The period of a particle executing SHM is `T`. There is a point `P` at a distance `'x'` from the mean position `'O'`. When the particle passes `P` towards `OP`, it has speed `v`. Find the time in which it returns to `P` again.

Explanation

Let the particle is at `P` at an instant `t` starting from mean position and it remains to `p` again after interval `t`.
Then `x=asinomegat`……….`(1)`
Also `x=asinomega(t+t')`…………`(2)`
Form equation `(1)`
`(dx)/(dt)=a omega cos omegat=v`
`rArr cos omegat=v//aomega` ..........`(3)`
from equation `(2)`
`x=a[sinomegat'cosomegat+cosomegat'.sint]`
`=a[sinomegat'(v)/(aomega)+(x)/(a)cosomegat']` using equation `(1)` and `(3)`
`t'=(2)/(omega)tan^(-1)(v)/(omegax)`