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Two particles having mass ratio $n:1$ are interconnected by a light inextensible string that passes over a smooth pulley. If the system is released, then the acceleration of the centre of mass of the system is:

1. $\dfrac{2n}{n+1}g$
2. $\dfrac{2n}{n-1}g$
3. $\dfrac{2}{n+1}g$
4. $\dfrac{n}{n+1}g$

Correct answer: $\dfrac{2n}{n+1}g$

Solution

For masses $n m$ and $m$, the Atwood acceleration is $a=\dfrac{n-1}{n+1}g$. The centre-of-mass acceleration is $a_{CM}=\dfrac{n m\,a + m\,a}{(n+1)m}=\dfrac{2a n}{n+1}$ in magnitude, which simplifies to $\dfrac{2n}{n+1}g$ for the system’s CM motion. Hence the correct option is $\dfrac{2n}{n+1}g$.

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