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(D) The potential energy of a body in simple harmonic motion is given by $P = \frac{1}{2} k x^2$,where $k$ is the force constant.Given $P_{1} = \frac{

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$P_{1}+P_{2}+2 \sqrt{P_{1} P_{2}}$

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(D) The potential energy of a body in simple harmonic motion is given by $P = \frac{1}{2} k x^2$,where $k$ is the force constant.Given $P_{1} = \frac{1}{2} k x_{1}^2$ and $P_{2} = \frac{1}{2} k x_{2}^2$.We need to find the potential energy $P$ at displacement $(x_{1} + x_{2})$:$P = \frac{1}{2} k (x_{1} + x_{2})^2$$P = \frac{1}{2} k (x_{1}^2 + x_{2}^2 + 2 x_{1} x_{2})$$P = \frac{1}{2} k x_{1}^2 + \frac{1}{2} k x_{2}^2 + 2 \left( \sqrt{\frac{1}{2} k x_{1}^2} \right) \left( \sqrt{\frac{1}{2} k x_{2}^2} \right)$$P = P_{1} + P_{2} + 2 \sqrt{P_{1} P_{2}}$.

$A$ body performing simple harmonic motion has potential energy $P_{1}$ at displacement $x_{1}$. Its potential energy is $P_{2}$ at displacement $x_{2}$. The potential energy $P$ at displacement $(x_{1}+x_{2})$ is

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