victor directed along a line, we know that this is equal to the magnitude of f multiplied by the position Victor. All right, so we want to calculate the unit vector. So that's simply the Cartesian victor divided by the magnitude that is then equal to what we have over here. This force, which is 10 i plus eight j minus seven K Killer Newtons, divided by the magnitude of the force, which you calculate with this 10 squid plus eight squared plus minus seven squared. All right. And that's also in Killer Newton's. So you'll notice the killer Newton's and killer, and you tend to cancel art leaving the unit. Victor Unit Lis, a unit victor, is always unit lists. You can then write you is equal to Tyne. I divided by that which will give you 0.685 to I Farraj, then eight divided by the magnitude of the force, which is equal to 0.5482 J, right, and then minus seven, divided by the magnitude of the force is equal to minus 0.47 96 K And then once again, no units or right. So now that we have calculated the unit vector. All right, which is you Over here. We can substitute it into this equation over here. So you is equal to our over our now we are looking to find the position victor of our because we want to get to this equation over here. All right, so our position vector is equal to the unit. Victor multiplied by the distance between A and B. All right, so that's what that order represents. And we know that distance between and be is equal to five meters. That's the length off the rope are stated in the question. So if we then say the unit Victor multiplied, well, that's gonna look like a cross product just multiplied by five all right, that's will give you five multiplied by 0.656852 That will give you 3.4 to 6 I All right. Five multiplied by 0.5 for a two is equal to 2.741 j five multiplied by minus 0.4786 is equal to minus 2.398 K rights and then a position Victor is always measured in units of length. In this case, it's meters because the length of the rope is five meters. So meters multiplied by no units or result in meters now that are right is equal to our A B , as mentioned earlier because it's from a to B. So when you say from a TBI, you're saying the cordant or B minus the coordinates off A. So if you continue, you can write out the formula for position Victor, which is XB minus X A. Because you're saying B minus eight because of forces going from a to B I plus why b minus y a j plus. Does it be minus zit? A. Okay, all right and meters. So now we have these two equations which represent the same thing. So if they represent the same thing, that's like saying XB minus X A. This value here is equal to this. Value here is equal to 3.4 to 6. All right, no, if we look, that's our diagram. XP is unknown, but X A is equal to 0.5 meters, which is half a meter. So XB it minus 0.5 equals 3.4 to 6. If you add 0.5 to 2, that you'll get a very XB equals 3.93 meters. All right, why be minus why a is equal to 2.741 All right, looking at the diagram we have, why be, um Why be over here, which is unknown, But we do know the value off. Why A and the Y coordinate or a zero because it's it's against, the said explain, which means the wife Hornet's era. So if the why couldn't why a is equal to zero that's equal to zero, that simply means that why be is equal to 2.741 meters then lastly, we have zit be minus zid a equal tu minus two point 398 Well, the rights if we look here, is that be the corn it will be is zero cause it's on the X Y plane. So there's a coordinated zero on, then the is it coordinates of a is unknown, its not given and they're also to calculate that so simply minus zero minus Visit A is equal to minus 2.398 when negative and the negative will cancel each other out. And you left with Zid a equal to 2.398 meters And there you have it. You can go relook at the question. Make sure you wanted everything. Calculate the X and Y locations off. Be on the zit off a length of the rope is five meters. So the x and y off B the X b y b on the off said on. There you have it. We have calculated all that is required 15. Dot Product and its Applications: In this video, I will be discussing the dot product on discussing its applications, as well as how to project, um, a force along a line parallel or perpendicular tubes. So, firstly, the dark product, if you have two position victors, are A A, B and R C, and it has an angle feet between them. You can use this formula the dot product between position victor or a be on our A C says from A to B and from A to C is equal to the magnitude off our A B multiplied by the magnitude of R A C multiplied by co sign off the angle between the two position vectors S O R B and a b r a c. You know how to work out the magnitude. We use the square root off the X squared plus y squared. Plus, that's weird. All right, but to do the dot product, use the following. So here's a little example. If you have a position Victor R. A. B, which is five i plus four j plus one K meters and our A C is three I plus two J plus six k um, meters, and then you want to do the dot product, you simply multiply the I could ordinance plus multiply the J cornets plus multiply the case so it would be five multiplied by three plus four multiplied by two plus one multiplied by six and that equals 29. All right, so you'll notice that the answer off the dot product will always be a value or a magnitude . It will never be a a Cartesian victor. All right, so now, to calculate the force projected along a c So now we want to find, if a c So if you have, um, a line A B all right on a C and the force acts a long line a be right and you want to calculate the force projected along another line. So we know the force acting on a B. But to calculate the force along a sea projected, um, along this line, you would then say the Cartesian victor off this force dot product the unit Victor A. C. All right, and then you'll notice there's no line on here, which means the answer will be a magnitude of value. That's not going to be a Cartesian vector. All right, so the line you want to project it along is the same as the line as the unit victor. And then And that will give you the force along that line. So these two will always be the same. A c A c Um, All right, so now to calculate, to calculate the force projected perpendicular to the line off the unit. Victor, you can use the following formulas. So here we have the line of the unit. Victor. All right, we'll call that, um, the line a see. All right. And then here we have the force applied, which is, if in the Cartesian Victor form on a perpendicular to the universal im you have if perpendicular, and you will see that 90 degrees on the along the line we have forced a c. All right, so if you want to calculate this force, all right. You didn't say force a C plus for a force perpendicular, and then you can changes Iran. You can say first force perpendicular equals the force. Minors force a C etcetera right in the magnitude off the perpendicular force is equal to the magnitude off this force. If multiplied by sine theater and theaters the angle between the unit Victor on the force. All right. Another way to calculate the magnitude off the fourth perpendicular is the square root off . If squid the magnitude of if squared, minus the magnitude off the Cartesian victor projected along the unit Victor line Um, a C squid or right, so that's the magnitude squared. So, um, if a C is the same as if parallel All right, that's what we were doing over here. When you calculate the force parallel to a C. All right, so a c is a force parallel. Andi, these are basically the main formulas you'll be using. It might seem very confusing now at first, because, like, it's very abstract away, you explain it, but when we do an example, it will make more sense. 16. Dot Product and its Applications Example: a force off 300 Newtons acts along the line as indicated in the diagram. So here we have the line of action off the force. 300 Newtons determine both the parallel and perpendicular components off segments BC off the piping. So he basically find the finding the force parallel to B. C. All right, on the force perpendicular BC. So you're going to be using the dot product to project this force along the line off B C, which is the same as parallel to B. C. So, um, in order to do this, we we say the force off B c equal to the Cartesian vector off the force dot the unit vector BC. All right, so if you remember from the previous video to calculate the force projected along a line a C in our case, it's beast. Um, bc but she has called a C. So this forces projected along this line. So then the force across the unit Victor is equal to the force dot the unit vector a C in our case bc quiz off the line, you're projecting it along. All right, so this is how you calculate, um the force BC parallel. Parallel. All right, so this means we have to first define the force as a Cartesian victor as well as defined the unit victor off B. C. So I'll start off by doing the Cartesian victor off this force if all right, So obviously you have to use these angles to find the i j and K components of this force. So I'll just redraw this a bit larger, so it's easier to visualize. So here we have the force. All right, here. We have a 90 degree angle in here. We have a 90 degree angle, then here we have an angle off 20 degrees. So here we have 20 degrees, and here we have an angle off 50 degrees. All right, we know the magnitude of the fourth is 300 Newtons, so then we cannot use trigonometry. So this line here would then be 300 multiplied by cause off 50. This line here is 300 multiplied by sine off 50. Um, we can then use this triangle over here to calculate this on this magnitude. So this one would be, um, the sign off 20 off 20 multiplied by the hype autumn use, which is this in this case is cause 50 times 300 right? And in this side, it would be the cause of 20 because of 20 multiplied by cause of 50 times 300. All right, so, um, we cannot write the magnitude of the force if magnitude of force f is equal to, um, now we want the I component. So if the eye is along the X axis and it's moving this way towards a negative X direction, so that means it is going to be a negative off this value here. So you know that value is 300 signed twin e this value because 50 All right, but it's the negative. All right, then, for the J components, we look at the Y direction, which is this way, and it's moving in the positive Y direction and it's along this line over here. So if we write that, it would be plus 300 cause off 20 cause off. 50 j. We're not on then for the K components. It's going up in the positive que direction, which is this way. Positives it direction, which is plus 300 sign 50. All right, so, um, this is all in, um, units off mutants. Because this forces in mutants. So if we could punch that in a calculator, he would get the following values for the Cartesian victor of the force minus 65.95 I +181 point to J plus 2 29.8 K. Mutants. All rights. Now we have the If not, we still needed coconut. The unit, Victor BC. So, to calculate the unit victor of BC, we need the coordinates off. See, so xy y si on's etc. And we need the coordinates off B. XB. Why be unsaid? C b. All right, So if we look at sea, this point here, the X coordinate is zero. All right, the why coordinates is seven. Because it's seven meters here, and the Z coordinate is this distance up till that point, which is five meters. All right. Looking at the cordant off be its X zero. It's y zero, but it zed is five meters. So if you could write that out, it would look like this. See? Has 075 and b has is there, uh is there, uh, five. All right, So now, because um we are calculating the segment BC that from B to C B C from B to C. And then you should remember that from B to to see means C minus B. All right, so basically, I could write if I could just write that out just to clarify from B to C. Therefore, you say C minus B. All right, so now, to calculate the unit victor off B. C, we use the formula. Are the position victor off B c divided by the magnitude off B. C. So that is then equal to, um So it's C minus B. So zero minus zero reminded zero I the, um c minus B seven minus zero plus seven minus zero J plus five minus five. Okay. All right, then you divide this by the magnitude off zero squid to remind a zero squid plus seven minus zero, which is seven squared plus zero squid. All right, um, with this, you will end up with a value off seven divided by 77 j, divided by seven. Which would just give you one, Jay. So the answer is simply jay, you and Victor off B c is equal to J So now if we were to do the dark product Force B C is equal to if dot UBC writes the if we calculated to be this year all right, this value here and then UBC recalculated to B J. All right, so basically, we dot the i we multiply the I components plus multiply the J component components plus multiply the cape components. So here we have minus 65.95 I off if multiplied by the eye off the unit Victor, which is some p zero plus 1 81.2 multiplied by the J components off the unit Victor, which is one plus two, not +212 Sorry to to 9.8 multiplied by is it or the cake components off the universe to BC , which is also zero. So then dessert times at zero his every times at zero, and you left with 181 mutants. And that is your value BC, All right, and that is the parallel component. All right, so we have not calculated the first part. Determined both the parallel and perpendicular components off segment B B C off the piping . So we have this value. All right. Now, to calculate the perpendicular components, we have the falling formulas we have if perpendicular there have perpendicular there and they so you could use anyone depending on the question, you must just look at what you have on the one that is most appropriate or easiest to use would be the best one for you to use for that problem. So in this case, it would be if perpendicular sorry, if perpendicular is equal to the square root off if squid minus if a sea squid or rights. So now, if a C is the same as if parallel. But we working with BC so it actually BC parallel, which we just calculated to be 181 mutants. The magnitude of the force if is given as 300 Newtons, so he could simply then do this. If perpendicular is equal to the square, root off 300 squared minus 1 81 squid and then if you plug it into your calculator, you'll get a final answer off. 200 59 mutants. That is the value for FBC perpendicular. And there you have your final answers. Okay, 17. Particle Equilibrium in Two Dimensions: In this video, we will be discussing particle equilibrium in two dimensions. So, um let me just explain what concurrent forces are on what complainer forces are. Just so you haven't understanding off the to So a concurrent force also called a linear force, is when the forces who line off action meet at one point. So basically, if you have these three forces, they are all concurrent or Colin AEA at point A. All right, we has co plainer means that if all the forces are acting on the body Andi are in the same plane then the forces or complainer So he has really example If you have the X y plane, the forces are all on the same plane which is the X Y plane. There are all co plainer. But if say you have a force which is facing outwards towards the same direction, then that force would not be complainer With these three forces just some hints for attacking particle equilibrium problems. You can use the brie abbreviation F B D in tests if BD stands for free body diagram, there's a video on free body diagrams. If you just want to have a look at that, then another tip is you must clearly fully label your free body diagram as the first step to answering the question. Why, that is because the market may refuse to mark a question, um, without and accompanied free body diagram. So it's very important that you draw the free body diagram not only because they weren't markets if you don't have it, but it also makes life so much easier for you. If you have a free body diagram to work from, then do not change the free body diagram directions or victors while um doing the question . So basically, if you you get a negative answer, the force chosen acts in the opposite direction to the direction chosen in the free body diagram. All right. And then these are your particle equilibrium equations, which is used to solve these problems. Basically that some of the forces in the X must equal zero and the some of the forces in the why must equal zero Andi, you treat to right as positive for the X direction, and you treat upwards as positive for the Y direction. All right, so in the next video, we will be expanding free body diagrams on. We will then do quite a few examples off to that 18. Equilibrium in Three Dimensions: in this video, we will be doing equilibrium in three dimensions on that is for rigid bodies. So basically, you will get a question. Andi, you will be. There will be either ropes, cables, bearings, beams, basically any mechanical or rigid body where there are forces acting on this party on what you have to do is solve for the unknowns in three dimensions. But before you could do this, you have to have some, um, memorized theory or background knowledge, which is the following. All right, if you have a cable all right, connected to a component some way, um, in the question, then a force act in the same direction of the cable. If you have a beam, we can call this a beam for now, all right, and it leans against a smooth surface in the force. Act perpendicular to that sort for surface. If you have a roller, the forceful act perpendicular to, um, the surface once again that it is really not. If you have a ball and socket, you'll then have an X Y and zed component. A journal bearing will have, ah, if X and if said, as well as a moment X and a moment, said notice. There's no if, Why, For a journal bearing there is, however, and if why andan m y for journal bearing with a square shaft. So in this case we have around Shaft, in this case, we have a square shaft. So when you have the square shaft will have if y and M y a thrust bearing is the same as a general bearing with a square shaft, with the difference that it doesn't have an M. Y all right. So basically by by if one m why we mean along the direction off the pole if X would mean perpendicular to the pole, all right, on, if they would also mean perpendicular to a pole but acting upwards all right, a pen would have. And if said, if, why am wine and said Andi know him? Why? So there's no moment acting in the X direction the hinge has If x wines it. It has an moments in the X direction a moment in the city but does not have an M y. And for a fixed support, it will have all three forces and all three moments, all right, it might be hard to visualize on these drawings. I have. But if you look into a textbook, um, you can I'm not sure which page exactly it is. But you can just flip through your textbook and you will find a summary off these drawings with the definitions Andi, the forces and moments that acts on those certain situations. So it's a lot easier to visualize if you look at your textbook images. All right, Andi. Vein of it. Here we have the equations of equilibrium that you should use. So you know, you obviously know that some of the force in the X Men sequel zero Some of the force in the UAE must equal zero and some of the forces in the third mass equals era. All right, And then we use a formula. Summer moments must equal zero. We need work with the i components. We work with a moment of X equals zero for the J eight. Why? And for the k it zed So and basically it's rather simple. All you need to know is thes rules. You just need to memorize these rules. If you have a journal burying, you must know there's no if wine in Why? If you have a thrust bearing and you know you have if y with knowing why, etcetera. So you just need to memorize those rules on, then apply your equations of equilibrium on. Then the problems for Equilibrium in three Dimension follows the same solving procedure as equilibrium in two dimensions. 19. Free Body Diagrams: in this video, I'll be teaching you everything you need to know for free body diagrams. All right, just know that when you draw a free body diagram, you must always label the free body diagram as a zoo much as possible. All right, you won't get marked on for over labelling, but you'll get marked down for under labeling. So just make sure you have all the details off the magnitude of the forces off the distances off the points. So basically everything you need for free body diagram. When you do a free body diagram, just know that if you have an object or point all right, which is represented as a dot, that's how you draw a free body diagram of the dots. Andi, If the forces acting outwards so it's moving away from the dot, then it his intention. And if the arrows going inwards towards the dot you're basically implying that it's being compressed all right. As mentioned in the previous video, um, if you choose an object, for example, hitting, um in the tension direction, so your state that's going outwards because you can't look at an image and just figure out if it's intentional compression, so you usually have to assume which, whether it's tension or compression. If you state for example, it's tension, and then you calculate the value of that force and then you get a negative answer. It simply means that that force is actually acting in the opposite direction, so it means it is actually being compressed. So that's what I wrote over here. If you get a negative answer as a force, it means you assume the wrong direction to the actual direction off the force. Therefore, the sort of the four simply acts in the opposite direction. All right, we'll do some examples with that springs if you have a spring in the problem. All right. You need to know this formula. Um, the force in the spring or yeah, the force in the spring is equal to the constant off the spring measured in Newtons, divided by meters multiplied by s where s is the elongation length off the spring. So basically, if you have the final length out, all right and you have the initial length all zero, then the change in the length is s So this is equal to old, minus all Zahra all right. And then this little diagram basically represents what I'm trying to explain in these formulas. Over here for weights. All right, if you have this problem, all right, which basically has a rope connecting to await. And then you have to forces acting upwards, which connect to the top of the rope. You could draw a free body diagram. That's a which is the connection point of the forces on the rope. And you could also draw a free body diagram at B, which is on the actual weight. All right, so if you draw the free body diagram at the weights at B, then it would look like this. So that dot represents be or this weight. And then there's a downward force, which is the gravitational pull, which is the 10 kilogram weight multiplied by nine. Common 81 the gravitational acceleration, and thus the downwards force is equal to 98.1 u turns. All right. And because some of the forces in the Y direction need to be equal, you know that the tension upwards is 98.1 mutants. All right, then, if you don't a free body diagram at a Then you would have a downwards force, which is the tension force as 98.1 mutants, which you calculated from this free body diagram. All right. And then and you'll have your force one, and you're forced to also both acting upwards. All right, the equations of equilibrium that you will be using. Um, from this moment forward for all your free body diagram problems are these the some of the force in the Xmas equals error, But some of the forces in the wireless equals era if you have a Cartesian plane with your X and your y there, then verse represent to the writer presenter positive X into the left represents one negative x above represent your positive y and below represents your negative. Why, all right, And just another little hints for pretty systems watching. You know, if if you have a pulley and you need to draw free body diagrams, just know that the tension in the same rope is equal everywhere on that rope. So the tension here is equal to the tension over there. All right. In the next video, we will be doing an example with free body diagrams 20. Free Body Diagrams Example 1: determine the UN stretched length offspring