Question

Assuming the general expression for the displacement of a particle performing SHM, obtain the expression for its velocity and acceleration as functions of the displacement​

Answer 1

The general expression for the displacement of a particle performing SHM is:

x=Asin(ωt+Φ) →(1)

[Φ=phase difference, A=amplitude]

Differentiating the expression (1), we will get the velocity of the particle.(dx/dt=v)

dx/dt=Aωcos(ωt+Φ)  

because [d/dx(sinx)=cosx]

v=Aωcos(ωt+Φ) →(2)

Now differentiating the expression (2), we will get the acceleration of the particle. (dv/dt=a)

dv/dt=-Aω²sin(ωt+Φ)

because [d/dx(cosx)=-sinx]

a=-Aω²sin(ωt+Φ)