Question

The displacement of a particle executing SHM is given by
$y=5\sin 4t+\frac{\pi }{3}$
If $T$ is the time period and the mass of the particle is $2$ g, the kinetic energy of the particle when $t=\frac{T}{4}$ is given by

• A. 0.4 J
• B. 0.5 J
• C. 3 J
• D. 0.3 J

Answer 1

Answer: (D)

0.3 J

The displacement of particle, executing SHM is
$y=5\sin 4t+\frac{\pi }{3}$......(i)
Velocity of particle
$\frac{dy}{dt}=\frac{5d}{dt}\sin 4t+\frac{\pi }{3}$
$=5cos4t+\frac{\pi }{3}$
$=20cos4t+\frac{\pi }{3}$
Velocity at $t=\frac{T}{4}$
${\frac{dy}{dt}}_{t=\frac{T}{4}}=20cos4\times \frac{T}{4}+\frac{\pi }{3}$
Or $u=20cosT+\frac{\pi }{3}$.......(ii)
Now, putting value of T in eQ. (II), WE GET
$U=20\cos \frac{\pi }{2}+\frac{\pi }{3}$
$=-20\sin \frac{\pi }{3}$
$=-20\times \frac{\overline{3}}{2}$
$=-10\times \overline{3}$
The kinetic energy of particle,
$KE=\frac{1}{2}m{u}^2$
$\because m=2g=2\times {10}^{-3}kg$
$=\frac{1}{2}\times 2\times {10}^{-3}\times -10{\overline{3}}^2$
$={10}^{-3}\times 100\times 3$
$3\times {10}^{-1}$
$KE=0.3J$