Torsion and Rotation

I haven't done any physics in a while, but a bowl suspended from a twisted length of thread shouldn't be too hard.

A twisted thread provides the force that spins the bowl. It should obey Hooke's law: . This just says the torque (twisting force) is opposite to the direction of the twist, and equal to the amount of twist (theta) scaled by a constant (k) that is a property of the thread. Interestingly, the force does not depend on the force the bowl is exerting on the thread (i.e. the weight of the bowl,) nor does it depend on the length of the thread.

Given the torque, we can derive the angular acceleration using Newton's laws:   This says the acceleration equals the torque divided by the moment of inertia of the bowl. I estimated the moment of inertia as 2 kg at 10 cm from the center of the bowl, or .02 kg-m^2. Of course, it's really not a constant, because it will increase as the mercury rotates and moves outwards from the center.

Integrating the acceleration over time gives the angular velocity. The plot above gives a sense of how the velocity of the bowl changes over time. Note that the shape of the curves should be reflective of reality, but the actual values (e.g. revolutions per minutes) should not be trusted at all: they are based on a complete guess for the term k / I. Unluckily, I can't find values for k for types of twine online. Luckily, if we actually put a bowl on a string, it should be easy to experimentally derive this term for the system.

Anyway, the system does spend a good portion of the time at a relatively constant velocity (the flatish peak in the velocity plots.) That's good, but I have an idea for how to get a much flatter, longer  peak in the rotational velocity...