Question

Q) the displacement of a particle along x-axis is given by x=a sin 2 wt . the motion of the particle corresponds to ?

Answer 1

X=a sin^2 wt
X=a×1/2×(1-cos2wt)
X=a/2-a/2coswt
Frequency of shm=2w/2 pi=w/pi

Answer 2

The motion of the particle corresponds to Simple Harmonic Motion.

We know, for a particle executing Simple Harmonic Motion (S.H.M), its acceleration (double derivative of displacement) ∝ -ω².

Given, displacement x = a sin 2ωt

Derivating x with respect to time, we get velocity v.

v = dx/dt = d(a sin 2ωt)/dt = 2ωa cos 2ωt             [2ω, a are constant]

Derivating v with respect to a, we get acceleration a.

a = dv/dt = d(2ωa cos 2ωt )/dt =  (- 2ω×2ωa) sin 2ωt

a ∝ -ω².

So, the particle executes S.H.M.