A small ball of mass `m` having charge `q` is suspended by a string of length `L`. Another identical ball having the same charge is kept at the point of suspension. Determine the minimum horizontal velocity which should be imparted to ball at lowest point so that it can make complete revolution.

Explanation

To complete vertical circle , the string should not slack. As the ball moves from A to B, tension in the string decreases and is minimum at the top.
To complete vertical circle , tension in the string at the top should be `ge 0`.
Apply energy conservation between `A` and `B`
`(1)/(2) m u^(2) = (1)/(2) mv^(2) + mg 2L`
`v^(2) = u^(2) - 4 gL` ....(i)
When the ball is at the top
`T + mg - F_(e ) = (mv^(2))/(L)`
`T = (mv^(2))/(L) + F_(e ) - mg`
` = (m)/(L) (u^(2) - 4 gL) + (1)/( 4 pi in_(0)) (q^(2))/(L^(2)) - mg`
`= (m u^(2))/(L) - 5 mg + (1)/(4 pi in_(0)) (q^(2))/(L^(2))`
`T ge 0`
`(m u^(2))/(L) - 5 mg + (1)/( 4 pi in_(0)) (q^(2))/(L^(2)) ge 0`
`u ge sqrt(5 gL - (q^(2))/( 4 pi in_(0) mL))`
`u_(min) = sqrt(5 gL - (q^(2))/(4 pi in_(0) mL))`