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(D) The potential energy of a body executing simple harmonic motion at displacement $x$ is given by $U = \frac{1}{2} kx^2$,where $k$ is the force cons

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$E_1 + E_2 + 2\sqrt{E_1 E_2}$

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(D) The potential energy of a body executing simple harmonic motion at displacement $x$ is given by $U = \frac{1}{2} kx^2$,where $k$ is the force constant.Given $E_1 = \frac{1}{2} kx^2$ and $E_2 = \frac{1}{2} ky^2$.We need to find the potential energy $E$ at displacement $(x + y)$:$E = \frac{1}{2} k(x + y)^2$$E = \frac{1}{2} k(x^2 + y^2 + 2xy)$$E = \frac{1}{2} kx^2 + \frac{1}{2} ky^2 + 2 \cdot \frac{1}{2} kxy$Since $E_1 = \frac{1}{2} kx^2$ and $E_2 = \frac{1}{2} ky^2$,we have $x = \sqrt{\frac{2E_1}{k}}$ and $y = \sqrt{\frac{2E_2}{k}}$.Substituting these into the expression for $E$:$E = E_1 + E_2 + 2 \cdot \frac{1}{2} k \left( \sqrt{\frac{2E_1}{k}} \right) \left( \sqrt{\frac{2E_2}{k}} \right)$$E = E_1 + E_2 + k \left( \frac{2}{k} \sqrt{E_1 E_2} \right)$$E = E_1 + E_2 + 2\sqrt{E_1 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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