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I was studying for the GRE and came across the problem below.

Problem

A small puck of mass $M$ is attached to a massless string that drops through a hole in a platform, as shown in the diagram above. The puck rotates at radius $R$ when the tension in the string is $T$. The string is pulled downwards until the radius of rotation is $r<R$. What is the change in energy of the puck when the radius is decreased? You may assume the puck is a point mass.

The book proceeds to solve this problem by calculating the velocity at $R$ using the given central force and then calculating the velocity at r using conservation of angular momentum before finally calculating the change in energy as the difference in kinetic energy (the answer is A). This is all fairly straightforward. However, isn't there usually also a reduction in potential energy associated with a reduced radius of orbit? I'm not sure how one would go about calculating the potential energy associated with a force like this. I'm not sure if it is even a conservative force. If not, is there a potential energy associated with non-conservative forces?

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Indeed, in the rotating frame, there is centrifugal force and Coriolis force. They are both inertial force. The former can construct a potential energy, and the latter is always perpendicular to velocity, which means it cannot do work.

So we consider the centrifugal force, $$\vec{F_c}=m\omega^2\vec{r}$$it is easy to get its potential energy,$$E_p=-\frac{1}{2}m\omega^2r^2$$It's obvious that $E_p=-E_k$, which $E_k$ is kinetic energy.

If you have more questions, let me know.

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  • $\begingroup$ Isn't the centrifugal force a fictitious force? I've never seen it considered when calculating potential energy. For instance, we don't use it when calculating the potential energy of an object in a gravitational orbit. $\endgroup$
    – Leon
    Commented Sep 1, 2021 at 7:40
  • $\begingroup$ Yes, centrifugal force is a fictitious force. In a rotating frame, it could correspond to a potential energy for convenience in energy calculating. But in a inertial frame, even centrifugal force doesn't exist. So it entirely depends on what frame you choose. As you said, when considering a gravitational orbit, we don't need to write down the potential energy that fictitious force causes, because we usually consider the problem in a inertial frame. $\endgroup$
    – Victor Zhang
    Commented Sep 1, 2021 at 10:27
  • $\begingroup$ If what you want to ask is the potential energy that tension $T$ causes in the inertial frame, well that's impossible to calculate, because the process to reduce the radius is casual. We can't tell if $T$ is conservative force, not to speak of potential energy. $\endgroup$
    – Victor Zhang
    Commented Sep 1, 2021 at 10:28

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