Unit 2 · Topic 2.8

2.8 Spring Forces

An ideal spring pushes or pulls back toward its relaxed length with a force proportional to how far it's stretched or compressed: $\vec{F}_s = -k\Delta\vec{x}$ . The spring constant k measures stiffness. Combining springs in parallel makes a stiffer system, and in series a softer one.

Key terms

Hooke's law
spring constant
ideal spring
equilibrium position
springs in series
springs in parallel

Hooke's law

An ideal spring is massless and obeys Hooke's law:

$\vec{F}_s = -k\Delta\vec{x}$

Here Δx is how far the spring is stretched or compressed from its relaxed (natural) length, and k is the spring constant, in N/m. A stiff spring has a large k. The minus sign means the force always points opposite to the displacement, back toward the relaxed length. That's called a restoring force.

A graph of the force needed to stretch the spring against the stretch is a straight line through the origin, and its slope is k. Real springs follow Hooke's law only for small enough stretches; past that they deform permanently. A spring whose mass matters, or whose force isn't proportional to its stretch, is called nonideal.

Hanging masses and equilibrium

Hang a mass m from a vertical spring and it stretches until the spring's upward force balances gravity: $k\Delta x = mg$ . This new equilibrium position is a stretch of $\Delta x = \frac{mg}{k}$ below the relaxed length. Measuring that stretch for several masses is a common way to find k in the lab.

Be careful about what "x" means. In Hooke's law it's always measured from the relaxed length, not from the equilibrium position of a hanging mass and not from the floor.

An ideal spring has the same tension all along it, like an ideal string. If you pull on both ends with 10 N, the spring force is 10 N, not 20 N.

Combining springs

Several springs can act like one equivalent spring with constant $k_{eq}$ . You only need the cases where all the springs are in parallel or all are in series.

Arrangement	What's shared	Equivalent spring constant
parallel (side by side, both attached to the load)	the same stretch	$k_{eq} = k_1 + k_2 + \cdots$
series (end to end, one after another)	the same force	$\frac{1}{k_{eq}} = \frac{1}{k_1} + \frac{1}{k_2} + \cdots$

Why the rules make sense

In parallel, each spring stretches the same amount, so their forces add: $F = k_1x + k_2x$ . Two identical springs side by side are twice as stiff.

In series, each spring feels the full force, and their stretches add: $x = \frac{F}{k_1} + \frac{F}{k_2}$ . Two identical springs end to end are half as stiff, because the total stretch is doubled. A series combination is always softer than its softest spring. (Cutting a spring in half doubles k for each half, for the same reason in reverse.)

Worked examples

Try each one yourself first, then open the solution.

Example 1Calculator allowed
Finding k from a hanging mass
A 0.50 kg mass hung from a spring stretches it 4.0 cm. Find the spring constant, and how far a 0.80 kg mass would stretch it. Use g = 9.8 m/s².
Show the solutionHide the solution
1. Step 1: At equilibrium, $k\Delta x = mg$ , so $k = \frac{(0.50)(9.8)}{0.040} \approx 123$ N/m (122.5 N/m).
2. Step 2: For 0.80 kg: $\Delta x = \frac{(0.80)(9.8)}{122.5} = 0.064$ m = 6.4 cm.
3. Step 3: Check: the stretch is proportional to the mass, and 0.80/0.50 × 4.0 cm = 6.4 cm. ✓
Answer: k ≈ 123 N/m; 6.4 cm.
Example 2Calculator allowed
Series versus parallel
Springs with $k_1 = 200$ N/m and $k_2 = 300$ N/m support a 60 N load. Find the total stretch when they're (a) side by side in parallel and (b) end to end in series.
Show the solutionHide the solution
1. Step 1: (a) Parallel: $k_{eq} = 200 + 300 = 500$ N/m. Stretch = 60/500 = 0.12 m.
2. Step 2: (b) Series: $\frac{1}{k_{eq}} = \frac{1}{200} + \frac{1}{300}$ , so $k_{eq} = 120$ N/m. Stretch = 60/120 = 0.50 m.
3. Step 3: Check (b) another way: each spring carries the full 60 N, stretching 0.30 m and 0.20 m. Total 0.50 m. ✓
4. Step 4: The trap is adding the constants for springs in series. That gives 500 N/m, the parallel answer.
Answer: (a) 0.12 m; (b) 0.50 m.

Common mistakes

Measuring Δx from the equilibrium position or the floor instead of the spring's relaxed length.
Adding spring constants for springs in series. In series, add the reciprocals.
Doubling the force when a spring is pulled at both ends. A spring pulled with 10 N at each end has a 10 N tension.
Dropping the restoring direction. The spring force always points back toward the relaxed length.

On the exam

Lab questions often give force-versus-stretch or mass-versus-stretch data and ask for k from the slope of a best-fit line. Check which variable is on which axis before deciding whether the slope is k or 1/k.
Expect derivations of an equivalent spring constant for two springs. Start from which quantity is shared: stretch in parallel, force in series.

Connected topics

Videos

Topic 2.8 - Spring Forces
Lessons With LondotWatch on YouTube (opens in a new tab)
Hooke's Law Introduction - Force of a Spring
Flipping PhysicsWatch on YouTube (opens in a new tab)
Intro to springs and Hooke's law | Work and energy | Physics | Khan Academy
Khan AcademyWatch on YouTube (opens in a new tab)
AP Physics 1 - Unit 2 - Lesson 12 - Spring Forces
Allen Tsao The STEM CoachWatch on YouTube (opens in a new tab)
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INTEGRAL PHYSICSWatch on YouTube (opens in a new tab)
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MIT OpenCourseWareWatch on YouTube (opens in a new tab)

Check yourself

4 questions on 2.8 Spring Forces. Pick an answer to see if you got it, and why.

Question 1 of 4Calculator allowed

A spring with k = 200 N/m is hooked end to end with a spring with k = 300 N/m, and the pair is stretched as one. What is the spring constant of the combination?

Question 2 of 4Calculator allowed

A uniform spring has spring constant k. It is cut into two equal halves. What is the spring constant of one half?

Force (N)	Stretch (cm)
0	0
1.0	2.5
2.0	5.1
3.0	7.4
4.0	10.0

Experimental data: students hang different weights from a spring and measure how far it stretches from its relaxed length.

Question 3 of 4Calculator allowed

Which graph would give a straight line whose slope equals the spring constant?

Question 4 of 4Calculator allowed

What is the best estimate of the spring constant from these data?

0 of 4 answered