W9_parallel_resonance.eps

Week 1: Discrete Charge and the Electrostatic Field 61

Advanced Problem 11.

+Q

+q,m

A small (point) massm, which carries a chargeq, is constrained to move vertically inside a
narrow, frictionless cylinder. At the bottom of the cylinder is a pointmass of chargeQhaving the
same sign asq.

a) Show that the massmwill be in equilibrium at a height:

y 0 =

√

kqQ

mg

.

b) Show that if the massmis displaced by a small amount ∆yfrom its equilibrium position and

released, it will exhibit simple harmonic motion with angular frequency:

ω= (2g/y 0 )^1 /^2

.

You will need to use expansions to solve this problem.