A body performing simple harmonic motion has potential energy ' $\mathrm{P}_{1}{ }^{\prime}$ at displacement…

A body performing simple harmonic motion has potential energy ' $\mathrm{P}_{1}{ }^{\prime}$ at displacement ' $x_{1}$ '. Its potential energy is ' $\mathrm{P}_{2}$ ' at displacement ' $x_{2}$ '. The potential energy 'P' at displacement $\left(x_{1}+x_{2}\right)$ is
  1. $\mathrm{P}_{1}+\mathrm{P}_{2}$
  2. $\sqrt{\mathrm{P}_{1} \mathrm{P}_{2}}$
  3. $\sqrt{\mathrm{P}_{1}^{2}+\mathrm{P}_{2}^{2}}$
  4. $\mathrm{P}_{1}+\mathrm{P}_{2}+2 \sqrt{\mathrm{P}_{1} \mathrm{P}_{2}}$

Solution

$\begin{aligned} P &=\frac{1}{2} k\left(x_{1}+x_{2}\right)^{2} \\ &=\frac{1}{2} k\left(x_{1}^{2}+x_{2}^{2}+2 x_{1} x_{2}\right) \\ &=\frac{1}{2} k x_{1}^{2}+\frac{1}{2} k x_{2}^{2}+2\left(\sqrt{\frac{k}{2}} x\right)\left(\sqrt{\frac{k}{2}} x_{2}\right) \\ &=P_{1}+P_{2}+2 \sqrt{P_{1} P_{2}} \end{aligned}$ :

Asked in: MHT CET 2020 (19 Oct Shift 1)

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