First slide

Simple harmonic 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

Difficult

A

Simple harmonic
B

Circular
C

Elliptical
D

Parabolic

Solution

If first equation is $y_{1} = a_{1} sinω t \Rightarrow sinω t = \frac{y_{1}}{a_{1}}$ ... (i)
then second equation will be $y_{2} = a_{2} \sin (ω t + \frac{π}{2})$
$= a_{2} [sinω tcos \frac{π}{2} + cosω tsin \frac{π}{2}] = a_{2} cosω t$
$\Rightarrow cosω t = \frac{y_{2}}{a_{2}}$ ... (ii)
By squaring and adding equation (i) and (ii)
$\sin^{2} ω t + \cos^{2} ω t = \frac{y_{1}^{2}}{a_{1}^{2}} + \frac{y_{2}^{2}}{a_{2}^{2}}$
$\Rightarrow \frac{y_{1}^{2}}{a_{1}^{2}} + \frac{y_{2}^{2}}{a_{2}^{2}} = 1$ ; This is the equation of ellipse.

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