First slide

Simple harmonic motion

Question

A particle starts simple harmonic motion from the mean position. Its amplitude is a and total energy E. At one instant its kinetic energy is $3 E / 4 .$ Its displacement at that instant is

Moderate

A

$a / \sqrt{2}$
B

$a / 2$
C

$\frac{a}{\sqrt{3 / 2}}$
D

$a / \sqrt{3}$

Solution

$\frac{K}{E} = \frac{\frac{1}{2} {mω}^{2} (a^{2} - y^{2})}{\frac{1}{2} {mω}^{2} a^{2}} = \frac{a^{2} - y^{2}}{a^{2}} = 1 - \frac{y^{2}}{a^{2}}$
So, $\frac{(\frac{3 E}{4})}{E} = 1 - \frac{y^{2}}{a^{2}} \Rightarrow \frac{y^{2}}{a^{2}} = 1 - \frac{3}{4} = \frac{1}{4}$ $\Rightarrow y = \frac{a}{2}$

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