# Pendulum and Ball on Track

This demo can be used as a simple pendulum, or as a demonstration of rotational and translational kinetic energy.

Equipment:

• Pendulum setup (support and metal ball on string)
• Optional second ball on string with a different mass
• Second metal ball identical to the one on the string

Demo:

Demo 1: Use pendulum without the ball on the track to demonstrate conservation of energy and resolution of forces in simple harmonic motion. A second ball with a different mass can be used to show that mass does not affect the period of oscillation. Likewise, the length of the second string can be shortened to show that the length does affect the period.

Demo 2: Simultaneously release the bob and the identical ball on the track. The track follows the path of the bob and is designed to minimize friction. The ball on the track lags the pendulum bob (see picture on the left).

Explanation:

For a pendulum, the angular frequency ω of the oscillation is described by: $\omega = \sqrt{\cfrac{g}{L}}$

and the period T is described by: $T = 2\pi \sqrt{\cfrac{L}{g}}$

While the mass of the ball does not factor into these equations, the length of the string does.

If we take the potential energy at the bottom of the swinging ball’s trajectory to be zero, we can see the conservation of energy as follows:

1. Initially, the ball is released from rest at a height h1, with potential energy U = mgh1 and kinetic energy K = 0. The total energy is E = K + U = mgh1

2. At the bottom of its trajectory, the ball has kinetic energy K = 1/2 mv2 and potential energy U = 0. The total energy is E = K + U = 1/2 mv2; all of the gravitational potential energy has been transformed into translational kinetic energy.

3. At the far end of its trajectory, the ball has once again returned to its original height (ignoring any dissipative forces from friction and air resistance). The gravitational potential energy is again U = mgh1 and the kinetic energy K = 0 (the ball is temporarily at rest at the peak of its trajectory). The total energy is E = K + U = mgh1

When comparing the swinging ball with the ball on the track, two things are obvious:

• The ball on the track lags the swinging ball.
• The amplitude of motion of the ball on the track decreases significantly over time.

Although the ball on the track starts out with the same potential energy as the swinging ball (now setting the low point of the track to U = 0), this energy is not all converted to translational kinetic energy. Instead, part of it is converted to rotational kinetic energy: the ball rolls down the track due to static friction between the ball and the track. (If it were all converted to translational kinetic energy, the ball would simply slide back and forth on the track). When it reaches the far end of its trajectory, the translational kinetic energy and the rotational kinetic energy have been converted back into gravitational potential energy.

If there were no energy loss due to friction (or air resistance), the ball’s amplitude would remain constant over time. However, there is a significant amount of friction between the ball and the track, and as such the total energy decreases with time – the ball does not rise as high as its original height.