Answer to Question #131953 in Quantum Mechanics for sione

Question

Two small metallic spheres each of mass m = 0.200 g are suspended as pendulums by light strings of length L shown. The spheres are given the same electric charge of 7.2 nC, and they come to equilibrium when each string is at an angle θ = 5.00o with the vertical. How long are the strings?

Answer 1

Let y-axis be directed upwards and x-axis be directed from 1 charge to 2 charge. Let us determine the forces acting on the each charge.

y-axis: $mg - T\cos\theta = 0$ , so $T = \dfrac{mg}{\cos\theta}$ .

x-axis: $T\sin\theta - F_e = 0.$

Here F_e is the electrical force between two charges. The distance between charges is $2L\sin\theta,$ so $F_e = k\dfrac{q^2}{({2L\sin\theta})^2}$ , from x-axis $F_e = T\sin\theta = mg\tan\theta.$

$k\dfrac{q^2}{4L^2\sin^2\theta} = mg\tan\theta \;\; \Rightarrow \;\; L = \sqrt{\dfrac{kq^2}{mg\tan\theta\cdot 4\sin^2\theta}} =\dfrac{q}{2\sin\theta} \sqrt{\dfrac{k}{mg\tan\theta}} \approx 0.296\,\mathrm{m}.$