Two small particles A and B having masses m = 0.5kg each and charge $q_{1} = (- \frac{155}{18} μ C)$ and $q = (+ 100 μ C)$ respectively, are connected at the ends of a non-conducting, flexible and inextensible string of length r = 0.5m. Particle A is fixed and B is whirled along a vertical circle with centre with centre at A. If a vertically upward electric field of strength $E = 1.1 \times 10^{5} N C^{- 1}$ exists in the space. $(g = 10 m s^{- 2})$ choose the correct statement(s)

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The minimum velocity of the particle B at the lowest position is $\sqrt{31}$ m/s so that if may just complete the circle.

The minimum velocity of particle B, required at highest point so that if may just complete the circle is $\sqrt{61} m s^{1}$ .

The minimum velocity of particle B, required at highest point so that if may just complete the circle is $\sqrt{67} m / s$ .

The minimum velocity of the particle B at the lowest position is $\sqrt{37}$ m/s so that if may just complete the circle.

answer is C.

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Detailed Solution

$F_{A B} = \frac{K q_{1} q_{2}}{r^{2}} = 31 N = F_{1}$
$F_{E O} = F_{2} = q_{2} E = 11 N$
$W = 5$
Tension can be zero at lowest point
$F_{1} + F_{2} - W = \frac{1}{2} m V_{\min}^{2}$
$V_{\min} = \sqrt{37}$
$\frac{1}{2} m V_{B \min}^{2} . = \frac{1}{2} m v_{\min .}^{2} + F_{2} (2 r) - W (2 r)$
$V_{B \min .} = \sqrt{61} m / s$