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(B) For a body executing $SHM$,the potential energy $U$ at displacement $d$ is given by $U = \frac{1}{2} k d^2$. Given: $E_1 = \frac{1}{2} k x^2 \impl

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$\sqrt{E}=\sqrt{E_1}+\sqrt{E_2}$

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(B) For a body executing $SHM$,the potential energy $U$ at displacement $d$ is given by $U = \frac{1}{2} k d^2$. Given: $E_1 = \frac{1}{2} k x^2 \implies x = \sqrt{\frac{2 E_1}{k}}$ $E_2 = \frac{1}{2} k y^2 \implies y = \sqrt{\frac{2 E_2}{k}}$ The potential energy $E$ at displacement $(x+y)$ is: $E = \frac{1}{2} k (x+y)^2$ $E = \frac{1}{2} k (x^2 + y^2 + 2xy)$ $E = \frac{1}{2} k x^2 + \frac{1}{2} k y^2 + 2 \left( \frac{1}{2} k x y \right)$ $E = E_1 + E_2 + 2 \sqrt{\left( \frac{1}{2} k x^2 \right) \left( \frac{1}{2} k y^2 \right)}$ $E = E_1 + E_2 + 2 \sqrt{E_1 E_2}$ $E = (\sqrt{E_1} + \sqrt{E_2})^2$ Taking the square root on both sides: $\sqrt{E} = \sqrt{E_1} + \sqrt{E_2}$

$A$ body is executing simple harmonic motion. At a displacement $x$,its potential energy is $E_1$ and at a displacement $y$,its potential energy is $E_2$. The potential energy $E$ at a displacement $(x+y)$ is

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