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## Homework Statement

I have a string(length= r) and negligible mass to which a uniform sphere of radius r is attached on one end. I fix upper end of the string to the ceiling and release the sphere when it made angle θ

_{0}with the vertical. I have to write the kinetic energy of the sphere when it makes an angle θ with the vertical.

## The Attempt at a Solution

Say, the angular velocity of the center of mass about the point of suspension be ω when it makes an angle θ.

I have a doubt:

Way 1:

KE=0.5*Iω

^{2}

Using parallel axis theorem;

'I' about the point of suspension= (2/5)mr

^{2}+m(2r)

^{2}

=4.4mr

^{2}

KE=2.2mr

^{2}ω

^{2}

Way 2:

KE=0.5*mv

_{cm}

^{2}

v=2rω

KE=2mr

^{2}ω

^{2}

Which of these is correct. My book gives the method 1 as a correct solution. But I cannot understand what's wrong with method 2?

What I think is that method 1 gives kinetic energy when the sphere itself is rotating about its own axis with the same angular velocity 'ω' as the center of mass has about the point of suspension. Am I right?

But in the case specified does the sphere rotate about its axis? If yes, from where does the torque come from? The tension and the mass both pass from the center of mass, so they will have zero torque.

Please help me. I am confused!