First slide

Simple harmonic motion

Question

A body is executing Simple Harmonic Motion. At a displacement x its potential energy is E₁ and at a displacement y its potential energy is E₂. 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

$\begin{array}{l} E_{1} = \frac{1}{2} K x^{2} \Rightarrow x = \sqrt{\frac{2 E_{1}}{K}}, E_{2} = \frac{1}{2} K y^{2} \Rightarrow y = \sqrt{\frac{2 E_{2}}{K}} \\ and E = \frac{1}{2} K (x + y)^{2} \Rightarrow x + y = \sqrt{\frac{2 E}{K}} \\ \Rightarrow \sqrt{\frac{2 E_{1}}{K}} + \sqrt{\frac{2 E_{2}}{K}} = \sqrt{\frac{2 E}{K}} \Rightarrow \sqrt{E_{1}} + \sqrt{E_{2}} = \sqrt{E} \end{array}$

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