$\mathrm{MV}_{0} \mathrm{L}=\mathrm{MV}_{1}(\mathrm{L}-\ell)$

$V_{1}=V_{0}\left(\frac{L}{L-\ell}\right)$

$\mathrm{w}_{\mathrm{g}}+\mathrm{w}_{\mathrm{p}}=\Delta \mathrm{KE}$

$-m g \ell+w_{p}=\frac{1}{2} m\left(V_{1}^{2}-V_{0}^{2}\right)$

$w_{p}=m g \ell+\frac{1}{2} m V_{0}^{2}\left(\left(\frac{L}{L-\ell}\right)^{2}-1\right)$

$=m g \ell+\frac{1}{2} m V_{0}^{2}\left(\left(1-\frac{L}{L-\ell}\right)^{-2}-1\right)$

Now, $\ell<<\mathrm{L}$

By, Binomial approximation

$=m g \ell+\frac{1}{2} m V_{0}^{2}\left(\left(1+\frac{L}{L-\ell}\right)^{-2}-1\right)$

$=m g \ell+\frac{1}{2} m V_{0}^{2}\left(\frac{2 \ell}{L}\right)$

$\mathrm{W}_{\mathrm{P}}=\mathrm{mg} \ell+\mathrm{mV}_{0}^{2} \frac{\ell}{\mathrm{L}}$

Here, $\mathrm{V}_{0}=$ maximum velocity $=\omega \times \mathrm{A}=(\sqrt{\frac{\mathrm{g}}{\mathrm{L}}})\left(\theta_{0} \mathrm{L}\right)$

$\mathrm{So}, \mathrm{w}_{\mathrm{p}}=\mathrm{mg} \ell+\mathrm{m}\left(\theta_{0} \sqrt{\mathrm{gL}}\right)^{2} \frac{\ell}{\mathrm{L}}$

$=m g \ell\left(1+\theta_{0}^{2}\right)$