Question

Three simple harmonic motions in the same direction having the same amplitude a and same period are superposed. If each differs in phase from the next by $ 45{}^\circ , $ then (1) the resultant amplitude is $ (1+\sqrt{2})a $ (2) the phase of the resultant motion relative to the first is $ 90{}^\circ $ (3) the energy associated with the resulting motion is $ (3+2\sqrt{2}) $ times the energy associated with any single motion (4) the resulting motion is not simple harmonic

• A. 1, 2 and 3 are correct.
• B. 1 and 2 are correct.
• C. 2 and 4 are correct.
• D. 1 and 3 are correct.

Answer 1

Answer: (D)

Let simple harmonic motion be represented by $ {{y}_{1}}=a\sin \left( \omega t-\frac{\pi }{4} \right);{{y}_{2}}=a\sin \,\omega t $ and $ {{y}_{3}}=a\sin \left( \omega t+\frac{\pi }{4} \right) $
On superimposing the resultant SHM will be
$ y=a\left[ \sin \left( \omega t-\frac{\pi }{4} \right)+\sin \omega t+\sin \left( \omega t+\frac{\pi }{4} \right) \right] $
$=a\left[ 2\sin \omega t\cos \frac{\pi }{4}+\sin \omega t \right] $
$=a[\sqrt{2}\sin \omega t+\sin g\omega t] $
$=a(1+\sqrt{2})\sin \omega t $ Resultant amplitude
$=(1+\sqrt{2})a $ Energy is $ SHM\propto (amplitude) $
$ \therefore $ $ \frac{{{E}_{\operatorname{Re}sul\tan t}}}{{{E}_{\sin gle}}}={{\left[ \frac{A}{a} \right]}^{2}}={{(\sqrt{2}+1)}^{2}}=(3+2\sqrt{2}) $
$ \Rightarrow $ $ {{E}_{\operatorname{Re}sula\tan t}}=(3+2\sqrt{2}){{E}_{\sin gle}} $