Transcripts
1. Circular Motion: Introduction and Class Content: We will review the basics
of circular motion, which are so useful in
other areas of physics, like for example, gravitation, magnetism, atomic
physics, mechanics, etc. We will approach notions as to the definition
of an angle, angular velocity, centripetal acceleration,
and centripetal force. We will discuss the relations between these quantities and how to apply these quantities to practical physics problems.
2. What is an Angle?: This video is about
circular motion. So let's draw a
circle of radius r. Now let us imagine
we have a body circulating around the circle. So in circular motion
and that it starts at point M and covers the arc. And these could be
called at the same time, it would cover an angle theta. Note that we have two ways
of knowing where N is. We could define
some coordinates, Cartesian coordinates by
the founding two axis x and y would have a coordinate
x and y coordinate y. We could also define
the position of point, angle and the radius
of the circle. Let's see the relationship
we have between R and S. This relationship is actually S equals r
multiplied by theta. Think about it. If I increase the radius
and I keep theta constant, what happens to the, well,
it increases proportionally. Now if I fix a radius
and I increase the NGO, what happens to the arc? Well, it increases also proportionally that why
we have S equals r theta, a very important relation. We can rewrite this relation, SLR, that gives us
information about Theta. Theta the angle is actually just a ratio of distances,
meters over meters. And NGO has no unit. We humans, we need units, we need to clean on something. We can define a unit. The AHRQ has the same
length as the majors. Then you can see that
to Tycho was 11 radian. That is the definition
of the radian. Radian is the angle covered by an arc S when it is
equal to the radius.
3. Angular Velocity: I suppose now that object is going around the circle at a certain speed. How could we calculate the speed? Well, we could consider the length of the Ark and divided by the time it takes for the object from him. Doing that would be the linear speed of the object that we've seen that the ark is equal to our by data So we can write our to talk over tea and we waited are mute, apply by deter over tea. What is that? This is I will covered any time. It's a velocity to the heads of the lost C for the angle. It's a number radiance per second. This is called the angular speed. Leading us with V equals R omega are very important relationship in circle emotion. Linear velocity is, he quoted Regis, were supplied by the angular velocity. Know that I'm talking about velocity now? Yes, because we could define the damage right. That would be the linear velocity. And this would be the angular velocity. The angular velocity vector would actually be directed out of the board. We know that the angular velocity is the amount of angle per unit Time object is in circular motion. How long does it take to complete its cycle? Yes, a circle emotion is a cycling motion. The object sausage and goes around and comes back. It n How long does it take to do them? It takes a period. The period is a duration for one cycle. So if the time was a period, what would be the angle covered? What will be the angle when the ark is the four perimeter well to pipe. So that leads to a direct relationship between the angular velocity on the period off the circular motion, we also can see it on another four. We know that the period is one of the frequency at so low frequency is a number of periods in the second. So we can by down that omega is will say cool to two pi f. This is a very important relationship in many areas in physics, so please remember
