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. circle
• C. ellipse
• D. figure of eight

Answer 1

Answer: (C)

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 $y_2=a_2\sin (\omega t+\pi /2)=a_2\cos \omega t$ $\Rightarrow$ $\cos \omega t=\frac{y_2}{a_2}$ ?(ii) By squaring and adding Eqs. (i) and (ii), we get ${\sin }^2\omega t+{\cos }^2\omega t=\frac{y_1^2}{a_1^2}+\frac{y_2^2}{a_2^2}$ $\Rightarrow$ $\frac{y_1^2}{a_2^2}+\frac{y_2^2}{a_2^2}=1$ This is the equation of ellipse.