Grade by Grade
C. Energy and Momentum · Work, Energy and Simple Harmonic Motion

Hooke's Law: Springs and Restoring Forces

C3.1 C2.1About 65 minutes
Learning goals
  • I can state and use Hooke's law, F_s = −kx, and explain the minus sign.
  • I can find a spring constant from data or from the slope of a force–stretch graph.
  • I can find the effective spring constant of springs in series and in parallel.
  • I can explain the elastic limit and what an ideal spring is.

Words to know

Hooke's law
The force exerted by an ideal spring is proportional to its displacement from equilibrium and opposite in direction: F_s = −kx.
spring constant (k)
A measure of a spring's stiffness, in N/m; the force needed per metre of stretch or compression.
restoring force
A force that always points back toward an equilibrium position.
equilibrium position
The position where the spring is at its natural length and exerts no force (x = 0).
elastic limit
The largest deformation from which a material or spring can return to its original shape.
ideal spring
A massless spring that obeys Hooke's law exactly.

Springs everywhere

The click of a ballpoint pen, the bounce of a trampoline, the smooth ride of a car over a pothole, the reading on a luggage scale at the airport: all of these depend on springs. A spring is special because the force it exerts depends on how far it is stretched or compressed, in a very simple way. That simple rule, Hooke's law, is the foundation for elastic energy and for the oscillations you will study in Simple Harmonic Motion of a Mass on a Spring.

Hooke's law

Let x be the displacement of the end of a spring from its natural (equilibrium) length: positive when stretched, negative when compressed.

To hold a spring stretched or compressed by x, you must apply a force

F_app = kx

By Newton's third law, the spring pushes or pulls back on you with the restoring force

F_s = −kx

  • k is the spring constant (force constant), in newtons per metre (N/m). A stiff spring has a large k; a soft spring has a small k.
  • The minus sign means the spring force always points back toward the equilibrium position: a stretched spring pulls back, a compressed spring pushes back. That is why it is called a restoring force.
  • Hooke's law holds only up to the elastic limit. Stretch a spring too far and it deforms permanently and no longer returns to its natural length.

A spring that obeys Hooke's law exactly is called an ideal spring. Real springs come close within their working range.

natural length: x = 0, no forcestretched by xcompressed by xx = 0F_appF_s = −kxxspring pulls back toward x = 0F_appF_sspring pushes back toward x = 0
A spring at its natural length, stretched and compressed. The spring force F_s always points back toward x = 0, opposite to the displacement, which is why Hooke's law has a minus sign.

The force–stretch graph

If you plot applied force F against stretch x for an ideal spring, you get a straight line through the origin. Its slope is k. This is how you measure k in a lab: hang known masses, measure the stretches, and find the slope of the best-fit line (in N/m, so convert cm to m).

stretch x (cm)F (N)24685101520elastic limitrun 2 cmrise 5 NStraight line through the origin:F = kxslope k = 5 N ÷ 0.02 m= 250 N/mPast the elastic limit thespring is permanently bent.
Force against stretch for a spring. Up to the elastic limit the graph is a straight line through the origin, and its slope is the spring constant k (here 250 N/m).
Worked example

Example 1: Finding k from a hanging mass

A 0.50 kg mass hung from a spring stretches it 4.0 cm and hangs at rest. Find k.

  1. At rest, the spring's upward pull balances the weight: kx = mg.
  2. mg = (0.50)(9.8) = 4.9 N; x = 4.0 cm = 0.040 m.
  3. k = 4.9 ÷ 0.040 = 122.5 N/m ≈ 1.2 × 10² N/m.

k ≈ 1.2 × 10² N/m.

Worked example

Example 2: Car suspension

A 1200 kg car rests on four identical coil springs, each with k = 3.0 × 10⁴ N/m. Assuming each spring carries a quarter of the weight, how far is each spring compressed?

  1. Weight per spring: (1200)(9.8) ÷ 4 = 2940 N.
  2. x = F ÷ k = 2940 ÷ (3.0 × 10⁴) = 0.098 m, about 9.8 cm.

About 9.8 cm each. Engineers choose k so the car sits at the right height and rides smoothly.

Worked example

Example 3: Springs side by side and end to end

Two identical springs, each with k = 200 N/m, support a load. Find the effective spring constant when they are (a) side by side (in parallel), (b) end to end (in series).

