Question

Two small metallic spheres, each of mass m = 0.25 g, are suspended as pendulums by light strings from a common point as shown in the figure below. The spheres are given the same electric charge, and it is found that they come to equilibrium when each string is at an angle of θ = 6.1° with the vertical. If each string has length L = 39.0 cm, what is the magnitude of the charge on each sphere? Give answer in nC

Answer 1

Please provide the figure in comment if there is something missing.

Suppose tension in each String is T and angle with the vertical is $\theta$

Now Using force balance on sphere

In vertical direction

Ty = W

T*cos $\theta$ = m*g

In horizontal direction

Tx = Fe

Fe = electrostatic force between both charged spheres

T*sin $\theta$ = k*q1*q2/r^2

given that q1 = q2 (Suppose Q)

T*sin $\theta$ = k*Q^2/r^2

Now by dividing both equation

T*sin $\theta$/(T*cos $\theta$) = k*Q^2/(r^2*m*g)

tan $\theta$ = k*Q^2/(m*g*r^2)

Q = sqrt [(m*g*r^2*tan $\theta$)/k]

Using given values:

k = electrostatic constant = 9*10^9

m = mass of sphere = 0.25 gm = 0.25*10^-3 kg

g = 9.81 m/sec^2

r = distance between both spheres

Using trigonometry

sin $\theta$ = (r/2)/L

r = 2*L*sin $\theta$

L = length of rope = 39.0 cm = 0.39 m

r = 2*0.39*sin 6.1 deg = 0.083 m

So,

Q = sqrt (0.25*10^-3*9.81*0.083^2*tan 6.1 deg/(9*10^9))

Q = 1.42*10^-8 C = 14.2*10^-9 C

Q = 14.2 nC = Charge On each sphere

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