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Chapter 11: Problem 8

A group of polar bears is standing around the edge of a slowly rotating ice floe. If the bears all walk to the center, what happens to the rotation rate?

Short Answer

Expert verified
When the polar bears move to the center of the rotating ice floe, the rotation rate of the ice floe increases. This is due to the conservation of angular momentum in the absence of external torques.

Step by step solution

01

Identify the initial condition

Initially, the polar bears are standing around the edge of the slowly rotating ice floe. This scenario gives us the initial moment of inertia, which is large due to the distance of the bears from the center. As the ice floe rotates slowly, the angular speed is low.
02

Understand the action

The action here is the polar bears moving to the center of the ice floe. When the bears move closer to the center, the moment of inertia of the system decreases as moment of inertia depends on the mass and distance from the pivot (center of the ice floe, in this case). The farther away the mass, the greater the moment of inertia, so as they move closer, the total moment of inertia decreases.
03

Apply the principle of conservation of angular momentum

\[LI = L_f\] This principle states that in the absence of external torques, the total momentum of a system remains constant. In this case, the initial angular momentum (LI) which is the moment of inertia multiplied by the initial angular speed is equal to the final angular momentum (L_f) which is the final moment of inertia multiplied by the final angular speed. If the moment of inertia decreases and no external torques are acting on the system (our assumption in this case because it鈥檚 not mentioned in the problem), then the angular speed or rotation rate must increase to keep their product constant.

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