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## Homework Statement

A solid sphere of mass M and radius R rotates freely in space with an angular velocity w about a fixed diameter. A particle of mass m, initially at one pole, movies with a constant velocity v along a great circle of the sphere. Show that when the particle has reached the other pole, the rotation of the sphere will have been retarded by an angle

[tex]\alpha = \omega T ( 1 - \sqrt{\frac{2M}{2M+5m}}) [/tex]

## Homework Equations

## The Attempt at a Solution

So I have a picture of a rotating sphere with a mass on it. I know that [tex]I_{total} = 2/5 M R^2 + mr^2[/tex] where r is the vector from the initial pole to the position of the particle on the sphere. I believe I also need to take into account the polar angle phi, and the azimuthal angle theta.

I know that angular momentum is conserved. So dL/dt = 0

I also know that L = Iw

I was thinking of doing the following: d/dt(Iw) = 0 = d/dt(I) w + I d/dt(w)

so far that hasn't worked, but I think it may be because I haven't been able to come up with a decent guess for the angular velocity w. My best guess so far is w = v/(r*sin(phi)), which is clearly incorrect.

Anyone have any hints?

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