Rolling Discs

Draw the three discs up and lock each of them with the hooks provided in the exhibit. Pull up the lever attached to the hooks gently. Observe that the three discs roll down at different speeds. The smallest disc moves forward the fastest but spins slowly. Whereas, the largest disc spins fastest but moves forward the slowest. Whether or not a disc spins fast or slow depends on its radius provided the axle on which it rolls is the same.

Rolling Discs.jpg

The model here has three wheels with axle, all made out of the same material.  The wheels are of different radii but the radius of the axle is the same in all of them.  All wheels have same thickness.

When all three of them are released from the same height at a time, we see that the wheel with large radius spins fast but moves down slowly and the wheel with smallest radii spins slow but moves down fast.

Here all the three discs or wheels lose their potential energy as they roll down. Since the total energy is conserved, the energy that is lost in the form of potential energy has to appear in some other form. 

Here, it gets transformed into kinetic energy. 

Again, there are two parts in this kinetic energy as the motion is of two kinds

  1. Rotation about the axis of the axle
  2. Forward motion down the rails

 

Therefore, the total kinetic energy is the sum of kinetic energy due to rotation and the kinetic energy due to forward motion.  And this sum will always be equal to the change in the potential energy.

If a large fraction of the potential energy is going into the rotation of the wheel, smaller will be the fraction going into the forward motion and vice-versa.

This distribution of potential energy is actually decided by the rotational inertia associated with the wheel and axle system which in turn is depends on

  1. Mass of the wheel    ii)   Mass of the axle     ii)  Ratio of squares of the radii of wheel and axle.

 

Here, the axles are identical. 

Therefore, the rotation inertia depends on mass and radius of the wheel.

When all the wheels are of the same thickness, the larger wheel will be having a greater radius and higher mass.  Therefore, its rotational inertial will be more.

Hence a large fraction of potential energy goes into rotation and smaller will be the fraction available for its forward motion down the rails.

Therefore, the larger wheel rotates fast and moves down slowly thereby taking more time to reach the bottom than the smaller wheel which rotates slow but moves down fast.