(Because, as anybody knows, that’s the answer to “Pop Quiz, Hotshot”…)
The answer to the pop quiz posted below is “v.” That is, the speed is unchanged between the start of the problem and the collision between the ball and the pole. There are several ways to see this– conservation of energy is my usual approach (the only energy at the start of the problem is the kinetic energy of the ball’s motion, and nothing else in the problem takes up any energy, so you’ve got to have the same kinetic energy at the end)., but I really like Ross Smith’s dimensional analysis argument. If I were giving prizes, he’d get one.
A couple of other comments on things that were said by various people:
- Angular momentum is a red herring. If you look carefully at the situation, you’ll find that the string is not quite perpendicular to the ball’s velocity (it might help to draw a better picture than mine), so there’s a small component of force in the backwards direction. This can be thought of as a torque that keeps the angular momentum from increasing as the ball is pulled in.
- If the force isn’t perpendicular, why doesn’t the magnitude of the velocity change? It would change, if the force was perpendicular– the work done moving the ball in to a smaller orbit would be converted to kinetic energy, and the ball’s speed would increase (the inward force is slightly greater than the required centripetal force). The fact that it doesn’t increase is due to the torque thing mentioned above.
- Doesn’t a tetherball speed up as it gets closer to the post? The linear speed doesn’t change, but the angular speed does. As the orbits get smaller, the ball takes less time to complete each revolution, so it appears to be moving faster.
Credit where due: I heard this problem from somebody on Usenet years ago, long enough that I’ve forgotten who it was. It was introduced as a problem from a test given to prospective TA’s at one of the Ivies, consisting of problems that could be solved using freshman physics principles that would induce the same sort of confusion in first-year grad students that the classic frictionless-block-sliding-on-an-inclined-plane problems induce in actual freshmen.
These sorts of problems are sort of hard to come by, so I doubt I’ll be making this a regular feature. If you know any good ones (I’m stealing the three-cylinders one from BioCurious for the next time I teach something having to do with thermodynamics), I’d love to hear them.