AP Physics C: Mechanics · Unit 2: Force and Translational Dynamics · Lesson 2.8

Deep Dive: Spring Forces

🔬 Deep Dive

This is your textbook for this topic. Take your time. Read it more than once.

2.8.A.1Concept

Ideal vs. Nonideal Springs

An ideal spring makes two simplifying assumptions, much like the ideal strings from Lesson 2.3: negligible mass, and a force that's exactly proportional to how far it's stretched or compressed from its natural (relaxed) length. A nonideal spring breaks either assumption — it might have significant mass, or its force might stop being proportional to displacement (real springs typically do this if stretched far enough).

💡Unless a problem tells you otherwise, assume every spring you encounter in this course is ideal — the same "assume ideal unless stated" convention you've been using for strings since Lesson 2.3.

2.8.A.22.8.A.3Math

Hooke's Law

The force an ideal spring exerts is given by Hooke's law:

Fs = −kΔx

Here, k is the spring constant (stiffer springs have larger k), and Δx is the displacement from the spring's equilibrium (relaxed) length. The negative sign is doing real work: it guarantees the spring force always points toward equilibrium — a restoring force, regardless of which direction the spring has been displaced.

🔑Stretch it, and it pulls back. Compress it, and it pushes back. There's no displacement direction where a spring "helps" motion move further away from equilibrium — that consistent restoring behavior is exactly what makes springs the engine behind simple harmonic motion later in the course.

Explore the relationship directly below.

Adjust the spring constant and displacement from equilibrium. Positive Δx stretches the spring; negative Δx compresses it. Watch the force respond — and always point the opposite way from the displacement.

k (N/m)40

Δx (m)0.15

Fs = −(40)(0.15) = -6.0 N

Spring is stretched — force pulls back toward equilibrium (negative direction).

2.8.B.12.8.B.1.i2.8.B.1.iiMath

Springs in Series

When springs are connected end to end (in series), the whole chain behaves like one single spring with an equivalent spring constant:

1/k(eq,series) = Σ(1/kᵢ)

⚠️This combination rule guarantees something that might feel backward at first: the equivalent spring constant of springs in series is always smallerthan the smallest individual spring constant in the chain. Chaining springs together end to end makes the combination easier to stretch, not harder — each spring gets to stretch its own share, so the total stretch (and softness) adds up.

2.8.B.1.iiiMath

Springs in Parallel

When springs are connected side by side (in parallel), sharing the same displacement, the combination rule flips:

k(eq,parallel) = Σkᵢ

🔑Parallel springs always combine into something stiffer than any individual spring in the group — every spring resists the same displacement at once, so their individual stiffnesses simply add together.

Explore both combination rules side by side below.

Two springs, combined two different ways. Adjust their individual constants and toggle between series and parallel — watch how differently the equivalent constant responds.

k₁30

k₂60

k(eq) = 20.0 N/m

Series always lands below the smallest individual spring (30 N/m here) — chaining springs end to end makes the combination easier to stretch, not harder.

ExampleGuided Example — Two Springs, Two Configurations

Two springs, k₁ = 40 N/m and k₂ = 60 N/m, are combined first in series and then in parallel. Find the equivalent spring constant each way.

Step 1 — Series formula

1/k(eq) = 1/40 + 1/60 = 3/120 + 2/120 = 5/120

ExampleWorked Example — Boundary Statement Check

A student is given a system with three springs: two in parallel with each other, and that parallel combination then placed in series with a third spring. Is this problem within the scope of AP Physics C: Mechanics?

← Back to Lesson 2.8Ready for 2.9? Resistive Forces brings calculus back in full force — differential equations for velocity-dependent drag.