Topic – Electrostatics/Mechanics

Q: Two small balls of mass m, bearing a charge q each, are connected by a non-conducting thread of length 2l. At a certain instant, the middle of the thread starts moving at a constant velocity v perpendicular to the direction of the thread at the initial instant.

Figure-1

Determine the minimum distance d between the balls.

This problem has a very elegant solution, if it strikes you to use the right frame of reference: that of the moving center of the thread. Note that this frame is inertial, which means there will be no pseudo-forces.

Figure-2

In this frame, at t = 0, both the two balls have a velocity of v. Also, since the two balls are charged, the initial energy stored in the system is

Now, carefully visualize the motion of the two balls from the chosen frame of reference.

Figure-3

As depicted, in this frame, the two balls have initial velocity v, which will cause the configuration of the system to change – the thread will form a “V”, with the two balls coming closer to each other. From our observation point, we’ll see that as the two balls come closer, the force of repulsion between them increases, causing the tension in the thread to increase, which in turn has an effect on the velocity of the balls, causing it to decrease.

Eventually, the increasing tension in the string will cause the velocity of the two balls (relative to our frame) to become zero – and that is the moment the balls come closest to each other. Visualize this description of the motion very carefully – that is the essence of this problem.

Figure-4

At this instant, the energy stored in the system (again, from our frame of reference, in which the balls now have zero velocity) is:

By energy conservation, we have

Elegant, isn’t it?