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Two masses in the ratio $n:1$ are connected by a light inextensible string over a frictionless pulley (Atwood machine). After the system is released from rest, what is the acceleration of the centre of mass of the two-mass system?

1. $\left(\frac{n-1}{n+1}\right)^2 g$
2. $\left(\frac{n-1}{n+1}\right) g$
3. $(n-1)^2 g$
4. $\frac{1}{2}\left(\frac{n-1}{n+1}\right) g$

Correct answer: $\left(\frac{n-1}{n+1}\right)^2 g$

Solution

For an Atwood machine with masses $n m$ and $m$, the magnitude of acceleration of each mass is $a=\frac{n-1}{n+1}g$. The centre-of-mass acceleration is $a_{cm}=\frac{n m\,a - m\,a}{(n+1)m}=\frac{n-1}{n+1}a$. Substituting $a$ gives $a_{cm}=\left(\frac{n-1}{n+1}\right)^2 g$.

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