Springs in series and parallel
You will be able to: Derive equivalent stiffness by identifying the shared force or shared extension.
Why do two springs together sometimes become softer?
Two identical springs end to end stretch more than one under the same pull. Put them side by side supporting the same block and they stretch less.
A useful starting point: Spring force and the reference length →
Words and symbols before equations
- Equivalent stiffness k_eq
- Stiffness of one ideal spring with the same total force–extension relation.
- Series
- Springs end to end with massless connections; force magnitude is shared.
- Parallel
- Springs attached so their extensions match and their forces add.
What this picture assumes
Ideal springs with massless junctions. Geometry is schematic, not a displacement scale. Only purely series or purely parallel arrangements are modeled.
Read the picture in three steps
- Locate the labeled sources, system boundary or graph axes. Read the units before comparing values.
- Series k_eq = 66.67 N/m; total extension 0.3 m under 20 N. Each spring carries the same force.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the physics
In series, x_total = x₁ + x₂ = F/k₁ + F/k₂. Comparing F = k_eq x_total gives 1/k_eq = 1/k₁ + 1/k₂. The same force acts through each spring while their extensions accumulate.
In parallel, F_total = F₁ + F₂ = k₁x + k₂x, so k_eq = k₁ + k₂. The same extension occurs in both springs, while each contributes part of the total force.
A series equivalent is less stiff than either constituent; a parallel equivalent is stiffer than either. These results assume ideal springs with the described geometry and negligible-mass junctions. This unit handles purely series or purely parallel arrangements, not mixed networks.
| Property | Series | Parallel |
|---|---|---|
| Shared quantity | Force magnitude | Extension |
| What adds? | Extensions | Forces |
| Equivalent stiffness | 1/k_eq = 1/k₁ + 1/k₂ | k_eq = k₁ + k₂ |
A worked example, step by step
Two springs of 100 N/m and 200 N/m carry a total applied pull of 20 N. Compare their total extension in series and parallel.
- Series: 1/k_eq = 1/100 + 1/200, giving k_eq = 66.67 N/m.
- Series extension = 20/66.67 = 0.30 m, from 0.20 m + 0.10 m.
- Parallel: k_eq = 100 + 200 = 300 N/m.
- Parallel extension = 20/300 ≈ 0.0667 m; the two spring forces add to 20 N.
Do not add spring constants in series. First identify what is shared and what adds.
Two identical springs in series have what equivalent stiffness?
Compare with an explanation
k/2, because each contributes the same extension under the shared force.
Predict. Change one thing. Explain.
Keep both spring constants and total pull fixed. Switch from series to parallel. Predict total extension and explain the change using shared quantities.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
Series k_eq = 66.67 N/m; total extension 0.3 m under 20 N. Each spring carries the same force.
Ideal springs with massless junctions. Geometry is schematic, not a displacement scale. Only purely series or purely parallel arrangements are modeled.
Explain what you noticed: Which quantity changed? Which stayed fixed? Use the relevant force, system boundary, acceleration 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 questionCombine two 60 N/m springs, first in series and then in parallel. For a total 12 N pull, calculate each equivalent stiffness and total extension.
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Compare with the answer and four-point rubric
- 1 point: Series k_eq = 30 N/m.
- 1 point: Series total extension = 12/30 = 0.40 m.
- 1 point: Parallel k_eq = 120 N/m.
- 1 point: Parallel extension = 12/120 = 0.10 m.
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 1What adds in series?
Extensions.
RECALL 2What adds in parallel?
Forces, at a common extension.
RECALL 3Can k_eq in series exceed both individual constants?
No for positive ideal spring constants.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Springs in series and parallel
- Series: 1/k_eq = Σ(1/kᵢ).
- Parallel: k_eq = Σkᵢ.
- Total extension = F_total/k_eq for the stated ideal setup.
Remember: Do not add spring constants in series. First identify what is shared and what adds.
Conditions: Ideal springs with massless junctions. Geometry is schematic, not a displacement scale. Only purely series or purely parallel arrangements are modeled.
Refresh Kid · AP Physics C: Mechanics Unit 2 (official Unit 2) · Objectives 2.8.B · Review edition
Framework, scope and review status
Mapped to College Board CED, Topic 2.8, objectives 2.8.B. CED effective Fall 2024, current PDF ©2026; checked September 16, 2026 alongside the Fall 2026 clarifications. This is Mechanics Unit 2: Force and Translational Dynamics. The unit covers Topics 2.1–2.10. Calculus is introduced where it is needed for continuous mass and velocity-dependent forces. Shell theorem is applied without requiring a proof; spring combinations are purely series or purely parallel. 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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