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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
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B
0.5 J
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C
3 J
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D
0.3 J
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Solution

The correct option is C 0.3 J
The 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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