Springs and Hooke's Law

A spring stretches in proportion to the load on it, $F=kx$ , up to the limit of proportionality. The spring constant $k$ tells you how stiff it is. Extension is measured from the original length, and a load–extension graph is a straight line through the origin until the limit is passed.

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Section 1 of 14

Learning objectives

Measure extension correctly from a spring's natural length.
Use $k=\dfrac Fx$ to find the spring constant, force or extension.
Interpret load–extension graphs and identify the limit of proportionality.
Describe an experiment to investigate how a spring stretches.

Pause and think

One $1$ N weight stretches a spring by $2$ cm. What would you predict for $3$ N? Would the pattern go on forever?

Section 2 of 14

Core idea

Start hereA spring balance

Hang 1 N on a spring and it stretches 4 cm; hang 2 N and it stretches 8 cm. This simple doubling is why a spring can be marked as a scale. It works until you pull too hard.

The big idea

Hooke's law: $F=kx$ up to the limit of proportionality. Extension = stretched length − original length.

1. Extension

Load–extension graph for a spring

In the straight part, doubling the load doubles the extension. Here

k=\dfrac{4.8}{8}=0.6

N/cm.

Extension = new length − original length

Always measure from the unloaded length, not from zero on the ruler.

F=kx

$F$: load (N).
$k$: spring constant (N/m): stiffness.
$x$: extension (m).

Worked exampleFinding k

Original length 12.0 cm; with 4 N it is 28.0 cm.

Goal: $k$ ; load for a 10 cm extension.
Strategy: Extension, then $k=F/x$ .

$x=28.0-12.0=16.0\text{ cm}=0.16\text{ m}$
Extension, in metres.
$k=\frac{4}{0.16}=25\text{ N/m}$
Rearrange.
$F=25\times0.10=2.5\text{ N}$
Use.

Answer

25

N/m;

2.5

✓ Check10 cm is

\frac58

of 16 cm, and

2.5

N is

\frac58

of 4 N ✓.

Common misconceptionLength instead of extension Use

28-12=16

cm, not

28

cm.

Edge caseBeyond the limit Past the limit of proportionality the graph curves; past the elastic limit the spring stays permanently stretched.

Pause and think

Two identical springs side by side share a load. Effect on extension?

Summary

Extension = new − original.
$F=kx$ , straight line through the origin.

Spring constant: Force per unit extension.
Limit of proportionality: End of the straight-line region.

HistoryHooke's anagram. In

1676

Robert Hooke published his law as an anagram, "ceiiinosssttuv", to claim priority without revealing it. Two years later he gave the answer: ut tensio, sic vis, "as the extension, so the force".

Fun factSprings in your bed and your watch. A mechanical watch keeps time with a tiny hairspring vibrating back and forth. Because of Hooke's law, every swing takes the same time, whatever its size.

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