A point mass is subjected to two simultaneous sinusoidal displacements in $x$-direction, $x_1(t)=A \sin \omega t$ and $x_2(t)=A \sin \left(\omega t+\frac{2 \pi}{3}\right)$. Adding a third sinusoidal displacement $x_3(t)=B \sin (\omega t+\phi)$ bring the mass to a complete rest. The values of $B$ and $\phi$ are

Resultant amplitude of $x_1$ and $x_2$ is $A$ at angle $\left(\frac{\pi}{3}\right)$ from $A_1$. To make resultant of $x_1, x_2$ and $x_3$ to be zero. $A_3$ should be equal to $A$ at angle $\phi=\frac{4 \pi}{3}$ as shown in figure. $\therefore$ Correct answer is (b). Alternate Solution If we substitute, $$ x_1+x_2+x_3=0 $$ or $A \sin \omega t+A \sin \left(\omega t+\frac{2 \pi}{3}\right)$ $$ +B \sin (\omega t+\phi)=0 $$ Then, by applying simple mathematics, we can prove that $$ \begin{array}{rlrl} B & =A \\ \text { and } & \phi & =\frac{4 \pi}{3} \end{array} $$ Analysis of Question (i) Question is simple. (ii) Question can be solved by applying mathematics also. (iii) Amplitudes of two or more sine or cosine functions of same frequency $\omega$ can be added by vector method.