Question #131953
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?
1
Expert's answer
2020-09-06T17:15:51-0400

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: mgTcosθ=0mg - T\cos\theta = 0 , so T=mgcosθT = \dfrac{mg}{\cos\theta} .

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

Here Fe is the electrical force between two charges. The distance between charges is 2Lsinθ,2L\sin\theta, so Fe=kq2(2Lsinθ)2F_e = k\dfrac{q^2}{({2L\sin\theta})^2} , from x-axis Fe=Tsinθ=mgtanθ.F_e = T\sin\theta = mg\tan\theta.

kq24L2sin2θ=mgtanθ        L=kq2mgtanθ4sin2θ=q2sinθkmgtanθ0.296m.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}.


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