AP® Physics 1: Algebra-Based review sheet from Aim for Five (aimforfive.com/physics/units/2/2-8)
Unit 2 · Topic 2.8
2.8 Spring Forces
An ideal spring pushes or pulls with a force proportional to how far it's stretched or compressed from its relaxed length. This is Hooke's law, F = kΔx, and the spring constant k measures how stiff the spring is. The force always points back toward the relaxed position.
Key terms
- Hooke's law
- spring constant (k)
- ideal spring
- restoring force
- equilibrium position
Hooke's law
An ideal spring has negligible mass and a force that grows in proportion to its change in length: |F_s| = kΔx. Here Δx is the stretch or compression measured from the spring's relaxed (natural) length, not the spring's total length.
The spring constant k has units of N/m. A spring with k = 200 N/m needs 200 N to stretch it 1 m, or 2 N to stretch it 1 cm. A larger k means a stiffer spring.
The equation sheet writes it as F_s = −kΔx. The minus sign means the force points opposite the displacement from equilibrium: stretch a spring to the right and it pulls left.
Direction of the spring force
A spring's force always points toward the equilibrium position of the object–spring system. A stretched spring pulls the object back in; a compressed spring pushes it out. That's why springs are called restoring forces, and why they produce the back-and-forth motion you'll study in Unit 7.
An ideal spring exerts the same size force at both ends, pulling inward on both objects when stretched and pushing outward on both when compressed. If you pull on both ends of a spring scale with 10 N each, it reads 10 N, not 20 N.
The spring force is a contact force like any other: draw it on the free-body diagram as one arrow pointing toward the relaxed position, with size kΔx.
Force–stretch graphs
Plot the spring force (vertical axis) against the stretch Δx (horizontal axis) and an ideal spring gives a straight line through the origin. The slope is k.
To measure k in a lab, hang different known masses from the spring. At equilibrium the spring force equals the weight, so F = mg. Measure each stretch from the unloaded length, graph F against Δx and find the slope of the best-fit line. If you graph Δx against F instead, the slope is 1/k.
On one set of axes, a stiffer spring gives a steeper line. If two springs are stretched by the same force, the one with the smaller k stretches more.
A real spring follows Hooke's law only up to a limit. Stretch it too far and the graph curves because the spring is permanently deformed. Rubber bands don't follow Hooke's law well at all, which is one reason labs use metal springs.
Hanging masses
When a mass hangs at rest on a vertical spring, the forces balance: kΔx = mg, so the stretch is Δx = mg/k. Double the mass and the stretch doubles. Use a spring twice as stiff and the stretch halves.
Worked examples
Try each one yourself first, then open the solution.
- Example 1Calculator allowed
Finding k from a hanging mass (classic trap)
A spring hangs with a relaxed length of 20.0 cm. When a 0.50 kg mass hangs from it at rest, its length is 24.9 cm. Use g = 9.8 m/s². Find the spring constant.
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- Step 1: Stretch: Δx = 24.9 cm − 20.0 cm = 4.9 cm = 0.049 m. The trap is using the full 24.9 cm length.
- Step 2: At rest, the spring force balances the weight: kΔx = mg.
- Step 3: k = mg/Δx = (0.50)(9.8) ÷ 0.049 = 4.9 ÷ 0.049 = 100 N/m.
Answer: k = 100 N/m
- Example 2Calculator allowed
Spring constant from data
A student hangs weights from a spring and records the spring force and stretch: 1.0 N at 0.025 m, 2.0 N at 0.050 m, 3.0 N at 0.076 m and 4.0 N at 0.099 m. Find k.
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- Step 1: Graph force (vertical) against stretch (horizontal). The points lie very close to a straight line through the origin, which confirms Hooke's law.
- Step 2: The slope of the best-fit line is k. Using two points on the line, such as (0, 0) and (0.100 m, 4.0 N): slope ≈ 4.0 ÷ 0.100 = 40 N/m.
- Step 3: A least-squares fit of all four points through the origin gives about 40.0 N/m. Use the line, not one data point, so random errors average out.
Answer: k ≈ 40 N/m
- Example 3Calculator allowed
Block released from a compressed spring
A 0.50 kg block on a frictionless table is pressed against a spring (k = 250 N/m), compressing it 0.040 m, and released. Find the spring force and the block's acceleration at the moment of release.
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- Step 1: Spring force: F = kΔx = (250)(0.040) = 10 N, pointing away from the wall (toward the spring's relaxed position).
- Step 2: That's the only horizontal force, so a = F/m = 10 ÷ 0.50 = 20 m/s².
- Step 3: As the spring expands, Δx shrinks, so the force and acceleration both shrink. That's changing acceleration, so the kinematic equations don't apply here; use energy (3.4) to find the launch speed.
Answer: 10 N; 20 m/s², both directed away from the wall
Common mistakes
- Using the spring's total length instead of its change in length from the relaxed position.
- Forgetting to convert centimeters to meters before using k in N/m.
- Using kinematic equations for a spring launch. The force changes as the spring moves, so the acceleration isn't constant.
- Mixing up the slope: force against stretch gives k; stretch against force gives 1/k.
On the exam
- Springs make a natural setup for the Experimental Design and Analysis question: plan how to measure k, then plot data and use the slope. Say which variable goes on each axis and what the slope represents.
- Ratio questions ask how the stretch changes if the mass or k changes. Start from kΔx = mg.
Connected topics
Videos
Check yourself
4 questions on 2.8 Spring Forces. Pick an answer to see if you got it, and why.
A 5.0 N force stretches an ideal spring 0.10 m from its relaxed length. How far will an 8.0 N force stretch it?
A 0.50 kg mass hangs at rest from an ideal spring with k = 200 N/m. How far is the spring stretched? Use g = 10 m/s².
| Hanging mass (kg) | Stretch (m) |
|---|---|
| 0.10 | 0.020 |
| 0.20 | 0.039 |
| 0.30 | 0.061 |
| 0.40 | 0.080 |
Experimental data: students hang masses from a spring and measure how far it stretches from its relaxed length. Use g = 9.8 m/s².
What is the best estimate of the spring constant?
To find k from the slope of a straight-line graph, what should the students plot?
0 of 4 answered