4. Centripetal Acceleration: let's read rule. Also here we would have our initial point on our body going around the circle. So here it would be an initial velocity. And let's say here that's considered to be lost you again and call it huh? A small. Maybe. Now you see that the magnitude would be the same because the object it in certain emotions . So the speed is the same. But the damage shin of the velocity has changed on If there is a change in velocity, there is an acceleration that is involved. The acceleration is equal to the rate of change of velocity. Delta V is actually the final manners be initial. So if you find the direction of Delta V, we find the election off a. Let's do that. So we've got the final. Let's find the initial would be something. Maybe like this. It's no perfect because my silk was not perfect. But what we see is that the Delta V will look like this will be along the Raiders elected to the center. Therefore, the acceleration is directed to with center. Such an acceleration is called a century pittle acceleration. Now we know the direction of the acceleration in circular motion. It is centripetal always directed towards the center of the circle. But what about its magnitude? How can I calculate its value for the proof I will develop now? I will use differentials. So just a quick reminder when I like Delta X that the X is an interval, right? It's an interval between two values. This X Final Man's ex initial. If I want to consider this interval like very, very small, I will write it instead. DX that's actually dead tax when it tends to work CEO, I will use this notion so back to finding the acceleration in circular motion. So let toe a circle. Hey, at this point, I would have a certain billows TV enoughto a time baseball a time DT Well, the velocity will have changed a little bit. All right. Hey, I would have V plus the change of velocity during that time duty, and it will have covered an angle theta for details. A tiny little so here. You see, it's becoming very messy because I'm talking about baseball intervals. So I went Rejoice its care. I have my videos TV here, my velocity V plus DV on this would be therefore Devi c v plus DVD because and here I would have my good They did that. I know that Devi will be in the same direction as the acceleration, therefore in the direction of the Raiders, while V Plus DV will be perpendicular to radios. So I know that here I have a look tangled triangle and I can use a little bit of trick. Dita We'll sign of data is opposite of the opera finished So Devi of people that digitize a very small angle. And I know that when I take the sign of a baseball angle, I find the same value as the angle itself so I can just write Dictator equals DV over meaning that DV equals V. Now I know that the acceleration is a rate of change of velocity, meaning V Dita of Teeth de Titova DT. What is that? Well, it's a rate of change off the yellow. It's the angular velocity. Second, why this V only I know that vehicles are only so I could replace. For example, the V here and get only that squid or I could replace the Omega by veal are giving me B squared of our consent allies by writing down that a equals the square of all O v. Omega. Oh, yes. Quick, super important relationship the most important into motion.
5. Centripetal Force: so to symbolize the magnitude of the velocity is a waste constant and its direction is always tangent to the path tangent to the circle. On the other hand, the acceleration perpendicular to the velocity is always directed towards the center of the circle. That's why we call it centripetal acceleration. Its magnitude is also constant. Seat depends only on the speed and the wages, which both their costs. But anak situation is a consequence. Yes, it's a consequence off the force. Remember that Newton taught us that the acceleration of a body is a consequence off the resultant force applied on the body f equals and a on these are vectors. Therefore the direction of the fools will be that off the acceleration tools, the center. This is why it is also called as centripetal force for the magnitude of the force. Well, it's just m v squared off because a is b squared. It are that you can also express and only r squared. I wish. Oh, view me depending on the situation. What formula used is dependent on what's more practical for the exercise. You are working, but be careful about something. I want to make sure that you understand something? Well, centripetal force can be returned. Envies quit of all When it causes a circle emotion it is just a mathematical consequence off the circle emotion This force could be gravitational, magnetic, electric, mechanical, whatever. If it causes circle emotion, you can also write it down like MV squared of our I want to illustrate this with an example . Imagine an electron going around the proton. We are going to calculate its kinetic energy.
