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(B) Let the initial angle between the strings be $2	heta$. The angle each string makes with the vertical is $	heta$.For a pith ball of mass $m$ and

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$60$

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(B) Let the initial angle between the strings be $2	heta$. The angle each string makes with the vertical is $	heta$.For a pith ball of mass $m$ and charge $q$, the forces acting are tension $T$, gravitational force $mg$, and electrostatic force $F_e = \frac{kq^2}{(2L \sin 	heta)^2}$.Equating forces in equilibrium:$T \sin 	heta = F_e = \frac{kq^2}{4L^2 \sin^2 	heta}$$T \cos 	heta = mg$Dividing the two equations:$	an 	heta = \frac{kq^2}{4mgL^2 \sin^2 	heta} \implies 	an 	heta \sin^2 	heta = \frac{kq^2}{4mgL^2} = C$ (constant).When charge becomes $3q$, the angle between strings becomes $2(2	heta) = 4	heta$, so the angle with vertical becomes $2	heta$.Thus, $	an 	heta \sin^2 	heta = 	an(2	heta) \sin^2(2	heta)$.Using $\sin(2	heta) = 2 \sin 	heta \cos 	heta$ and $	an(2	heta) = \frac{2 	an 	heta}{1 - 	an^2 	heta}$:$	an 	heta \sin^2 	heta = \frac{2 	an 	heta}{1 - 	an^2 	heta} \cdot (2 \sin 	heta \cos 	heta)^2$$	an 	heta \sin^2 	heta = \frac{2 	an 	heta}{1 - 	an^2 	heta} \cdot 4 \sin^2 	heta \cos^2 	heta$$1 = \frac{8 \cos^2 	heta}{1 - 	an^2 	heta} = \frac{8 \cos^2 	heta}{1 - \frac{\sin^2 	heta}{\cos^2 	heta}} = \frac{8 \cos^4 	heta}{\cos^2 	heta - \sin^2 	heta} = \frac{8 \cos^4 	heta}{\cos(2	heta)}$.Alternatively, using the ratio method:$\frac{	an 2	heta}{	an 	heta} = \frac{q'^2}{q^2} \cdot \frac{\sin^2 	heta}{\sin^2 2	heta} = 9 \cdot \frac{\sin^2 	heta}{4 \sin^2 	heta \cos^2 	heta} = \frac{9}{4 \cos^2 	heta} = \frac{9}{4} \sec^2 	heta$.$\frac{2 	an 	heta}{1 - 	an^2 	heta} \cdot \frac{1}{	an 	heta} = \frac{9}{4} (1 + 	an^2 	heta) \implies \frac{2}{1 - 	an^2 	heta} = \frac{9}{4} (1 + 	an^2 	heta)$.Let $x = 	an^2 	heta$: $\frac{2}{1-x} = \frac{9}{4}(1+x) \implies 8 = 9(1-x^2) \implies 9x^2 = 1 \implies x = 1/3$.$	an^2 	heta = 1/3 \implies 	an 	heta = 1/\sqrt{3} \implies 	heta = 30^{\circ}$.The initial angle between the strings is $2	heta = 60^{\circ}$.

Two pith balls, each carrying charge $+q$, are hung from a hook by two strings of length $L$. It is found that when each charge is tripled, the angle between the strings doubles. What was the initial angle between the strings (in $^{\circ}$)?

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