First slide

Oscillation motion

Question

The displacement of a particle is represented by the equation $y = \sin^{3} ωt$ . The motion is

Difficult

A

non-periodic
B

periodic but not simple harmonic
C

simple harmonic with period $\frac{2 π}{ω}$
D

simple harmonic with period $\frac{π}{ω}$

Solution

Given equation of motion is
$y = \sin^{3} ωt$
= $\frac{(3 \sin ωt - 4 \sin ωt)}{4}$
[ $∵ \sin 3 θ = 3 \sin θ - 4 \sin^{3} θ$ ]
$\Rightarrow \frac{dy}{dt} = \frac{[\frac{d}{dt} (3 \sin ωt) - \frac{d}{dt} (4 \sin 3 ωt)]}{4}$
$\Rightarrow \frac{4 d y}{d t} = 3 ω \cos ω t - 4 \times [3 ω \cos 3 ω t]$
$\Rightarrow 4 \times \frac{d^{2} y}{{dt}^{2}} = - 3 ω^{2} \sin ωt + 12 ω \sin 3 ωt$
$\Rightarrow \frac{d^{2} y}{{dt}^{2}} = - \frac{3 ω^{2} \sin ωt + 12 ω \sin 3 ωt}{4}$
$\Rightarrow \frac{d^{2} y}{{dt}^{2}}$ is not proportional to y.

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