The figure shows a long, nonconducting, massless rod of length L, pivoted at its

Question

The figure shows a long, nonconducting, massless rod of length L, pivoted at its center and balanced with a block of weight W at a distance x from the left end. At the left and right ends of the rod are attached small conducting spheres with positive charges q and 2a, respectively. A distance h directly beneath each of these spheres is a fixed sphere with positive charge Q. (a) Find equation for the distance x when the rod is horizontal and balanced. (b) What value should h have so that the rod exerts no vertical force on the bearing when the rod is horizontal and balanced? Use permittivity constant epsilon_0.

Explanation / Answer

(a)
Balance the torques. Let the point of rotation be the left end. Since the rod is horizontal, the distance between Q and 2q is h. Set the clockwise torques equal to the counterclockwise torques:
W*x = (k*Q*2q*L)/h^2 => x = (k*Q*2q*L)/(W*h^2) <------------------ (a)

(b)
Balance the vertical forces
k*Q*q/h^2 + k*Q*2q/h^2 = W
h = sqrt[3*k*Q*q/W] <-------------------------- (b)

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