Question

A $20gm$ particle is subjected to two simple harmonic motions, $x_1=2\sin 10t$;
$x_2=4\sin (10t+\frac{\pi }{3}).$ Where $x_1\&x_2$ are in metre $\&$ $t$ is in sec.

• A. The displacement of the particle at $t=0$ will be $2\sqrt{3}m$.
• B. Maximum speed of the particle will be $20\sqrt{7}m/s$.
• C. Magnitude of maximum acceleration of the particle will be $200\sqrt{7}m/s^2$
• D. Energy of the resultant simple harmonic motion will be $28J$.

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A The displacement of the particle at $t=0$ will be $2\sqrt{3}m$.
B Maximum speed of the particle will be $20\sqrt{7}m/s$.
C Magnitude of maximum acceleration of the particle will be $200\sqrt{7}m/s^2$
D Energy of the resultant simple harmonic motion will be $28J$.

At $t=0$
Displacement $x=x_1+x_2=4\sin \frac{\pi }{3}=2\sqrt{3}m$

Resulting Amplitude $A=\sqrt{2^2+4^2+2\left(2\right)\left(4\right)\cos \pi /3}=\sqrt{4+16+8}=\sqrt{28}=2\sqrt{7}m$

Maximum speed $=A\omega =20\sqrt{7}m/s$

Maximum acceleration $=A{\omega }^2=200\sqrt{7}m/s^2$

Energy of the motion $=\frac{1}{2}m{\omega }^2{A}^2=28J$