Question

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

• A. Simple harmonic
• B. Circular
• C. Elliptical
• D. Parabolic

Answer 1

Answer: (C)

Elliptical

Answer 2

If first equation is $y_{1} = a_{1}\sin\omega t$ ⇒ $\sin\omega t = \frac{y_{1}}{a_{1}}$ ... (i)

then second equation will be $y_{2} = a_{2}\sin\left( \omega t + \frac{\pi}{2} \right)$

$= a_{2}\left\lbrack \sin\omega t\cos\frac{\pi}{2} + \cos\omega t\sin\frac{\pi}{2} \right\rbrack = a_{2}\cos\omega t$

⇒ $\cos\omega t = \frac{y_{2}}{a_{2}}$ ... (ii)

By squaring and adding equation (i) and (ii)

$\sin^{2}\omega t + \cos^{2}\omega t = \frac{y_{1}^{2}}{a_{1}^{2}} + \frac{y_{2}^{2}}{a_{2}^{2}}$

⇒ $\frac{y_{1}^{2}}{a_{1}^{2}} + \frac{y_{2}^{2}}{a_{2}^{2}} = 1$; This is the equation of ellipse.