Bug on Globe

A toy globe rotates freely without friction with an initial angular velocity ω₀. A bug starting at one pole N travels to the other pole S along a meridian with constant velocity v. The axis of rotation of the globe is held fixed. Let M and R denote the mass and radius of the globe (a solid sphere, moment of inertia I₀ = 2MR²/5, m the mass of the bug, and T the duration of the bug's journey (see Figure 1.1).

Figure 1.1

Show that, during the time the bug is traveling, the globe rotates through an angle

A useful integral is

SOLUTION

The angular velocity of the globe is always in the same direction (along the fixed axis, see Figure 1.2). Since the angular momentum 1 is constant

Figure 1.2

And |1| = Iω we may write

(1)

Initially I₀ is just the moment of inertia of the sphere (the bug is at the pole), so I₀ = (2/5)MR². Substituting this into (1) we obtain

where  so

We used here sin² (x/2) = (1/2)(1-cos x) and the integral given in the problem:

If the bug had mass m = 0, the angle ∆θ would be

which corresponds to the free rotation of the globe with angular velocityω₀.