A. C. If a force F P equals 60 Newton's causes an angle off theatre equal steady degrees for equilibrium. Court A B, which is this court from A to B or write A B. Um is 0.3 meters in length. The spring constants is 300 newtons per meters on a B. C has a length off half meter, 0.5 meters. So where do you start with a problem like this? That's the nice thing about these type of problems. It's because you always know where to start, and that is always with a free body diagram. You start drawing your free body diagram. All right, So, um, you would look where most of the forces interlinked and you draw as many free body diagrams as you can. In this case, you'll just have one at this point here, all right, at point A. So we have a force in this direction, which is force A C. Then we have a force in that direction. Force A B, and then we have the downwards force off if P, which is equal to 16 Newton's. All right, So now what we want to do is calculate the on stretched length off a C. All right, So in order to do that, we need to use the spring formula where if a C is equal to okay, ISS, where s is equal to the original length minus the initial length in this initial length is the and stretched length. This is what you are looking for? No, que is given in the question as 300 Newton meters. So we knows 300 and then, you know, we want to know the final length, minus the initial length on Force A C. All right, So, in order to calculate this stretch length, which is Oh, um, we can use geometry, all right, because you know, the length off a B, you know, the link off B c. And we know this angle so we can then use geometry. So let me just read or that for you, we have BC as 0.5 meters. We have an angle of 50 degrees on a B. Has a length, um, off 0.3 meters. All right, so now, to calculate this length of here, we use this formula. All right? This formula when you have two links on an angle, so to calculate this length will cope it out. Oh is equal to 0.5 squid plus 0.3 squid minus two times 0.5 times every 0.3 multiply by cause off city. All right, so then the stretched length is equal to 0.2 0.28 32 meters. All right, so now we have 300 multiplied by 0.283 to minus all Zahra. All right, so we still have the unknown off if a c in order to calculate the UN straight length. So to calculate, if they see we can do this some off the forces, All right in the X on the Y direction. All right, so if we start in the why direction we know this angle here is equal to 30 degrees, all right, because we quite alternating angles. So this angle, they this angle here is equal to deter because off the alternating angles, cause this line that line or parallel All right, so if we choose up as positive and you do the some of the forces in the UAE Sequels era then we have, Which is this Is there Jason side on? Then we would use the opposite to calculate the why. So then it's sign off. 50 multiplied by force A B. All right, um, on. Then we still need to calculate. And the angle over here. All right. In order to to calculate if a c So to do that, we can use geometry. Once again, I'm going to use the sign rule. All right, so the sign looks like this. We'll have with each of the triangle over here. All right, so we know this length is equal to 0.2832 meters. So we want to find art this angle here. Right. Um So what we do is we Let's call this angle offer. All right, then. Sign of Alpha over 0.3 meters is equal to sign off 30 degrees, divided by 0.2832 meats is so we can plug it into a calculator and get the value of offer. So shift sign. Sign off. 50. Divided by 0.2832 Multiply by 0.3 off. The has a value off 51 0.98 degrees. All right, So that means if this angle here is study 1.98 degrees, then it means this angle here is 51.98 degrees. So you can then get back to our some of the forces in the UAE. Andi, once again, we working with the opposite side. So it's sign serving, you say, Plus, if a C sign off 31.9 eight on that minus the downward force that's upwards at upwards and then downwards Force is equal to 60 cents servicing minus 60. I'm just going to say equals 60 mutants. All right, so you'll notice. Here we have one equation and two unknowns. Which is if a b andi, if a c All right, so we need to equations, cause we need to calculate. We need to calculate if a c in order to plug it into this equation. So now we can do the some of the forces in the X Alright, Cheetah, right? As positive. Some of the forces in the X equals zero. All right, so you look here, so lift is negative. So then we have, um, negative cause off city multiplied by if a B then right, is positive. So we have plus cause off 51.98 If a c A b right on, then that is literally equal to zero. Those are all the forces in the X direction, this force and that force. So, um, now we have two equations and two unknowns, so you can solve simultaneously. Or you could just solve it on your calculator. I'll show you how you could do that. You go mode. You kick number five for equations number five. All right. When you click on the first option, number one right, and then you can flag in your values. All right, so here we have signed city equals sign 31.98 equals 60 and we have minus cause off city equals positive cause 51 point 98 and zero. Then you could equal again. And that means if a B is equal to 57.65 mutants and if a C it's equal to 58.86 mutants. All right, so now that we have calculated are two forces, we can come back to this formula. All right, I'll just read, write that. Okay, So we have If a C equals 300 multiplied by Sarah 0.283 to minus on stretch length. If a C, we just calculated as 58.86 All right, then we can just divide by the 300 and he convinced subtract 0.2 832 equals all zero And this one. You should be negative. And then also would then be equal to minus 58.86 over 300 minus 0.2832 All right. And then you have your own strictly on stretched length off 0.87 meters and that is your final answer. 21. Free Body Diagrams Example 2: a 3.6 meter rope passes through to police B and C The mass off box that B is unknown, but the mass off the box A T is the 18 kilograms. Determine the mass off the box that be on the system isn't equilibrium. So now there are three things you need to know. In order to solve this problem. Number one you know that you have to draw the free body diagram at BNC because B is where the unknown force acts. All right, And then the second thing you need to know is that when you use your equilibrium equations because it is a system in equilibrium. But you should assume that anyway because, um, for this module, all the systems will always be in equilibrium, which means that some of the forces in the X is equal to the sum of the forces in the why. All right. And the third is that the tension in the same rope is always equal. So the tension throughout this entire Apia is going to be the same. All right, so you can start off by drawing a free party diagrams. I'll start off doing the free body diagram that see at the police E. So the police see if the start represents the police. We will then have the following. We would have it down its force. All right. Which is a 13 kilograms multiplied by gravity. 9.81 which will give us a value off 100 and 27.5 mutants. All right. So we could give that the name. We can call this tea for tension in the rope. All right, then. We know this forced down this way is the same tension. All right, so we can call that see as well. And then we have an upwards force acting at this poorly by that little string there. So we can call that the force I see. All right, so we cannot do our free body diagram at B. All right on that be we have a tension force upwards. All right? You know, this tension is the same as this tension going down because it's the same road. Let's have attention going upwards here. And then we have the mass off this object going downwards the force due to gravity. So this doctor presents be all rights. We have this force are foods, which is attention, that other force, which is attention. And we have the downwards, which is the mass call, that the mass at be multiplied by gravity, which is 9.81 all rights. Ah, we cannot do our equilibrium equations. So what you'd usually need is these angles that angle they that angle with that angle what much? Whatever you I feel you want to use but in this case will probably just be easier to use the the lengths since this length is given and that link. You can just use Pythagoras to calculate this link here so the length of a B is equal to the length off B C. Because you'll notice that link. The adjacent and the opposite sides are both the same for these two triangles. All right, so if you wanted to calculate a B o B c, we know they will be the same. So a b c equal to the square root off 0.16 squared plus 1.2 squared, which is equal to 1.21 one but meters, which is equal to the length off B C. You could have done this a different way. You could have just used this length, which is 3.6 meters. So you know, the total length off the rope is 3.6 meters. You subtract the length off CD 1.2. So we know 2.4 meters. Is this length over here? All right, then. 2.4 divided by two because you know it's symmetrical is equal to 1.2 meters, which is the same as a value in kwacha. All right, so now we can do our some of the forces by looking here. You can tell that if you do that some of the forces in the X we are just going to have the same force to left and to the right, so it'll will just get a value of t is equal to t that we already know. So if you do the some of the force in the why, we will be able to calculate, um, the mass off B which is unknown. So some of the forces in the Y direction equals zero. So, um, here we have the opposite side, divided by the high party news, right to get the this forced upwards for the components. A B. All right, so that's 1.2 1.2 divided by high Potter news, which is also 1.2 All right, multiplied for attention. Plus. So we just calculated this component that components of this force, and I'm going to calculate this component off this force. So it's the same thing because it's symmetrical, multiplied by t, then minus the mass off, be acting downwards times 9.81 And that'll equals zero some of the force in the Y equals era. So we know the tension, which we calculated area is 1 27.5 Newtons. All right, so, um, we could rewrite this just so for NB So we'll have 1.2 of ah, 1.211 multiplied by attention, which is 1 27 0.5. All right, um, remember, for significant digits, because it's an intermediate step. If you add this and add that, you'll just get two times this so I'll just multiply that by two just hard enough to write the whole thing out. And then that equals the massive be divided by 9.81 All right, so you could plug this into our calculator. Andi, we'll get the falling answer. 1.2 of a 1.211 1 27.5 comes to devoted by 9.81 equals. And then the mass off B is in equal to 25 points. Um, eight kilograms. I remember three significant digits for your final answer. All right, so that is how you calculate the mass of B. Let me just show you what would have happened if he does the some of the forces in the X direction. So if you do this, some of the forces in the X equals zero. Then you would get the adjacent divided by the hype autumn use. All right, so it is minus because now we acting to the left, minus 0.16 divided by I'm the hot parts in use, which is 1.211 All right. Just making sure. Multiplied by attention, the into the right, we have plus 0.16 over 1.211 Public attention equals era, right? Because if you look at our free body diagram, that's the only force acting to the right and that any force acting to the left, which you did find over here on the If you try, figure out what this equation will tell you, it's just going to tell us that T is equal to T. So what's useful about this is if you didn't know that the, um or the forces in a report equal, which is just something theoretical, you should know. If you didn't know that, then you would have maybe defined verse force if one and this force, if to not knowing that there are equal in this equation would have told you that Force One is equal to force to. All right. But that is the final answer over here, where the mass off B is 25.8 kilograms. 22. Particle Equilibrium in Three Dimensions: In this video, we will be doing particle equilibrium in three dimensions. So basically you solve these problems the same way you did particle equilibrium in two dimensions where your first step would be drawing the free body diagram. Second would be using your equations of equilibrium, which is some of the forces Equal X equals error in the X equals zero and some of the force in the Y equals era. But now that we will walk working in three dimensions X y and zed dimensions, we add an additional equilibrium equation, which is the sum of the force in those it equal zero. All right, you can solve these problems using and scaling method. If you want to use vector, then you must just know that if you're working along the positive X axis you use Plus, I hat along the positive y axis. It's plus J hat and along the positives eggs. Is it access? It's plus K hat. All right, so here we have an example. Which tells us the three cable support the 500 muting. Wait. Yeah, we have a 500 meter rates. Determine the force in each of the three cables, all right. So we have our free cables a C, A B and A D. So first step, as always, drawing a free body diagram. So you look where all your force, the acting and where your unknowns or which is that point A To enjoy a free body diagram at a All right. So you could draw your Cartesian plane if you like. All destroyed in a different color here. Just so it's easier for you guys to visualize here. We have visited access. There we have this. The X axis on the Y axis became in his point A on our free body diagram rights. We have acting along this direction. Fourth, a B acting along the X Direction Force A C Andi along that direction. All right? Yeah, Force they and then you will sell about downwards. 500 Gittins. All right, so this set off cables has now Bean rejoined as a free body diagram. We assumed that all the ropes, our intention, all right, if we get a negative value for one of our answers, we simply know that that means we assume their own direction on that force should actually be a compression force. Instead of a tension force. All right, so now we can start off by doing the some off the forces. All right, But before you do that, you need to know the length off the high part news off this rope, which is the length of a D. Um, you need to know the length of a deep because you needed to calculate the X y and zed components off force 80. So to calculate that you will notice that we have the following, Um, we have this dotted line here, right, which is the distance from origin to a. So if you could get that length, we can use Pythagoras with this five meter length to calculate our party news off the length a d. All right, so to get this link here, we simply use path accurate with verse link six meters and this length one meter. We have the adjacent and adjacent and opposite sides to calculate the heart bottom use. So you're being have the falling corporation. Um, we will called. Since this is the origin, we'll just call that 0.0 point So they in length Oh, a is equal to the square root off six squared plus one squared. All right, um, screwed of six quid plus one squared. That is 6.8 three meters. All right. Not to calculate length a D Over here, we can use length O A on the length of five meters to do Pythagoras to calculate 80. So then I d equal to the square it off. Six points there a three squared plus five squared. All right, that is 7.8 74 meters. All right, so we have that. We have this length 80 now. So now we can go along and do our equilibrium equations. So the sun off the forces in the Z direction must equal zero. Raj can start any of them. I'm just doing what? Said, Um So how you do that is way. Look at this diagram, right? And we see which forced her acting in the Z direction. All right. This way on. That would be the component of force A D. In is the direction as well as the force off this weight acting downwards, which is a 500 newtons. So to do that 4th 80 you're right. Force a d multiplied by visit components length five divided by the high part in use, which is 7.874 All right, so he said five divided by this length. All right pick was that will give us the angle. All right, so we have that acting in the positive direction. Upwards positive. The direction minus 500 Newtons, she's acting downwards equals zero. So then the force A D is equal to see the following You can just like the Intrakota Later, for 100 times. 7.8 and four divided by five equals 787.4 medications, which is 787 mutants to three significant digits. All right, so that is the force, um, off a d. All right now, we still need to calculate the force a see and A B all rights. We can't do the some of the forces in the X direction equals zero. All right, so here we have plus force. They see acting in the X direction minus. Now we have this component this that x component of force a D. And you know, there are four a d multiplied by one meter over the high part in use, which is the length. 87.874 equals zero. So not calculate force. Hey, see, we simply say that Force a D, which we previously calculated as 787.4 mutants times 1/7 0.874 That gives you an answer off. 787.4, divided by 7.874 I could give early or 100 mutants force A C. All right, um, and that's so that we get a positive act. I sorry. A positive value. Which means we assumed its intention. And therefore, it is, in fact, intention. If you got a negative answer, we that would mean have assumed the wrong direction on a free body diagram. And it's actually acting that way. But everything is good at the moment, so we can continue. We go. The some of the forces in the UAE must equals era. All right, sir, looking here, we have forced a B in the Y direction. All right, so that's a positive force, a b. All right. And then acting in this direction, we can look here. We have the force a D as a component, acting in a negative Y direction All right, so right. Negative force a D multiplied by its components, which is the y Direction six divided by the high party. News six divided by 7.874 equals era. We know the value. Our force a D, which we have previously calculated, are 787.4. So 787.41 supplied by 6/7 0.874 equals the Force A B All right then Force A B is thus equal to the following 7 87.4 multiply by six divided by 7.874 Andi, have your final answer as 600 Newtons for force, baby. 23. Moment of a Force: in this video, we will be discussing the moment off the force on, then do an example. So, um, this is basically all you need to know for hard to calculate the moment off a force, Um, using the scale of method rightful for Victor's. You'd be using the cross product. But this is how you do it. The scale away. All right, which is seen as easier. All right. So the formula simple. The moment about a point in this case zero is equal to the force multiplied by the perpendicular distance. So now this perpendicular distance represents the distance between that point with a moment is acting or about the point where you want to calculate it on where the forces acting. All right, So this will become appear to you when we do the example to calculate the resultant you some the force and the distances. Andi here we have notation or some sign convention which states that anti clockwise is always positive and clockwise is always negative. So just memorized that you always need to know that anti clockwise positive clockwise is negative. And then, um, you could also use a right and rule. So a moment acts perpendicular to the force at the points where you are calculating the moment. That sounds a bit confusing for now. But basically, if you have a 0.0 on, do you use the right? And also your fingertips represent the direction which the moments traveling, which is anti clockwise in the moment it is perpendicular to that which is coming out the page positive. All right, whereas if it's going the other way around you spending that way or right and then your thumbs pointing downwards, um, into the page. So the US it is negative. All right, So that's what the positive and negative actually mean. Aren't the pages positive into the page is negative? All right, so if we have a little example here, we I would say the moment about 0.0 All right is equal to 100 mutants. The magnitude of the force multiplied by the perpendicular distance between the force. So on the forces acting here in this direction to the perpendicular perpendicular to this force 2.0 is four meters. That's 100 multiplied by four. So then you have your answer as 400 mutant meters, so just note that its mutant multiplied by meter not, um, mutants per meters because a lot of people make that mistake. But it's actually Newton times meters. All right, because obviously your multi plane, the two. So in this case, um, it would be minus 100 times for which is equal to minus 400 meters. Now, how do you know which direction this is acting? Because I won't be. I'm told what direction the moment is acting clockwise or anti clockwise. You have to figure that art by just looking at the picture. So basically, if we calculating the moment of our 0.0, you look at where the forces acting and then you try to figure out in what direction that object would rotate about that points due to that force. So if we have an upward force here, then obviously it would rotate this beam in an anti clockwise rotation. Whereas in this picture, we're pushing downwards would rotate this beam about 0.0 in a clockwise, um, rotation. So that's just something you have to figure out. And by looking at that, you can then add your sign, whether it's positive or negative or right. So in the next video, we will be doing a more complicated example off the moment of 1/4 year. We will have to calculate the result in moments. 24. Moment of a Force Example: determine the result in moments off the five forces acting on the road about point so point . Oh, is this point here, which is the center off this bar? All right, So, um, we have in blue the dimensions. Will the, um the distances between the forces and 0.0 on then we have an orange, the actual forces measured in mutants. So to calculate the resultant moment, m resultant is equal to the sum off the forces multiplied by the distances. All right, So, um, like I mentioned in the previous video, be sure to remember that this distance represents the perpendicular distance. That's very important. So about perpendicular distance, I mean the distance between points on the force, the perpendicular distance between those. So if we start to do this with right, some off some off the moments of our points Oh, all right, which is some off the forces multiplied by the distance, which is equal to the resulting moments. We will then have, um, six meters right, multiply by three meters, so you'll have six multiplied by three force times distance. That's the perpendicular distance between oh in the force. And then to get the sign you notice little rotated clockwise. It will rotate of our point of a clockwise if our forces applied over there, which means it is negative. Remember the sign. Convention states. Clockwise is negative. Anti clockwise is positive. Then we do this force here. Four Newtons on the perpendicular between distance between the four Newtons on Dpoint. Oh, is three meters all right, because if this line of action of the force extends to hear the distance between that, that line of action and points by is three meters. We have three meters, then you look how this will rotate point. Oh, if it's because it's above point. Oh, and it's pushing that way. It would then rotate a in an anti clockwise rotation, which means it's positive. Then we have the five mutants, all right, but you'll notice there's no perpendicular distance between the five mutants and point because it acts along the same line so the distance is equal to zero. All right, so then this entire thing just equals era. Then we have this to Newton's perpendicular distance between the team Newton's and 0.0 is 10 meters. All right, so you have 10 meters over there and it'll rotate about point, or if you pushing up here on the beam, it will irritate about 0.0.0 in an anti clockwise rotation. So it's positive. And then we just left with one lost force year. But notice there's no perpendicular distance between zero and seven on the seven you 10 force, so you have to break it up into its components. So to break it up into its components, we could simply redraw it. So what it says we have is, um if we have a Cartesian plane, something like this and we have the force acting here, all right, you will have a wine, i X. This force is equal to the seven mutant force, and we are given this 230 degrees. So that is 250 degrees, which means this angle here is equal to 230 minus 180 degrees, which is 50 degrees. All right, so we can then use silent cause to break up the seven you 10 force into its two components . So if we have the seven you 10 force we have, it's X component acting this way to the right, and it's why component acting downwards. All right, so here we have seven mutants. I came here because this is the adjacent side. This is the adjacent This is the opposite. And that's the high part news. And we want to calculate the adjacent we use cause or 50 multiplied by the seven u turns. Andi. So here we have our 50 degree angle. And to calculate this components, we using the opposite overhear part news, which is sign. So it's seven sign 50 degrees. All right, so if we re look at a picture over here, the ex components off, the seven mutants will act along the line off 0.0, which means they won't be a perpendicular distance between this force on the point. Which means you can simply it. No. Seven cause 50. But there will be a perpendicular distance between the wire components acting downwards on point, which is, in fact, seven signed 50. All right, So because it's acting downwards or right, this forces acting this way, which is on the wire components acting downwards. Twitter, Rotate point. Oh, this way, which is clockwise, all right. Which means we have a negative moment. So then it could write minus so It's the negative moment seven signed. 