# Rotational kinetic energy

Tags: energy, kinetic, rotational
 P: 9 I'm afraid I need some help on this one. We are studying rotational dynamics and I read in my textbook, the explanation of a common classroom deonstration. A person sits on a rotating stool and holds heavy weights in their outstretched hands. Someone sets the individual into rotational motion and when they pull their hands in, the speed of rotation increases. This is easily understood with the conservation of angular momentum. We are asked to imagine that we actually perform this demonstration and that pulling our arms in results in halving our total moment of inertia. This obviously doubles our angular speed according to the law of conservation of angular momentum. If we consider the rotational kinetic energy, we quickly see that we have doubled the energy associated with that rotation. Again, not a difficult concept because we have done work on the weights (The source of energy being food that we ate earlier that day, or perhaps even days ago.) But here's the problem I have. Suppose we do a DIFFERENT experiment. Suppose we tether a volleyball on a pole. Now we strike the volleyball perpendicularly to the string that holds it on the pole. We have given the ball a certain amount of energy and it begins to rotate at a given angular speed. The string wraps around the pole and in a period of time, the radius will have decreased just enough to make the moment of inertia half of its orignal value just as in the previous demonstration. When that happens, again the angular speed will double and the kinetic energy will also double. But this time, NO ADDITIONAL ENERGY WAS ADDED AND THE POLE DIDN'T HAVE LUNCH TODAY... OR YESTERDAY EITHER! How do we reconcile the fact that the ball seems to have gained energy with no additional energy input here? I really need help here. Thanks, in advance, for whatever you can supply.
Emeritus
PF Gold
P: 29,238
 Quote by leebenjamin@adelphia I'm afraid I need some help on this one. We are studying rotational dynamics and I read in my textbook, the explanation of a common classroom deonstration. A person sits on a rotating stool and holds heavy weights in their outstretched hands. Someone sets the individual into rotational motion and when they pull their hands in, the speed of rotation increases. This is easily understood with the conservation of angular momentum. We are asked to imagine that we actually perform this demonstration and that pulling our arms in results in halving our total moment of inertia. This obviously doubles our angular speed according to the law of conservation of angular momentum. If we consider the rotational kinetic energy, we quickly see that we have doubled the energy associated with that rotation. Again, not a difficult concept because we have done work on the weights (The source of energy being food that we ate earlier that day, or perhaps even days ago.) But here's the problem I have. Suppose we do a DIFFERENT experiment. Suppose we tether a volleyball on a pole. Now we strike the volleyball perpendicularly to the string that holds it on the pole. We have given the ball a certain amount of energy and it begins to rotate at a given angular speed. The string wraps around the pole and in a period of time, the radius will have decreased just enough to make the moment of inertia half of its orignal value just as in the previous demonstration. When that happens, again the angular speed will double and the kinetic energy will also double. But this time, NO ADDITIONAL ENERGY WAS ADDED AND THE POLE DIDN'T HAVE LUNCH TODAY... OR YESTERDAY EITHER! How do we reconcile the fact that the ball seems to have gained energy with no additional energy input here? I really need help here. Thanks, in advance, for whatever you can supply.
First of all, let me just say that I WISH, when I was teaching, that all of my students are like you. It is a JOY to teach students who are are thinking like this, and stumble upon things that don't quite jell with what they understand.

Back to you question. The thing with both moving your arm inwards and with the string coiling in is that there is a force within the system that is doing work. When the centripetal force is applied to move the object either inwards or outwards, work is either done onto the rotational system, or done by the rotational system (remember that work requires a force AND a net displacement of the object in question).

In the case of a person pulling in the weights, the centripetal force is provided by the arm/hand, while the rope+pole provide this force in the other example. This "external" force (external to the rotation motion) does change the angular momentum because it is always perpendicular to the motion (orthorgonal to the direction of the angular momentum vector), and therefore adds no net torque to the system to affect angular momentum conservation.

So moral of the story: the source of the force can be anything. As long as they act the same way, the result will be the same.

Zz.
 P: 9 Mr. Zapper... I think you misinterpreted my question. I understand fully that work is done by the tension in the string. And that the work done is Force times the distance. My question is, WHAT IS THE SOURCE OF THE ENERGY GIVEN TO THE VOLLEYBALL? I only gave it so much energy and the pole or the string don't have any food (or fuel of any kind) to provide more energy, yet at a later point in time, the ball now has twice the energy it did after I hit it. Where did this extra ENERGY come from? Lee
Emeritus
PF Gold
P: 29,238
Rotational kinetic energy

 Quote by leebenjamin@adelphia Mr. Zapper... I think you misinterpreted my question. I understand fully that work is done by the tension in the string. And that the work done is Force times the distance. My question is, WHAT IS THE SOURCE OF THE ENERGY GIVEN TO THE VOLLEYBALL? I only gave it so much energy and the pole or the string don't have any food (or fuel of any kind) to provide more energy, yet at a later point in time, the ball now has twice the energy it did after I hit it. Where did this extra ENERGY come from? Lee
The energy came from the tension in the string - the EM force that's holding the molecules of the string together. This energy is external to the rotational energy.

