Hooke's Law
Hook a bag of apples onto a kitchen spring scale and the spring stretches. Add a second bag —
twice the pull — and the spring stretches twice as far. Add a third and it
stretches three times as far. The spring is doing something wonderfully simple and reliable:
the harder you pull, the more it stretches, and it stretches in exact proportion to
the pull.
That neat straight-line relationship is Hooke's law, and it is the reason a
spring can be used to measure a force at all. Bathroom scales, kitchen scales, the
springs inside a car's suspension, a trampoline, a diving board, a set of luggage scales at the
airport — every one of them turns a force into a stretch (or a squash) you can read off,
because for a spring the two are locked together in a straight line.
Force is proportional to extension
The key measurement is not the spring's total length — it is how much longer the
spring has become. That extra length is called the extension, written
e: if an unstretched spring is 10 cm long and you pull it out to
14 cm, the extension is e = 4\ \text{cm}, not 14 cm.
Double the force and you double the extension; treble it and you treble the extension. Two
quantities that grow in lockstep like this are called proportional, and we tie
them together with a constant. That constant is the spring constant
k — a number that says how stiff this particular spring is.
A stiff spring (big k) barely budges; a floppy spring (small
k) stretches easily.
The force needed to stretch (or compress) a spring is proportional to the extension it
produces:
F = k\,e.
- F — the force applied to the spring, in newtons (N)
- k — the spring constant (stiffness), in newtons per metre (N/m)
- e — the extension, i.e. the extra length, in metres (m)
Read the units straight off the formula: k = \dfrac{F}{e} is
newtons divided by metres, so a spring constant of 200\ \text{N/m}
means it takes 200 N of pull to stretch that spring by one whole metre — a fairly stiff
spring. The same law works when you squash a spring instead of stretching it: then
e is the compression, and the harder you push the more it compresses.
Play with a spring
Below hangs a spring from a fixed support with a mass on the end. Turn up the force
and watch the spring stretch — the extension e grows in exact
proportion. Now change the spring constant k: a
stiffer spring (larger k) gives less extension for the same force, a
floppier spring gives more. At every setting the numbers obey e = F / k.
The straight line: a force–extension graph
Plot the force F up the side against the extension
e along the bottom and Hooke's law draws itself: a
straight line through the origin. Straight, because doubling
e doubles F; through the origin, because
with no force there is no extension.
The gradient of that line — how steeply it climbs — is the spring constant
k. A stiff spring makes a steep line; a floppy spring makes a shallow
one. Slide the spring constant below and watch the line pivot. Notice too that the line only
stays straight for a while: past a certain point (the limit of proportionality) it
starts to bend, and Hooke's law stops holding — more on that next.
Worked examples
Because F = k\,e links three quantities, knowing any two gives you
the third — just like rearranging
any equation.
Example 1 — find the force. A spring has spring constant
k = 30\ \text{N/m} and is stretched by
e = 0.2\ \text{m}. What force is pulling on it?
F = k\,e = 30 \times 0.2 = 6\ \text{N}.
Example 2 — find the spring constant. A force of
12\ \text{N} stretches a spring by
e = 0.08\ \text{m}. How stiff is it? Rearrange to make
k the subject:
k = \frac{F}{e} = \frac{12}{0.08} = 150\ \text{N/m}.
Example 3 — find the extension. A spring of constant
k = 25\ \text{N/m} has a 10\ \text{N} weight
hung on it. How far does it stretch?
e = \frac{F}{k} = \frac{10}{25} = 0.4\ \text{m}.
Watch your units every time: the force must be in newtons and the extension in metres, or
k comes out wrong. If a question gives an extension in centimetres,
convert to metres first (8\ \text{cm} = 0.08\ \text{m}).
The four traps that catch out nearly every student:
-
Hooke's law only holds up to the elastic limit. Stretch a spring gently and
it springs right back — that is elastic behaviour, and the force–extension
line is straight. But pull too far and you pass the limit of proportionality
and then the elastic limit: the line bends, proportionality fails
(F = k\,e no longer works), and the spring is permanently
stretched — it stays longer even after you let go. That kind of permanent, non-springy change
is called inelastic (or plastic) deformation.
-
Extension is the EXTRA length, not the total length. A spring stretched from
10 cm to 14 cm has an extension of 4 cm. Putting 14 cm into
F = k\,e is a classic wrong answer.
-
k is a property of the spring. It doesn't change
when you pull harder — a stiffer spring simply has a bigger k.
Pulling with more force gives more extension, but the ratio F/e
stays the same (as long as you're below the elastic limit).
-
Force and extension must be in newtons and metres. Centimetres or grams sneak
in and wreck the arithmetic — convert first.
Past the limit: elastic vs inelastic
Every real spring has a breaking point to its good behaviour. While you keep the force small,
the spring is elastic: let go and it returns to exactly its original length,
and its force–extension graph is a straight line obeying F = k\,e.
Pull past the limit of proportionality and the graph stops being straight —
the spring extends more than its share for each extra newton. Push on past the
elastic limit and the damage becomes permanent: the spring has been
inelastically (plastically) deformed and will not return to its
starting length. You have seen this if you have ever over-stretched a Slinky or a biro spring —
it comes back longer and floppier than before, ruined. Below the elastic limit: elastic and
reversible. Above it: inelastic and permanent.
Energy stored in a stretched spring
Stretching a spring takes work, and that work is stored inside the spring as
elastic potential energy — the energy that fires a nerf dart, snaps a mouse
trap shut, or flings you off a trampoline. Where does the amount come from? It is the
area under the force–extension graph.
Because the graph (in the elastic region) is a straight line from the origin, the area beneath
it is simply a triangle with base e and height
F:
E = \tfrac{1}{2}\,F\,e = \tfrac{1}{2}\,k\,e^{2}.
The second form uses F = k\,e to write it with just
k and e. Notice the
e^{2}: stretch a spring twice as far and it stores
four times the energy — which is why a bow drawn back that little bit further sends the
arrow so much faster. For example, a spring with
k = 200\ \text{N/m} stretched by
0.1\ \text{m} stores
E = \tfrac{1}{2}\times 200 \times 0.1^{2} = 1\ \text{J}.
Roughly, yes — and that is exactly why bungee jumping works (and is survivable). A bungee cord
is a giant spring: the further it stretches, the harder it pulls back, following
F = k\,e for most of the fall. At the bottom of the leap the cord is
stretched to its longest, so its pull is at its greatest — enough to overcome your weight,
halt you, and throw you back up. All the energy of your plunge has been stored in the cord as
elastic potential energy (that triangular area under the graph), then handed straight back.
The same physics fills a mattress with springs, gives a car its comfy suspension, cocks a
stapler, and lets a trampoline hurl a child skywards: a spring quietly banking your energy on
the way in and repaying it on the way out. The engineers just have to make sure the force never
pushes the spring past its elastic limit — otherwise it would sag permanently and never bounce
back.