A small sphere of mass m = 0.5 kg carrying a positive charge q = 110 μC is connected with a light, flexible and inextensible string of length r = 60 cm and whirled in a vertical circle. If a vertically upward electric field of strength E = 105 NC-1 exists in the space, calculate the minimum velocity of sphere (in m/s) required at highest point so that it may just complete the circle (g = 10 ms-2).

Weight mg (: 0.5 x 10 = 5 newton) of sphere acts vertically downward and electric force qE (=110 x 10-6x 105 =11 newton) vertically upward.Since upward force qE is greater than downward force mg,  therefore 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 vo. Considering free body diagram of sphere at A. T+qE−mg=mv02r where T=0⇒v0=65ms−1But 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 and gravitational potential energy of sphere increases. Hence, according to law of conservation of energy, minimum required velocity v at highest point is given by, 12mv2=12mv02+qE⋅2r−mg⋅2r⇒v=6ms−1