The general expression for the displacement of a particle in SHM at time t is
x = A sin(ωt + α) ….. (1)
where A and ω is a constant in a particular case and α is the initial phase.
The velocity of the particle is
\(v=\frac{dx}{dt}=\frac{d}{dt}[A sin(ωt + α)]\)
= ωA cos(ωt + α)
= \(ωA\sqrt{1-sin^2(ωt + α)}\)
From Eq. (1), sin(ωt + α) = x/A
\(v=ωA\sqrt{1-\frac{x^2}{A^2}}\)
∴ \(v=ω\sqrt{A^2-x^2}\) …… (2)
Equation (2) gives the velocity as a function of x
The acceleration of the particle is
a= \(\frac{dv}{dt}=\frac{d}{dt}\)[Aω cos(ωt + α)]
∴ a = − ω2A sin(ωt + α)
But from Eq. (1), A sin(ωt + α) = x
∴ a = − ω2x ……. (3)
Equation (3) gives the acceleration as a function of x. The minus sign shows that the direction of the acceleration is opposite to that of the displacement.
