QR: Bear on High Wire

Due to a request, the demo today became an instant quiz question to discuss. I like you to think about this carefully. This is a great problem.

NOTE: If you submitted a late quiz report, you have gotten a 50 % credit for this one by default, without your doing anything. In order to get the full credit, you need to write a report about this problem. I strongly recommend you do it!!

We had a little bear riding a small unicycle (pulley). The pulley was put on a "high wire". The bear happily went back and forth on it! It was thanks to the balance due to the heavy weights at the end of the bear's extremely long plastic arms, (one of which I, not any student, helped break!!).

A student also pushed the bear from the side gently and then let go. The bear was seen to oscillate back and forth, and did not fall down, as long as the angular displacement of the bear was small.

Based on these observations,
(1) where was the center of mass of this bear (including the long arms and weights of course) -- was it above the wire, on the wire, or below the wire?
(2) how can you explain the (in)stability of the system in all three above cases of the position of the center of mass, in terms of the torque that the center of mass motion of the bear experiences?
(3) how can you explain the (in)stability of the system in all three above cases of the position of the center of mass, in terms of the potential energy function U(theta), where theta is the small angular displacement?

Additionally, how can you verify where the center of mass of the bear is, by using a string? Ans: hold the bear twice at two different orientations, and each time draw a vertical line using the string. The crossing point of the two lines is the center of mass.

Lastly, how can you measure the rotational inertia of this bear? Ans: Measure the center of mass position for this bear, and measure the frequency of the small angle oscillation. The angular frequency should be sqrt (g / (gamma L)) from our physical pendulum discussion, where L is the distance of the CM to the center of the rotation. So, from the measured values of L and omega, one can extract gamma. Finally, if one measures the mass of the bear using a scale, then we are done. The rotational inertia is then given by I = gamma ML² (from our definition during our discussion of the physical pendulum).