A body is executing a linear S.H.M. Its potential energies at the displacement ' \(x\) ' and ' \(y\) ' are ' \(E_1\) ' and ' \(\mathrm{E}_2\) ' respectively. Its potential energy at displacement \((\mathrm{x}+\mathrm{y})\) will be

We know,
Potential Energy \(\mathrm{E}_{\mathrm{P}}=\frac{1}{2} \mathrm{Kx}^2\)
\(\therefore \quad \mathrm{E}_1=\frac{1}{2} \mathrm{Kx}^2 \Rightarrow \mathrm{x}=\sqrt{\frac{2 \mathrm{E}_1}{\mathrm{~K}}}\)... (i)
and
\(\mathrm{E}_2=\frac{1}{2} \mathrm{Ky}^2 \Rightarrow \mathrm{y}=\sqrt{\frac{2 \mathrm{E}_2}{\mathrm{~K}}}\)... (ii)
Given, total displacement \(=(x+y)\)
\(\therefore \quad\) Potential energy at displacement \((x+y)\),
\(\mathrm{E}\) is \(\frac{1}{2} \mathrm{~K}(\mathrm{x}+\mathrm{y})^2\)
\(\begin{aligned}
& =\frac{1}{2} K\left(\sqrt{\frac{2 E_1}{K}}+\sqrt{\frac{2 E_2}{K}}\right)^2 \\
& =\frac{1}{2} K\left[\frac{2 E_1}{K}+\frac{2 E_2}{K}+2\left(\sqrt{\frac{E_1}{K}}\right)\left(\sqrt{\frac{2 E_2}{K}}\right)\right] \\
& =\left(\sqrt{E_1}+\sqrt{E_2}\right)^2=\left(E_1+E_2+2 \sqrt{E_1 E_2}\right)
\end{aligned}\)