50. So that is the magnitude of the force multiplied by the distance between point. Oh, Andi, the y component off this force, which is four meters. So there you have it. We have 12345 We have all five forces. Um, and now we can some them. So, using our calculator, we can just plug in all these values minus six times three plus four times three plus five times zero, which is just there. Plus two times 10 minus seven. Sign 50 and multiply by four equals on the Your answer is minus 7.45 mutiny meters. All right, so that is your final answer on Because the answer is negative. It means the resultant moment or force well, cause this beam to have irritation in the clockwise, um, direction. All right, because it's negative. Remember? Negative. Likewise positive anti clockwise 25. Moment of a Force Using Cross Product: In this video, we will be discussing the moments of a force using the cross product. So now you can calculate the moment or force using the cross product for both two dimensional and three dimensional problems. Usually the question should state how they want the answer. Whether it's using scale analysis, will victor analysis. If they don't state you could choose which method you prefer on a scale, it would probably be easier. But for more complicated examples, using Victor analysis would be the better choice. So by Victor analysis, I mean, you have i, j and K components off the forces. So you working with Cartesian vectors and not just magnitudes and distances, sir, um, to calculate the moment about a point. So in this case, 0.0.0, you do our So our is the position victor between points. Oh, Andi, Any points on the line of action of off the force? All right, cross that cross product, the Cartesian victor off the force, or right, And then your answer will be a Cartesian victor as well. So no, um very important. The moment is equal to our cross. If it's not, if cross our that will give you a different answer. So always remember, our comes first. And then if our across if very, very important A lot of people make the mistake where they swapped it to Iran cause scale analysis. You can write distance multiplied by force or force multiplied by the distance. But in victor or yeah, Invicta analysis. It has to be our cross if all right and the in the moment of the force would act perpendicular, um, to to our if which is at 0.0, all right. So you could use a right and rule in order to do the cross product. But basically, when you multiply that I jen k components, it will result in either and I j OK, which is positive or negative or some please error. So if we have a quotation plane with ex excess, where access is it access, then we'll have a positive K and a negative positive and the negative a positive jay and a negative. So if you're multiplying to Cartesian, Victor's using this formula then and you multiply and I and A J, then you get a positive K A, J and K. You'd get a positive I Okay, I positive. Jay, I crossed K is negative j So I noticed. I'm Kate. Cross eyes positive. Jay and I cross K is negative J. That is why it's so important that our comes first or else you're onto it. Be in the opposite direction, all right? And then and I times I j times J or K times k hole equal to zero. So if you to lady to memorize these, you can use a right and rule. So I cross j he used all right, and we'll see Go from I to j and then your thumb will tell you the direction So from I positive I two j So if you have so the fingertips are where the arrow is and you going from I to j i two j the, um, que is acting upwards. So it's positive k or right then Jay across cases from J to K. All right, so from j tu que It would then act in the positive direction if you do from K t. J. So from K to J, when your time is acting in the negative I direction and then care cross Jay's negative. I So that's just a way to figure it out. If you too lazy to memorize this, um, in a few videos off to this, I'll show you how to do the cross product on your calculator. But if you don't show your working art, then they will mark you down. So that's why I am teaching you this method off using the cross product. In the next video, we will be doing an example using victor analysis to find the cross product. 26. Moment of a Force Using Cross Product Example: determine the moment off the force if a bar point. Oh, using victor analysis. So here we have a point. All right. Andi, here we have the force off magnitude 300 Newtons on. It wants us to calculate using victor analysis. So, you know, to calculate the moment using victim analyses, you use the cross product. So the moment about points oh, is equal to our cross. If where r represents the position Victor from the points over here, which is a point you doing the moment about in this case, it's 0.0.0, Andi, Any points on the line of action off the force? So you the best would be the easiest or to calculate um, in this case, it would be this point over here because we have the dimensions to that point. All right, so to get our the position, Victor are we write it in terms of I and J and K, but we just working two dimensions, so it's just I and J. So you have in the X direction 1.2 meters, so at one point too high. And if we have the Cartesian plane about 0.0 and going towards the right. So then it's positive. 1.2 I All right? And then we have in the positive jay direction. So it's positive. I positive, Jay, because it's this point here. So our would actually be a line from here to there. All right, J would be this distance from O to that distance, which is 1.4 minus 0.2. So that would be left with 1.2 as well, J All right. And then that is all in meters. Okay, so that now we have the position victims are. Now when you calculate the position victor off. If so, we have the force, if off 300 Newtons rights. And if we have a question plane here we have this as 50 degrees, then 90 minus 30. We have 60 degrees over there, so no to calculate. Um, the components, we say if is equal to the thank you it Jason's over High Potter news, which is the same as cause off. 60 cause off 60 times, 300. All right in the eye direction because it's moving to the right. All right, it's to the positive X direction. But then now it's moving downwards right. So it's in the negative y direction. So then you drive negative 300. And now we working with the opposite of a heart. Broten, you. So it's sign our 60 degrees in the J direction. So versus all measured in mutants. All right, so now we have Or in if defined as our Cartesian victors so he could be and right. The moment is equal to 1.2 I plus 1.2 j, right. Cross 300 cause off. 60 I, you could have written on Sign off 30 and cause off city. But, um, you know, whatever works for you is how you should just define it. Um, so here we have 300 cost 60 minus 300. Sign 60. All right, So if we were you write that we have one points. You I cross 300 cause off 60 i so that I multiplied by that in this I multiplied by that. So you just basically multiplying after brackets? Um, plus 1.2 I times. Miners 300 sign 60. All right. And now we working with this jay plus 1.2 j cross. 300 cause 60 I plus 1.2 J. He's all have little hats on top of them. All right, um, this waas a J over here. All right. One point to J multiplied by across products 300 sign 60. Jay. All right, so, um, if you could just group all of them all your cross product to have that that that and that . All right, so now we can do our actual cross product calculation. So m zero is equal to 1.2 times, 300 cause of 60. 180 1 80 So now I cross I is equal to zero. So we basically should remember from here I across eyes equal to Zahra. So basically, this is then equal to zero. So here we haven't across equities era, and we also have a We should have. Ah, here, J cross J. So that is also equal to. So you could simply just scratch these two out because they're not going to influence the answer. All right, so here we all have 1.2 1.2 multiplied by minus 300. Sign off 60. All right, and you'll get a value off minus 311 point eight and then I cross J If you look at the nature I cross J is positive K. So you're right k for positive care. The this one over here, we rights. Um, 1.2 times, 300 cause of 60. The answer is 180. So we have plus 180 and then Jay cross I we look at across parts rules J cross eyes Negative k. All right. Jkos I Negative k. So, um, this is actually a negative, and then you can write k like that. All right, So you can, then, right. The final on supik was not both in K minus 311.8 minus 1 80 vehicles. Miners for 91 0.8 is equal to minors for 90. Sorry. Minus full 92. Okay. New10 meters. All right. So let me just rewrite that. Me miners for 92 okay? Mutiny meters. So that's basically saying that it is acting into the page. Um, because it is negative. It is acting clockwise, which is into the page 27. How to do the Cross Product on your Calculator: in this video, I'll show you guys how to calculate the cross product on your calculator. So you're going to need a Casio FX 991 years. Plus, which is this calculator over here, This silver one. If you don't have this calculator, you cannot do, um, Victors on your calculator or the cross product. So to calculate the moment m o about 00.0 If the position Vector z could do this and the force is equal to this, you would do it like this. So demo, all right, Should actually have a line on top of it. M o is equal to our cross if or rights. So remember, that weighs our first cross if it's never, ever cross card. All right, So what we'll do when you're on the calculator will define the c T. Victor A as our on the city be as if So this is how you do it for make space in the calculator on the Roger. All right. You do it like that. So first you click the mode button a thea mode and you click number eight to get it into victor eight. No, you want to define Victor a which is on. So you kick number one now it'll give you ah, um option to choose whether you want it in three dimensions or two. So three would mean I j k and two would simply mean I and J all right, So we would click number one for three dimensions because we have Vijay's on case, and then you simply plug in your values so we'll have positive 0.4 and you click equals, and we'll take you to an extra to fill in. The J components. A. J is minus 0.2 equals on the K is minus 0.5 equals. All right, so you have now defined your victor A. You can clear it by clicking a C. All right, so the court later will now remember that you have defined Victor a U N Click shift 54 vector. Where I receive it is Victor five. Then you want to know I define the second victor. So you click number one for dimensions number one on. Then you click on number two, Sir. Victor one or Victor A. We already defined, which is number one there. So now we want to define victor be for the if we click to once again, we wanted in ideas in case to its option number one. So you plug in our values. 300 I minus 4 56 points, you one J aceveda to four significant digits for the intermediate steps on 1 33 points era for significant digits. All right. And then we have not defying thick to be all right. So became click clear again A sea of it here then you go once again shift five for Victor Then we've defined Victor A. So what you want to do is we want her out. V c t Victor a cross v c t be which will give us Oh, answer. So we click number three over there for Victor A. Then we kicked the multiplication sign on the calculator. Will know he talking about cross products since we in the victim mode of the calculator Cross shift five and then number four for Victor Be. You know, we have Victor a cross big to be not Victor. Be cross big toe because that will be f cross are so it. Just remember it's our across if and then you click the equals button and the is your answers, for the moment is equal to minus two. 50 five in the eye direction minus to are three in the J direction, minus 1 to 2 in the K direction. And then you can put these brackets and writing you the units. It's a mutants and meters sort. Mutants multiplied by meters. And there you have the cross products off are in. If so, I just took for when you, um in the test on and you have to do the cross products, they do mark your calculations for the cross product, so make sure you're right out the calculations. Um, but you don't have to then calculate each step. What is our times? J. What direction does it act? What's J times K etcetera. All you need is right out of formula. Then you can simply plug in your vectors into your calculator, get the answer, and will save you a lot of time in the test. 28. Moment of a Force About a Point: determine the moment off the force at a about point B, give the answer as a Cartesian victor. So as you can see, the force is given as a Cartesian victor. All right, um, and then we have the coordinates off points A and B. That's basically what we have to work with. So the question tells us to determine the moment off the force. We're right. So to calculate the moment. And it says off the force at a about point B. So we calculating the moments about this point over here. All right, so the moments of our point B is equal to our the position Victor are from B to a cross, the Cartesian victor of the Force, which is this over here. All right. So just note when you calculating the moment of a force about a point in the first letter off the position, Victor should be the same as the point about which are calculating the moment. So in this case, it's B and B. All right. In the second letter off the position, Victor should be a point which lot which lies along the line off action off the force. So could be any points on this line off the force. So that's what point A represents or the second letter off the position vector. All right, So, firstly, we have to define the position Victor R b A To do that, we can first, right down the coordinates of being a So the coordinates off B are equal to. So the X is a positive three meters. All right, cause it runs along the positive X axis. So it's plus three. Why is plus seven? Because it runs along the positive y axis. And then is it is a negative one meter minus one, because it runs along the negative. Said access, or right now for the coordinates off a The X coordinates off A is my minus two over here, which has the X access and then negative two writers minus two. Um, or negative, too. The y coordinate is negative. Six on the their coordinates is positive. Three. All right, So if we calculating the position Victor from B to A says from B to a, we then say a minus bi. All right, so it's gonna be a minus bi. So then, um, because rights are be a equals a minus, B squared, minus two, minus positive. Three. I plus minus six, minus seven or positive. Seven J plus three minus minus one. Okay. And that is all measured in meters. All right, we cared. Simply rewrite this like this. We're just minus five. I minus 13 J plus four. Okay. All right. So this is now our position. Victor R B A. So we can now Then calculate the moment them be equals R B a cross if which is equal to minus five. I minus 30 and J plus four k cross product with the magnitude off the force. All right, which is 58. I minus 28 J minus 10-K All right, so we could know calculate the cross product. But if you just simply punch this inter calculator and get your answer. Um, it depends on the market. They might not give you the mark for the final answer. Chris, you didn't show You're working out. So what I would recommend is you showed the working out, and then you do it on your calculator to make sure you get the correct answer. So this is how I used to, right? Art? My answer. When I used to do the subject on, they would give me my calculation mark on the final mark where the people who had just used a calculator will not get marks for calculations. And some markers might not even give you, um, the mark for the final answer if you didn't show waking out. So this is how you would write it out to dry minus five. I across 58 I so that times that in that times that in that times, that's then you then you repeat, you just expanded like he used to do in high school. So then, plus minus five, I cross minus 28 j plus minus five. I cross minus 10-K plus minus 13 J Cross 58 I plus minus 13 J cross minus 28 j plus minus 13 J Cross minus 10-K Mustn't forget all these little hats on top of the victors. All right. Plus four. Okay, across 58 I plus four K cross minus 28 J plus four K cross minus 10-K All right, too. Now, obviously, if you then have to multiply each one of these art and then determine the directions on each of them that will be very time consuming. So then what I do? Is I just some P punch into my calculator? All right, so you go into victim mode number eight, click. It's at number one, Andi. Then we select three dimensions, so R is equal to minus five minus 13 plus four. Rights shift Victor dimensions number two 58 minus 28 minus 10. Sorry about that. 58 equals minus 28 equals minus 10 equals shift victor number three, Cross shift victor on before equals. And then your final ounces 2 42 I plus 1 82 j plus 8 94 k And then that's all measured in mutiny meters. And that is your final answer. If you just want some clarity on how to, um, calculators on your calculator there is a video, um, where I explain to you how to do the cross product on the calculator so you can go check that out where I explain it in more detail. 29. Moment of a Force About an Axis or Line: in this video, I will be discussing the moment of a force about a specified access or line. So you might receive a question where they will ask you to calculate the moment or the force about a specific line in this case, line a B. All right, so and be represent two points on the line. So from a to B represents the line. All right, so yeah, this m maybe without a line at the top represents the moment along the line off the unit. Victor, you all right? So, verse is the unit victor off the line, which they request. So he our rights unit vector off line or access? As mentioned in the question, if they say they want to you to calculate the moment off a force about line a B, then you would write a be here on a B, then. All right, then. Over here, we have a dot product, and then you are doing a dark product. With this moment, this position vector multiplied by the force. So the position victor over here refers to the position vector from a where a represents a point on the line off the unit. Victor two points X anywhere along the line off action off. If so, x represents a point on the line off If and a represents a point on the line on the unit Vector a B All right. And then this cross represents the cross product. And this, if, with a line at the top three presents the Cartesian victor off the force. So basically, you do the cross product here, you're going to get an answer as a Cartesian victor in this unit. Vector on verse are birth Cartesian victors. And then when you do that, the dark products between the two, you will get a, um, single value answer because you are doing a door product so it won't be a Cartesian vector . All right, so over here I drew out what the formula actually means. So here we have two points on the line A and B, so the unit vector would be along this line from a to B. Then here we have R a ex r a X represents the position vector from a point on the unit Victor to a point on the force. All right, then, um, yes, sir. The moment over here acts in the same direction as the unit Victor, A B All right, and then yet is note that point A is a point on the line off the unit vector. All right, so, basically, as long as you know how to apply this formula, um then then you will be able to do the problems for the section. So what you need to know is when you know how to calculate unit victors on position Victor's Andi Just how to calculate Cartesian victors in general. In the next video, we will be doing an example, um, where you have to calculate the moment off a force about a specified access or line. 30. The Moment of a Couple (Scalar): In this video, I will be discussing the moment off a couple and how to do a problem worth a couple moments using scaler. I will also have a video to come with, calculating the moment of a couple using vector analysis. So, basically, what is a couple? A couple moments is formed when to Colin ear forces. So that is to forces which lie in the same plane, for example, the same X y plane. So it's you calling your forces or equal in magnitude, but opposite in direction, it creates a tendency for irritation so that basically what a moment of a couple does. So every here, if you have a steering wheel right, these are two separate examples acting in opposite directions. So here we have the moment, which is equal to force times distance with the represents, the perpendicular distance between the forces. All right, so in this case we have the fitting Newton force. So we have to force is equal in magnitude, right? But opposite in direction. This one's acting the one direction, and that was acting 180 degrees in the opposite direction at a distance d away from it. So you're I 30 multiplied by 0.2, which is a perpendicular distance, and that'll give you answer off six Newton meters on their acts clockwise. So, you know, acting clockwise. Because if there were this, if the steering or where to turn from these forces, that force it irritates that way and then on the moment acts downwards the direction of the thumb, right, unless it's negative. So that's why clockwise is always negative and anti clockwise is always positive. In this case, we have 30 multiplied by 0.2 same thing, except this one's acting the other way around. So now the moments acting, the positives it direction. So, um, anti clockwise means positive. Six Newton meters. All right, now, just something you need to know. What is a an equivalent couple? An equivalent couple is basically, um, two different couples which have different forces and different distances between them. But the value over the moments is the same. In this case, we have the moment, which is a force multiplied by the distance. All right, Andi, that's 20 Newtons multiplied by three meters, the perpendicular distance between it and we'll get 60 mutants. But as you notice this is acting in the clockwise direction, right? So it's in a clockwise direction. So that is equal to minus 60 mutant meters. So and that's mutant meters, not mutant. That's awesome, You two meters. In this case, we have the force of 6.66 mutants multiplied by the distance. The perpendicular distance between them, which is 369 meters on that is 60 Newton meters acting in that direction. Once again, it's clockwise direction. That means clockwise is negative and that is your final onset. She'll notice. He's two answers all the same, which means these two couples, our equivalence couples. 