A) The resultant amplitude is \[(1+\sqrt{2)}a\]
B) The phase of the resultant motion relative to the first is 90°
C) The energy associated with the resulting motion is \[(3+2\sqrt{2)}\] times the energy associated with any single motion
D) The resulting motion is not simple harmonic
Correct Answer: A
Solution :
Let simple harmonic motions 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, resultant SHM will be\[y=a\ \left[ \sin \,\left( \omega \,t-\frac{\pi }{4} \right)+\sin \omega \,t+\sin \,\left( \omega \,t+\frac{\pi }{4} \right) \right]\] \[=a\ \left[ 2\sin \omega \,t\cos \frac{\pi }{4}+\sin \omega \,t \right]\] \[=a\ [\sqrt{2}\sin \omega t+\sin \omega t]=a\ (1+\sqrt{2})\sin \omega \,t\] Resultant amplitude =\[(1+\sqrt{2})a\] Energy is S.H.M. µ (Amplitude)2 \ \[\frac{{{E}_{\text{Resultant}}}}{{{E}_{\text{Single}}}}={{\left( \frac{A}{a} \right)}^{2}}={{(\sqrt{2}+1)}^{2}}=(3+2\sqrt{2})\] Þ \[{{E}_{\text{Resultant}}}=(3+2\sqrt{2}){{E}_{\text{Single}}}\]
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