Question

A $20gm$ particle is subjected to two simple harmonic motions
$x_1=2\sin 10t,x_2=4\sin \left(10t+\frac{\pi }{3}\right).$ 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 motion will be 28 J.

Answer 1

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

At $t=0$
Displacement $x=x_1+x_2=4\sin \frac{\pi }{3}=2\sqrt{3}m$
Resulting Amplitude $A=\sqrt{2^2+4^2+2(2)(4)\cos \pi /3}$
$=\sqrt{4+16+8}=\sqrt{28}=2\sqrt{7}m$
Maximum speed $=A\omega =20\sqrt{7}m/s$
Maximum acceleration $=A{\omega }^2=200\sqrt{7}m/s^2$
Energy of the motion $=\frac{1}{2}m{\omega }^2{A}^2=28J$