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(III) On a level billiards table a cue ball, initially at rest at point O on the table, is struck so that it leaves the cue stick with a center-of-mass speed $v_{0}$ and a "reverse" spin of angular speed $\omega_{0}$ (see Fig. 40$)$ . A kinetic friction force acts on the ball as it initially skids across the table. (a) Explain why the ball's angular momentum is conserved about point O. (b) Using conservation of angular momentum, find the critical angular speed $\omega_{C}$ such that, if $\omega_{0}=\omega_{C},$ kinetic friction will bring the ball to a complete (as opposed to momentary) stop. (c) If $\omega_{0}$ is 10$\%$ smaller than $\omega_{\mathrm{C}},$ i.e, $\omega_{0}=0.90 \omega_{\mathrm{C}},$ determine the ball's $\mathrm{cm}$ velocity $v_{\mathrm{cM}}$ when it starts to roll without slipping. $[$Hint. The ball possesses two types of angular momentum, the first due to the linear speed $v_{\mathrm{cu}}$ its an relative to point $\mathrm{O},$ the second due to the spin at angular velocity $\omega$ about its own $\mathrm{cm} .$ The ball's total L about $\mathrm{O}$ is the sum of these two angular momenta.]

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Physics 101 Mechanics

Chapter 11

Angular Momentum; General Rotation

Moment, Impulse, and Collisions

Rotation of Rigid Bodies

Dynamics of Rotational Motion

Equilibrium and Elasticity

Rutgers, The State University of New Jersey

University of Michigan - Ann Arbor

University of Washington

Simon Fraser University

Lectures

02:21

In physics, rotational dynamics is the study of the kinematics and kinetics of rotational motion, the motion of rigid bodies, and the about axes of the body. It can be divided into the study of torque and the study of angular velocity.

04:12

In physics, potential energy is the energy possessed by a body by virtue of its position relative to others, stresses within itself, electric charge, and other factors. The unit for energy in the International System of Units is the joule (J). One joule can be defined as the work required to produce one newton of force, or one newton times one metre. Potential energy is the energy of an object. It is the energy by virtue of an object's position relative to other objects. Potential energy is associated with restoring forces such as a spring or the force of gravity. The action of stretching the spring or lifting the mass is performed by a force which works against the force field of the potential. The potential energy of an object is the energy it possesses due to its position relative to other objects. It is said to be stored in the field. For example, a book lying on a table has a large amount of potential energy (it is said to be at a high potential energy) relative to the ground, which has a much lower potential energy. The book will gain potential energy if it is lifted off the table and held above the ground. The same book has less potential energy when on the ground than it did while on the table. If the book is dropped from a height, it gains kinetic energy, but loses a larger amount of potential energy, as it is now at a lower potential energy than before it was dropped.

