Question

The acceleration of a particle, starting from rest, varies with time according to the relation $a=-s{{\omega }^{2}}\sin \omega t$. The displacement of this particle at time t will be
A. $s\sin \omega t$
B. $sw\cos \omega t$
C. $sw\sin \omega t$
D. $-\dfrac{1}{2}(s{{w}^{2}}\sin \omega t){{t}^{2}}$

Answer 1

Hint:Here we have given a particle which was starting from rest and then changes with the time and we have to calculate the displacement of particle after a time t, as we know that the acceleration is the rate of change of velocity with time and given expression is in the terms of t so by integrating it, we can find v at a specific time t.

Formula used:
$a=\dfrac{dv}{dt}$
Where a is the acceleration, $\dfrac{dv}{dt}$ is a very small change in the velocity and time.

Complete step by step solution:
Given that the acceleration of a particle starts from rest and it varies with time
Given relation is
$a=-s{{\omega }^{2}}\sin \omega t$…………………………… (1)
We have to find the displacement of the particle at time t. We know acceleration is the rate of change of velocity w.r.t time. So,
a = $\dfrac{dv}{dt}$ ……………………………………… (2)
Comparing equation (1) and (2), we get
$\dfrac{dv}{dt}$ = $-s{{\omega }^{2}}\sin \omega t$
$\Rightarrow dv=-s{{\omega }^{2}}\sin \omega tdt$

Integrating both sides, we get
$v=s\omega \cos \omega (t-0)+{{c}_{1}}$
$\Rightarrow v=s\omega \cos \omega t+{{c}_{1}}$
We know $v=\dfrac{ds}{dt}$
$\dfrac{ds}{dt}=s\omega .\cos \omega t+{{c}_{1}}$
$\Rightarrow ds=s\omega .\cos \omega tdt+{{c}_{1}}$
We integrate again both sides and we get
$S=s.\sin \omega t+{{c}_{1}}+{{c}_{2}}$
$\Rightarrow S= s.\sin \omega t+c$
Eliminating integration constant, we get
$S = s.\sin \omega t+c$

Hence, option A is correct.

Note: As the particle starts from rest, so its initial velocity is zero. So we integrate it from the time when the particle was at rest to the time when it undergoes acceleration. Acceleration of a particle is the time derivative of the velocity. Hence the velocity of the particle at any moment would be the integral of its acceleration.