A small sphere of mass `m = 0.6 kg` carrying a positive charge `q = 80 muC` is connected with a light, flexible, and inextensible string of length `r = 30 cm` and whirled in a vertical circle. If a horizontal rightward 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

When a particle having no charge is whirled in a vertical circle, the only force (other than tension in thread) acting on the particle is its weight. It is vertically downward and tension is minimum at the highest point, which is vertically above the center of the circle.
In the pervious illustration, there were two forces mg and qE (other than tension in thread) on the particle were responsible for analysis of critical condition. Their resultant was vertically upward. In that case, tension was minimum at the lowest point of vertical circle, which was vertically below the center of the cercle. It means tension is minimum when the resultant force (other than tension) acting on the particle is toward center.
In the present illustration, weight is `mg = 0.6 xx 10 = 6 N` (downward) and `q E = (80 xx 10^-6)(10^5) = 80 N` (horizontally rightward).
Resultant force F of these two forces is at `theta = [ = tan^-1 (6//8) = 37^(@)]`, with the horozontal as shown in (Fig. 3.82.) Hence, tension is minimum at A, as shown in figure.
Let critical velocity at A be `v_0`. Considering free body diagram of sphere at A,
`q E cos 37^(@) + mg sin 37^(@) = (mv_0^2)/( r)`
or `v_0 = sqrt(5) ms^-1`
As the sphere moves from A to B work `q E(r cos 37^(@))` is done on the sphere by the electric field and the gravitational potential energy increases by `mg (r - r sin 37^(@))`
We can find the required minimum velocity V, at the highest point B, by using work energy theorem between points A and B.
`W_("total") = Delta K rArr q E r cos 37^(@) - m g r (1 - sin 37^(@)) = (1)/(2)m v^2 -(1)/(2) m v_0^2`
which gives `v = 3 ms^-1`.
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