Question

A simple harmonic oscillator of angular frequency 2 rad/s is acted upon by an external force

Answer 1

your question is incomplete. A complete question is ----> A simple harmonic oscillator of angular frequency 2rads−1 is acted upon by an external force F=sint N. If the oscillator is at rest in its equilibrium position at t=0, its position at later times is proportional to

( A ) cost−1/2sin2t

( B ) sint+1/2cos2t

( C ) sint+1/2sin2t

( D ) sint−1/2sin2t

solution : we know, from Newton's 2nd law, F = ma = m(dv/dt) = sint

so, m dv/dt = sint

or, $m\int{dv}=\int\limits^t_0{sint}\,dt$

or, $mv=[-cost]^t_0$

or, mv = 1 - cost

again, v = dx/dt

so, m(dx/dt) = 1 - cost

or, $m\int{dx}=\int\limits^t_0{(1-cost)}\,dt$

or, $mx=[t-sint]^t_0$

or, $mx=t-sint$

here, at t = 0, x = 0 and v = 0

so, now check from which option it is possible.

option (D) x = sint -1/2 Sint follow the above conditions .

hence, option (D) is correct choice.