Question

A body of mass $1 kg$ is executing simple harmonic motion (SHM). Its displacement $y$ (in $cm$ ) at time $t$ given by $y=\left[6 \sin \left(100 t+\frac{\pi}{4}\right)\right] cm$. Its maximum kinetic energy is

• A. $1.8 J$
• B. $18 J$
• C. $180 J$
• D. $0.18 J$

Answer 1

Answer: (B)

Given, mass of body, $m=1 kg$
Displacement equation of SHM,
$ y=\left[6 \sin \left(100 t+\frac{\pi}{4}\right)\right] cm$
$ \text { Let } v_y=\text { velocity Hence, } v_y=\frac{d y}{d t} $
$ v_y=\frac{d}{d t}\left[6 \sin \left(100 t+\frac{\pi}{4}\right)\right] $
$=6 \times 100 \cos \left(100 t+\frac{\pi}{4}\right) $
$ =600 \cos \left(100 t+\frac{\pi}{4}\right) cms ^{-1} $
$ =6 \cos \left(100 t+\frac{\pi}{4}\right) m s^{-1}$
Maximum velocity, $\left(v_y\right)_{\max }=6 \,m / s$
$\therefore$ Maximum kinetic energy,
$ K_{\max }=\frac{1}{2} m\left(v_y\right)_{\max }^2 $
$ =\frac{1}{2} \times 1 \times 6^2 $
$ =\frac{1}{2} \times 1 \times 36$
$=18 J$