Energy Conservation for Pendulums

A pendulum exhibits periodic energy exchange between gravitational potential energy and kinetic energy. Because tension acts perpendicular to the motion, only gravity does work. Thus mechanical energy $E = K + U$ is conserved.

Energy at Key Positions

At the highest points of the swing, the bob momentarily stops. All energy is gravitational potential energy:

$$K = 0, \quad U = mgh$$

where $h$ is measured above the lowest point.

At the lowest point, the bob moves at maximum speed. Potential energy is minimized and kinetic energy is maximized:

$$U = 0, \quad K = \frac{1}{2}mv^2$$

if we set the reference level at the bottom.

Throughout the entire swing, total mechanical energy remains constant:

$$E = K + U = \text{constant}$$

As the bob descends, gravitational potential energy decreases while kinetic energy increases by exactly the same amount. As the bob ascends, kinetic energy decreases while potential energy increases. The sum $E$ never changes.