7.2 Energy of a Simple Harmonic Oscillator

Energy trades between PE and KE

In simple harmonic motion, total mechanical energy stays constant (no friction). Energy oscillates between kinetic energy (KE = ½mv²) and potential energy stored in the spring (PE = ½kx²). At maximum displacement (amplitude A), velocity = 0, so all energy is potential. At equilibrium (x = 0), all energy is kinetic and speed is maximum.

Total energy depends on amplitude

The total mechanical energy of a spring-mass oscillator is E = ½kA², where A is the amplitude. This value never changes if the system is ideal (no non-conservative forces). Since E is constant, ½kA² = ½kx² + ½mv² at any point in the motion — letting you solve for velocity at any position.

Graphing energy vs. position

A graph of PE vs. x is a parabola centered at x = 0, while KE vs. x is an inverted parabola, also centered at x = 0. Their sum (total energy) is a horizontal line. As amplitude increases, both max KE and max PE increase, since E = ½kA² grows with the square of amplitude.

Key terms

Total mechanical energy of SHM oscillator

E = ½kA², constant throughout the motion (no friction).

Kinetic energy formula in SHM

KE = ½mv², maximum at equilibrium (x = 0).

Potential energy formula in SHM

PE = ½kx², maximum at maximum displacement (x = ±A).

Energy at maximum displacement

All energy is potential; velocity and KE are zero.

Effect of doubling amplitude on total energy

Total energy quadruples, since E is proportional to A².

Try an AP-style question

Where is the speed of a mass–spring oscillator greatest?

1. At the equilibrium position
2. At maximum displacement
3. It is constant
4. Halfway to maximum displacement