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Energy in Simple Harmonic Motion

A spring oscillator has two ways to hold energy. The block has kinetic energy when it’s moving. The spring has potential energy when it’s stretched or compressed. In ideal SHM, the two trade places while their sum stays locked at $E = \dfrac{1}{2}kA^{2}$. Doubling the amplitude quadruples that total.

OSCILLATOR x=0 x = A/sqrt(2) v a ENERGY BAR E U = E/2 K = E/2 x = A/sqrt(2) AT THIS INSTANT K = E / 2 U = E / 2 E = (1/2) k A^2 held constant for the full cycle HALF-ENERGY SPLIT
Fig. 7.4.0   A single moment in the cycle: the block at $x = A/\sqrt{2}$, where $K$ and $U_s$ each take half of $E$. As the block moves, the two segments swap heights inside the white frame; the frame itself doesn’t move.
EnergyLab · Open the sandbox →

Three traps to watch for. First, scaling: energy feels linear in amplitude (it isn’t, it’s quadratic). Second, the turning point: the block stops, so it looks like the energy disappeared (it didn’t, it’s sitting in the spring). Third, the spring PE formula itself: $U_s$ isn’t $F \cdot x$, because the force grew the whole way. The lesson works through each.

The work

3 ways in · any order
Lesson
Energy in SHM ›

Five sections: spring PE as a quadratic, conservation of total mechanical energy, and three pitfalls. Closes with a ten-scenario applet that drills the headline misconceptions.

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

Ten single-select items across the three pitfalls. Wrong answers are sorted by which trap you fell into, so the Targeted Practice card knows where to focus next.

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

Pick one of the failure modes you've missed and grind it on its own. The round is adaptive: two correct in a row clears the misconception and you move on.

Take the diagnostic to identify your misconceptions
Diagnostic first