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so before doing Ah, the parts off the problems with me, the mind they would act for a complete rule in motion. We can write the velocity that is the velocity of the fender off Mars to be equal to the radius off the balls. Brian's the angular velocity Onda. This is only valid when the motion is completely ah, rolling motion. But in this problem, since the ball is initially slipping as well while it's violet rolls. Therefore, this condition or this relation between we not an Omega, not for the initial motion off the ball is not valid. Therefore, we won't be applying this relation at all. So whenever we have to use the initial velocity, we will be using be not on. Whenever we have to use the initial angle velocity, we will be using homemade. Not without using this tradition. No, you must see that from the free body diagram. Ah, so basically this ah diagram the Bodi Graham off the ball is after it has moved away from the initial point where all is the initial point. Now there are three forces acting on the ball. As you can see, the fictional force in which I have drawn Toby opposite to the direction off Motion Off the ball. Direction of motion off the center of mass. Off the vault are to be specific on there is a normal force from the ground at the point off contact off the ball and the ground. And there is a week Ah, at being from the sentinel. The blood No, if in an MD, as you can see are in opposite directions And each has the same liver arm about an axis passing through the point. Oh, opened the collected the plane off this paper. Now, because of that, there will be no dark, no next dark. This is because now, if you can call it the talk about point or so let me just to that PSINet dark about point or will be equal toe I fr Or you can write that that means the other condition. So you know that next talk can build an exchange in Angola. Memento. Well, like I am, let's find that talks about this point, which is the initial point of the book. So since this'll frictional force passes through that point, so we have no talk. Do tow this frictional force Not if you think clockwise stocks to be positive. Then we have effin giving a count of look we're stopping for the negative sign on the distance is our on. We have n g giving a clock way. Start on the distances are less with the top of the distance are is the perpendicular distance off either force from and that is it going to do the l over D D, which is the right side that we look? This means that if in negative offensive, let me destroyed the ability. So this gives our times n G minus f n to be equal to the L over DT. Now, if you light Newton's second law for equilibrium, you have. So there is no acceleration off the ball in the vertical direction. Therefore, we can say that the ball is an equilibrium in the vertical direction. It's moving, but it's moving to the right. So it's moving holy, gentle horizontally. But typically there is no acceleration, so we can write the net force in White Election tau equals zero. So this gives S n to equal toe mg which are the true pope toe vertical forces. If you applied this condition or will heal it. We'll get that this stone blows away most to zero. This means we can write the hell of a day to be equal to see you, and therefore and is a constant. Now they're done with the first part. So for the second. But we follow the hint this thing on. Dhe hopes the fall of the hint. A given in the problem on expressed the total angular momentum as it's some off two domes. So it is angular momentum with respect to the sink that off Marsellus angle a moment down the rotational angle. A moment of a boat? No. So for angry, the angular momentum for the center off mass will be, and I times the velocity off the center of Mass. So they have Maas tanks, the radius and the velocity of center of mass. On the angle of momentum off. This will do to the rotational motion off Omega. So that was the equal. Do I dines Dominga? Now the angle of momentum is a constant, so we equipped the angular momentum at the initial position. But the center off must equal toe. We not on omega is only go not on the gallows and call it Toby when they got seats on dhe final angle. A memento is Siegel, velocity of Center of Marcis, Edo and only guys you know as well. So if we do that so let's night since angle a moment of Ms Constand. So initial angular momentum is final, angular momentum. Now we used this relation over here. So we have m r the c m bless Iomega, initially equal to M AJ. You see him less Iomega with the final well, angry the momentum. Now it's like the values for the initial motion. Sylvia, I am, are they not? Plus, I Omega or not on this is equal to So finally we have this storm velocity of Santa Macedo and also my guys. You know, this means that the right hand side will be zero. So from here we can express omega not odd Omega Psi to the equal do and on Be not was I on guy for goto, right, That this is no moment off initiated the center of mass. So basically, if you're getting confused about how to ally at this relation, remember the velocity related velocity innovation. So let me Just try that over here. So velocity according to the letter velocity addition, Velocity off. Hey, with respect 00 can build it in us. Velocity off. Gave it respect, Toby. Less velocity off Being the respectable Now, considering this is yours point Oh ah nde This is be Let's say this is a so we can drive that velocity off So we will apply this velocity extent this velocity relation to other angle a moment so we can write angular momentum off Oh, I guess I marked the imitation strong So let's call it here. Let's call this point B on DDE this boy Andy. So let me destroyed the coalition for this over here and just forget about this. This is true, in fact, but ah, the e m o here don't got respond to the piano here, So let me destroy their relation. So began, right, The velocity off. Be with Sorry. You have to write for Angela Momentum so we can write angle a moment. Um, off be with respect to a we can write that to be equal dough angular momentum must be with respect. Oh, last angular momentum off full with respect to e on angular momentum off. All with respect to a is the angle of momentum Off the center. Off mass on angular momentum off. Be with respect. Duel is I times Omega. There is the center off Mass off the ball. Yeah, So now we have Yeah, that's how we arrive. This relation? No. We have the initial angular velocity on guys ascended off Moss. Ah, moment of initial the center of mass. Now we know that the ball is spirit ical. So sending off mask in a shop, a moment up in ourselves, in bed of masses through by five capital Small M That's good. Now, like this expression toe over here, you'll find Omega Psi, Toby equal Do Fife the knot over to wash. And this is the expression off Omega Psi that we will be using in the other parts of the problem. So this is the answer, not with Bart, See? And the momentum is against guns oft so. But in this case, velocity off center of mass history Not, but the initial velocity is cereal 0.9 times on my gusty on DDE In the final state, the ball is purely rolling. Therefore, we have the center off must equal. The Omega Times are and therefore only God is the absent bit of muscle or not. So here didn't destroyed, though condition for constant, angular momentum. Now the values dIsa question again over here. So unusually we have and ah, velocity of center, off mass less I thanks or make or not with this zero point nine Omega Psi. Then for the final motion, we have an odd velocity of Santana massless I thanks Omega, I thanks to find the language of the loss of villages, it's gonna be expressed as little steps and then off muscle. What off? Now we can simplify the seclusion in order to get the knot. So first began substitute the value of Omega Psi that we can't related here over here and also express I in terms of Mars and Radius. So basically used this relation on this relation Over here on dhe soul for the velocity off center off muss. So doing that, you'll get the velocity of center of mass to be equal to one over 14. We not now, this answer is pretty reasonable, but there is not enough backspin. Since now Omega not is less than Omega Psi. Therefore, the Bulls final State is ruling forward now for the last part again bang government of its guns off. So dissolution on for the initial motion, we have velocity of sending off Master Beagle Duveen out. And only the nod is now grated by a factor of 1.1 on traffic duty to the case in part C will only got noticed less than our legacy. So that's for the initial motion. And for the final motion we have again. The ball is rolling. So we have Omega Tau, the equal to the intensity of center of muscle. Where are the same relation that we used here? Now let's blood the secretion over here. Sorry. Practice a question. Gooding, this over here on DDE you will arrive at I am oddly not. Yes, I dying. So on 0.1 on my gusty you called toe em all. We see him plus high times Omega. It's going to get another specie and over us again began blood. I will be equal to do by five. AM I scared on Omega Psi, which we found in part B. This expression we can plug these two expressions over here when find velocity of center off Mars, which took about Toby Equal, do negative one over floating. Thanks. We knock. And this expression or this answer is also reasonable because there is more than enough backspin. Now that only guy only God, not this greater than Omega Psi there is more than enough. Backspin. Therefore, this answer is reasonable, and so the ball's final state is rolling backwards.

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