It has been given that a particle is subjected to two simple harmonic motions in the identical directions which have similar amplitudes and identical frequency. When the resultant amplitude will be equivalent to the amplitude of the individual motions. Calculate the phase difference between the individual motions.
$\begin{align}
  & A.\dfrac{3\pi }{2} \\
 & B.\dfrac{2\pi }{3} \\
 & C.2\pi \\
 & D.\pi \\
\end{align}$

Hint: The resultant of the amplitude of the oscillation can be found by taking the square root of the sum of the square of the amplitude of first oscillation, square of the amplitude of the second oscillation and twice the product of the amplitudes of the oscillations and the cosine of the phase difference between the individual motions. This will help you in answering this question.

Complete answer:
The resultant of the amplitude of the oscillation can be found by taking the square root of the sum of the square of the amplitude of first oscillation, square of the amplitude of the second oscillation and twice the product of the amplitudes of the oscillations and the cosine of the phase difference between the individual motions.
This can be written as,
$A=\sqrt{{{A}^{2}}+{{A}^{2}}+2A\times A\times \cos \phi }$
Let us solve this equation now,
$\begin{align}
  & A=\sqrt{2{{A}^{2}}+2{{A}^{2}}\cos \phi } \\
 & \Rightarrow A=\sqrt{2{{A}^{2}}\left( 1+\cos \phi \right)} \\
 & \therefore \cos \phi =-\dfrac{1}{2} \\
\end{align}$
The inverse of the cosine can be written as,
$\begin{align}
  & \phi =120{}^\circ \\
 & \therefore \phi =\dfrac{2\pi }{3} \\
\end{align}$
Therefore the phase difference between the individual motions has been calculated.

The correct answer has been obtained as option B.

Note:
Phase Difference can be helpful in defining the difference in degrees or the radians if two or more alternating quantities which will reach their maximum or zero values. A sinusoidal waveform will be an alternating quantity which can be presented in the time domain along a horizontal zero axis graphically. The angle between the zero points of the two alternating quantities will be referred to as the angle of phase differences.