Zz.
 P: 9 But I don't think that works. This string applied its force without outside influence. In every case that I can think of, some source of energy needs to be responsible for causing a rope to exert tension. For example, an electric or a gas motor can cause a tension in a rope which can then lift a load or give it kinetic energy. I am not comfortable with a rope spontaneously exerting energy and furthermore, even if it did, wouldn't it have to decrease in its own energy? I, for example, use up some of the energy from the food I ate, when I exert the force on the masses. And if I give the masses kinetic energy using a motor, the motor has to use up fuel or utilize electric energy which results in a battery having less energy than before, etc., etc. I don't see the reduction in the rope's energy like these other examples. I still fail to see the source of the energy. Can you be more specific? Thanks! Lee
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P: 29,238
 Quote by leebenjamin@adelphia But I don't think that works. This string applied its force without outside influence. In every case that I can think of, some source of energy needs to be responsible for causing a rope to exert tension. For example, an electric or a gas motor can cause a tension in a rope which can then lift a load or give it kinetic energy. I am not comfortable with a rope spontaneously exerting energy and furthermore, even if it did, wouldn't it have to decrease in its own energy? I, for example, use up some of the energy from the food I ate, when I exert the force on the masses. And if I give the masses kinetic energy using a motor, the motor has to use up fuel or utilize electric energy which results in a battery having less energy than before, etc., etc. I don't see the reduction in the rope's energy like these other examples. I still fail to see the source of the energy. Can you be more specific? Thanks! Lee
OK, let's go back to the human example. Forget about the details for the moment. All you care about is that SOMETHING is causing the radius of rotation to become smaller, ya? I don't really have to tell you what is causing this. All I need to tell you is that the radius got smaller without any application of an external torque on the rotational system.

When we account for both L and KE, we see that L is conserved, but KE is not. How do we account for the non-conservation of KE? We say that, well, there is a radial force from SOMEWHERE that is doing work on the system (the relevant system here is a mass at in a rotational motion). Work (and energy) is transfered between the origin of this force and the rotational system, when this radial force does something that causes a radial displacement of that object.

Now so far, I still have not invoked any need for explaining the origin of this force. All I have done is explain this purely via mechanics. At this point, mechanics stops here and the explanation (at least as far as mechanics is concerned) is complete. Up to this point, I can easily use the same explanation for the ball+string system.

But since we are trying to fully account of what comes from where, we then invoke biological explanation for the human pulling the weight in - we say the muscles burn some energy that he ate, and this allows him to use this energy to pull the object in, etc.... You want an explanation similar to this for the ball+string system. OK, let's try this:

As the string is wrapping around the pole, it's radius is of rotation is getting smaller. It means that the rope is PULLING THE ball inwards radially. What is doing the work?

Recall that when you put a charged particle in an electrostatic field, and the charged particle moves due to the electrostatic force, the field is doing the work on to the particle. The energy of the field is transfered into the translational energy of the particle. I can invoke the same thing with the string pulling the ball inwards. The immovable pole (that is attached to the "immovable" earth) transfers the EM force to the string, which then transfers the force to pull the ball in. That's why I said that it is the EM field that's doing the work. One can easily argue that it is the whole earth that's providing the support for the pole+string+ball, so it is the earth that's doing the work. I would have no argument with that. It depends on how far up the chain of events you want to pursue.

Zz.
 Mentor P: 41,449 I think the tetherball problem is a bit subtle. It is different from the example of a person pulling in the weights, which you seem to understand. In that example, there is a force doing work on the weights, so you can see that mechanical energy is not conserved. And you recognize that the energy must come from the person's internal energy (food!). Also, there is no external torque on the system, so angular momentum is conserved. So what's different with the tetherball winding around the pole? For one thing: the rope is not pulling in the ball (although it is pulling on the ball), it is merely winding around the pole. I believe the tension in the rope does zero work on the ball and thus there is no need for an energy source: Mechanical energy is conserved. Note: The tension in the rope exerts a torque on the ball. (The pole has thickness.) Angular momentum is not conserved! (If I'm wrong, Zapper, please correct me. )
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 Quote by Doc Al I think the tetherball problem is a bit subtle. It is different from the example of a person pulling in the weights, which you seem to understand. In that example, there is a force doing work on the weights, so you can see that mechanical energy is not conserved. And you recognize that the energy must come from the person's internal energy (food!). Also, there is no external torque on the system, so angular momentum is conserved. So what's different with the tetherball winding around the pole? For one thing: the rope is not pulling in the ball (although it is pulling on the ball), it is merely winding around the pole. I believe the tension in the rope does zero work on the ball and thus there is no need for an energy source: Mechanical energy is conserved. Note: The tension in the rope exerts a torque on the ball. (The pole has thickness.) Angular momentum is not conserved! (If I'm wrong, Zapper, please correct me. )
But mechanical energy cannot be conserved if L is conserved. This is because L has a linear dependence on angular velocity, while KE has a quadratic dependence on angular velocity. They cannot be satisfied simultaneously unless w=0 or 1.