31. The Moment of a Couple (Scalar) Example: determine the result in couple moment off the three couples acting on the object to calculate the resulted moment moment resultant that is equal to the sum off the couples. So we can call that the force times distance with the represents the perpendicular. That's a symbol for the perpendicular distance between couples. All right, so couple obviously means to. So that's why there are two forces and that's why it's called a couple. All right, so if we do so, we have to do the couple for force 12 and three. So we'll leave for Force one for loss because it's the most complicated. So start with forced to so forced to has a magnitude off 50 Newtons. So it's 30 right on then. These two will cause this object irritate in this direction result, which is anti clockwise. Therefore, it's positive. So we have a positive city multiplied by the perpendicular distance between them. Is that this distance here? We're out the distance between this line here on this line, which is six meters minus three meters and then you'll be left with this distance between the two forces, which is three meters. Six minus three is three. All right, now we will look at Force number three over here on as it rates makes this object irritate . It will rotate in this direction, which means it is clockwise and clockwise means negative to have negative force. Tree were forced through his 40 newtons. Negative 40 multiplied by the perpendicular distance between them. So this distance from here to here is equal to the total distance, which is eight meters minus that four, which will be four meters. All right, so, no, we have force one. All right, which can be broken, be broken up into its components. So if we redraw the object, something like this, Force one is broken up it into its wife components like this. Andi. Then along this axis here. All right. And then this force acting upwards and then too. Right? So notice how it creates two additional couples or as we have this couple here and we have this couple here, so if this is 20 degrees, we can then calculate using Trig here we have 20 degrees. Then this is the Jason side. And that's the hard part news. Which means we using cause off theatre. So this is cause off 20 multiplied by force one which is 60 Newtons 60 cause tweeting Here we have 60 Scient Winnie along the X axis. Here we have the same thing 60 cause 20 and here we have 60 silently. All right, so if we look at the y component off Force one, you hear the white components are forced one, we have 60 signed 20 as a magnitude. So 60 sign 20 and the the distance between them is four meters right, four meters. But the perpendicular distance between this force and this force. All right, so that's multiplied by full Andi you'll notice as it rotates, would rotated. He look at these two forces would rate it our object in this direction which is antique clock. No, which is clockwise, which means it's negative goes clockwise, always negative. So we have a negative or right. Then we can look at the X components off Force one, which is 60 cause 20 All right, multiplied by the distance between them, which is six meters. If you look here six meters. All right, that's the distance between the X components off Force One force ones, couple to the right, six meters And then if you figure out how how it were, take the object, it would rotate it in this direction, which is once again negative on. There you have it. So we have all we have. I equation set out. We could then simply punching into our calculator. 30 times three minus 40 times four minus 60. Sign 20 multiplied by four minus 60 caused 20 multiplied by six. All right, and then that will give you a final answer. Your result in moments as minus full 90 you 10 meters. All right, Andi, That could also be written as 490 butin meters clockwise. Andi, that is your final answer. 32. The Moment of a Couple (Vector): in this video, I'll be discussing the moment of a couple on how to calculate the moment of a couple for victor analysis. So a couple is when there are two forces that are equal in magnitude, but opposite in direction. All right, so, as you can see here, um, to force is equal in magnitude by opposite in direction, separated by a certain distance. That is what a couple is a free victor. Um, this is defined as a couple that causes rotational defict regardless of where the couple moments is applied. All right, so that's just the definition of a free victor. Just something you need to know. I mean, this is the formula to calculate the moment off a couple. All right, The couple moments, all right, as a Cartesian victor is equal to the position Vector crossed the couple force or art. In this case, the couple force would simply be 600 Newtons. But you would write that 600 Newtons as a Cartesian vector so you can use unit the unit vector formula to get it from a value such as a magnitude into a Cartesian victor force. Now, our this is very important the position. Victor is defined as, um a position position vector from any point on the line of action, off one force to any point on the line of action off the other force. All right, so if you're doing scale analysis and using force times distance, then that distance has to be the perpendicular distance between the two forces. So it would have to be perpendicular distance between the two forces. Right. But when you're working with Victor analysis, it doesn't have to be the perpendicular position, Victor. It can be any position vector from any points along this line to any point along this line or the other way around any points along this line to any point along this line. So, for example, the moment here with our from A to B across the force of B. All right, we'll give you the same answer as a moment from far from B to a across the force at a All right. So basically, the second letter off the position vector is the the direction off the force you should use . All right. So, basically, if you're going from from B to A from B to A in the force as a Cartesian victor must be in the direction off a Not in the direction off beat. All right, so that's just a very important thing to remember. You can just remember, the second letter is the same as the letter for the force. All right. In the following video, we will be doing an example where we will calculate the moment off a couple using victor analysis. 33. The Moment of a Couple (Vector) Example: determine the couple moments on the assembly and give the answer as a Cartesian victor. All right, So to calculate the moment of a couple as a Cartesian victor, we know we have to use this formula the moment off, a couple Invicta analysis. The moment a couple moments is equal to the position Victor across the couple force. All right, So if I were to write that out, a moment is equal to our cross. If now we could, either. At the moment is equal to the position victor from a to be. So it's from a to B being cross the force off, be all right. Or you could write. The moment is equal to our from B to a across the force off a So from B to a cross the force off A. So you could either use any of these two formulas if you remember what I said in the previous video. The second letter of the position vector needs to be the four letter for the force. All right. So you could use either of these two to calculate the moment of the couple. I will just use this formula over here. All right? so I'll start off by calculating the position victor from B to a So now we are hitting from B to a. So that is, is there a 0.6 meters in the J direction just right? 0.6 j. All right on, because it's from B to A from B to A, it's hitting in the direction off the negative y axis. So that means we have a negative J that is negative 0.6 j. We can add these brackets and put the new the units that you are working with, which is meters. All right. So just remember position Victor's always have units off meters. Now. What we want to do is we want to calculate if a all right, So if a is given as its magnitude 20 Newtons, but we need to calculate its Cartesian victor. All right, so how we can do this is to draw it out, that always nice to just re troll the plane at which the forces acting because it's easier to visualize. So if we have our Cartesian plane with the positives, it access on the positive X on the negative X and then negative said all right. We have a force acting along the Does it explain? All right, So if I withdrew this force to look like this, this is at a distance off 0.5 meters above his it access. And then we have 72 and five. All right, so, 72 on five. And then we have the magnitude of 20 Newtons so you can use these components in order to find the Cartesian victor off the force. All right, so the force off a is then equal to its 20 mutants. That's equal to if a all right without the line at the top, because it's the magnitude. But this one has the line at the top. So we will say to over seven to over seven, which is the opposite over there. Hi Potter news. So to over seven multiplied by 20. And that's in the K direction rights. In here we have seven five of the seven multiplied by 20 Newtons, 5/7, the Jasons. So we are waking with the negative X axis. So it's an I, but in the negative direction, because this is hitting in the negative high direction. But it's acting in the positive K direction. All right, so that's what that formula means. And since it's in the exit plane, that means they is a zero J components. This is all measured in mutants. So you can then continue and do the cross product. So the moment is equal to R B A. Across the force of a we are B A is equal to minus 0.6 J cross, and then we have minus 5/7. All right, if you just calculate that minus 5/7 times 20 you will be left with minus 14 points to nine . All right, And then for the 2/7 times 20 you'll be left with plus 5.714 in the K direction. All right, so we have this position Victor across this Cartesian victor of the Force. All right, so you can right this out. So that's minus 0.6 J cross minus 14.29 I plus minus 0.6 J cross 5.714 Okay. All right. So if you work that out, you can either just do that. I'm using the right hand rule, or you could plug it into a calculator. If you plug it into a calculator, you will kids and answer as follows minus 3.43 I plus zero j. All right, so you can just leave the J component art because it's equal to Sierra, then minus eight point 57 Okay, all right. And then you can write it the units for this and that's in Newton meters because you have meters multiplied by Newton's. You're left with mutant meters that it is your final onset. 34. Reduction of Force and Couple system to its Simplest Form: in this video, I'll be discussing the reduction off a force and couple system to its simplest form at a point so you can reduce a four cent couple system in scale analysis or in vector analysis. So we here have the formulas written in black. Those are the formulas to do it in scale analysis, and the formula written in blue is for vector analysis on same applies over here for when you're reduced to arrange. So to reduce to a single result in force using scale analysis, you sum all the forces in the X direction or plane. All right, then, um, you do that to find the result enforcing the Y direction you some of the forces in the Why , All right, then, to get your final result in force your roots the result enforcing the X direction squared, plus a result in force in the Y direction squared. All right, so then you'll have your magnitude of the force or the result in force, and then to get the direction off this result in force, you then calculate the angle theater with the two represents the direction as, or Qattan, which is on the calculator when you kick shift. Town becomes 10 to the power of minus one. Shift 10 to minus one off the result in force in the Y direction or the why access divided by the if our x All right for victor analysis, it's a lot easier. All you do is you some or the forces at all the I component or the J and all the k and in your get your result in force in order to reduce something to arrange. What does this mean? Arrange is made up of both a single result in force as well as the result in force off all moments. All right, so when you read you something to arrange your reducing her to two things, the first thing you reducing it to is the single result in force or right, so you'll do this as step number one and then a step number two when you're reducing to arrange, is you calculate the result in moment about a point. All right, so how you do that in scale analysis, you had some all the moments off all forces about a point. Oh, all right. And then you add the some off the couple moments, but instead off calculating it with a couple moments, you could determine the moments off the components off each force instead of doing it together with the two, um, forces acting opposite directions, you could just treat each force as its individual force or right they intend to do it. In Victor analysis, you simply do you use the same formula. You just some all the moments off the forces, plus some all the couple moments, all right, and then you're Kitchell result in moments about eight points. So let me just mention again, because it's very important. If you reduce something to arrange, you are reducing it to two things. You're reducing it to a single result in force, which he used, which you calculate using these formulas over here. Andi, you then calculate the result in moments about a points. All right, so in the next video we will be doing an example 35. Reduction of Force and Couple system to its Simplest Form Example: replace the forces on moments acting on the pipe assembly with a result in force and couple moments at 0.0 give the answer as a Cartesian victor and draw the range on the X Y Z airplane. All right, so we are granted kits a results in moments on a result in force, which we will then draw on the X y Z airplane. So here we have the X y Z airplane. They have a pipe OBE Oh, ABC, All right. And we have two forces, if one and if to given to us in Cartesian Victor form and then we have a moment over here and then we have all these dimensions. All right. So in order to calculates the resultant force, we use the formula for victor analysis. All right, which is the force resulting T is equal to the sum of the forces. So force resultant equals some off the forces which is equal to force one plus force to all right so forth when you have 16 I minus nine j plus 15 k Sorry. That's if to write, plus if one which is minus eight. I plus 21 j plus five k so where I like to do is I like to use Tyler's. Teoh highlights the different components to avoid making mistake. So have the eyes, then the jays on then the case, so it's easier to see. It's always good to take colors into an exam so you could use for things like this. All right, so 16 minus H is equal to eight. I minus nine minus eight minus nine plus 21 is equal to plus 12 J. Andi 15 K plus five is 20 k and that is all measured in mutants. We're right, so there we have our result in force. If you're working in scale, we would have to calculate the direction angle. But because you're working and victor by Jay's in case, um, give the direction for us. So now we need to calculate the resultant moment on. The result In moments can be calculated using this formula over here, where the result in moment about a point on in this case, it's a bark point. Oh, because that's what the question Austin and couple moments Barbara Block at point. In order to do this, we would then right out the formula to some of the moments. But when her plus the some off the moments, These are a couple moments, these other forces. All right? We then have or across if All right, So, um, our cross if, where r is equal to or off a across the force number two. All right, it crossed, forced to. And then we have Oh, see Cross Force One. All right. And then we have this moment over here, which is acting in the negatives, their direction. This moment has acted in the negatives, that direction. So that is automatically minus 600 mutant per meter. It's minus 600. Okay. All right. So where we can do now is define our A and R O C. And then do the cross product. So are a is equal to the idea Rations. Air 0.35 I Alright, suit 0.35 i and then their points to J. So this is in the units off meters now. We can define our, uh, see from ER to see. All right, we have the x distance from oh, to see which is 0.6 plus 0.2. So that's their 0.8 meters. So 0.8 i looking at the y direction. Sorry, that's in. That's in the J components here. Point eights j. All right, Looking at the I, we have this distance here which is 0.35 plus 0.35 I. And in the K direction, we have 0.6 meters Andi, that is plus 0.6 Okay, that once again measured in meters. So what we can do, we can calculate the cross product are a cross if, to which is 0.35 i plaster of 0.2 j cross product with force to alright forced to which has a value off 16 I minus nine j plus 15 k So this will give us a K value, which is the moment so we can plug this into the calculator. Victor Torn, you have 0.35 I place 0.2 j zero k 16 I minus nine J plus 15 k Victor. A cross victor be is equal to three. I minus 5.25 j minus 6.35 Ok, All right. So that's these are moments for this cross product here, then full across products are. Oh, see Cross Force One. We then have 0.8. It's just for your iPhone. The order I j k 0.35 I plus 0.8 j plus 0.6 k cross product with Force One on Force One has the following Cartesian Victor, which is given that is equal to minus eight. I plus 21 J plus, five k. All right, so, once again, we can do the cross product. We go define Victor. A 1350.8. So put in six minus aids. 21 five on There. You have your answer, which is minus 8.6. I minus 6.55 j plus 13.75 Okay. Alrighty. So now we can simply add them all together using this formula. All right, so then you just highlights what we have. So this year represent that over there represents this over here. Okay, this here represents this over here. All right, so we can then plug in on our values. All right? So result of moments, the bark points. Oh, all right. Four point. Oh is equal to. Now we add all the eyes, so it's three minus 8.6. That's minus 5.6 I. Then you have minus five minus 5.25 J minus 6.55 That's minus 11.8 J. And we have minus 6.35 plastic 18.75 minus 600. Be sure not to forget that minus 600 equals plus 5 92.6 K. And that is equal well, in the units mutiny meters. Another question also asked us to drool The result in a moment on the results in force on the X Y plane X y z airplane. So, yeah, we have the positive Why Positive X and the positives it access. All right, so, uh, all right. Okay. And it's all of our point. Oh, which is the origin in the result in force equal to eight. So in the X direction. So it will be somewhere like here. All right, Jay is positive. 12. It will be somewhere about the and K is equal to 20. So it's some we really high, so it's hard to draw, but basically it would look something like this all the way up to points 20 and in that is your result in force, which is equal to it's I plus 12 j plus 20 k Then we can offer draw our moments. We are moments is minus 5.6 I. So it's somewhere there minus 11.8 j. So something like that and then plus 592 which is a really, really high. All right, let's just say 592 is. It is. It is somewhere here. I don't really draw it properly. Let's say that's 592 then of moment like that, where that is the moment result into bark point. Oh, and then you could even ride are to this entire value which is on us for a point. Six I minus 11.8 j plus 5 92.6 K and that all in mutiny meters or rights. Ah, there you have it. It's just make sure we have wanted everything. The question Ost give the answer as a Cartesian victor and draw the range on the X Y Z airplane on. That is the final answer 36. Further Simplification of a Force and Couple System: in this video, I will be discussing further simplification of a force and couple system. So say you have an object. In this case, it's a circular disk in an X y Z airplane, and you have forces acting on the disk. You could then simplify Oh, reduce these forces to a single result in force. All right, worth a moment perpendicular to this force. If you ever asked to draw the moment, be sure to draw the little arrow acting in the direction. So you point your thumb in the direction of the moment on, then the error spends in that direction around that moment. So if you were to further simplify this, um, you would get rid of the moment on, then move the force a distance d away from the origin. So then you're be left with a result in force at a specific distance. And then, obviously, you know, if you have the four something distance, then you'll be left with a moment. So these two things actually mean the exact same thing. This is just simply simplifying its even more. So how do you know when you have to do this? Um, or reduce it to something like that. The question will ask something like replaced the loading system by an equivalent result in force and specify the X Y and Z coordinates off its line of action. All right, so when you see something like about it, equivalent result in force. And that's when you know you have to simplify the this system. So in the next video, we will be doing an exam pool, Um, worth replacing the loading system with an equivalent result in force. 37. Further Simplification of a Force and Couple System Example: replace the loading system by an equivalent result in force and specify the X and Y and Z coordinates off its line of action. All right, so usually the the easiest way to go about solving this problem would be scale analysis. But I am going to solve this problem using victor analysis because most people struggle with victim analysis. So it's just, um just to show you guys how to solve this problem using victor analysis so we can start off by finding the result in force as a victor and that is equal to the sun off the forces . All right, so that's some off 4th 1 plus forced to plus force three, Force One is equal to positive 50 k forced to equal to minus 40 cave because it's acting in the negative direction. So the negative K direction all right on then, forced three is also negative minus 50 k So if you some all that up going to be left with minus 60 K and that's all measured in mutant. So now we can calculate the moment. So the result in moment bought a points. We will do it about the origin about points have been. When he calculates how distances X wines it it will be measured from the origin. So that's why I always quick to use the origin as your reference point. So that is equal to the sum off the moments. All right, so that is equal to, um the moment off. 4th 1 force to enforce three. So we'll do the cross product. Oh, position vector. Are are a from two. A cross 4th 1 we're not. Then we do it for Obi Cross. Forced to plus R V cross forced to plus I see our mercy across Force three. All right, so now what we can do is we can calculates are a all right. I'll just right around the put the equal sign on the left here because we're running out of space. So this is equal to are a We're so it's from oh to a So that is five in the eye direction. Positive. Five. All right. All right. Across with 50 k We then have OBE from A to B. All right, but we don't have a distance. Be. Although we do have this distance here, which is six meters. So what we can do is you can we can say six meters minus this distance over here in order to calculates, I ve so there's no ways to calculate this distance here unless you can assume that this triangle here is in fact equilateral. So if we assume it's an equilateral triangle and that means the high part in use length here, his six meters. So then we can calculate this length of here, which is six squared, minus five square, and then you're left with a distance or 3.31 over there. So then that distance from OTP is six, minus that answer, which is 2.683 So you'll have plus 2.683 acting in the positive J components. Sorry. So it's 2.683 j direction. I've just put brackets just to clarify. So that's or maybe across forced to, which is minus 14 care plus r o c. So are our c has from O to see has our components off six meters Sorry. Five meters and then a J component off six meters. So if you write that out, we will have the following five. All right. Last six j and in this is crossed with 43 which is minus 50 K All right, so you can then do the cross product of all these. You can go watch the cross product video on how to do the cross product if you I forgot. But basically, if you do across practice with that on your calculator, you should get the following answer minus full, uh, 7.2 I plus 100 j All right. And that's measured in Newton meters. So no, um, the resultant position victor crossed with They result in force if on should be equal to minus 47.2 I plus 100 j so we could rewrite this as it's all right. Plus why j plus zk across the result in force which calculated as minus 60 K cross minus 60 . Okay, as equal to minus 407.2 I plus 100 j. So we could then spit up the eyes and chase in case so, x I cross product minus 60 k is equal to We know an eye on a K will make a J. So that will be 100 j. All right, so you could Just look at this. And you and you calculate. For what? Value off X? Well, verse cross product equal. Positive. 