First slide

Simple harmonic motion

Question

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 displacement $(x + y)$ is

Moderate

A

$\sqrt{E} = \sqrt{E_{1}} - \sqrt{E_{2}}$
B

$\sqrt{E} = \sqrt{E_{1}} + \sqrt{E_{2}}$
C

$E = E_{1} + E_{2}$
D

$E = E_{1} - E_{2}$

Solution

$E_{1} = \frac{1}{2} {Kx}^{2} \Rightarrow x = \sqrt{\frac{2 E_{1}}{K}}$ , $E_{2} = \frac{1}{2} {Ky}^{2} \Rightarrow y = \sqrt{\frac{2 E_{2}}{K}}$
and $E = \frac{1}{2} K {(x + y)}^{2} \Rightarrow x + y = \sqrt{\frac{2 E}{K}}$
$\Rightarrow \sqrt{\frac{2 E_{1}}{K}} + \sqrt{\frac{2 E_{2}}{K}} = \sqrt{\frac{2 E}{K}}$ $\Rightarrow \sqrt{E_{1}} + \sqrt{E_{2}} = \sqrt{E}$

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