6. Circular Motion: Solved Example: to illustrate how to apply the principles of circle emotion to practical problems. I selected a hydrogen atom hydrated at all as a poor tone is its nucleus on on electron in circular motion around it. Now we are going to use a principal circle emotion to deter line the kinetic energy off the electron. Yes, the electron has a certain velocity like this, so it has kinetic energy. But before I do that, I would like to remind you about the force that exists between two charges. I have two charges Q one and Q two distance by a distance R from each other. There will be a force between them, which is proportional to the product of the charges and inversely proportional to the square of the distance between them. It is called a cool in force. If the two charges off the same sign, the charges will repel. So the forces will be directed like this. And if the charges are off opposite sign, that will attract each other. Okay, so in our situation we have the election, which is attracted by the proton. So there's a force on the electron dow it'd to center. Therefore, it's a century. The fourth on the electron is an electrical force, says a cool force here. Second, write it down. F equals Okay, it's a charge off the leash on chart of the proton, divided by the distance between them squared, which is the radius of the hydrogen atom. I can be right. It's because the charge of the electron is the element charge dynasty on for the poor. Tone is plus element charge divided by the square off the distance between them. We're interested in the 19 truth, actually. So I'm just going to put us here on we write it again. Okay, East, glad of all squared. Good. So that is an expression off the force that post off applies on the electron, which forces it to go in circular motion. Therefore, we can also I did. And these quit of all, giving us any relation on which we can work on. I want the kinetic energy of the electorate. So that's 1/2 m of the electron. He's quit, right? So here I have a wedding in the square there. So I just need to rearrange this a little bit. I can kill one of the arts and write down case Quit of R equals and B squared If I divide by two on both sides Well, I've got my kinetic energy So that's que ese quit of a too long I can just like in the numbers The kinetic energy of the electron will be nine by 10 to the nine, which is K by the limit Recharge squared If I did by twice the wages off the hydrogen mental on the value I find is, uh, you remember Yes, 2.304 by tens of minus 18 jewels, which was around 2.3 by tens, minus 18 jewels. Applying circle emotion, principles. Specific cases like here allows to define new relationships between the quantities which are involved on. This new relationship allows you to find whatever you're looking for. So here it was a kinetic energy of the ledger. This is how you can use circle emotion. And we are going to use these principles a lot within this course because we're talking about gravity on what is one of the most known effects of gravity. All wakes which are sacred emotions now jump to the next video it contains for exercises are increasing levels of difficulty. I prepared these for you so that you can train using concepts related to emotion.
7. Training Exercise 1 (Easy): this video is a training video on circle emotion that full exercises the two first exercise is actually straight applications off the formula. Yes, just to get used with numbers and with different quantities involved. The third exercise requires a little bit more reflection. You would have a problem that we need to solve using circle emotion, principles. And finally, the fourth exercise would be more challenging, more theoretical. Give it a shot. We have a ball of mass 200 grams will. Which path? It opposed me to of a circle. So it is in SoCal emotion. On the radius of this circle is 40 centimeters. The speed of the ball is given two meters per second. Question number one, determine the distance traveled per revolution. A revolution is actually one circle one cycle. That's question. Okay, well, it will correspond to the perimeter off this circle, so d equals two pi R, which is two pi 5.4 meters, giving 2.51 meters question. Be determine the time taken for one revolution. Actually, this is one period, so we know that the speed is equal to the distance denied by the time said that the time is a distance divided by the speed and just rearrange the equation. The distances to put 51 divided by two which is a speed giving me 1.26 seconds. How was questioned? I turn It'd Lee. You could have used a circular motion for me Now equals Omega are only guys to pie Divided by the period. Give you a range is and you find a period two pi r divided by the speed which is this actually d of a V d being to pay off. Okay, question. See, they don't mind the angle. Slept out for revolution. Well, dangle in one cycle here is just to pipe. It's de deter mined the angular velocity. We have the period so we can find the angle of velocity to buy over tea, which is two pi over 1.26 getting you 4.99. That's per second. No, that here. I'm always writing down my whistles with three significant figures. While it is clear with my data that I don't have three significant figures, I should that she used to Why am I doing this? Is because I know that these results will be used in future calculations. In the next questions, all intimated calculations need to be performed in more significant fears than the final result. So here, if I was at the test, I would keep these numbers. But then I would add 2.5 one point B on the 5.0 went per second. When using the data for future questions, I would use this. Okay, let's go to question E the centripetal acceleration. The acceleration is a centripetal one sits just feet square law and I'm using this formula because I wouldn't have V and I have. So it's two squared, divided by 40 centimeters 400.4 meters four divided by 0.4, giving me 10 meters per seconds. Quit on. Finally question f the centripetal force. Well, this acceleration is caused by force. This acceleration is centripetal. So the force will also be centripetal is just a man 0.3 kilograms by 10. Giving me three Newtons
8. Training Exercise 2 (Easy): - this exercise requires that we calculate the angular and linear velocity off the moon while it is in circular motion around the earth. We know the distance between the moon and the earth. 284,000 kilometers on. We also know the period off the circle emotion 27.3 days. Well, this is a straight application off the formulas. The angular velocity is two pi divided by the period. So two pi divided by 27.3 days. Remember to put it in seconds. So there 24 hours in the day and there are 3600 seconds in and now find 2.66 full buy tenders of minus six imagines the second, which gives me to 0.66 by 10 0 minus six wagons. I just welded up to keep three things difficult numbers, and I need to calculate the Linear Spiegel's so again straight application of the formulas . V equals R. So 2.664 by tens of minus six multiplied by the distance in meters 3 84,000 and thanks for 1000. And if I plug this in my calculator, I found 1023 meters per second. That is 1.2 by 10 to the three meters per second, 1.2 kilometers per second.