100 k So then you could say 100 divided by 60. And the notice that X is equal to 1.667 meters. We can do the same for the white components. Why J across minus 60 k So the J and K, you're gonna be left with an eye component. So that's minus for 7.2. I all right. So you could think what value off. Why will result in this Craddock crocked cross products being equal to minus 407.2. So you said I, minus 407.22 thought about 60 on rye is then equal to 6.787 meters. All right, And if you try to cross product off this K cross that care, you're going to be equal with zero. So that means zid is, in fact, equal to zero. And then that is your value. So just for interest say they might even actually ask you Detroit's if it's all you replace the loading system by an equivalent result in force and space for the X Y and Zed coordinates off its line of action and drew it on the ex wives. Did, um, Cartesian plane? It would look like this. So here we have visit access. Here we have the X and the Why all right. We calculated recalculated X to be 1.667 We should actually answer to three significant figures, which is 1.67 meters and 6.79 meters. All right, but X is equal to 1.67 So let's say that's over. They 1.67 on Y is equal to 6.79 It's about 6.79 All right, then. This point here is the distance from the origin zero On our results in force, we calculated as negative 60 k So we know it's actually the negatives, that direction. So that's acting don words. And you can label that as 60 mutants. Andi, there you have. You're simplified. Result in force 38. Rigid Body Equilibrium in Two Dimensions: equilibrium in two dimensions. All right, so you will be given a rigid party on. Then there will be different things connected to it. For example, a cable, um, a weightless link, a rolla, a ruler in a slot rocker contacting surface as well as a pin connected to a collar. A panel hinge. Um, remember, fixed to a collar on a fixed support. So basically, you have to memorize, um, what would happen in these cases? So if you have a cable that you'll have a force acting along the same direction as a cable if you have a weightless link and you'll have a force acting along the direction off that weightless link if you have a roller, then you have a force acting perpendicular to the surface. The rollers on etcetera. So you have to remember the effect of all of these, um, as a force. All right, If you have a pen connected to a collar, it would just be a force perpendicular to the bar at which the pen is connected to the color. If you have a pen or a hinge, you then have to forces a wire component of the force, an X component of force. If the member is fixed to the color, it has a moment as well as a perpendicular force. And if you have a fixed support, which is very commonly used, then you would have an F wine and if X as well as a moment. So what you have to do is and you assume the direction off if y and if x and you assume whether the moment is positive or negative, all right. So you always just assume the directions, whether it's going to the rights effects, whether it's going to the left and then you do your calculations. And then if you get a negative for your answer, you know that your answer should simply be, um or you Assumption was simply the opposite off. What is the correct direction? So here's a few hints. Um, always use your equilibrium equations to answer these questions. Which means, um, some of the force in the X must equal zero sum over forces in the UAE must equals R zero, and the some of the moments, at a certain point, must equal zero. Now where should you usually do this? Some of the moments at which point. Well, here's a little hint. Do the some of the moment equals zero at the position, often unknown force to be able to solve for another unknown. For example, here you haven't unknown A Y and B why, If you do the some of the moments that point A. It means you can't calculate. The moment at eight was a white times a distance to the perpendicular distance between a y on that 80.0. So then you wouldn't have a Y in that formula, and you only have B Y, and it's perpendicular distance. So with that one equation, you'd be able to soar for B Y. And then, if you do some of my moments that be, you are would cancel our because there's no perpendicular distance between B y and the point B, and then you'd be able to calculate a what. Or you could just do some of the forces in the UAE, and then you could solve for the other. But that's just a very useful hints. You do some of the moments at the position, often unknown force. In the next video, we will be, I'm doing an example for equilibrium in two dimensions where you have to know how to apply these difference conditions 39. Rigid Body Equilibrium in Two Dimensions Example: calculate the horizontal and vertical components off reaction at the support A and B. All right, so basically, the key to solving these problems is having enough equations to solve for the unknowns. So first step to these equilibrium problems is drawing the free body diagram. So if we have this beam here, what, uh, we have a downwards force over here off seven killing mutants, all right? And we have a moment acting in the clockwise direction, which is three 100 Newton meters. Then we have a pen at B, all right? And a, we know gets broken up into its if X And if y components So then we can assume that we have Ah, a B Y over here, acting upwards and real human direction, right? As be eggs. All right, Been at point A. We have a hinge is the same thing that's broken up into if X and y components, so you can assume it's directions. I assume it's directions, same direction. So that's a why on And, um, it's assumed right is a X. So that's all free body diagram. That's quite a free body diagram. Number one Raj. We could also have a free body diagram. Number two off this components over here, all right, or we could even have a free body diagram off the entire assembly. So let's try that out. Free body diagram of the entire assembly would look something like this, but we have the Donald. It's seven Killer Newtons. We have our moments off 300 Newton meters. Then we have over here our link. And along here we have a force. It's cool it if b All right, then we have our hey, why and a X components at this hinge of their And then at this hinge at D, we also have its components. Do you? Why? It's a shame that acts in this direction de eggs. All right, So to solve these problems, we use the equations off equilibrium, all right? And as a question stated, it wants us to find the reactions at Supports A and B, which is a y XB dry and B X. So we can start off by doing the some off forces in the Y direction so human up as positive some of the forces in the wire must equal zero. So that a y plus here, why plus B Y minus 7000. Mutants must equal zero. All right, so there we have an equation. Now we can do the some of the force in the X. Assuming right is positive. Some of the forces in the X must equal zero. So, plus a X plus B X must equal zero. All right, So, uh, that tells us that X is equal to minus be X. Uh, here we have an equation. Here. We have an equation. Now we can do a moment about point a miraculously, to the moment about point B, but I'll just start off over here. So the some off the moments of our point a must equal zero. So then we have b y at a perpendicular distance away off 1.5 meters. All right, so it's b y. There's a free body diagram. You are multiplied by the 1.5 meters, and because it's acting in this direction, it's anti clockwise, and therefore it's positive we then have a seven killing you to enforce. All right, so that's 7000 multiplied by the total distance off 1.5 plus 2.5, which is four meters. Right. So that's 7000 times four meters on. Because it's rotating this way, it will be a negative answer. Negative rotation clockwise always negative on the We also have this clockwise 300 meters per meter moment over there. So it's minus 300 Newton meters equals era, all right, because be off X runs along the same line of action as point A. There is no moments created. We cannot calculate the value of B Y, which is equal to 7000 times full, plus 300 divided by 1.5. That is 18,866 0.67 Mutants Swing killer Newtons, We have 18 0.87 killer new terms, right? I notice how we get a positive value for B Y, which means we assume the correct direction off B Y. So be why does, in fact act upwards? All right, so that be why now we can use this formula to calculate a y. So then a wise equal to 7000 minus B y All right. So 7000 minus 18 866.67 we have minus 11 866.67 u turns, which is the same as minus 11.87 killing U turns. All right, so notice how we get a negative answer for a why a y is a negative answer. So that means over here we assumed the force of a Y is acting upwards, but in fact, the force of a wise actually acting downwards. All right, so if we originally assumed that they were I was acting downwards, we would have got a positive answer over here. All right, But don't change this to positive people is you did it wrong? If you have your free body diagram drawn like this in your marker will market correct because it follows your free body diagram and never change a free body diagram while he busy with a question because then they will mark it down rights because obviously your calculations linked your freedom body a free body diagram. So we have completed a Y b y, but we haven't calculated a X and B X. All we know is that a X is equal to negative BX in order to solve that. All right, we're gonna have to do a moment. Some of the moments somewhere which is at a perpendicular distance away from P X and B Y. So that leaves us with point D. All right, so if we do the some of the moments, Appointee, they must all equal zero. So you'll notice the force A Y lank runs through 20 so you can ignore a Y you notice force off be also runs three pointy. So there's no perpendicular distance to that. Although there is this this force creating a moment, Andi, is this moment all right. And of course, a X. So if we have a X multiplied by perpendicular perpendicular distance between D n a X, which is two meters, we're right to be here. So a exit acting here and then this is a perpendicular distance too pointy. So then we can have our two meters over there. We think have looking at our free body diagram. We then have our negative 300 meters so we can say minus 300 on. Then. We also have our seven killer Newton force, creating a negative moments about point D. So that's minus 7000 multiplied by the distance between them. Well, the perpendicular distance, which is four meters. All right, if you look here. This distance here is four meters along the line of action. Off the force. Perpendicular distance is in fact, four meters. All right, so this is then all equal to zero on. We can now solve for X. So I X is equal to 300 plus 7000 multiplied by four. And that's all divided by two. So that is 14,150 riches, 14.15 Killer Newtons. So you'll notice we get a positive answer, which means we assume the correct direction for a X, which is to the right B X. However, B X is equal to negative a X. As you can see here, X is equal to negative B X. So, therefore be X is equal to negative eggs. So then we have negative 14.15 Killer Newtons, all right, And because be X's negative, that means we assume the wrong direction for B E X. P X is actually acting to the left. All right, but that's how you leave your answers, rides or just underline a y b y b x on day X. That is the answers we are looking for, and then of course, you can just write them in three significant digits because that's how they want the answers in this module that would be equal to minus 11.9 Killer Newtons. And then here we have 14 points to Killer Newtons on D minus 14.2 Karen mutants. 40. Two Force Members: to force members to force members are two forces that are equal in magnitude but opposite in direction. So if you have an assembly, all right, connected with a pin over there, then you can separate these two Andrew a free body diagram. Zoff. Both then And it'll look like this. So this component here will have to forces that are equal in magnitude, but opposite in direction. All right. And then that is what you call the to force members in these forces can then be broken up into the X and Y components. All right, then, if you look at the free body diagram off this part, then the force would be acting in the same direction as from B to C. Between the connection points. That's the direction that the force will act. And then, obviously, here at the bottom, we have the components of the hinge a y in a eggs. All right, so that just shows you how you should draw your free body diagrams. Um, when you know it's us, you have a to force member in your question. 41. Method of Sections: in this video, I'll be teaching you guys the method off sections. So what is the method of sections? It is when you need to find the forces in only a few members off the trust and you can use the method off sections. All right, so there's a general three step procedure how to do the method of sections. So step number one is decide where to cut or section the members. All right, Step number two is draw the free body diagram of the trust with the least amount off forces acting on it. All right, we'll see what I mean by number two when we do the example and the number three apply the equations of equilibrium to solve for the unknowns i e. The some of the forces in the X direction Some of the forces in the why and then the some of the moments about a point must equals ever. So a lot of the time people struggle to know which method to use to solve different problems. So, um, Edelnor chur hard. You know when to use the method off sections. So just a general rule you could follow the question will ask you to find the forces of members that you can cut with one single section. So, for example, if you tell you to find B C, C, H and H G, that is B, C, C, H and H G with a single section, you can cut through all three that they lost you too. All right. And then those three off for three forces you are looking for and here I have a little hint when applying the some of the moments about of points equals era. The point p must be on a point that lies at the intersection off to unknown forces so that the third under enforce can be can be determined. Now that is very important, because that is how you actually solve for the unknown. You have to know that you have to do it by the point of intersection, off to unknown forces or just any amount of men forces. All right, so yeah, another little hints. So, as previously mentioned, when using the method off sections using some of the forces equals in the X equals zero and some of the forces in the Y equals era then uses some of the moment about point equals era for as many points p that is required to have enough equations for solving the unknowns. All right, so, uh, basically, the number of equations needs to equal the number of unknowns, and then you will be able to solve for the unknowns. If you have, for example, t equations. I have three unknowns you will not be able to solve for with unknowns. So just remember, you can keep applying this formula in order to have enough equations for your innards. 42. Method of Sections Example: in this video, we will be doing an example where we have to use the method off sections. All right, so here's the question and it asks, Find the forces in members D F, D, G and G. That's D F, D, G and G at ST with. The members are in tension or compression. We're right. So, like I mentioned to you earlier, how do you know to use the mythical sections? Basically, the question will ask you to find forces of members that you can cut with one single section. So, for example, find B C, C, H and H G B C C h h g. We can cut through order them with one single section, as in with this problem here, D F d g on G can cut through all three with one single section just like that. All right, so order to solve this problem, we can follow our general three step procedure over here. So step one is the side Great yukata section the members so that we know. All right, Step number two is drawing a free body diagram off the trust with the least amount of forces acting moments. All right, so we know we cutting it by the f, B, G and G e. So it's a line through here, and then we must draw a free body diagram off this. All right, so this is partly has been sectioned. Here we have the pen. All right, then you have a fortunate muting force acting time with you for killer from u turns. Then here we also have a five killer of U turns. All right, This arrow should actually be longer in the fortune. You didn't because the magnitude is bigger. But just for the sake of solving the problem, you can just continue. All right? And here we have force D if at an angle here will just assume the all intention for now. Fourth d g. Andi Force g all rights because the arrows looking outwards be assuming they are all intention your eyes. If you assume the arrows pointing inwards, then we are assuming those components are in compression. All right, so if we get a positive answer for forces, we know that that those members, our intention If you get a negative answer, then you know those members are in compression. All right, so we can start off by. Well, that's step number three. Over here. Apply the equations of equilibrium to solve, for the unknowns are used force in X y and moments. So we know we haven't angle on. You can enable this angle as 30 degrees, 30 degrees. So if we are doing some of the forces in the X direction, All right. And just call this free body diagram. Okay? Being with some enforcement X, we have minus Deif been here. We have the Jasons opposite in the hard part. Amused. So we're working with adjacent side. It's adjacent Everhart parts amused to be working with a cause that's minus cause off. 50 multiplied by forced e g minus force G equals zero. All right, so there we have are one equation. We can now move along and do the some of the forces in the Y direction. All right, so we have no force in the Y direction of the here. Here. We do have a force acting upwards. So we were working with sign off 30. So if gg multiplied by sine off duty All right, um, minus four. Killing mutants to put out 4000 newtons minus 5000 Newtons equals zero. All right, so now we actually have an equation here where we can so full these unknowns. Oh, just for F g. So let's solve for the f d. G. If DJ is equal to minus 4000 minus 5000 which is minus 9000. If you take it over, you're going to get positive. 9000 survived by sign off City. All right, so you can 9000 divided by the sine off city, all right? And we get a value of 18,000 mutants. I thought the same as 18.0. Killing you turns to three significant digits. All right, on because our answer is, in fact, positive. We know that forced GG is in tension. All right, So we still have if Deif And if g e to find we've just found f d g over here, so they will be left with one equation with these two unknowns. So this means we need one more equation. So you can do this Some, um, of the moments around The point on I was mentioned in the previous video when applying the some of the moments of our point, the point p must be on a point that lies at the intersection off to unknown forces so that the third honor and Force can be determined. So now we can choose our point Pete to be either this point here. All right. And then we can get any sorry this point yet. And then you can get an equation with unknown forces if d g and F g and then you can solve for if g Because we know the value of here, which you just calculated. All right? Or we can choose this point here and then do the some of the moment about this point and find the force off F d g. Sorry, ftf, And then you can substitute it back into this equation on. So, for our I knows So this year I called this DF It should actually be death. DF represents simply the member. All right, all the distance on the member where ftf would represent the force acting on that member. All right, so there was a city mistake on our behalf, so just met. Make sure you doing write it out in creative because you will get knocked down. All right, so now we can do that. Some of the moments equals era, all right? And we're doing it now about this point here, which is point G. Some moments about 20 plus equals zero. All right, looking at for G. Over here we have the one for the four killing yet in force acting on this point. All right, so the distance between verse Force and 20 is zero, so you could ignore this force. Then you have this five killer new10 force, which is acting one meter away from punchy. So, you know, to calculate every moment it's force times distance. So we then have five 1000 multiplied by one meter. All right, because I would rotate point G in this direction, which is clockwise or I g to force. It'll make it rotate. This way, we have a negative movements so we can put in negative in front with us here. All right, then we have force G, which is once again lying along the same line of action as Point G so you can ignore it and the same with f d. G. But if DF does have a perpendicular distance on that distance, according to this, it is the length off member if g All right, because at the distance from point G to the force acting along this member here. All right, so to do that, we can use Pythagoras. Oh, no, not actually actually have the link yet. You have the two meter. Oh, all right, so it's just simply two meters. 60 meters multiply. But if if because it bore, rotate would rotate 20 in this direction, which is that way, all right, because the forces acting to the left Ron said we gonna rotate point G in that direction. We didn't have a positive anti. Clockwise is always positive. All right. On that equals there. So now we have when equation One unknown. That means 50 if sequel to 5000 my times one taken over. It becomes a positive as we have 5000 divided by this to here. Take to the bottom. And then we have 2500 mutants, which is the same as two point five. They're killing Newton's three significant digits on because our values positive it is in tension. All right? No, we have one unforced left. Which this Force G All right, So you can We have forced the f We have forced e g, and we just need calculate force G. All right, so we can rewrite equation force you is equal to and equal to minus force the F minus Force G cross or 50. All right, so we cannot substitute. I would calculated forces into this equation minus force. Deif is minus 2500. Mutant last 2500 minus on force DJ is 18,000 Newtons, 18,000 cause for 15. All right, minus two Foncier's air price. 