(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A ring of radius 18 cm that lies in the yz plane carries positive charge 6x10^-6 C uniformly distributed over its length. A particle of mass m that carries a charge of -6 [tex]\mu[/tex]C oscillates about the center of the ring with an angular frequency of 29 rad/s.

Find the angular frequency of oscillation of the mass if the radius of the ring is doubled while keeping the linear charge density on the ring constant. Answer in rad/sec.

(I was confused about the phrase "oscillates about the center" so I asked my professor, and he clarified that the charge is moving back and forth along the positive/negative x-axis through the center of the ring)

2. Relevant equations

The electric field due to a ring with linear charge density [tex]\lambda[/tex] at point x above the radius is:

k[tex]\lambda[/tex](2*pi*R)x[tex]/[/tex](R^2 + x^2)^(3/2)

I'm expressing the equation in terms of charge density rather than Q because the problem conserves charge density rather than Q. The charge density is easily obtained: Q/(2*pi*R) = 5.306x10^-6 C/m.

Acceleration in harmonic motion can be described:

a = [tex]\omega[/tex]^2*x

3. The attempt at a solution

(i'm just going to work with magnitudes here to avoid negatives)

F = qE = kq[tex]\lambda[/tex](2*pi*R)x[tex]/[/tex](R^2 + x^2)^(3/2)

F = ma = m[tex]\omega[/tex]^2*x

Combining these equations:

kq[tex]\lambda[/tex](2*pi*R)[tex]/[/tex](R^2 + x^2)^(3/2) = m[tex]\omega[/tex]^2

Now I'm stuck - I don't know the mass of the particle, and I have no idea how to model the displacement (which varies over time, and I don't know the amplitude). It seems as though with the given charge and frequency, one could come up with variable masses m and displacement x which fits the equation, and the variables aren't related in such a way that I could just double R and obtain a solvable system of equations. Is there another approach to this problem I'm not seeing?

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# Homework Help: Harmonic motion through a ring of charge

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