9. Training Exercise 3 (Moderate): in this exercise, we have a car going towards a bridge at the constant velocity. The bridge is seen like the arc of a circle of radius 20 meters. You can see that if the car goes really fast when it reaches this point, it might lose contact with the road. We're required to find the maximum velocity at which a car can go without losing contact with the road. So where were the action of? Well, yeah, it would be useful to find out what happens to the card this point five during a few bloody diver. What other forces when we have the weight of the call and G and we would have the normal force and with dealing with forces, we need to define an axis. I'm going to choose. A quarterback sees dumbs. From that, I can calculate the net falls. This would be N G minus end on the fly. New tendrils, they said. Force is equal to and a okay, that's good. But let's not forget that the car, when it's on the bridge will be in circular motion. Yes, we've seen the bridge like the arc of a circle. Therefore its acceleration will be centripetal. A is V's quit of our, therefore can write that that force is actually a centripetal force value MV squared of all . Now I got a cool relationship that shows up. Let's relight it and b squared of our equals m g minus. And why's that equation cool? Because it introduced V and I'm going to re arrange for V. The is quit off our M fact of M G minus spend. We are required to find the maximum values of the car can have without losing contact with the ground, meaning the maximum value that the car can have while staying in circle emotion. They're full. We're trying to find the maximum value off the speed of the car for this to be true, because this was derived from the fact that we considered a full centripetal i e. That the car was in circular motion. So let's look at what's inside. Can all change? No, it's a radius of the bridge can ever change. Now that's a mass of the car. Can g change? The gravitational field tracks Well, that's one of the Earth it can change, can end the normal force change. Yep, this one can change. What would it be to have the maximum? The normal force cannot be negative. It zero to positive have anything to sign here. So the maximum value for V would be obtained when an equal zero giving you v equals squared off M g r of n Kili EMS and you get skirt of Geo. What does this really mean? But that its reasoning with a normal force being zero mean the best way to see it is to remove the book. Yeah, we'll go. So suppose to car now being really fast. So when it lives at this point, well, it's tragic treat be something like this. Suppose now that is going really slow as soon as it passes this point with full like this, it would fall across a position where the road was. There's no more road to push it back. When the road is there, the car falls on load and the load pushes Basque giving you and normal force. But what if the car had a specific speed so that it's tragic? Treatment is falling is exactly that that road had I in circular motion when you put the road back, you could imagine the car. She's sliding just one millimetre above the vote and keeping the circular motion keeping this form and that true. And here the world wouldn't be in contact with the car. It's just a car following the path of the road naturally, so there would be no normal force. But the car was still being self promotion. We need that. I can do this. That is the idea behind the fact that the normal force is zero and that would be the maximum value of the car. Can have a speed so that culture news it's circle emotion without touching load. This put numbers in the equals credit off 9.8 by 20 Well, that's abound. Square root of 200 giving me around 14 meters per second. Jeremy, just plow I'm enterprises by 3.6. Get something out of 15 meters pile
10. Training Exercise 4 (Difficult): - In this exercise, you have a bowl end connected to a group of length L, which itself is connected to the ceiling. The bull M is made to swell, such as it is in circular motion in the office. Enter plain, and when this happens, the road makes an angle deter with vertical. The first question was to draw the forces that are acting the ball on deduce if the ball is in the killer beer more, not less. Well, the bull will feel Wait right MGI and a bowl will feel tension from the look. So you see that the two forces I feel the bulls are not on the same axis, meaning that there will be a net force, so the ball is not in the Caribbean. Another way to answer this is to say that the Mass M is actually in circular