18,000 of course of 50. All right, so you just plug this in the calculator, we get value minus 18. There are eight fate mutants, all right? And because the answer is negative, it means we assumed the wrong direction for four straight. So here we assumed before she was acting art words on output means tension. So, in fact, forced. She's actually acting towards the right, which is compression. So we can then rewrite this answer as positive 18.1 killer Newton's arrives. I just wrote This are 2 to 3 significant digits because that's how they want the final answer, Andi. Then your art compression. All right, and that is your final answers. Or are they have the value for force DJ for Force DF and for force G E. 43. How to Determine if a Member is in Tension or Compression: in this video, I'll teach you guys hard to determine if a member is in tension or compression. So how did you do this? You basically look at your free body diagram. When you drive a free body diagram, you usually don't know in which direction the forces acting, whether it's in or art or compression attention. So in this video, I'll teach you guys how to figure out whether it's compression attention. So if you do, you offer you a free body diagram. You didn't do your some of the forces or your moments off your equations of equilibrium. You will then get your answers all right for the different forces you were working with. So let's assume the forces around a point where Force A Force B and four C all right. It doesn't necessarily have to be around a point, but just, um, anyway, within your free body diagram. If you then, um, render a positive answer for force a Force B and four C or right or just a positive on send general for any of your forces on. You assumed your force was acting in words towards the point, Then it is in compression. All right. So in this case, forced Air Force B R birthing compression, whereas for C acting outwards is intention. All right, so, um, opposite applies if you get a negative answer. All right, so here we haven't have inexhaustible. All right, so, um, if you're negative answer, obviously art means compression and in means tension. So here we have fourth a right, which is positive. 60 Newton's. All right. So it trended a positive answer. So fourth, avery assumed was going in. And because it's going in with a positive answer, it means it's in compression. So therefore, force A's and compression force be alright. We render a negative answer. So a negative answer. All right, on because force be is going in in equals attention for negative answer. All right, so that means force be is intention four c. All right is equal to Nick and negative answer. All right, Andi, because its a negative answer and reassumed its going out negative answer out equals compression. So therefore, FC is in compression. All right, um and so is usually easy to just remember that too. This is what I do. But I always do this. I always draw all my forces acting outwards. All right, So basically, just like FC, assume they are all intention. All right? And then if I get a positive answer, I know the answers. Tension. If I get a negative answer, then I know the answer is compilation, So that's what I usually like to do. So you assume it's going out all right? That's And if you get a negative Ansett's compression and you get a positive answer, it is attention for going out. You could do the same thing. You could assume the or go in, and then if you get a positive Ansett's compression, etcetera. But whatever works for you would probably the best method to father. 44. Zero Force Members: in this video, I'll be discussing zero force members. So there are two scenarios where you can determine whether or not the members are zero force members or not. So the first scenario is this one here if two members former trust joints. So here we have Member A, B and member A c connected to the rest of the trust and they form a trust joint. All right, so that's only two members forming a trust joint and no external forces acting on the joint . Then both the members are zero force members. So, as you can see at this joint, we have no external forces acting on it. So therefore, the force off the member A B and force at a C R. Birth equal to zero. So therefore, they are the zero force members. If we had the situation where we had an external force acting at the trust joint, then forced a B is not equal to their and force A C is not equal to zero as it stated him and no external forces acting on the joint. Then both the members are zero force members. All right, so, um, this is the first scenario you need to know. Just note that if this external force was acting say at this point here, then these would still be zero force members. The external force only applies, um, at the trust joints. That is where the two members are connected. All right, so the sick and scenario is as follows If three members form a trust joint, So yeah, we have Member A, B, A, C and A D forming a trust joint here a on day two of the members or Colonia. So, Colin, your means, like along the same line. So here we have a B and A D. So those are colonia because they basically you could grow must say there. Yeah, they Well, they're not parallel to each other, but they're the run along the same line. All right, then, The third member that is not Colin. Yah is the zero Force member provided no external forces act on the trust. So here we have the third member, which is a c All right on a C is not Colin year with a B and A D. So that means that this third member, which is not cool in here, is in fact, zero force men. But so if a C is equal to zero, So what is there a force member actually means it basically just means the force in that member zero. So it's as if that member shouldn't even be there because to not making a difference to the forces anywhere in the trust. All right. And then, as it's stated here, um, provided no external forces acts in the trust. So here we have, um, external force acting at the trust joint. Not just anyway on the trust. It has to be at the trust joint. So if we have this external force acting at the trust joint, then we know, um, that if a c is in fact not Is there a force member? All right, Andi, um, it's it's mentioned over here that if a f a c is a zero force member, then if a B is equal to if a d Alright, so f a B is equal to f A D. And they are not equal to zero. All right, Only this third member that is not Colin ear with other two is the zero force number. All right, so you might ask yourself, eh? So what is the points off a zero force member like, Why are we even learning about this? Um, this Because zero force members are used to increased ability off the trust during construction. So as air Force members are added to a trust while constructing it just so it doesn't for the part. And then once everything has been constructed, those air force members will then be removed. So and that's basically the purpose of the Air Force members. But in the next video, we will be doing an example off identifying the zero force members from a trust. 45. Zero Force Members Example: in this video, we will be evaluating every single member in the trust. Andi, I'll be explaining why or why not. It is a zero force member. All right. So, as mentioned in the previous video, we have scenario one and scenario to for identifying a zero force member so we can start off by looking for any scenario, one where we have to members that are connected at a joint. So if we look at this diagram here, um, we have the situation of here where we have forced Member F G and member G E that are connected at a point. All right, obviously, this point here doesn't count because there are other members connected to it. So we're just looking forward to members are connected at a point. So here we have this scenario. But force members F G and G E are not zero force members because we have an external force acting at the joint. All right, sir. Number if two members, former trust joints and no external forces act on the joint, then birth. The members are zero force members. So if we had no external force, then these two members would be zero force members. But because we have external force, they are not there a force members or not. So we then no force forces If G or members F g 4th 50 is equal to is not equal to zero on force Member G E is also not equal to zero. All right, So, um, we have forces acting at this rock and at this hinge. So because there's a rock and the hinge, we know there's going to be forces acting in these members and all the members connecting to it, because these forces actually act as external forces. So that means that members f e e g e i on e. D. All not equal to zero. So then he could not arrive force if e is not equal cheese era force e I is not equal to zero, and force e d is not equal to sever. All right, so what we can do? We can look here. We know this forces and equal to zero Andi. Now we want to know if I j is a zero force member on i d. If I ds is Air Force member, so we will look at scenario to all right over here. We have to Caroline, your members. And then you have 1/3 member That is not clearly near to the to. So that's what we have here. E i and I j or Kaleena. And then we have the third member i D. So I d would be a zero force member if there weren't an external force acting at the joint . All right, so remember, be here. We have the two, uh, members that are Colin here we have the third member. That is not Colin. Yeah, a C would be. Is there a force member? Um, if there's no external force acting at the joint, but because we do have a external force acting at the joint, then if a C is not equal to zero along with if a b and a d So we have this external force. So you know that Force E Eyes and Article 20 Force I J is not equal to Zahra and force I d is not equal to zero. All right, so we can then move on over here to this, um, section. So we know Force E. D. Is an article 20 Andi, we then follow the same rules as the scenario to over here. So as you can see because we know if I d has a force, then that means there's an external force acting at this joint. She are working with this this member on this quill in your member, these talks bologna and then we have the third member D. J. So we know that D J is not equal to zero because we have d I, which is an external force which is not equal to zero. All right, so once again, we are looking at this scenario here where the external forces acting at the joint. So then the third member is nor equal to zero. So then force deejay the not equal to Zahra and Force D. C is not equal to Zahra. We cannot look at, uh, this section here We have members D C, D B and C J. All right, Andi, by looking at this, there is no external forces acting at the joint the trust joint. So because there's no external force acting here, we know that force CJ is in fact zero force member, So Force CJ is equal 20 all right. So why is that here We have that's equal in your members A B and A D. We have the third member that is not Catalunya. And there's no external forces. Um, that acts on the joint trust. Therefore, that third member, if a C is equal to zero, all right, But then also, force A B is equal to force a d, which is not equal to zero. So we know that Force DC's an article jazira on, you know, forced CB CB is not equal to zero. We already wrote that D C is not equal to zero over here. All right, so moving on, we could look at maybe this joint here we have J H Member, we have Member H A and you have member HB. So we have the two color linear forces acting along this line, and we have the third member at the trust joints HB. All right, so there are no for external forces acting at point age, So we can vin state that the Force HB is equal to zero. So this may be here is zero force member cancer gave me applying scenario to all right this year, so this may be here. Is these Air Force Member moving on? Um, we could also state that members H J and H A are not equal to zero because off this scenario writes a B and a D on Article two. Zahra So therefore Force H J is in article 20 and force A H is not equal to their. So now we are left with three more members that we haven't evaluated, which is this member here. This may be here on this member here. That's, uh because this member be age, force or force. HB is a zero force member. It means there's no external forces acting at this trust joint. So that means we have C B and A B A which are Kaleena running along the same line. And then we have the third external force beat J on. Therefore, that means B J is a zero force member because we have no external forces acting at the joint. So Force BJ is equal to zero and then once again, we can state that force CB is not equal to zero and force be a is not equal to zero for specie is not equal to the error and force be a not equal to their. You could have worked backwards as well on Noticed that Force H ah and Force A B are not zero force members because we have a rocker acting at the trust joint. So if there's a rocket here means there's an external force acting at this point. And because it's an external force acting at that joint, it means that the two members connecting that joint are both not equal to zero. So that means Force H and B A or not equal to his error. And you can confirm this by looking here for age. Andi Force A, B or B A are both not equal to zero. So you know that is correct. All right, so and that is how you identify the zero force members in the trust below. We just need to apply these two scenarios. Andi. It will. It is easy marks to get in any exam 46. Frames and Machines: in this video, I'll be teaching you guys what you need to know for frames and machines. All right, So what are frames and machines? It's when the force is common to two members that are in contact, usually with a pen act with equal magnitude but opposite directions on the free body diagrams off the members. So you basically have, um, for example, two members connected Bipin. Then, if you draw that the free body diagram separately, you will then have, or you're then draw the components off the forces acting at that point, both in equal magnitude, but in opposite directions. This will come a lot more clear to you when we do the example. So there are three things you need to know. Um, we're just three useful tips for frames of machines. Basically, the number of equations need to equal the number of unknowns. That is only where you will be able to solve for unknowns. The second thing is that some of the moments about a point must be done at the point of intersection off as many unknowns as possible. This is basically the key to being able to solve these problems, so that's very important. We will be looking at this point in the following example on, if the solution of a force is found to be negative, it means the force acts in the opposite direction to that chosen on the free body diagram. All right, so the next video, we will be doing an example with frames and machines on show to you step by step on how to go about solving such a problem. 47. How to Calculate the Centroid: in this video, I'll be teaching you guys hard to calculate the central off a work, please. So the first step you need to do is define your X and y axis. That is way is zero point off your Cartesian plane on that two dimensional work piece. All right. You don't really need to stress too much about that because this should be given in the question they'll tell you calculates the X and Y c enjoyed about points on. Then they'll mention the points. All right, So how do you do this? You use these two formulas. If the Letter X has little squiggly line on top, that means it is the central it writes about represents the X enjoyed and that represents the Y. Essentially, the Inter calculated you some areas times the same choice off the shape. So and then you divide by the sum of the areas. If the area is, um, an actual part off the metal, where peace, it's positive. But if it is simply negative space, you then subtracted that will become a lot more clear to you when we do an example. All right, so you should just remember that on the right off the X access the EC central. It would be positive on the left, it would be negative. The same applies to why, except above the, um X excess. We have the positive Y Centro roids and below the y axis we have negative. All right, so when we do the example, you'll see exactly what I mean. How the signs come into play when calculating the same droid. All right, Um, I mentioned here again at positive space and subtract negative space. All right, so this would be represented positive space. And wherever there isn't the metal work piece that would represent the negative space, Alright. And just the quick thing to notice. Um, it is possible to calculate the central of the piece that where the century does not lie on the work piece itself. So in this case, it's possible that the central it could lie at this point a point in mid air that's not actually on the work piece itself. So that's just something to note how to calculate the central's. Basically, the work piece would be a combination off several common shapes, and then you can use the rules off Central. It's off those common shapes on apply it on a larger scale. So over here we have the central of common shapes. Basically, you'll have a plane or composite area, and then you can use these different rules. I'm don't. You don't need to worry about memorizing all these formulas, although it does help if you just know them off by heart. But you will be given a formula sheet in the tests, where they have all these shapes with the dimensions, and they are all labeled as well as the areas. All right, so, to calculate the century of a triangle, it is the distance off the length of the triangle to the point of the right angle divided by three. And then the same applies in the Y direction that high divided by three are writing in. This point here would represent the central read off the shape to calculate the area of a triangle. It's half base times height so that you have the base times height divided by two, all right to calculate the century off a semi circle. It is for over three pie, and that will give you the distance from the the halfway off the circle to the points off the central aid. All right, we are represents the radius off the semi circle, um, the area of the semi circle. It's just the area off a circle pirate squared, divided by two. All right, and then the area for 1/4 circle is power squared, divided by four. All right to calculate the central oId off a semi circle. It's for our divided by three pie from the distance of the right angle on a gain in the Y direction for of a three pie on. Then that point, they would represent the central or your shape. This one's a little bit more complicated, but it's not too complicated. If you understand it to calculate, the century often arc. All right, so basically, it's not a semi circle, not 1/4 circle. It's just a certain, um, piece off a circle or rights on and the central. It would always lie mid points between the the Ark or write to You had have an angle often offer, so there's two angles are equal, and along that line at a distance off to our sign offer, divided by three offer that would give you the central read off the ark, where R represents the radius of the circle, and A represents the entire angle between the arc, divided by two. So that's often offer. It's basically the entire angle between the two sides, divided by two and in the area to calculate the area of her arc offer, which is this angle multiplied by the radius squared right. So in the next video, we will be doing an example off, hard to calculate the same toyed off a worth piece or a compass. It's area. 48. Centroid of a Wire: in this video, I'll be teaching you guys all you need to know for how to calculate the central off a wire . So what we did before is we were calculating the same Troy off complicit areas. So of a solid item started shapes. We're in this case, it is a wire. So it's century off this wire, acting along the line of a semi circle off this quarter circle off this park. All right, so just a little hint. Um, it hasn't been tested before, but it's just good to know if the wire were complete. So this semi circle, um, then had a wire all along here. All right. That would then mean that you would have to use the central off a common shape. All right, So if this quarter circle had a wire acting along that line and that line, you would then use these formulas to calculate the same story because you're working with empty space there, and it's just the wire acting along these lines. You were then used the formulas given on this page. You don't necessarily need to worry about memorizing all these formulas. They will be given to you in the test. All right, so just a little hint for calculating the same trade off a wire. I'm with an arc. This also applies to the same toyed of a common shape. So this formula on this formula must be calculated using radiance. That's very, very important. All right, so we will do an example, right to calculate the century of a wire often arc, so I can show you how to calculate its using radiance. So here we have the question of states. Find the X and Y century it off the wire. So there are two ways you could go about doing this. You can either calculate the century off the quarter circle on the small arc and then add them together. Or you could simply just do it for this entire arc and then calculate the answer. So I will do it that way because it will be a lot quicker. So if we were to redraw this circle? No, this talk. Rather, we have 50 degrees here. All right, which is that city. And then here we have 90 degrees, cause it's a right angle. All right, so then this entire angle between this line on this line would be equal to 90 plus 50 degrees. So this angle here is 90 plus city, which is equal to 120 degrees. All right, so if you look out the formula for the ark all right. The central it runs along the center off the ark. As you can tell by that often that offer being equal. All right, so we need to call Kate offer. All right, so it's me that off away. So you know, the same toyed runs along the centre line, all right, between the ark with this angle is equal to offer, and this angle is equal to offer. All right, so now we know this is 120 degrees, but for Arc formula, we have to work with radiance, as I had mentioned. So, you know, pie is equal to 180 degrees. All right, this is an old crisscross productor I've been using for years, and it's very useful for converting anything into anything. Basically, as long as the relationships Alenia So pi is equal to 180 degrees, then 120 degrees is equal to home match radiance. So it's a little crisscross method So you say 1 20 multiplied by pi, divided by 1 80 1 20 multiplied by pi divided by 1 80 All right, that is equal to 2/3 pie on. We know offer is equal to half of that 120 degrees. So it is then to over three pie divided by two the twos or cancel art and will be left with pi over three as offer. All right, so we cannot calculate the radius or not necessarily the radius, but the distance from the points. Oh, or the the center off the ark To this point, which is the point way the century is acting. So use the formula Our sign offer over over. So our sign Alfa of offer is equal to R is equal to four meters. All right, so it's four meters Sign pi over three divided by pi over three. Now you can ensure your calculator is in radiance by doing the following you click shift mode. Oh, set up the option number four is radius or so A reindeer radiance. So you click four and then you'll see at the top over there. It will tell your calculator is in radiance are. And if you can see that with it isn't radiance. So we cannot plug this into our formula and also into a calculator four multiplied by sine pi over three, divided by pi over three equals. And that is equal to 3.3, uh, eight meters. All right, so that basically means central. It is acting along this line at 3.38 meters from the center. So this distance here all right, is three points, 308 So if we redraw that, yeah, 3.38 meters at an unknown height. All right, on. We know this angle here is over, all right? And offers pi over three. So, you know, this is pi over three, and we know the entire 90 degrees 90 degrees as pi over two, because it won 80 divided by two. So that's 90 degrees to get this angle over here, we say pi over two minus pi over three. All right, because remember, we know offer is pi over three. So pie of a to minus pi over three. That gives us a pie of a six. So this angle here is pi over six on we are. What we trying to calculate is the X and Y central. Eat right X and Y century off the wire so we can then calculate the X coordinate all rights using python gris. So here we have the adjacent side, the opposite side and the high party news off the triangle. Onda, let's say we want to find the ex central at first we are then looking for the adjacent So adjacent over hard party news is cause off eater. So the Jasons, which is the essentially, is equal to cause off theatre cause off price of a six multiplied by the high part. Amuse you just taking that H up there which is 3.308 and that is equal to cause of pi over six. Multiply by 3.38 which is two point 86 meters rights to calculate the wife's enjoyed. You do the same thing except working with the opposite of a high party news. So there are calculating the sign off. Pi over six multiplied by 3.308 All right, so you could just save time itself typing that aren't you just go like this? Delete sign equals, and that is 1.6 five meters. All right, so here we have our ex central rate 2.68 Sorry. 