Question

A $20\,gm$ particle is subjected to two simple harmonic motions
$x_{1}=2 \,\sin \,10\, t, x_{2}=4 \sin \left(10 \,t+\frac{\pi}{3}\right) .$ 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 $=\frac{1}{2} \,m \omega^{2} A^{2}=28 \,J$