First slide

Simple hormonic motion

Question

A particle is subjected to two simple harmonic motions in the same direction having equal amplitudes and equal frequency. If the resulting amplitude is equal to the amplitude of individual motions, the phase difference between them is:

Moderate

A

$\large \frac{\pi }{3}$
B

$\large \frac{{2\pi }}{3}$
C

$\large \frac{\pi }{6}$
D

$\large \frac{\pi }{2}$

Solution

A₁ =A ₂=A A_net=A
$\large {A_{net}}\, = \,\sqrt {A_1^2 + A_2^2 + 2{A_1}{A_2}\cos \theta }$
$\large {A_{net}}\, = \,\sqrt {{A^2} + {A^2} + 2{A^2}\cos \theta }$
$\large A_{}^2\, = \,2{A^2} + 2{A^2}\cos \theta$
$\large 2{A^2}\cos \theta \, = \, - {A^2}\, \Rightarrow \,\cos \theta \, = \, - \frac{1}{2}\,\,\theta = {120^0}$
$\large \theta \, = \,\frac{{2\pi }}{3}$

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