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Mass over a Smooth Pulley

Try this problem, useful for Physics Olympiad, based on Mass over a Smooth Pulley.

The Problem:

One end of a string is attached to a rigid wall at point O, passes over a smooth pulley and carries a hanger S of mass M at its other end. Another object P of mass M is suspended from a light ring that can slide without friction, along the string, as is shown in the figure. OA is horizontal. Find the additional mass to be attached to the hanger S so as to raise the object P by 10cm.

Solution:

Let us denote the tension in each string as T.
$$2Tcos\theta=Mg$$
$$\Rightarrow2(Mg)cos\theta=Mg$$
$$\Rightarrow cos\theta=\frac{1}{2}$$
$$ \Rightarrow\theta=60^\circ$$
$$ \Rightarrow tan60=\frac{\frac{40\sqrt{3}}{2}}{PQ}$$
$$ \Rightarrow tan60^\circ=\sqrt{3}$$
Hence,
$$ PQ=20cm$$
Now, when an additional mass m is hung from the pulley, the length of PQ changes to P'Q'
P'Q'=PQ-10=20-10=10
Hence, P'Q'=1cm.
$$ Q'S'=\sqrt{P'Q'^2+P'S^2}=\sqrt{1300}$$
Now, again considering the force equation
$$\Rightarrow 2Tcos\theta=Mg$$
$$\Rightarrow 2(M+m)g\times\frac{10}{\sqrt{1300}}=Mg$$
$$\Rightarrow 2(M+m)\times\frac{1}{\sqrt{13}}=M$$
$$ \Rightarrow (M+m)=\sqrt{13}M$$
$$\Rightarrow 2m=M(\sqrt{13}-2)$$
$$\Rightarrow m=\frac{M\times(\sqrt{13}-2)}{2}=0.9M$$

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