  1. (a) Parallel: both stretch by the same x, and each pulls with kx, so the total force is 2kx. k_eff = 2k = 400 N/m.
  2. (b) Series: the full force F passes through each spring, so each stretches F/k and the total stretch is 2F/k. k_eff = F ÷ (2F/k) = k/2 = 100 N/m.
  3. In general: parallel, k_eff = k₁ + k₂; series, 1/k_eff = 1/k₁ + 1/k₂.

(a) 400 N/m (stiffer); (b) 100 N/m (softer).

Just for fun

Robert Hooke was so worried someone would steal his idea that in 1676 he first published it as a scrambled anagram, ceiiinosssttuv. Two years later he revealed the answer: ut tensio, sic vis, Latin for "as the extension, so the force."

Hands-on activity

Rubber bands in series and in parallel

You need: 2 identical long rubber bands, a plastic grocery bag, 4 identical cans of food, a kitchen scale, a ruler, tape, and a cupboard door handle or a sturdy hook. Safety: keep your face away from stretched bands, and add cans gently.

  1. Weigh one can and calculate its weight (mg) in newtons.
  2. One band: hang the bag from one band. Record the bottom position with the bag empty, then add the cans one at a time, recording the stretch each time.
  3. Series: loop the two bands end to end and repeat.
  4. Parallel: hang the bag from both bands side by side and repeat.
  5. For each setup, plot force against stretch (in metres) and find the slope k of the straight part.

What to look for: compared with one band, the series pair should have about half the k and the parallel pair about double the k. Rubber bands are not perfect springs: their graphs often curve, so use the most straight-looking part. That curve is a real result, not a mistake.

Your turn

1

A spring has k = 250 N/m. What force is needed to stretch it 12 cm?

2

A 15 N force stretches a spring 6.0 cm. Find k.

3

A 0.40 kg mass hangs at rest from a spring with k = 80 N/m. How far is the spring stretched?

4

A spring is compressed. Which way does the spring force point?

5

What does the slope of a force–stretch graph for a spring represent?

6

If the stretch of an ideal spring doubles, what happens to the force?

7

What happens if a spring is stretched past its elastic limit?

More practice: warm-up

1

Write Hooke's law for the force a spring exerts.

2

What are the units of k?

3

Find the force to compress a 100 N/m spring by 0.20 m.

4

Is a spring with k = 1000 N/m stiffer or softer than one with k = 50 N/m?

5

What does the minus sign in F_s = −kx tell you?

More practice: on your own

1

Forces of 0, 2.0, 4.0 and 6.0 N stretch a spring 0, 2.5, 5.0 and 7.5 cm. Find k.

2

A spring's constant is given as 5.0 N/cm. Express it in N/m and find the force to compress it 3.0 cm.

3

A 70 kg person stands on a bathroom scale whose spring compresses 2.0 mm. Estimate k for the scale's spring.

4

Two identical springs (k = 150 N/m) side by side support a 3.0 kg mass. How far does each stretch?

5

The same two springs are hooked end to end with the same 3.0 kg mass. Find the total stretch.

Challenge

1

Show that for two springs in series, 1/k_eff = 1/k₁ + 1/k₂.

2

A spring with constant k is cut exactly in half. What is the spring constant of each half? Explain.

3

With a 200 g mass hanging, a spring is 24.0 cm long. With 500 g, it is 30.0 cm long. Find k and the spring's natural length.

Check your understanding

Quiz

1.

Hooke's law states that the force exerted by an ideal spring is:

2.

A spring stretches 5.0 cm when a 20 N force is applied. Its spring constant is:

3.

On a graph of applied force against stretch for an ideal spring, the slope is:

4.

Two identical springs with k = 60 N/m are connected side by side. The effective spring constant is:

5.

A spring is compressed 3.0 cm from its natural length. The spring force on the object pressing it:

6.

The elastic limit of a spring is:

For parents and teachers

Big idea (C3.1): an ideal spring exerts a restoring force proportional to how far it is stretched or compressed: F_s = −kx. The spring constant k measures stiffness and is the slope of a force–stretch graph.

How to help: look for springs at home (pens, mattresses, door closers, a car's suspension) and ask your teen which have a large or small k. Do the rubber-band activity together and compare one band, two in series and two in parallel.

Common difficulties:

  • Using the total length of the spring instead of the stretch x.
  • Forgetting to convert centimetres to metres.
  • Confusing the direction of the spring force (it always points back toward equilibrium).
  • Mixing up series (softer) and parallel (stiffer).