There is an analogous experiment to this, which is a rope attached to a ball that is rotating on a frictionless table, but the other end of the rope passes through a small hole in the table and is attached to a weight. Here, the gravitational force is doing work by pulling onto the weight, and this is transfered to the tension in the string to pull in the ball. I see no difference between these two other than the origin/source of the work done.

Zz.
 P: 9 Well, okay, but in the case of the person pulling the weights inward, the internal energy of the person is DECREASED by an amount equal to the increase of KE for the mases at the ends of his arms. In the case of the volleyball, where is the corresponding DECREASE of energy of the pole/string system that would account for the increase of energy of the ball? Lee
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 Quote by leebenjamin@adelphia Well, okay, but in the case of the person pulling the weights inward, the internal energy of the person is DECREASED by an amount equal to the increase of KE for the mases at the ends of his arms. In the case of the volleyball, where is the corresponding DECREASE of energy of the pole/string system that would account for the increase of energy of the ball? Lee
That's difficult to account for. I mean, when you drop a mass in a gravitational field, and the field is doing work and transfering energy to the mass, where is the corresponding reduction in energy? We can do this ad nauseum and consider the source of the gravitatioinal field and say that it is also being pulled in (miniscule) etc. etc. But at some point this no longer becomes practical.

Zz.
 P: 9 To Dr. Al... If there is no need for a mechanical energy source, then how do we account for the fact that at a later point in time, the kinetic energy of the volleyball is TWICE as much as it was at the beginning of the problem? That cannot happen without an energy source. (Or if it can, everything I learned about conservation of energy was false.) Lee
Mentor
P: 41,449
 Quote by ZapperZ But mechanical energy cannot be conserved if L is conserved. This is because L has a linear dependence on angular velocity, while KE has a quadratic dependence on angular velocity. They cannot be satisfied simultaneously unless w=0 or 1.
Right. Angular momentum is not conserved.

 There is an analogous experiment to this, which is a rope attached to a ball that is rotating on a frictionless table, but the other end of the rope passes through a small hole in the table and is attached to a weight. Here, the gravitational force is doing work by pulling onto the weight, and this is transfered to the tension in the string to pull in the ball. I see no difference between these two other than the origin/source of the work done.
Here the rope pulls in the ball, performing work on the ball. (That energy comes from the change in gravitational energy of the descending weight, of course.) In this case the force is radial, so it exerts no torque on the ball; Angular momentum is conserved.
Mentor
P: 41,449
 Quote by leebenjamin@adelphia If there is no need for a mechanical energy source, then how do we account for the fact that at a later point in time, the kinetic energy of the volleyball is TWICE as much as it was at the beginning of the problem? That cannot happen without an energy source. (Or if it can, everything I learned about conservation of energy was false.)
What makes you say that the kinetic energy is twice as much at a later point in time? (Neglect any change in height of the ball.) You are assuming conservation of angular momentum. I don't believe that's true in this case: the rope exerts a torque on the ball.
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 Quote by Doc Al Right. Angular momentum is not conserved. Here the rope pulls in the ball, performing work on the ball. (That energy comes from the change in gravitational energy of the descending weight, of course.) In this case the force is radial, so it exerts no torque on the ball; Angular momentum is conserved.
But where is the torque in this set up? Ideally, I can make the string as thin, and the pole as thin as I want to (after all, we neglect the string's mass). So I don't see how you can insert an angular component to the force from just the string.

Zz.
Mentor
P: 41,449
 Quote by ZapperZ But where is the torque in this set up? Ideally, I can make the string as thin, and the pole as thin as I want to (after all, we neglect the string's mass). So I don't see how you can insert an angular component to the force from just the string.
The problem assumes the rope winds around the pole. You cannot neglect the thickness of the pole, else the problem disappears. The rope is not aligned in the radial direction; it acts at an angle, thus exerting a torque on the ball (with respect to the center of the pole).
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