Question

How was Rotational Dynamics concept “discovered”? What is the background/history of Rotaional Dynamics?

Answer 1

Before analyzing rotational motion, it's worth considering what patterns of forces cause rotation. For example, suppose you and I face each other, standing on a slippery floor. We both hold our hands up at shoulder level, say. Then you hold my right hand with your left hand, and my left hand with your right hand. Then at the same time you begin to push on my right hand and pull on my left hand with the same force. What happens? If the floor is slippery enough, we'll both begin to rotate.

Notice, though, that the two forces you're exerting on me are equal and opposite, so you might think Newton's Laws would predict nothing at all would happen, even though we know better. But if you look back at the notes on static equilibrium, the statement was that there will be no change of motion if the forces add up to zero and if the forces act at the same point. Even if the forces don't act at the same point, if they add to zero, the body acted on won't move away - that is to say, its center of mass will stay put. But it will, in general, begin to rotate, unless frictional forces come into play to balance the applied forces.

This is the principle of the lever, first spelled out by Archimedes. The two downward forces, the weights of the two children, both tending to rotate the horizontal seesaw, will cancel each other, so there will be no rotation, if they have the same amount of leverage, defined as the magnitude of the force multiplied by the distance from the center of the point where it operates. This leverage goes under several names: it is also called the torque, and sometimes the moment of the force. Actually, this definition needs to be made a little more precise, in that we need to keep track of which way the forces tend to rotate the seesaw. It won't balance if both children sit on the same side of the middle. We could count the leverage as positive if the force is tending to rotate the seesaw clockwise. It doesn't matter what convention you choose, as long as you stick to it when adding up the effect of all the forces involved.

The simplest way to make all this clear is to hang weights from a meter stick (or any uniform rigid stick), supported at its midpoint, perhaps by having a rod through a hole in it, so that it's free to rotate in a vertical plane. Several weights could be hung at different distances from the center. In this way, it can be verified experimentally that when the beam is at rest, the total moment-the sum of weights multiplied by their individual leverage distances-tending to rotate the stick clockwise just equals the total moment tending to rotate the stick anticlockwise.

There's also a simple extension of this experiment to include upward forces on the meter stick. We can pull upwards on it with a piece of string tied around it, using a spring scale to measure the tension in the string.

In the case illustrated, with the string twice as far from the middle as the 2 kg mass, the tension in the string is equivalent to 1 kg weight, that is, 10 newtons (taking g = 10). A more interesting case arises when the string is not perpendicular to the meter stick.