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ritwik06

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## Homework Statement

There are two spheres of equal radii. The bottom sphere is fixed in space. The other sphere purely(i.e. without slipping) rolls over the fixed sphere with constant angular velocity [tex]\omega 1[/tex] about its axis of rotation. Find the angular velocity of the axis of the rotating sphere about the center of the fixed sphere [tex]\omega 2[/tex] . (Assume a gravity free environment)

http://img201.imageshack.us/img201/6729/sphrert9.jpg

## The Attempt at a Solution

The problem is that I get two different and inconsistent answers from two different methods.

Method 1:

I follow a basic approach:

The sphere has a constant angular velocity about its own axis. The linear speed of the center of mass will be R[tex]\omega1[/tex]. The center of mass will move in a circle of 2R. Its velocity = R[tex]\omega 1[/tex]

The angular velocity of the center of mass of the rotating sphere(wrt to centre of fixed sphere) is =(R[tex]\omega1[/tex])/2R

[tex]\omega 2[/tex]=[tex]\omega 1[/tex]*0.5Method 2:

As the radii of the two spheres is the same. When the rolling sphere rotates through an angle of 2[tex]\pi[/tex] along its axis, it would have reached its original position. It implies that the centre of mass of the rotating sphere has rotated through an angle 2[tex]\pi[/tex] about the centre of the fixed sphere (in the same time interval)

2[tex]\frac{\pi}{\omega 1}[/tex]=2[tex]\frac{\pi}{\omega 2}[/tex]

[tex]\omega2[/tex]=[tex]\omega1[/tex]

Now which of the results is plausible. Both seem right to me. Please help!

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