The question: \"A ring of radius 18cm that lies in the yz plane carries positive

Question

The question:
"A ring of radius 18cm that lies in the yz plane carries positive charge of 9 microcoulombs uniformly distributed over its length. A particale of mass m that carries a charge of -9 microcoulombs executes small oscillations about the center of the ring on its axis with an angular frequency of 21 rad/s.

Find the angular frequency of the oscilation of the mass if the radius of the ring is double to 36 cm and all the other parameters above remain unchanged.

Hint: it is useful to draw an analogy between this problem and the 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 omega = sqrt(k/m).

Answer in units rad/s."

not even sure what is going on here, any help would be much appreciated.

Explanation / Answer

themass oscillates along x direction

The restoring force on the particle , if it is displaced by x (which is too small) is given by

=[[k q*Q] /[R+x]^2] *cos theta

wherer theta is the angle the force makes with x axis

and Q= 9 micro C

and q = -9 micro C

k= 9*10^9

cos theta = x /R

when R>> x then (R+x)^2 = R^2

hence the force equation becomes

F = -k*9^2*10^-12*x/R^3

with anaolgy F = -kx

we get w as sqrt (k/m)

w = sqrt (C/R^3)

where C = 9^3*10^-3/m

so when the radius is doubled

w' = w * sqrt(R^3/8*R^3)

w' = 0.35*w=7.42 rad/sec

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