A point charge $q$ is fixed at point $A$ . Two identical balls of mass $m$ having charge $+ q$ and $- q$ are attached to the ends of a light rod of length $2 a$ and are free to rotate about their centre of mass which is fixed. The system is released from the situation shown. The angular velocity of the rod when the rod turns through $90 °$ is

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$\frac{q}{\sqrt{6 π ε_{0} m a^{3}}}$

$\frac{\sqrt{2} q}{\sqrt{3 π ε_{0} m a^{3}}}$

$\frac{\sqrt{2} q}{\sqrt{6 π ε_{0} m a^{3}}}$

$\frac{q}{2 \sqrt{π ε_{0} m a^{3}}}$

answer is D.

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Detailed Solution

Apply principle of conservation of mechanical energy.

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$Δ U + Δ K E = 0 \frac{1}{2} (2 m a^{2}) ω^{2} = \frac{2 k q^{2}}{3 a} ω = \sqrt{\frac{2 k q^{2}}{3 m a^{2}}} = \frac{q}{\sqrt{6 π ε_{0} m a^{3}}}$

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