We are studying rotational dynamics and I read in my textbook, the explanation o

Question

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.

Explanation / Answer

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.

Related Questions

Navigate

• Browse (All)
• Browse W
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.