A small sphere of mass `m = 0.5 kg` carrying a positive charge `q = 110 mu C` is connected with a light, flexible, and inextensible string of length of length `r = 60 cm` and whirled in a vertical circle. If a vertically upward electric field of strength `E = 10^5 NC^-1` exists in the space, calculate the minimum velocity of the sphere required at the highest point so that it may just complete the circle `(g = 10 ms^-2)`.

Explanation

Weight `mg ( = 0.5 xx 10 = 5 N)` of sphere acts vertically downward and electric force `q.E (= 110 xx 10^-6 xx 10^5 = 11N)` vertically upward
Since upward force q E is greater than downward force mg, critical condition corresponds to the tension in the thread to be zero at lowest point A of the circle instead of highest point B,
Let velocity at lowest point be `v_0`. Considering free body diagram of sphere at A.
.
`T + q E - mg =(mv_0^2)/(r)` where `T = 0` or `v_0 = (6)/(sqrt(5)) ms^-1`
But corresponding velocity at highest point is to be calculated. As sphere moves from A to B, work is done on it by electric field. Due to this work, both kinetic energy and gravitational potential energy of sphere increase. Hence, according to law of conservation of energy, minimum required velocity v at highest point is given by
`(1)/(2) m v^2 = (1)/(2) mv_0^2 + q E. 2r - m g. 2 r` or `v = 6 ms^-1`.