motion. Therefore, there needs to be a centripetal acceleration and worth a centripetal force. So the net force on the mass M cannot be zero. So that's M is therefore not in IQ Libya. Question. Be find an expression for V as function off the gravitational field strength, the legs of the rope and the angle the rope makes with vertical. So how do we do that? Let's look at what we know. We know that the Mass M isn't circle emotion. Therefore, we know that it has a centripetal force. On the centripetal force will be the net force. If the Net force a centripetal it will be equal to envy is quite a bar, and this is how we can introduce. Three. It might be a good idea to express the Net force. The Net force will have two components X and why. So let's define some access. Let's use an access for X, which is horizontal positive that way and access for why which is vertical positive upwards . And now let's find a contribution off forces on the need access. So let's start with why. Well, we have minus envy and the contribution of the tension. So here we have the angle deter. So it's going to be plus T cause now for the X axis, where there's only the contribution of detention because N g is perpendicular to the X axis . So this would give us tea, signed it. Let's apply now new tunes laws on the Y axis. You see that the mass M is not going up or down. Therefore, it is balanced. The Y component of the net Force Zero on the X axis is going to be equal to men and they x But there's no component on why. So we can actually write down. The F net is equal to t sign titter which is also equal to envies. Quit of our because we know that the net force is a century mental falls So we nearly there . We got V here and we want to express it as function of Twitter Jnl But he has got in our so I need to figure out how to change these. Well, look at the white component here This why component allows me to express tears Function of liturgy an M So let's do that. Got t cost eta because MGI so t would be equal to N g over cost data on this I can plug in here. What about all well Oh, is this I got l and I've got better so I can find out. Oh, here. Ah will be l scientific. The off equals l sign. Ditto. It's up with this into that and this to that leading to n g cost iter sine theta equals And this squared of l signed. I cannot wait to see that I can kill the EMS. Hanley arranges to get his quest so b squared equals G l sign square deter, of course. Data. That's it on. If I take the square root, I should find this. No, that sign on course is time so I could even simplify further g l turn to duck sign. Did that questions see? Find an expression for the angle of velocity as function off the gravitational field Strengths The length of the rope and the angle the work next with vertical. So it's the same para meters. Okay, so the new space some good two days days Try to remember the formula by heart. Okay, Same question be we had V equals square it off G l sine squared editor of custody. And now I want to find only got I I know that the relation between the angular velocity and the linear velocity is vehicles only Ourself, v equals are so oiga is just v of r second plate things in one of the ah g l sine squared it up Cost Itta Well. Ah, what is all right? We calculated before ah was equal to l sign Did them seconds that like it here. And we're not integrated in the square root by scrubbing it started giving me We got calls G l signed Square meter of, uh l squared sine squared. Theta costed A lot of things cancel Here I get one of the elders cancels the sine squared. It cancels giving me square root off g over l course I did them questioned the the angle titter is 30 degrees. So this is 30 degrees using your answer to see, find an expression off the centripetal acceleration and determine its magnitude. Well, I know that the exploration is equal to Omega squared off. I could have done this, whereas this one was the squid are. But they asked us specifically to use the onset. Sees Well, I just like it in. I get g cost ETA, miss Code goes way because of the square here by our but our is l signed it. So the outs goes away and silent cause his town giving me a simple equation a equals G tell him to do pretty simple. So if I like the numbers in 1981 by turn off 30 I get a equals 5.7 meters per second. Squared Ebola. That's it for this training video on Circle Emotion. Now we we take what you've just learned insecure emotion blended with what you have previously seen in gravity. The result will be orbital motion. So next videos, we'll deal with Orbitz. See that?