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A small point mass m has a charge q, whi...

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?

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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))`.
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