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

The correct option is C Elliptical

If first equation is $y_1=a_{1}sin\omega t\Rightarrow sin\omega t=\frac{y_1}{a_1}$ ...(i)
then second equation will be $y_2=a_{2}sin(\omega t+\frac{\pi }{2})$
$=a_2[sin\omega tcos\frac{\pi }{2}+cos\omega tsin\frac{\pi }{2}]=a_{2}cos\omega t$
$\Rightarrow cos\omega t=\frac{y_2}{a_2}$ ...(ii)
By squaring and adding equation (i) and (ii)
$si{n}^2\omega t+co{s}^2\omega t=\frac{y_1^2}{a_1^2}+\frac{y_2^2}{a_2^2}$
$\Rightarrow \frac{y_1^2}{a_1^2}+\frac{y_2^2}{a_2^2}=1;$ This is the equation of ellipse.