A) The resultant amplitude is\[\sqrt{2}a\]
B) The phase of the resultant motion relative to the first is \[90{}^\circ \]
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: C
Solution :
[c] 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. \[\propto {{(Amplitude)}^{2}}\] \[\therefore \frac{{{E}_{\text{Resultant}}}}{{{E}_{\text{single}}}}={{\left( \frac{A}{a} \right)}^{2}}={{(\sqrt{2}+1)}^{2}}=(3+2\sqrt{2})E\] \[\Rightarrow {{E}_{\text{Resultant}}}=(3+2\sqrt{2}){{E}_{Single}}\]
You need to login to perform this action.
You will be redirected in
3 sec
