First slide

Simple hormonic motion

Question

The resultant of two rectangular simple harmonic motions of the same frequency and unequal amplitudes but differing in phase by $\frac{π}{2}$ is

Moderate

A

Simple harmonic
B

Circular
C

Elliptical
D

Parabolic

Solution

If first equation is $y_{1} = a_{1} \sin ωt$
$\Rightarrow \sin ωt = \frac{y_{1}}{a_{1}}$ -------------------(i)
then second equation will be
$y_{2} = a_{2} \sin (ωt + \frac{π}{2})$
$= a_{2} [\sin ωt \cos \frac{π}{2} + \cos ωtsin \frac{π}{2}] = a_{2} \cos ωt$
$\Rightarrow \cos ωt = \frac{y_{2}}{a_{2}}$ --------------(ii)
By squaring and adding equations (i) and (ii)
$\sin^{2} ωt + \cos^{2} ωt = \frac{{y^{2}}_{1}}{{a^{2}}_{1}} + \frac{{y_{2}}^{2}}{{a_{2}}^{2}}^{}$
$\Rightarrow \frac{{y^{2}}_{1}}{{a^{2}}_{1}} + \frac{{y_{2}}^{2}}{{a_{2}}^{2}}$ = 1. This is the equation of ellipse.

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