Derive elastic energy from a force-area construction

1. Orient: Derive elastic energy from a force-area construction

Loading an ideal spring slowly from equilibrium requires an external holding force whose magnitude rises from 0 to kx. The force-versus-extension region is a triangle. Its area is½×x×kx=½kx², the stored elastic potential energy when zero is chosen at equilibrium. The factor½ follows from the increasing force; using final force times displacement would overcount.

2. Explain and derive

Equal-magnitude compression and extension store the same positive energy because x is squared. Their restoring forces point in opposite directions. Force depends linearly on signed displacement, while energy depends quadratically on its magnitude. Doubling compression doubles initial force but quadruples stored energy.

For a frictionless horizontal spring–object system released from rest, initial spring energy becomes kinetic energy as the spring reaches equilibrium. Set½kx₀²=½mv² to obtain speed|x₀|√(k/m). The force is zero at equilibrium, yet speed can be maximum there because energy has accumulated during the preceding motion.

The energy relation predicts speed magnitude at a position, not which way the object is moving. Direction requires initial conditions and motion history. The ideal model omits spring mass, internal damping and the elastic limit. A real stiffness inferred at small extension should not be extrapolated to a large stretch without evidence.

3. Work an example

For k=60 N/m and compression 0.20 m, energy is½×60×0.20²=1.2 J. With mass 1 kg and no losses, speed at equilibrium is√2.4≈1.55 m/s. Initial force magnitude is 12 N. Doubling compression to 0.40 m doubles the force to 24 N while energy quadruples to 4.8 J.

4. Predict and investigate

Predict first: Will doubling compression double or quadruple the speed at equilibrium? Separate the energy scaling from the square root used for speed.

Compare compressions 0.10,0.20 and 0.40 m at fixed spring constant. Record force, stored energy and release speed. Then double stiffness while holding compression fixed. Explain why speed scales with the square root of stiffness rather than stiffness itself. Use the lesson controls and retain the settings with your evidence.

5. Transfer and explain

Try a new practice case

A spring with k=80 N/m is compressed by 0.10 m. Find its stored elastic potential energy.

Your numerical result (J) Check number

Reveal worked check (practice, not mastery)

This extra practice is not saved and does not award mastery. Record your first answer, explanation and whether you revealed help in the lesson notebook. Use the lesson’s saved question separately.

6. Reflect and retain evidence

What changed measurement would reveal energy loss while leaving a static spring-force calibration nearly unchanged?

Retain a first attempt, signed ledger, units, system boundary, assumptions and a changed-case prediction. Public worked checks are practice, not independent mastery evidence. Synthetic outputs are not physical measurements.

Framework and source notes

Original teaching prose and numerical cases. College Board AP Physics 1 CED, Unit 3 topic mapping: 3.3; 3.4. Checked 30 October 2026; this is a content cross-reference, not AP authorization.