Rotational kinetic energy

aaaa202

Okay thanks! Very helpful! Though one question: Why is it that the rotation MUST be around the center of pole, if the balls angular momentum is to be conserved?

Philip Wood

There is rotation about the centre in the case we're considering, isn't there? Angular momentum is not conserved, though. Admittedly it's not a pure rotation,because the ball is also moving radially inwards. But this isn't relevant, either!

When you talk about the ball's angular momentum (that is the angular momentum of a body acted upon by an external force) you have to specify about what point you're calculating that angular momentum. You need a fixed point (certainly not, for example, the point of run-off of the string from the circumference of the pole, because that point keeps moving round the pole, and is therefore accelerating towards the centre).

Exactly the same goes for torque. If you choose the same fixed point about which to calculate torque, G and angular momentum, L, a very simple law applies: G = dL/dt.

Michael C

Angular momentum is always conserved for the whole system (ball + pole + Earth), no matter which axis is chosen to measure it.

In this case angular momentum is transferred from the ball to the Earth via the pole: there is a torque on the pole, and therefore on the Earth. This means that if we consider only the pole and the ball we come to the conclusion that angular momentum is not conserved: in fact the momentum has been transferred elsewhere.

The ball would have a constant angular momentum if it did not exert a torque on something else. This would be the case if the centre of rotation of the string always stayed at the same point, for instance in the examples where the string is being pulled through a hole.

aaaa202

hmm it's just that when you see the ball for the point of contact between string and pole it makes a uniform circular motion. So can't you say that the angular momentum is conserved in this frame for the ball? And why does that not qualify to the ball's angular momentum being conserved like if the rotation was around the center of mass? :)

Philip Wood

Michael C. Agreed. Though, of course, there are interesting cases, such as a system of two charges moving at an angle to each other. [That's not the complete system, I hear someone say.]

aaaa202 As I said, the point of contact of string and pole is accelerating. The laws of Physics need modifying somewhat for use in an accelerating (non-inertial) reference frame. That's why I'm choosing to take our torque and angular momentum about the still centre of the pole.

Michael C

The point of contact is changing all the time: it's turning in a circle around the pole, so (as Philip pointed out) it's constantly accelerating. If we fix one point on the surface of the pole and measure the angular momentum around this point, we'll see that the momentum of the ball must be changing, since there is only one instant when the ball exerts no torque in this frame: the instant when the centre of rotation is at the point we have fixed. For the rest of the time, the centre of rotation is not at the point we have fixed, so there is torque around this point.

aaaa202

yes okay, I should have realized that. But doesn't there exist conservation laws in non inertial reference frames?

Philip Wood

I expect so. Indeed I expect we could easily find such a frame in which angular momentum is conserved for the ball. But that would not be anything to be especially pleased about. The ball's angular momentum would be conserved simply because we've chosen a special frame in which it is conserved. In this frame, things whose angular momentum we'd normally expect to be conserved won't have it conserved... The laws of Physics are usually easier in inertial frames.