# Rotational kinetic energy explaination

## Main Question or Discussion Point

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.

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ZapperZ
Staff Emeritus
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.

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

ZapperZ
Staff Emeritus
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.

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

ZapperZ
Staff Emeritus
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.

Doc Al
Mentor
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. )

ZapperZ
Staff Emeritus
Doc Al said:
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.

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

ZapperZ
Staff Emeritus
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.

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

Doc Al
Mentor
ZapperZ said:
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.

Doc Al
Mentor
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.

ZapperZ
Staff Emeritus
Doc Al said:
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.

Doc Al
Mentor
ZapperZ said:
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).

ZapperZ
Staff Emeritus
Doc Al said:
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).
Yes, but this is enough to truly violate conservation of angular momentum, and produce a conservation of energy? I also cannot make the "hole in the table" case arbitrarily small and frictionless, yet even rudimentary intro physics lab on this can produce quite a good agreement with conservation of angular momentum principle, which implies that this conservation is actually quite robust.

Note that due to the quadratic dependence on w, to be able to conserve KE, L has to be severely violated no matter how large (or small) we make the pole to be. So if I give the same initial condition to "mass attached to a string+pole" and "mass on frictionless table pulled by a vertically hanging mass", I should see severe difference in the angular velocity between the two if L is violated by KE isn't. I find this difficult to accept especially since I can make the rope VERY long when compared to the diameter of the pole.

Zz.

In response to Dr. Al's answer:

The reason why the kinetic energy is twice as great is this: The moment of inertia becomes one half of the original while (according to the law of conservation of angular momentum) the angular velocity doubles. Now KE (rotational) is 1/2 I (omega)^2 So since I becomes 1/2 as great and omega becomes twice as great, omega squared becomes 4 times as great and 4 times 1/2 = 2. Kinetic energy doubles.

All of these responses are not giving me what I need. The example in the book said that the reason why the kinetic energy could double like this was that the person spent energy from food previously eaten to increase the KE. I understand that. And the person's energy would certainly decrease by an amount equal to the increase in kinetic energy (That's easy to see and understand.) We could draw an analogy by cranking the masses in with an electric motor instead of a person. Clearly, the amount of energy in the battery would be reduced by an amount equal to the increase in kinetic energy of the masses. I do not see such a decrease in the energy of anything related to the problem if we do the volleyball thing. It seems as if we are CREATING energy here and I don't think that's possible. I haven't heard anything so far, however, that says it isn't so.

lee

Doc Al
Mentor
I'll have to give this one some more thought to give you a good answer. But here's something for you to ponder: If you think the energy is increasing, where's it coming from? (This was the original question.) I think we can both agree that the pole isn't moving--so it can't be doing any work on the rope. And the rope itself doesn't do work (unless it contracts); it merely transfers force.

Doc Al
Mentor
In response to Dr. Al's answer:

The reason why the kinetic energy is twice as great is this: The moment of inertia becomes one half of the original while (according to the law of conservation of angular momentum) the angular velocity doubles. Now KE (rotational) is 1/2 I (omega)^2 So since I becomes 1/2 as great and omega becomes twice as great, omega squared becomes 4 times as great and 4 times 1/2 = 2. Kinetic energy doubles.
My response to this remains the same: You are assuming conservation of angular momentum. I say it's not conserved in this case, because of the torque exerted by the rope.

All of these responses are not giving me what I need. The example in the book said that the reason why the kinetic energy could double like this was that the person spent energy from food previously eaten to increase the KE. I understand that. And the person's energy would certainly decrease by an amount equal to the increase in kinetic energy (That's easy to see and understand.) We could draw an analogy by cranking the masses in with an electric motor instead of a person. Clearly, the amount of energy in the battery would be reduced by an amount equal to the increase in kinetic energy of the masses. I do not see such a decrease in the energy of anything related to the problem if we do the volleyball thing. It seems as if we are CREATING energy here and I don't think that's possible. I haven't heard anything so far, however, that says it isn't so.
I agree with you! If the energy increases, then it must have a source. In my understanding of the problem, the energy does not increase.

I think this is an excellent and subtle problem. Mr. Zapper...

You said:

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.

As I see it, there is a HUGE difference between these two experiments and it underlies the problem I have been asking. If you do the one with the mass hanging through a hole in the table, the increase in kinetic energy of the spinning mass is EXACTLY balanced by a decrease in gravitational potential energy of the falling mass (at the center of the table) It is just that kind of a reduction in energy that I am looking for. It was the origin of my original question.

Lee

To Dr. Al:

You said:

My response to this remains the same: You are assuming conservation of angular momentum. I say it's not conserved in this case, because of the torque exerted by the rope.

But if you apply the law of conservation of angular momentum to the situation and if you use high speed photography, the numbers bear out very nicely that angular momentum is conserved. And as Zapper said in an earlier message, suppose we think of the diameter of the rod in millimeters and the length of the string in METERS. Then clearly, making the approximation that this is circular motion and that the tension in the string is providing a centripetal force + a force to do the work to increase the kinetic energy of the spinning volleyball doesn't seem like it should be much of a stretch of the imagination. I have never questioned the conservation of momentum. I am sure that it is true. The problem still remains that I cannot find a source of energy for the increase in KE of the volleyball.

Lee

Doc Al
Mentor
But if you apply the law of conservation of angular momentum to the situation and if you use high speed photography, the numbers bear out very nicely that angular momentum is conserved.
Well, now we're getting somewhere! An experiment will settle things. Do you have a reference for this experiment?

No. This is just an experiment you can do with a video camera and a volleyball on a pole. And you don't really have to DO it anyway! Just do it as a thought experiment like Einstein would have done. Use a VERY thin pole and strike the volleyball perpendicularly to the outstretched string. The ball now has energy mv^2/2.

The string continues to wrap around the pole until its length is such that the moment of inertia is half of what it was at the beginning. Since the pole is SOOO thin, the force the tension applies is radially inward and the tension does work on the ball bringing it closer to the center of the circle. The result is that while momentum is conserved (The angular velocity is now twice as great as it was, while the moment of inertia is half of its original value) the kinetic energy (since it depends on the SQUARE of omega) is now twice its original value. Well work had to be done, so that had to cost some energy. But where did that energy come from? What energy has been reduced as a result of the kinetic energy of the ball going up? That's the essence of my problem, but I don't seem to be getting any answers here. I REALLY hope that you can help!

Thanks!
Lee

Wow, this is a neat problem! I wish I had thought of stuff like this when I was taking physics. I had trouble keeping up with the homework, so I didn't have much free time to ponder. This thread is going so fast that I can't read all of every post, so forgive me if I'm repeating someone.

Here's my two cents:

I agree with Doc Al that no work is being done on the ball by the string or the pole. I also agree that imagining an infinitely thin pole doesn't help much, because the string would not wrap around it. Instead, imagine a very fat pole. At all times (after the initial punch given to the ball) the string is tangent to the pole, and the ball is accelerating toward that instantaneous point of tangency. But it is moving at right angles to that direction of acceleration, so no work can be done (edit: by the string).

Therefore, I would say that the angular velocity changes in such a way that energy is conserved. Also, as Doc Al pointed out, the string exerts a torque on the ball (about the center of the fat pole), so angular momentum of the ball need not be conserved. In fact, if you look at the whole system of the earth, the pole, and the ball, I believe that some angular momentum is transferred to the earth.

Last edited:
krab