A ring of radius R carries a uniformly distributed charge +Q. A point charge `-q` is placed on the axis of the ring at a distance 2R from the centre of the ring and released from rest. The particle executes a ismple harmonic motion along the axis of the ring.

Explanation

Force on charge `-q` due to small charge `dq` situated at length `dl` is
`dF=k(qdq)/(5R^(2))`
Resolving this force into two parts `dF cos theta` and `dF sin theta` as shown in Fig. SAI.126.

If we take another diametrically opposite lenght `dl`, the charge on it being `dq`, then the force on charge `-q` by this small charge `dq` will be
`dF=k(qdq)/(5R^(2))`
Again, resolving this force, we find `dF sin theta` cancels out with `dF sin theta` of the previous force and `dF cos theta` components add up.
`F=underset(0)overset(2piR)intdFcostheta=underset(0)overset(2piR)int(kqdq)/(5R^(2))xx(2R)/(sqrt(5)R)`
Charge on length `2piR=Q`
Charge on lenght `dl=(Qdl)/(2piR)=dq`
`F=underset(0)overset(2piR)int(2kq)/(5sqrt5(xx2piR^(2)))xx2piR=(2kQq)/(5sqrt(5)R^(2))`
This is not an equation of simple harmonic motion. Therefore, the statement is false.