Question

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

Answer 1

The general expression for the displacement of a particle in linear SHM at time t is
x=A sin `(oemgat+alpha)" "`…..(1)
where A is the amplitude, `omega` is a constant in a particular case and `alpha` is the initial phase.
The velocity of the particle, as a function of time, is
`v=(dx)/(dt)=(d)/(dt)[ A sin (omegat+alpha)]=omegaA cos (omegat+alpha)`
`=omegaAsqrt(1-sin^(2)(omegat+alpha))" "`....(2)
From Eq. (1), `sin (omegat+alpha)=x//A`
`:. v=omegaAsqrt(1-(x^(2))/(A^(2)))`
`:. v=omegasqrt(A^(2)-x^(2))" "`........(3)
The acceleration of the particle, as a function of time, is
`a=(dv)/(dt)=(d)/(dt)[A omega cos(omegat+alpha)]`
`=-omega^(2)A sin (omegat+alpha)" "`.....(4)
But from Eq. (1), `A sin (oemgat+alpha)=x`
`:. a=-omega^(2)x" "`.....(5)
Equation (3) and (4) give the velocity and acceleration as functions of displacement.