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 45⁰, then

Moderate

A

The resultant amplitude is $\large \left( {1 + \sqrt 2 } \right)a$
B

The phase of the resultant motion relative to the first is 90°
C

The energy associated with the resulting motion is $\large \left( {3 + 2\sqrt 2 } \right)$ times the energy associated with any single motion
D

The resulting motion is not simple harmonic

Solution

Let simple harmonic motions be represented by
$\large {y_1}\, = \,a\sin \left( {\omega t - \frac{\pi }{4}} \right);\,\,{y_2}\, = \,a\sin \omega t\,\,\,and$
$\large {y_3} = \,a\sin \left( {\omega t + \frac{\pi }{4}} \right)$ On superimposing,resultant SHM will be
$\large y\, = \,a[\sin \left( {\omega t - \frac{\pi }{4}} \right) + \sin \left( {\omega t + \frac{\pi }{4}} \right)]$
$\large = \,a\left[ {2\sin \omega t\cos \frac{\pi }{4} + \sin \omega t} \right]$
$\large = \,a\left[ {\sqrt 2 \sin \omega t + \sin \omega t} \right]\, = \,a\left( {1 + \sqrt 2 } \right)\sin \omega t$
Resultant amplitude = $\large \left( {1 + \sqrt 2 } \right)a$
Energy is S.H.M. (Amplitude)²
$\large \therefore \frac{{{E_{\operatorname{Re} sul\tan t}}}}{{{E_{\operatorname{Sin} gle}}}}\, = \,{\left( {\frac{A}{a}} \right)^2} = \,{\left( {\sqrt 2 + 1} \right)^2} = \,\left( {3 + 2\sqrt 2 } \right)$
$\large \Rightarrow \,{E_{\operatorname{Re} sul\tan t}}\,\, = \,\,(3 + 2\sqrt 2 ){E_{\operatorname{Sin} gle}}$

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