A small point mass m has a charge q, which is constrained to move inside a narrow frictionless cylinder, At the bae of the cylinder is point mass of charge Q having the same sign as q. If the mass m is displaced by a small amount from its equilibrium position and released. it will exhibit simple harmonic motion. Find the angular frequency of the mass?

Explanation

In the equilibrium position, gravitational force is balanced by Coulomb repulsive force
`mg=(Qq)/(4 pi epsilon_(0)y_(0)^(2))`
If charge q is displaced in positive y-direction such that `yltlty_(0)`, from Newton's law
`(Qq)/(4 pi epsilon_(0)(y_(0)+y)^(2))-mg=ma`
or `(Qq)/(4 pi epsilon_(0)y_(0)^(2))[(1)/((1+y//y_(0))^(2))]-mg=ma`
or `mg[1-(2y)/(y_0)]-mg=ma`
or `a=-(2gy)/(y_(0))` or `(d^(2)y)/(dt^(2))+(2g)/(y_(0))y=0`
Which is the equation for SHM with `omega = sqrt(2g//y_(0))`.