17.2 Energy in Simple Harmonic Motion

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What Is Energy in SHM?

In simple harmonic motion (SHM), an object oscillates — it moves back and forth repeatedly around a central point called the equilibrium position (the resting, middle point). As the object moves, two types of mechanical energy are constantly swapping between each other:

• Kinetic energy (KE) — the energy an object has because it is moving.
• Potential energy (PE) — stored energy that depends on the object's position or shape.

The key idea is this: as one type of energy increases, the other decreases by exactly the same amount. The total energy of the system stays the same throughout the motion (as long as there is no damping — no energy being lost to friction or air resistance).

Types of Potential Energy in SHM Systems

The type of potential energy involved depends on the system:

• A swinging pendulum — the energy swaps between gravitational potential energy (GPE) and kinetic energy. Gravitational potential energy is the energy stored in an object because of its height above the ground.
• A horizontal mass on a spring — the energy swaps between elastic potential energy (EPE) and kinetic energy. Elastic potential energy is stored in a stretched or compressed spring.
• A vertical mass on a spring — the energy involves both gravitational potential energy and elastic potential energy alongside kinetic energy.

Energy Interchange: The Simple Pendulum

Imagine a pendulum bob (a weight on a string) swinging from side to side.

• At the maximum displacement (the highest point on either side — called the amplitude, x₀), the bob is momentarily at rest. Its velocity is zero, so its kinetic energy = 0. It is at its greatest height, so its gravitational PE is at its maximum.

• As the bob swings towards the centre (the equilibrium position), it speeds up. Gravitational PE is converting into kinetic energy. KE increases; GPE decreases.

• At the equilibrium position (the lowest, central point), the bob is moving at its maximum speed. Therefore, KE is at its maximum and GPE is at its minimum (zero).

• After passing through the centre, the bob slows down again as it rises. KE converts back into GPE. This process repeats with every swing.