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SHM in Vertical Spring Block

In the figure...

Question

In the figure shows, pulley and spring are ideal. Find the potential energy stored in the spring $(m_{1} > m_{2})$

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Solution

we know $2 T = k x$
$T = \frac{k x}{2}$
for man m $T = m_{1} g = \frac{k x}{2} = x = \frac{2 m_{1} g}{k}$
$P E = \frac{1}{2} k x^{2}$
$= \frac{1}{2} (\frac{2 m_{1} g}{k})^{2} = \frac{1}{2} \frac{k 4 m_{1}^{2} g^{2}}{k^{2}}$
$∴ P E = \frac{2 m_{1}^{2} g^{2}}{k}$

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