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Where the energy goes

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Simple harmonic motion is energy in constant exchange. A frictionless oscillator carries a fixed total, the sum of kinetic and potential energy, equal to ½kA². As the block moves the two forms trade back and forth without the total ever changing: at the center the energy is all kinetic, at the turning points it is all potential. Because the total is set by A², doubling the amplitude does not double the energy, it quadruples it.

ENERGY OF SIMPLE HARMONIC OSCILLATORS E = ½kA² = ½mv²max The total energy stays fixed as kinetic and potential energy trade off through the cycle. x total E = ½kA² U = ½kx² KE PE -A 0 +A E ∝ A² E 4E A 2A kinetic energy peaks at equilibrium, potential energy at the turning points; the total never changes
Energy in a fixed well. The parabola is the potential energy U equals one half k x squared; the flat line is the constant total E equals one half k A squared. The gap between them is the kinetic energy, largest at the center and zero at the turning points. Raise the amplitude and the whole total line climbs as the square of A.
Energy of simple harmonic oscillators · Open the sandbox →

The trick is to hold the total fixed while the two pieces move. The most common slip is letting the total energy rise and fall with the block instead of staying constant. The next is scaling the energy with the amplitude when it really scales with the amplitude squared. The last is swapping where each form peaks: the kinetic energy is largest at equilibrium, where the speed is greatest, and the potential energy is largest at the turning points, where the displacement is greatest.

Energy of simple harmonic oscillators

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Lesson
Energy of simple harmonic oscillators ›

How energy moves through a simple harmonic oscillator: the total is one half k A squared and stays constant while kinetic and potential energy trade off, why the energy scales with the square of the amplitude so doubling A quadruples E, and where each form is largest, kinetic energy at equilibrium and potential energy at the turning points. Closes with a ten-scenario skill check.

Skill check · 10 scenarios
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Diagnostic
10-item topic check ›

Ten items across the three Topic 7.4 mistakes: treating the total mechanical energy as varying through the cycle instead of staying fixed, scaling the energy with the amplitude rather than the amplitude squared, and swapping where the kinetic and potential energy reach their maxima. Take it cold to see what still trips you up, or after the lesson to confirm it does not.

Not started · 10 items · ~15 min
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Targeted Practice
Drill a single misconception ›

Pick one mistake you keep making and drill it on its own. Two correct in a row clears it and you move on.

Take the diagnostic to identify your misconceptions
Diagnostic first