Question

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

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

Solution

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

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