AP Physics C: Mechanics · Unit 7: Oscillations · Lesson 7.4

Deep Dive: Energy of Simple Harmonic Oscillators

🔬 Deep Dive

This is your textbook for this topic. Take your time. Read it more than once.

7.4.A.17.4.A.2Concept

Total Mechanical Energy

The total energy of a system exhibiting SHM is the sum of its kinetic and potential energies:

Etotal = K + U

Conservation of energy indicates that this total is constant throughout the motion. As the system oscillates, energy continuously shifts between kinetic and potential form — but the sum of the two never changes.

Slide through the motion of a spring–object system (A = 2 m, k = 5 N/m) and watch U and K trade places while their sum stays flat — the same energy-vs-position graph shown in this unit's header art.

x (m)1.00

U = ½kx² = 2.50 JK = Etotal − U = 7.50 JEtotal = ½kA² = 10.00 J (constant)

🔑This is exactly why energy conservation is such a powerful tool for SHM problems: you can relate speed at any displacement to speed at any other displacement without ever needing to know the time in between.

ExampleGuided Example — Finding Speed at a Given Displacement

A 0.5 kg mass on a spring (k = 20 N/m) oscillates with amplitude A = 0.3 m. Find its speed when x = 0.15 m.

Step 1 — Write the total energy from amplitude

Etotal = ½kA² = ½(20)(0.3)² = 0.9 J.

7.4.A.37.4.A.47.4.A.4.iConcept

The KE/PE Trade-off

Because the total is fixed, kinetic and potential energy trade off in a completely predictable way. Kinetic energy is at a maximum exactly when potential energy is at a minimum — and potential energy is at a maximum exactly when kinetic energy is at a minimum, which for SHM is always zero:

Same system, viewed as a split of the total energy. At equilibrium the bar is all kinetic; at the extremes it's entirely potential — and kinetic energy never goes below zero, exactly as 7.4.A.4.i requires.

x (m)0.00

K

K = 10.00 J   U = 0.00 J

💡At equilibrium (x = 0), the system moves at its fastest — all of Etotal is kinetic, and U = 0. At the turning points (x = ±A), the system is momentarily at rest — all of Etotal is potential, and K = 0. Kinetic energy can never go negative, so K = 0 is the true minimum, not just a low point.

This trade-off is the same idea you've now seen play out across every energy-based topic in this course — just replayed here with a spring or pendulum instead of a hill or an orbit.

7.4.A.4.iiMath

Amplitude and Total Energy

Changing a system's amplitude changes the maximum potential energy it reaches — and since Etotal equals that maximum PE (all energy is potential at the turning points), changing amplitude changes the total energy too. For a spring–object system:

Etotal = ½kA²

For a spring–object system with k = 20 N/m, slide the amplitude and watch how fast total energy grows — it's proportional to A², not A.

A (m)0.30

Etotal = ½kA² = 0.900 Jat 2A = 0.60 m: Etotal = 3.600 J — 4.0× larger

⚠️Total energy scales with A², not A. Doubling the amplitude quadruples the total energy — a detail that's easy to miss if you assume a linear relationship.

ExampleGuided Example — Doubling the Amplitude

For the same spring system (k = 20 N/m), compare the total energy at A = 0.3 m to the total energy at A = 0.6 m.

Step 1 — Compute Etotal at A = 0.3 m

Etotal = ½(20)(0.3)² = ½(20)(0.09) = 0.9 J.

← Back to Lesson 7.4Ready for 7.5? Simple and Physical Pendulums closes out Unit 7.