Question

The displacement of a particle executing SHM is given by $y = 5\sin 4t + \dfrac{\pi }{3}$.
If T is the time period and the mass of the particle is 2g, the kinetic energy of the particle when $t = \dfrac{T}{4}$ is given by
$
  A. 0.4J \\
  B. 0.5J \\
  C. 3J \\
  D. 0.3J \\
 $

Answer 1

Hint:Here we will proceed by using the formula of Velocity of particle i.e. $v = \dfrac{{\partial \delta }}{{\partial t}}$. Then we will use the formula of $KE = \dfrac{1}{2}m{u^2}$ using the given displacement, mass and time to get the required answer.
Formula used-
1. $v = \dfrac{{\partial \delta }}{{\partial t}}$
2. $KE = \dfrac{1}{2}m{u^2}$

Complete step by step answer:
Here we are given that-
The displacement of a particle executing SHM is given by $y = 5\sin 4t + \dfrac{\pi }{3}...........\left( 1 \right)$
Velocity of particle,
$v = \dfrac{{\partial \delta }}{{\partial t}}$
Where $\delta $is the particle displacement.
Now substituting the value of particle displacement,$y = 5\sin 4t + \dfrac{\pi }{3}$
We get-
$\dfrac{{\partial y}}{{\partial t}} = \dfrac{{5d}}{{dt}}\sin 4t + \dfrac{\pi }{3}$
$ = 5\cos 4t + \dfrac{\pi }{3}$
Or $20\cos 4t + \dfrac{\pi }{3}$
Also given that velocity at$t = \dfrac{T}{4}$.
$\dfrac{{\delta y}}{{\delta t}}$ where $t = \dfrac{T}{4}$$ = 20\cos 4 \times \dfrac{T}{4} + \dfrac{\pi }{3}$
Or $u = 20\cos T + \dfrac{\pi }{3}..........\left( 2 \right)$
( u is initial velocity)
Now putting value of T in equation 1,
We get-
$U = 20\cos \dfrac{\pi }{2} + \dfrac{\pi }{3}$
Or $U = - 20\sin \dfrac{\pi }{3}$
Or $U = - 20 \times \dfrac{{\sqrt 3 }}{2}$
Or $U = - 10 \times \sqrt 3 $
So, the kinetic energy of particle,
As we know the formula of$KE = \dfrac{1}{2}m{u^2}$
Where m is the mass of the particle.
$\because m = 2g = 2 \times {10^{ - 3}}kg$
$KE = \dfrac{1}{2} \times 2 \times {10^{ - 3}} \times {\left( { - 10 \times \sqrt 3 } \right)^2}$
Or $KE = {10^{ - 3}} \times 100 \times 3$
Or $KE = 3 \times {10^{ - 1}}$
Kinetic energy = 0.3 J
Note- While solving this question, we must take care that we should not forget to put the SI unit of energy i.e. Joules with the answer otherwise the answer is incomplete. Also one must know that to find kinetic energy, we have to calculate initial velocity first using the formula of velocity.