Electric field of ring causing oscillation

[ponyhunter]

Homework Statement

A ring of radius 18 cm that lies in the yz plane
carries positive charge of 5 µC uniformly distributed over its length. A particle of mass m
that carries a charge of −5 µC executes small
oscillations about the center of the ring on its
axis with an angular frequency of 12 rad/s.
Find the angular frequency of oscillation of
the mass if the radius of the ring is doubled to
36 cm and all other parameters above remain
unchanged.

Homework Equations

{Hint: it is useful to draw an analogy between this problem and a mass-spring system.
For the latter, the force on the mass is given by
F = ma = −kx and produces 1d simple harmonic oscillation with an angular frequency
of ω = (k/m)^(1/2)}

Electric field of a uniformly charged ring: (1/(4*pi*epsilon))*q*z/(R^2+z^2)^(3/2)

The Attempt at a Solution

Well I took the electric field of the ring and multiplied it by the charge of the particle mass to get the force. I also took the angular frequency and squared it to get k/m, but after that I'm just stuck. I really just wish I knew what to do. Thanks for your help. I really appreciate it.

 

[Saitama]

Homework Statement

A ring of radius 18 cm that lies in the yz plane
carries positive charge of 5 µC uniformly distributed over its length. A particle of mass m
that carries a charge of −5 µC executes small
oscillations about the center of the ring on its
axis with an angular frequency of 12 rad/s.
Find the angular frequency of oscillation of
the mass if the radius of the ring is doubled to
36 cm and all other parameters above remain
unchanged.

Homework Equations

{Hint: it is useful to draw an analogy between this problem and a mass-spring system.
For the latter, the force on the mass is given by
F = ma = −kx and produces 1d simple harmonic oscillation with an angular frequency
of ω = (k/m)^(1/2)}

Electric field of a uniformly charged ring: (1/(4*pi*epsilon))*q*z/(R^2+z^2)^(3/2)

The Attempt at a Solution

Well I took the electric field of the ring and multiplied it by the charge of the particle mass to get the force. I also took the angular frequency and squared it to get k/m, but after that I'm just stuck. I really just wish I knew what to do. Thanks for your help. I really appreciate it.

The question says that charge performs oscillations near the centre of the ring. This means that z (assuming it is the distance of charge from centre of ring) is much smaller than the radius of ring (R) i.e z<<R. Use this approximation and find the force on the charge.