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
\(B =A\)
\(\text {and } \phi =\frac{4 \pi}{3}\)
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.