First slide

Simple hormonic motion

Question

A particle is subjected to two simple harmonic motions in the same direction having equal amplitude and equal frequency. If the resultant amplitude is equal to the amplitude of the individual motions, what is the phase difference between the two simple harmonic motions?

Moderate

A

$\frac{2 π}{3}$
B

$\frac{π}{3}$
C

$\frac{π}{\sqrt{3}}$
D

$\frac{2 π}{\sqrt{3}}$

Solution

Let the amplitudes of the individual motions be A each. The resultant amplitude is also A. if the phase difference between two motions is $δ$ .
$A = \sqrt{A^{2} + A^{2} + 2 A . A c o s δ} = A \sqrt{2 (1 + c o s δ)}$
= 2 A $\cos \frac{δ}{2}$
$\cos \frac{δ}{2} = \frac{1}{2} \Rightarrow δ = \frac{2 π}{3}$

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