2.86 And our wives enjoyed 1.65 So if he were to draw this somewhere on this diagram, it would be at about this point here with this. X points is equal to three. Sorry. 2.86 So that is 2.86 This why is equal to 1.65 and they have your final answer for the central aid off the wire. 49. Reduction of a Simple Distributed Loading: in this video, we will be doing the reduction of a simple distributed loading. So the area of a distributed loading will determine the magnitude of the force. So you basically need to know how to calculate the area off a rectangle and a triangle, which is length times, breadth. Andi, half length times breath, respectively. Andi, In order to calculate the area off a rectangle, for example, you would say the killer Newton per meters, which is, um, the units off a distributed loading multiplied by the length at which it's distributed across in meters, and that will give you the final magnitude of the fourth in Killer Newtons. All right. You could have done this with just knew temper meters, and then get Carter answer in mutants. All right, so what do you need to know? You need to know. Um, central rates. You have to use knowledge off Centrowitz to position the forces. So we know the sentry off a rectangle is in the middle, and we know the central it off. A triangle is 1/3 from the right angle off the base. All right? And then use knowledge or moments to position the result in force. So let's do an example. All right. This example in all six s to determine the result in force and specify where on the beam it acts measured from a So here we have point A Here we have a distributor loading 10 cologne . You temper meters across four meters and five killer Newton per meters across six meters so we can calculate the magnitude of the force by calculating the area. So let me just write that the area equal to the magnitude Okay, off the force. So for this section here we have 10 killer Newtons. The meter multiplied by four meters. You'll be left with 40 Killer Newtons. All right, that's the magnitude. And then for this section will have five killer Newtons per meter multiplied by six. And we work and have 30 killer Newtons. So where do you position these result in forces that with this part comes into play, use knowledge off Centrowitz to position the forces. So this is a rectangle. So the central oId off the rectangle is in the middle and the century off this rectangle is in its middle. So if I were to redraw this beam all right, you have point a over here, then halfway through this point is two meters because you know this link is formulas. So at two meters, we have the downwards 40 killer Newton Force and then halfway through this section or this area is three meters. So this lengthy is three meters plus four meters, which is seven meters. So we could label this distance here as two meters and then five meters after that which will make it seven meters from this point to halfway, we will then have 30 killer Newtons. All right, so the question asked us to determine the result in force and specify were on the beam. It acts. So to calculate the result in force, you simply add the forces. So force resultant is equal to 40 that 40 Killer Newtons plus 30 killing mutants. That's 70 Killer Newtons acting downwards. All right. But now you don't know in what direction or sorry, not which direction in which position reverse result in force acts. So that's where the third point comes into play. Use knowledge off moments to position the result in force. So in order to do that, you do this. Some of the moments abouts points a must equal zero. So you're doing the moments about this point so you could do that. Or you could simply just say the result in moment is equal to the sum off the other moments . All right, so this is the method will be using because we know the result in moments. So now we know that this is acting in a downwards directions. It's negative. 70 Killer Newtons, 70 killer mutants, Um, acting at a certain distance, x multiplied by X equals. So that's the moments of the force multiplied by the distance not supplied by 40 also acting in a negative way in a clockwise direction. So it's 40 killing mutants multiplied by and two meters Andi. Then once again, we have minus 30 killing mutants multiplied by seven meters, so we could then solve for unknown with just six so X is equal to minus 40 times tu minus 50 times seven. Divided by minus 70 they're full X is equal to full 0.1 four meters. So therefore, if you would redraw the result in force at the specific distance, if this is the beer and here we have point A on the entire length of the beam is 10 meters then at if this represents five, then this would represent about 4.1 acting downwards. This distance here is 4.14 in this distance here would equal 10 minus 4.14 which would be equal to 5.86 meters. And then the magnitude of this result in force as equal to the calculated 70 killer in mutants. And they have it. So that is how you determine the result in force. Andi specify were on the beam it act measures from a 50. Reduction of a Simple Distributed Loading Example: determine the result in force and specify way on the beam. It acts measured from a. So, as you can see in this diagram, we have a distributed loading. All right, you can just ignore the red. For now, we have a distributed loading Onda. We need to calculate the result in force. So in order to calculate the results in force, we first need to convert these distributed loadings into the magnitude off the force. How do you that How do you do that? You simply calculate the area. So the area is equal to the magnitude off the force. So in this case, we can calculate the area off this triangle over here. That could be our first step. Our second step would be to calculate the area off this triangle, all right. And in the third step would be to calculate the area off this rectangle and then by adding those treat areas together, you'll results in the result in force. All right, so let's start off by calculating the magnitude off this triangle. We can label them one to on three. So for one area off a triangle is based times high two divided by two so base is equal to five meters multiplied by the height and the high tier is four killer Newtons per meter. So that four killer Newtons per meter divided by two who will then have 20 divided by two which is 10 killer Newtons. All right, it's mutants because we have killing Newtons per meters multiplied by meters in the numerator and they end the meters. Cancel out and you're left with killer mutant. All right, so that's a magnitude full Section one for Section two, we use the same formula base times height divided by two because it's also a triangle. The base in this case is four meters. That length is four meters, four meters multiplied by the height which is this height over here. So we know the height at this point is four killer Newton per meter on the height at this point is to killing Newtons per meter. So basically the height over here from this point to this point is four minus two. All right, which will be to killer Newtons per meter? Divide that by two, you're going to have four times to which is eight and then you are going to be left with four killer Newtons. All right, we can then calculate third area. And you know the area off a rectangle is the length times breath. So here we have, um, length off particular Newtons per meter and a breath off four meters. All right. So you can just write four meters multiplied by two killer Newtons per meter. Then you'll be left with eight killer mutants. The result in force is equal to the sum off the areas which is equal to 10 plus four, which is 14 plus eight, which is 22. And there we have the magnitude off our result in force. All right, so we have answered the first part of the question. Determine the result. Enforce the second part. It's specify way on the beam. It acts measured from a. So in order to do that, we need to use knowledge off Centrowitz to position the forces so we can draw the Centrowitz of these forces. All right, so let's redraw all this beam neurons that we have the beam here. We have point a All right, Andi, The central oId on this triangle would be about here. All right, way. It's 1/3 off the distance off this length. So this distance here is equal to 5/3 that if you remember from the central videos how to calculate the central it off common shapes. All right. Just to recap here we have the base length to that Central is based of our by three. All right, and then the same with the heights. But in our case, you're only working with the base. So this lengthier is 5/3. So, basically this length here, from this point to the center, oId is five minus 5/3. All right, so we can draw that. And five minus 5/3 is equal to three point triple three. All this distance here is 3.333 meters at a magnitude off 10 killer mutants. All right, so now we can continue and do the same for the other two magnitudes we have. And this triangle living here, right? The central oId is once again 1/3 off this length. So four divided by three. Is this distance over here? All right, so then basically the distance from a to this point, it equal to five plus 4/3 search this distance. Plus that distance five plus 4/3. So I could do that. Five plus for over three. On day we have the distance. All right. And then if we simply subtract this distance minus 3.333 we are left with three meters from this point. So that is three meters, or you could have simply written the entire length, All right, as 6.333 meters. All right. And then the magnitude here, this arrow length should be longer because it's a larger magnitude. So 10 killer Newtons, Larger error or right? So Ohio or taller. And then here we have, um, a smaller error for killing Newtons and then lost way have a killer. Newton's force. All right, which is due to the rectangle and essentially off the rectangle is in the middle of it. So from this point to this point, we have two meters. All right, so five meters plus a two meters equal to seven meters. So we can then draw our eight killer Newtons force at the distance off seven meters or out . So notice how this magnitude has the longest error. This one has a shortest, and this one is in between by looking at the magnitudes off the force. All right, so what we can do now is calculate the position off the result in force. How do you do that? You need to use knowledge off moments to position the result in force. So you say the result in moment is equal to the sun off the moments about point a. All right. So you could also say that some of the moments about point A equals zero and then include the result in force in your calculations. But I'm just going to do it like this because it's easier to visualize. So the resultant moment is equal to the result in force multiplied by its distance. So the result in forces truly 22 killer Newtons 22 multiplied by certain distance X that we are trying to calculate. All right, because that's the position off the result in force on because it's acting downwards. It's rotating point a in a clockwise direction, so that is negative. We then do the same thing for this 10 continued, since it's once again rotating this in clockwise direction. So that's minus 10. Killer Newtons multiplied by the distance. 3.333 Here we have the four Kilo Newtons once again also rotating clockwise finest four killer Newton's multiplied by its distance from Point A, which is 6.333 And then once again we have this eight minus eight at a distance off seven meters. All right, so we can then calculate the right inside of this equation minus 10 times 3.333 Remember we working with four significant digits because it's an intermediate step minus four multiplied by 6.3. He three minus eight times seven. All right, and you lift with minus 114 point seven for significant digits, divided by minus 22. All right, we just bringing this minus 22 Killer Newton's to the bottom on. We are left with five point 21 meters. All right, so they have your final answer. If you would like to redraw this on the beam, So here we have deem. All right, we have points. A acting here entire length is nine meters point a. Then here we have the nine meter mark on 5.21 would be just the right, onside off the mid point. And then we have that as 22 Killer Newtons. This distance here. I don't really our space here, but that distance is 5.21 meters. On this distance. Here is nine minus 5.21 meters, which is 3.7 3.79 meters on. That is how you had to draw your final answer. 51. Moments of Inertia: in this video, we will be doing moments of inertia otherwise known as second moments off area. So in the test, they could I either ask you to calculate the moment of inertia, or they can ask you to calculate the second moment of area. Just know that those two things mean exactly the same thing. First thing you need to know is that different shapes have different moments off inertia. So basically the different shapes such a circular sectors, quarter circles, semi circles, circles, triangles and rectangles all used different formulas in order to calculate that shapes, moment of inertia, moments of inertia or calculated about her line. All right, Andi, in order to calculate the moment of inertia bottom line, you use the parallel access there. Now, this is a very, very, very important formula You need to know. All right, So this is the formula. Um, Andi, I have listed or described what each components of this formula comprises off. So here we have I with Artur line on top. This I represents the moment of inertia about an axis. So that is what you're actually looking for. All right. And then this line is parallel to the axis off I hat to here. We have I hat with the line on top on this represents the moment of inertia about an access through the central right off the shape. All right, so this is the access through the central it, and this is the access that is not through the central it but, um, still parallel to this line. Then you add the area off the shape multiplied by the perfect perpendicular distance between the axis off I on the access off I hat. So, as mentioned earlier, I and I had or both parallel to each other, all right? And they are separated by a distance. D All right, so then that distance would be inputted into the equation as de here. And just remember to always add the squid sign. You must always square it. Lots of students tend to make the mistake with writers formula plus a d, and then they would forget the squid, and then the answer will be wrong. All right, So to get your final answer, you must some All of these inertia is that you calculate all the moments of inertia Barton access. So you some all of them and remember positive space. You add it on negative space, you subtract it. Then another thing to note here is if they ask you to calculate the polar moment of inertia or right that's denoted as J zero, Is there a represents origin off the axes that is equal to I X plus I y. So you add the X and the Y components off the second moments off area. So in the next video, we will be doing an example where we have to apply the parallel access their, um all right. But before we do that, I will just briefly show you the different formulas you can use to calculate the moments of inertia over here. Said, In order to do moment of inertia problems, one needs to know how to calculate the century off the shape as well as apply the inertia formula based on the position off the axes. All right, So if you don't remember how to calculate the century of the shape, um, maybe go briefly review those central oId videos, Um, in this course. So here I have a few examples off how to calculate the area or moment of inertia here I say see formula sheet for all shapes. So you don't need to worry about memorizing all these formulas because they will be given to you on the formula sheet. But basically, if you're given a circle and you want to calculate I X notice, I has a hat on top of it because this is a moment of inertia about the access through the same Troy off the shape or right, so it has the house twin. It's through the same Troy. So this is I X through the centuries of the circle 1/4 pi r to the power of four. I wy is you use the same formula. All right, I'm for rectangle over here. You would with a central in the middle off the shape, calculate the moment of inertia using this formula where the highest cubed. But when you calculate I y hat, then the bases cubed all right. Same applies to when you do it with a triangle. The height would be cubed on the base would be cubed for the why. All right, so it's really not that difficult to calculate these, Um, it's just mainly about identifying where the access you're working with is on being using the parallel access their, um in order to calculate your moments off inertia. All right, so the next video, we will be doing a nice example that will be worth watching. 52. Moments of Inertia Example: determine the moment of inertia about the Y axis. So here we have an object, um, which is a rectangle connected to a triangle. And then there is a circular hole through the center off the rectangle. So to calculate the moments of inertia bottle, why access? We are going to have to use the parallel access theory, which is the following formula over here. All right, so what? We need to calculate the moment in off inertia about the y axis. Um, which in this case would be I Why we need to calculate the moment of inertia through the central. It's off each of the shapes, plus the areas off each of the shapes times the part of the perpendicular distance between the access off I and the access off I hat. All right, so this I represents the notion about a line where this one is the moment of inertia through the X or through the X and Y Central right off that shape. All right, So to get started, we can calculate the he nourishes through the state choice using the following formulas. Now, these formulas will be given to you on the formula sheet, but I just drew them art over here for the sake off the videos so we can start off by calculating a moment of inertia for the rectangles. All right, so we want to calculate I y through the central off the rectangle. So here we have the rectangle. We use this formula, so that is 1/12 height base cubed. All right. No base is the one that is cubed for I Y and height is the one that's cubed for I X. So just know that the base refers to the horizontal length off the rectangle. So if you were to write that out, it's one of the 12 times 1 40 millimeters times, 200 millimeters cubed. If you plug that into a calculator, you'll get a value off. 9333 3333.33 All right, on that is in millimeters to the power off four or aren't always remember that inertia? The units is in, um, meters or millimeters to Power Force. It's just the length unit. So could be centimeters. It could be millimeters, but whatever it is, it's always to the power off full. All right, so we can now calculate the inertia through this enjoyed off the triangle. So here we have the triangle formulas on D I Y is 1/36 multiplied by the height multiplied by the base. Cute. All right, so that's one of the 86 at times 1 40 All right, because the height is once again 1 40 This height here on the base length is 200. Cute. All right, because that's the length off the base of the triangle if you plug that into a coach later located value off 31111111 0.11 millimeters to the powerful. And once again, we will do this full with this circle. So I wy through a circle is one of the four pi r to the powerful. All right, so that's one of the full pie are to the part of full Andi. In our case, the radius is equal to 50 millimeters Saudati 50 to the part of four on that will give you an answer off. 636172.51 millimeters to the power of four. All right, so now that we have calculated the inertia through the centuries, we can calculate the areas. All right, so the area off the rectangle is high times base that simply 1 40 times 200 which is equal to 28,000 millimeters squared. All right, here we have the 200 the 1 40 that is we are got those two values from the area of the triangle is high times base over to all right. And that is equal to, um, the height 1 40 the based 200 divided by two. So that is 1 40 times the signal that times 200 divided by two to the 28,005 by two, which is 14,000 millimeters squared. Right for the circle pyre squared is the formula for the area pie. The radius of a circle is 30 millimeters. So that's 30 squid on. That will give you an answer off 28 to 7 millimeters square. We're not, so we can No, um, try and calculate the nurses, all right? Using the parallel access there. All right, so we want to apply this formula now, So I is equal to I had to plus a d squid. So for the rectangle. We have eyes equal to I had, which is the falling value. 93333333 All right, plus the area, which is calculated as 28,000 28 1000 rights and and that is then multiplied by the distance squid, where d represents a perpendicular distance between the access of I. In our case, it's the Y axis on the access off I had, which is the access through the central right off the shape as defined over here. All right, so if we look at our question, um, but the perpendicular distance between the central off the rectangle the why Andi, the Y axis is 100 millimeters or rights. Ah, that is times 100 millimeters squared. All right, on. Don't forget this squared over there, that is an equal to 3 73 times 10 to the sixth. We can say 373.3 times 10 to the six millimeters to the power of full. All right, because we have millimeters squared plus millimeters squared are all millimeter to the power of four, because areas millimeters squared distances millimeters Your squaring that. So you're gonna have, um millimeters to the power off. Four. All right, so we can continue and calculate the inertia for the triangle. So and always with a little triangle. So I is equal to I threw the central rate, which is the following 31111111 All right, Andi. Plus the area of a triangle, which is 14,000 millimeters squared, plus 14 1000 millimeters squared, multiplied by perpendicular distance between the central right off the triangle. So the century of the triangle would lie somewhere over here. All right, so it's basically the distance from the Y axis to the point of the centrally. All right, So this distance here from here to here is equal to 200 divided by three rights. And if you forgot about that, you can go revises central videos. So 200 divided by three. That is equal to 66.66 millimeters. So that is that length over there, Plus this entire length off the breath off the rectangles. That plus 200 we'll give you 266. So then we have the length off 266.7 squid. All right, so therefore we have the eye of the triangle, equating to the following 31111111 plus 14,000 times to 66.7 squid. All right, so that is ones. There are 26 whole. 