Question

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

• A. $\sqrt 2 A , \frac{ 3 \pi }{4}$
• B. $A , \frac{ 4 \pi }{3}$
• C. $\sqrt 3 A , \frac{ 5 \pi }{6}$
• D. $A , \frac{ \pi }{3}$

Answer 1

Answer: (B)

$A , \frac{ 4 \pi }{3}$

Answer 2

Resultant amplitude of $x_1 \, \, and \, \, x_ 2$ is A at angle $\bigg(\frac{\pi}{3}\bigg) \, \, from \, \, A_1$.
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
It we substitute, $x_1 +x_2 + x_3 =0$
or $Asin \omega t + A sin \bigg( \omega t + \frac{ 2 \pi}{3} \bigg) + B sin (\omega t + \Phi ) =0$
Then by applying simple mathematics we can prove that
$B= A \, \, and \, \, \Phi = \frac{ 4 \pi}{3}$