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A ball of mass M is attached to the end of a string...
My attempt (if it matters):
The initial period is given by $T_X = \frac{2\pi X}{v}$ for some $v$.
The new period is given by $T_Y = \frac{2\pi Y}{v}$ for the same $v$.
$Y = \frac{X}{2}$, so $T_Y=\frac{\pi X}{v} = \frac{T_X}{2}$

However, the correct answer is $\frac{1}{4}T$. Can somebody explain this to me?

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up vote 2 down vote accepted

OP writes in the question formulation(v2):

[...] for the same $v$.

Hint: The speed of the ball is not conserved. Think instead about

  1. why the angular momentum of the ball is conserved, and

  2. what angular momentum conservation implies.

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1  
$\omega$ is not angular momentum – Bernhard May 1 '12 at 14:57
    
Many thanks; I got it now. – Brandon Amos May 1 '12 at 15:03

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