7.2 Energy of Simple Harmonic Motion

Energy Trades Between KE and PE

In simple harmonic motion (SHM), mechanical energy continuously converts between kinetic energy (KE) and potential energy (PE). For a mass-spring system, KE = ½mv² and elastic PE = ½kx². At maximum displacement (amplitude A), velocity is zero, so all energy is PE. At equilibrium (x=0), all energy is KE.

Total Energy is Constant

Assuming no friction, total mechanical energy E = ½kA² stays constant throughout the motion, where A is amplitude. This gives E = ½mv² + ½kx² = ½kA² = ½mv_max². Energy conservation lets you find velocity or displacement at any point without solving differential equations.

Graphing Energy vs. Position

Plotting KE and PE versus displacement x shows PE as a parabola (½kx²) and KE as an inverted parabola, summing to a constant horizontal line (total E). This visual reinforces that energy shifts smoothly between forms as the oscillator moves, peaking oppositely at x=0 and x=±A.

Key terms

Total mechanical energy in SHM (spring-mass system)

E = ½kA² = ½mv_max², constant throughout motion (no friction).

Kinetic energy in SHM

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

Potential energy in SHM (spring)

PE = ½kx², maximum at x=±A, zero at equilibrium.

Energy conservation equation in SHM

½mv² + ½kx² = ½kA², relating speed and position at any point in the motion.

Where KE = PE in SHM

At displacement x = A/√2, kinetic and potential energy are equal, each half of total energy.

Try an AP-style question

A spring (k = 200 N/m) oscillates with amplitude 0.1 m. What is its total energy?

1. 2 J
2. 20 J
3. 1 J
4. 10 J