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.