1027 times 10 to the six millimeters to the power of four. All right, we can they in court later, the notion for circle so I hat for the circle was calculated as the following value. All right, sir, I'll just write that that 636172 0.51 All right, so just write that out over there, plus the area all right off the circle, which was calculated as the following 28 to 7 millimeters squared, plus two eight to seven millimeters squared, multiplied by the distance or the perpendicular distance squared. So that would be, um, the distance from sent off the circle to the y axis. So that is 100 millimeters. So then we'll have 100 millimeters squared. All right, so you can plug this into a calculator, and then I get equal to 636172.51 plus two eight to seven times 100 squared. All right equals 28 point 91 times 10 to the sixth millimeters to the powerful or rights. Now the trip comes in. When you look at the shape and you see the rectangle, will this area here it's positive. Space Triangle is positive space, but the circle is a hole through the objects, so that's negative space. So then you define this inertia as negative, all right, because it triangle is positive space. It's positive on the rectangle is positive as well. It's only negative for the circle because it's a hole through the object. It's not actually solid space right to the Inter. Calculate the the nausea or the moment of inertia about the Y axis. Um, we could then some all of the nurses about the why these are actually all about the Y axis . All right, that is an equal to this in Isha. Plus this scene OSHA minus this inertia. All right, so we'll have a following 373.3 times 10 to the six +1027 dumped into the six minus 28.91 time since of the six, and that will give you this large value off. 1371390 Reserve there. Zero millimeters to the power off full. All right. So it usually want you to get your aunts and for significant digits so that you could click this engineering button all rights. Ah, let's say we answered like this. 1371 times, 10 to the six millimeters to the power off for all right. And there you have your final answer. 53. Internal Forces in Structural Members : In this video, we will be doing internal forces in structural members. So to calculate the internal forces, the method off sections must be used. So what you actually going to be doing now is you'll be given a beam and your section it or cut it at a certain point along the beam, and then you would then draw the free body diagram off that cut peace within the being all right or a part off the bean. So just to visualize this for you, when sectioning a beam, you can use the falling sign Convention. If you have this beam A and this section it along the beam at this point here at a all right, it will be broken up into two Hoffs. All right, the left hand side, on the right inside. So now this is a sign convention you must follow. Um, every single time you do one of these problems. So if you want to work with the left hand side off the beam that you cut, then the sheer force which is V must be acting downwards. We're arts, But if you want to work with the right inside, the sheer force must be acting upwards. The normal force will always act out off the beam or rights. So that's perpendicular to the surface, which is section attracting outwards on the left. Our tweeds on the right, All right. On day, in the bending moment on the left hand side, off the section bean, The moment must act in the anti clockwise direction and for the right inside, the moment must act in the clockwise direction. So this you just need to memorize. All right, Andi, a question you might have Ask yourself is hard. You know which side you need to work with. Do you work with the side, or do you work with this side? The answer to that is you can work with either one. Usually the best side to work with would be the one that would be the easiest. All right, so here, have a little note. Draw the free body diagram off the part of a beam that would be the most simple to solve, right, If there's less forces on the left hand side off the beam, then you would choose the left hand side. If there's less forces acting on the beam on the right hand side. Then you would work with the right inside in just an extra note. Here, use the equilibrium in equilibrium equations. I eat some of the forces in the Xmas equals era. The some of the forces in the why must equal zero. And the some of the moments about a point must equal zero over here, our roads about point A because most of the times you are solving four points. And if you choose point A, you are then eliminating the need to calculate the moments off these forces at this point. And then you can calculate the forces at other points along the beam. All right, this may seem a little bit confusing at first, without a practical example. So in the names video, we will be doing an example where you apply this sign convention. Andi, I'll show you guys how to choose the best side to solve the problem. All right? 54. Internal Forces in Structural Members: determine the normal force. Sheer force on moments, a point B. All right, so this means we have to section point B, and then you can choose whether you want to work with the left hand side or the right inside. All right, I will choose in this case, you could choose any because both sides are fairly simple. But I work with a right and side because this side only has X and y four sets c as compared to this side which has this force Onda force at point A. All right, so the first step, like any problem, is to draw the free body diagram. All right, so a free body diagram, we have the beam as a whole. Then we have a force acting ill suited acting in the Y direction. And we'll in the positive y direction. And I assume here, acting in the positive X direction. Then he had a We have a force acting upwards again. Call there. It's actually from this point here, we can call that force a and then we have a downwards forced you to the distributed loading acting downwards over here. Right on the magnitude of that force is the area should eight killer Newtons per meter multiplied by three, divided by two. That's simply the area of a triangle off base times height. So eight times three defied by two. That is 12 Killer Newtons. Acting don words. That's a distance off to meters. All right, we know this is two meters because the central off this triangle is over there. So this distance here is the length or the based of other by three. So it's three divided by three. So that length is one. So therefore three minus one is equal to the distance from this point, which is two meters. All right, we know this distance here is three meters. All right, Andi, we know this distance from here to here is a total off four plus two, which is six six meters. All right. So we wanted became choose whether we want to work with the right or left inside. So, as I mentioned before, I'm gonna work with the right inside. So that means we need to know the value off, see why? All right, so we can do the moment about point A because you don't have the value off the force. So the moments some of the moment about point a must equal zero. All right, So firstly, we have this 12 killing it and force acting in the anti clockwise direction. All right, that's 12 killer mutants multiplied by the distance between it, which is three minus two. That one meter. And it's positive because it's acting anti clockwise. And then here we have see why, which is acting in the once again, the anti clockwise directions that's plus see why multiplied by the distance, which is six meters, and that must equal zero. All right, so see, why is then equal to 12? Divide 12 Killer Newtons. All right, so you take out of this. It's minus 12 divided by six, and you're left with minus two minus two. Kill. Oh, mutants or right? So, um, notice how we get a negative answer, which means see, Wise, in fact, acting downwards, we are previously assumed it Accent upwards. All right. So, to calculate the value of C X, you start to do that, some of the forces in the X direction must equal zero motor. See, X is the only force acting in the X direction. Therefore, see X is equal to zero. All right, so just another thing I want to mention we needed that some of the moments about a equals era. We didn't include C X because a line of action off C X runs through point A All right points a on. Therefore, there's no perpendicular distance because we're assuming that this, um, this road dozens have a length. All right, if it told us it had a length. All right, Um, then there would be a certain distance from this points at the bar here to the sent off the bar, but because they don't give us a length or thickness off the bar, we assume that it runs along the same line off action as a all right. I didn't confuse you too much, but, um, we cannot move onto calculating the internal forces. All right, so we cannot draw the right hand side of the spot. So we are sectioning it, and we are working with the right hand side where we had that force. See why you get a negative answers. And now we can draw it in the correct direction. So that means see, why is equal to positive two killer in mutants. All right, and then see X. We simply don't even need to draw it because its magnitude Azera All right, So he section did at this point here. All right, Andi, due to the sign convention, if you choose the right inside over every section, it's in V A, which is the sheer forces acting upwards. So here we have v A. The normal force is acting outwards in a on the moment is acting clockwise. So just roll their in a different color. That's a moment at a all right? So no, it wants us to calculate e a m A. And in a all rights on do we know the length off? This party is two meters, right? Because that's where resection dudes. So we have a to meet a long bar. Rights of EA is simple. We do the some of the forces in the why must equal zero so upwards were V a down with we have to kill a new tenants of that minus two killer Newton's equals zero. Therefore V A is equal to two killer mutants. All right, in a normal force, you the some of the force in the X must equal Zahra on in a only force acting in the X direction. Therefore, in a equal to Zahra, then to do the moments we do the some of the moments about point A that must equal zero. Here we have the moments acting clockwise. So you know that's negative. M A. Forces via an n a act at this point A. So we don't include those forces in the moment formula. But we do include this force. See why which will make it rotate in the clockwise direction. So once again, see, y's negative says minus, see why which is equal to to kill a Newton. So you could simply just right minus two killer mutants multiplied by the distance perpendicular distance From that point, which is two meters, that must equal zero. All right, so I m a is then equal to full kill. Oh, mutiny meters on its negative. All right, so that means I am a is actually acting in the anti clockwise direction. All right, so they have your final answers. Don't change your free body diagram. You could leave it like this in a is negative, as long as you drew it clockwise on your free body. Diagram on that is your final answer 55. Shear Force Diagram Example: draw the shear and bending moment diagrams off the beam. All right, so this question will be broken up into two separate videos. This first video will be we will be doing the sheer diagram. And in the following video, she time purposes. We will then do the bending moment diagram. So before you start with one of these questions, you need to stall for all the unknowns in the beam. All right, so once again, you start off by drawing the free body diagram so you can call this a free body diagram off the beam. At a what? At a We have a hinge. That means we have both a y and X components so human upwards direction for a why and also humor left direction for a X. Then we have, um we have this distributed loading, acting over four meters. So four multiplied by 1.5 is six, six kill. Oh, mutants. All right, acting downwards. And that is at the center off these four meters. So it would be two meters into two meters from this point. So it's four meters from the start. Can it be? We have this, um, little rocker. So that causes a perpendicular force, and we'll call that B y. All right. So, two, if you do this, some of the forces in the why we will obviously have to unknown, so that wouldn't work. So you'll have to do the some of the moments either point A will be. So start off by doing the some of the moments at point A that must equal Sarah. So you have this six killing mutants acting in the anti clockwork in the clockwise direction, which means it's negative. So have negative. Six. Killer Newton's multiplied by four meters because at a distance from a to the central oId off the distributed loading. So here we have two meters plus two, so that's simply for all right. Then we can se plus B y multiplied by four plus two, which is six pizzas equals zero. All right, it's plus because it acting in the anti clockwise direction be. Why will make point a rotate anti clockwise? So therefore we saw for B y who will then have a answer off four Killer Newtons because the answers positive, it means we assumed the correct direction for P Watt. Now we can do this some of the force in the UAE in order to solve for a y. So we will then have a Y in the positive Y direction minus six Killer Newtons plus four Killer Newton's equals era therefore a wise equal to two killer mutants. All right, so because you get a positive answer once again means we assume the correct direction for a why now, to calculate a X, you do this some of the forces in the X direction and because a X is the only force acting in the X direction, you know, a X is equal to zero, all right, But not that X is really required because we are working with the sheer diagram. So we only need the forces in the Y direction. All right, so, no, we have a x a why and b y we can continue on draw our sheer diagrams. So when you section a problem like this, you need to see where there is a change in force. So basically we have a force acting over here, right? And in that force will be even along this distance, all rights. And then there's a change in force over here by the distributed load. That means we are going to have to section this beam twice. All right, so we're gonna have to section eight year when you're going to have to section it here. All right, so let's say, for arguments sake, this distributed loading act stopped off the two meters and then it was just normal beam again. That would mean you'd have to section it three times. All right, so in this case, it's only twice so let's start off by doing that. So we're going to draw the free body diagram. Onda, we are sectioning it. All right. And we have the the internal four city we have the normal force, which will ignore for now and then we have the anti clockwise moment because he's section it from the left. Remember the sign? Convention left hand side internal sheer force acts downwards. If we sectioned on, we kept the right inside of the beam. We would then say v a act upwards. All right, so we also have this a y. And then this is our free body diagram. So we want to calculate V V is equal to a y. All right. You're simply doing the some of the forces in the wire that must equal zero. So that's why he is equal to on a why. And a why we know is equal to two killer mutants, All right? And so that the first part off off the sectioning done. Now we need to section the second part off the beam, which is three here. All right. So filled with a free body diagram of that. So you are sectioning. It's some way within the boundary or length off the distributed learning. We have a Y as our two killer Newtons rides. We have I was, uh, downwards sheer force on. We also then have our distributed loading acting downwards. All right, so we can assume that this distance here is our two meters, which is this distance here. And then we can assume the full length off the beam because we don't know where we sectioned it. We can assume a length off X. All right, so this is a general procedure to solve these problems. You define the length as X, which would then mean at this distance here is equal to X minus two or rights. So therefore, the magnitude of are distributed. Loading is 1.5 killer Newtons per meter multiplied by X minus two. So 1.5 multiplied by X minus two. All right, and this is all measured in killer in mutants. This is killing Newton's as well. So now we can do this. Some of the force in the UAE equals era upwards. We have to killing Newtons. Downwards, we have minus 1.5 X minus three, and then we also have a minus. V equals Zira. All right, so if we solve, we have two minus three. So you have minus one K minus 1.5 x minus V equals era. Therefore, the is equal to minus 1.5 X minus one, and that is all majored in Killer and Houston's. All right, so now we can go ahead and draw, Uh, sheer force diagram. All right, so along 0 to 2 meters. All right, so that's from zero two meters. First, we should actually label our axes. So that's the X. So that's the distance along the beam. And here we have the which represents the sheer force in Killer Newtons. It's just right that killer mutants, all right, So from 0 to 2, we had calculated that V is equal to two killer Newtons. So that's like saying Y is equal to two. So if you would you draw it on our sheer force diagram between zero and two, we have to Killer Newtons. All right, then. Between two and six meters, right at four plus two between two and six meters, we have V is equal to the falling, So it is a function of X. So, um, we are working from zero. All right, because we are free Body diagram started from zero to this point. That means we can substitute the values from zero to the points. Four X. All right. So basically, that just means you can substitute minus 1.5 when X is too minus one. That means wise equal to negative full. So, yeah, we would have minus full killer Newtons. All right, on you can see this is a straight line. So you can then just calculate the points at the end off the line, which is at six minus 1.5 times six minus one. And that gives you minus 10. All right, so minus 10. That six meters minus 10 at six meters. Then you can draw the line. All right. On. There you have. The is equal to minus 1.5 X minus one killer Newton's. And here we have the is equal to two Killer Newtons on that is your final answer for the sheer moment diagram. Also another Shia moment diagram. Just the sheer diagram. All right. In the next video, we will be doing the bending moment diagram. 56. Bending Moment Diagram Example: in this video, we will be drawing the bending moment diagram off the beam. So in the previous video we had drawn this year diagram of the bean. All right, so in this video, we will be drawing the bending moment off being so we can do that by storm. Firstly, finding the book bending moment between a and this point. And then secondly, the bending moment within the section off the distributed learning So you can start by drawing the free body diagram. All right. Off sectioning the beam at a point between a and over here. So if we taking the left hand side off section beam, then the sign convention means the moment acts in the anti clockwise direction. All right, Andi A why we had previously calculated to be two killer Newtons. Yeah. All right, so we section this on, we don't know at which points it's section. So we call this length picks. All right, so, no, we can do this some of the moments, and we can call this point. Oh, about 00.0 equals zero. So if the moment m is acting anti clockwise, you know, aim is positive, all right? And they in this to kill mutants act clockwise. Which means it's negative. Negative two k multiplied by the distance. X equals zero. All right, so that means Emma's equal to two X Killer Newtons. All right, so basically, the moment is a function of X, which is two x between a and two meters. All right, this second part, we can, um, draw the free body diagram. All right, so it had previously being joint drawn over here, so just redraw that. All right, so we have the beam section. At that point, we have our particular Newton force acting at a Then we have a downwards force acting at as the magnitude off a distributed loading. All right, this was calculated in the previous video as 1.5 X minus three Killer Newtons. All right. And now this is in the center, off the distributed loading. So the distributed loading would be acting here. That means it is somewhere between the we know this distance here is two meters that distance there's two meters on this distance. Here is X minus two meters, the distance off the entire bar. His ex means is all right. So because we know This is in the central oId off the rectangle. That means the distance they is equal to X minus 2/2. So it's this length divided by two, which will give you X minus 2/2. All right, we have a moment acting at this point. Once again, it acts in the anti clockwise direction because we sectioned it and keep the left hand side of the beam. So the moment act anti clockwise, and we cannot calculate the moment about this point so some of the moments must equal zero . All right, we have I m over here, so I am acting anti clockwise. So you have positive M. Then we have Verse two. Killer. Newton's acting in the clockwise direction, which means it's negative. So minus two K multiplied by the distance, which is X all right, and here we have a force which will make it rotate in the anti clockwise direction, so it's positive. Positive 1.5 x minus three multiplied. Buy the dip perpendicular distance to that line of action of the force, so that is X minus 2/2 meters. So that's multiplied by X over two minus two of the two, which is one and that all equals zero. Right? So we could No, simply five years. This is all in kilo. So moment minus two x plus 1.5 divided by just plus 1.5 X Then we have It's actually in brackets like that. It's 1.5 x multiplied by X over two. So that'll be X squared over two. Right, They will have minus 1.5 x. So that times that, then we have this minus three of a two X plus three. So what if somebody did? They was just expand these brackets. All right, so we can then simplify this further by writing em. So we have the, like terms with the X over there, over there and over there. So minus two minus 1.5 minus 3/2 equals minus five minus five x. I only have one of five divided by two that is equal to plus 0.75 X squid plus three equals era. So therefore em is equal to mine. Is there a 0.75 X squared plus five X minus three. All right, so this is the formula for the moment between two meters. Onda six meters, where this formula is the moment between zero meters and two meters. So if we were to then draw this thing, this bending moment diagram, it would look as follows. Here we have the Y axis and the X axis on the Y axis. We have the moment. All right. I should have actually mentioned that this moment is in killing Newton meters, not in Newton's. And here this is also in Killer Newton meters. All right, so the units off this moment is Killer Newtons meters. All right, on here we have X, which is the distance off the beam at 0.2 meters, which is here two meters. We have the function two x so to at X equal to zero. The aim is equal to Zahra. All right, which is here and then act two meters. Two times two is four. That mean the moment is equal to four. So if this value here is full in the graph, books like that between his error on two meters now between two. Andi not to plus four six between two and six meters, it gets a bit trickier. Now we have this squared function. Um, a good way to plot this graph would be to use a table function on your calculator so you could kick on mode their number seven for table. Now it says f of X is equal to now you write the function of X. We have miners air 0.75 Oh, for X squid plus five x minus tree veracity. We have the function on we do you want to start? We want to start at two meters. All right, start to and end at six meters because you are ending at six equals at increments off one. Yeah, so you could say one equals So basically you have to. You're 3456 se to 3456 All right are too. If of X is equal to four. So we start at this point at three f of X is equal to 5.25 so called up a bit. Then at four, it goes to five. So it goes down a little bit from 5.25 to 5 at five. It goes further down at about 3.25 somewhere there. And that's six. It's equal to zero. All right, so then he plot this. It would look like this. All right, so here we can write that the moment the moment is equal to two X killer Newton meters, and here we can, right at the moment is equal to minus 0.75 X squared plus five x minus three killer Newton meters. All right. And there you have your final answer plotted on the graph. And this is what you call a bending moment diagram.