Calculating Total Energy from Combined Rotational Motion of Two Spheres

[faklif]

Homework Statement

I have a sphere on top of another larger fixed sphere. The sphere on top rolls on the fixed one without sliding. The moving sphere is rolling and therefore has an angular velocity, it also moves giving it an angular velocity around the center of the fixed sphere. To calculate the energy from the rotation I need the rotational speed of the moving sphere.

Homework Equations

The Attempt at a Solution

I know that the answer is supposed to come from adding the angular velocity due to the moving sphere spinning and the angular velocity seen from the center of the larger sphere but I can't really see why. My initial thought, which is wrong, was that the total energy would come from the rotation of the moving sphere and the angular velocity around the center of the fixed sphere would only contribute as "regular" kinetic energy.

 

[Redbelly98]

The Attempt at a Solution

I know that the answer is supposed to come from adding the angular velocity due to the moving sphere spinning and the angular velocity seen from the center of the larger sphere but I can't really see why. My initial thought, which is wrong, was that the total energy would come from the rotation of the moving sphere and the angular velocity around the center of the fixed sphere would only contribute as "regular" kinetic energy.

Why do you think that is wrong? It seems right to me. The angular velocity around the center of the fixed sphere is the same as motion of the center-of-mass of the moving sphere, which is "regular" (i.e. translational) kinetic energy.

 

[faklif]

Thanks, I don't know what I was thinking. ;)