Question

Consider a Simple Harmonic Oscillator of mass m moving on a frictionless horizontal surface about mean position 'O'. When the oscillator is displaced towards right by a distance 'x' , an elastic restoring force F = -kx is produced which keeps it oscillating. Keeping in view equations of Simple Harmonic Motion show that the displacement x of the oscillator at time t is given by; x =ASin (ωt + θ). (1) where A is the amplitude of oscillation and (ωt + θ) is the phase angle. Find equations for velocity and acceleration during this cycle oscillation.

Answer 1

Displacement   x =ASin (ωt + θ)

differentiating x we get velocity v

v=dx/dt=d(ASin (ωt + θ) )/dt=Acos(ωt + θ)Xω=Aωcos(ωt + θ)=velocity     [d(sint)/dt=cost]

velocity can also be deduced to

v=Aω=ω=ω=velocity

Differentiating velocity , we get acceleration a

a=d(Aωcos(ωt + θ))/dt=-Aωsin(ωt + θ)Xω=-Aωsin(ωt + θ)= -x=acceleration           [d(cost)/dt=-sint]