Infer a spring constant and retain the restoring sign

1. Orient: Infer a spring constant and retain the restoring sign

An ideal spring near its equilibrium length exerts a restoring force Fₛ=−kx, where x is signed displacement from the equilibrium configuration and k is positive. Extension to the right produces a leftward force; compression to the left produces a rightward force. The minus sign describes direction, not a negative stiffness.

2. Explain and derive

A graph of signed spring force against extension is a line with slope−k. A graph of the external force needed to hold it in equilibrium has slope+k, because that force balances the spring force. Label whose force you measured before interpreting the slope. Total spring length is not the same variable as extension.

Calibration requires a known equilibrium length, a force reference and several points within the elastic range. A residual pattern can reveal that the linear law is inadequate even when one point appears consistent. A constant length-zero error shifts the apparent intercept; it does not justify reporting a different stiffness from a single F/x ratio.

The bounded model uses an ideal massless spring within±0.20 m and does not model oscillation, damping, buckling or failure. A spring can store energy, but the present force analysis does not yet establish the later work–energy relation. Use supplied values rather than stretching an unknown physical spring to a predicted failure point.

3. Work an example

For k=60 N/m and x=+0.10 m, the spring force is−6 N and the external equilibrium holding force is+6 N. At x=−0.10 m the signs reverse. Comparing spring forces−3 and−9 N at extensions 0.05 and 0.15 m gives slope−6/0.10=−60 N/m.

4. Predict and investigate

Predict first: If extension reverses sign without changing its magnitude, which of force direction and force magnitude changes?

Compare positive, zero and negative extension at fixed k. Then compare two spring constants at the same extension. Fit a slope from two table values and distinguish the spring’s force from an external holding force. Explain why a line through total length would need a nonzero intercept. Use the lesson controls and record the settings with your evidence.

5. Transfer and explain

Try a new practice case

A spring-force graph changes from−2 N at x=0.02 m to−8 N at x=0.08 m. Find positive spring constant k.

Your numerical result (N/m) 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 measurements would distinguish a wrong equilibrium-length zero from a genuinely nonlinear spring response?

Retain your first answer, force diagram, equations, units, assumptions and a changed-case prediction. Public worked checks are practice, not independent mastery evidence. Supplied synthetic outputs do not certify actual laboratory skill.

Framework and source notes

Original teaching prose and numerical cases. College Board AP Physics 1 CED, Unit 2 topic mapping: 2.8. Checked 30 October 2026. This is a content cross-reference, not AP authorization.