Build spring energy from tiny stretches
You will be able to: Integrate Hooke’s law and distinguish spring work from stored energy.
Why is spring energy ½kx² rather than kx²?
Stretch an ideal spring from 0 to 0.20 m. If k = 100 N/m, force magnitude increases from 0 to 20 N. The average opposing force is 10 N, so the spring gains 2 J rather than 4 J.
A useful starting point: Potential energy belongs to an interaction →
Words and symbols before equations
- Deformation x
- Signed stretch or compression measured from the spring’s relaxed length, in m.
- Spring constant k
- Stiffness in N/m; an ideal spring obeys F_s = −kx.
- Elastic potential energy U_s
- Energy stored by reversible deformation of an ideal spring.
- Quasistatic stretch
- A sufficiently slow change that kinetic energy remains negligible.
What this picture assumes
Ideal Hooke’s-law spring with negligible mass. Deformation is measured from relaxed length, where U_s = 0. The energy is stored in the spring interaction; no cart motion is simulated.
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- x = 0.2 m; U_s = 2 J; spring force = -20 N. Equal stretch and compression magnitudes give equal U but opposite forces.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the physics
The spring’s force points opposite its deformation, F_s = −kx. Therefore ΔU_s = −∫F_s dx = ∫kx dx = ½k(x_f² − x_i²). Choosing U_s = 0 at relaxed length gives U_s = ½kx².
Compression and extension by the same amount store the same energy in an ideal Hooke’s-law spring. Their restoring forces point in opposite directions. The U graph is a symmetric bowl while the F graph is a line with negative slope.
For a slow stretch with negligible losses, an external hand supplies ΔU_s. If a block speeds up during stretching, external work can also change K; it is not automatically all stored in the spring. The coordinate x must be deformation, not total spring length or distance from an arbitrary wall.
A worked example, step by step
A spring with k = 200 N/m moves from x_i = −0.10 m to x_f = +0.20 m. Calculate ΔU_s and work done by the spring.
- Choose the relaxed spring as zero energy and keep the signed deformation coordinate.
- U_i = ½(200)(0.10²) = 1 J; U_f = ½(200)(0.20²) = 4 J.
- ΔU_s = 4 − 1 = 3 J, even though the path passes through zero deformation.
- W_s = −ΔU_s = −3 J. The spring initially does positive work toward zero, then negative work during the stretch; the net is negative.
Spring work is minus the change in spring energy. Do not replace Δ(x²) with (Δx)².
If extension doubles, what happens to U_s?
Compare with an explanation
It quadruples because U_s is proportional to x², provided the spring remains ideal over that range.
Predict. Change one thing. Explain.
Keep k fixed. Compare x = −0.20 m and +0.20 m: predict the forces and energies before moving the control. Explain why one graph changes sign and the other does not.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
x = 0.2 m; U_s = 2 J; spring force = -20 N. Equal stretch and compression magnitudes give equal U but opposite forces.
Ideal Hooke’s-law spring with negligible mass. Deformation is measured from relaxed length, where U_s = 0. The energy is stored in the spring interaction; no cart motion is simulated.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant work, system boundary, energy or calculus relationship to justify your prediction.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionA 50 N/m spring relaxes from x = 0.40 m to 0.20 m. Calculate initial and final U, spring work, and its physical interpretation.
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Compare with the answer and four-point rubric
- 1 point: U_i = ½(50)(0.40²) = 4 J.
- 1 point: U_f = ½(50)(0.20²) = 1 J.
- 1 point: W_s = U_i − U_f = 3 J.
- 1 point: The spring releases 3 J from its stored energy; that transfer can increase an attached object’s K if no competing work removes it.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1Where does the one-half in spring energy come from?
Integrating a force proportional to x, or taking the triangular force–deformation area.
RECALL 2What is x in ½kx²?
Deformation from the relaxed length.
RECALL 3Is spring energy always all the external work?
Only when kinetic-energy change and other transfers or losses are negligible.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Build spring energy from tiny stretches
- F_s = −kx for an ideal spring.
- U_s = ½kx² with U_s(0) = 0.
- W_s = ½k(x_i² − x_f²).
Remember: Spring work is minus the change in spring energy. Do not replace Δ(x²) with (Δx)².
Conditions: Ideal Hooke’s-law spring with negligible mass. Deformation is measured from relaxed length, where U_s = 0. The energy is stored in the spring interaction; no cart motion is simulated.
Refresh Kid · AP Physics C: Mechanics Unit 3 (official Unit 3) · Objectives 3.3.A · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 3.3, objectives 3.3.A. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026 alongside the Fall 2026 clarifications. This is Mechanics Unit 3: Work, Energy, and Power. The unit covers Topics 3.1–3.5. Calculus connects work to force integrals, force to potential-energy derivatives, and power to the rate of energy transfer. Models distinguish object-only and multi-object systems; translational models exclude rotational energy unless explicitly noted. The lesson breakdown and questions are original Refresh Kid work, not official topic subdivisions.
Implementation and automated checks are separate from independent teacher review and observation of students. Both human review stages remain pending. This is a review edition, not a certified or validated assessment.
Optional further resource: College Board’s released questions and scoring guides. Papers can combine units; this link is an archive, not an assignment of every question to this lesson.
Our learn, explore, practice and recall sequence is informed by the IES learning guide. The exact Refresh Kid implementation has not been evaluated for learning effectiveness.
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