First slide

Simple harmonic motion

Question

The potential energy of a simple harmonic oscillator when the particle is half way to its end point is (where E is the total energy)

Moderate

A

$\frac{1}{8} E$
B

$\frac{1}{4} E$
C

$\frac{1}{2} E$
D

$\frac{2}{3} E$

Solution

$\frac{U}{E} = \frac{\frac{1}{2} {mω}^{2} y^{2}}{\frac{1}{2} {mω}^{2} a^{2}} = \frac{y^{2}}{a^{2}} = \frac{{(\frac{a}{2})}^{2}}{a^{2}} = \frac{1}{4}$ $\Rightarrow U = \frac{E}{4}$

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