a
Angular momentum conservation

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

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

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

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

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

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

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 \(=\omega \times \mathrm{A}=(\sqrt{\frac{\mathrm{g}}{\mathrm{L}}})\left(\theta_{0} \mathrm{L}\right)\)

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

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