- #1
bbhill
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1) Consider negatively charged -Q ring with radius R and a particle with charge +q and mass m. Particle is confined to movement along the axis that bisects the ring which also points along the force of gravity. Taking into account the force of gravity,
a) What condition must q/m meet so that the particle is stable, and
b) Suppose the above condition holds and the system undergoes small oscillations. Show that the equilibrium oscillation freq is identical to the case where gravity is ignored and the equilibrium position is shifted from the center of the ring.
F(ring on particle)= qE = [tex](-kQqz)/R^{3}[/tex] where z is distance of particle from the center of the ring.
F(gravity on particle) = -mg
Fnet (I think) = Fring - Fgravity
U(potential of ring on particle) = [tex](-kQq)/\sqrt{z^{2}+R^{3}}[/tex]
So what I did at first was just set the Fring equal to Fgravity and I came up that q/m = (gR^{3})[tex]/[/tex](-kQqz) where z is the distance from the ring.
Now I know there must be some way to get this in simpler terms, so I'm wondering if I am even going about this the right way. . .
Any help would be appreciated.
Thanks!
a) What condition must q/m meet so that the particle is stable, and
b) Suppose the above condition holds and the system undergoes small oscillations. Show that the equilibrium oscillation freq is identical to the case where gravity is ignored and the equilibrium position is shifted from the center of the ring.
F(ring on particle)= qE = [tex](-kQqz)/R^{3}[/tex] where z is distance of particle from the center of the ring.
F(gravity on particle) = -mg
Fnet (I think) = Fring - Fgravity
U(potential of ring on particle) = [tex](-kQq)/\sqrt{z^{2}+R^{3}}[/tex]
So what I did at first was just set the Fring equal to Fgravity and I came up that q/m = (gR^{3})[tex]/[/tex](-kQqz) where z is the distance from the ring.
Now I know there must be some way to get this in simpler terms, so I'm wondering if I am even going about this the right way. . .
Any help would be appreciated.
Thanks!