Hooke’s Law and Springs
A spring stretches in proportion to the force pulling it — up to a point. The physics is one equation; the marks are in extension versus total length and in the experiment.
1. Hooke’s law
The extension of a spring is DIRECTLY PROPORTIONAL to the force applied, provided the limit of proportionality is not exceeded.
F = k x force (N) = spring constant (N/m) × extension (m)
Force and extension are DIRECTLY proportional, not inversely. Thinking force is inversely proportional to extension was a recorded error — doubling the force doubles the extension.
Example: a spring with k = 25 N/m stretched by a 5 N force.
- x = F/k = 5 ÷ 25 = 0.2 m (20 cm)
2. Extension is not length
Extension = stretched length − original (natural) length
This is the most-tested subtlety on the page.
Example: a spring is 12 cm long unstretched and 18 cm with a load.
- Extension = 18 − 12 = 6 cm = 0.06 m
Always subtract the original length. Using the total length in F = kx is the standard error and gives a badly wrong spring constant.
Convert to metres — mm and cm must become m for the spring constant to come out in N/m. Uncertainty converting millimetres to metres was recorded.
3. The spring constant
k = F / x, measured in newtons per metre (N/m)
It measures stiffness:
A large k means a stiff spring — a big force produces only a small extension. A small k means an easily stretched spring.
The unit of the spring constant is N/m, not grams. A recorded error gave “2.5 grams” — k is a force per unit length.
Example: a 4 N load produces a 0.08 m extension.
- k = 4 ÷ 0.08 = 50 N/m
4. Force–extension graphs
Plot force (y-axis) against extension (x-axis).
A straight line through the origin shows Hooke’s law is obeyed — force is proportional to extension. The gradient of that line is the spring constant k.
The line must pass through the ORIGIN — zero force gives zero extension.
The limit of proportionality:
The limit of proportionality is the point beyond which extension is no longer proportional to force — where the graph stops being a straight line and curves.
Beyond it:
- The spring extends more for each extra newton
- It may pass its elastic limit, after which it will not return to its original length — it is permanently deformed
The graph curves AWAY from the force axis (towards the extension axis) beyond the limit. Identify the limit as the point where the line stops being straight.
Elastic vs inelastic (plastic) behaviour:
Elastic — returns to its original shape when the force is removed. Inelastic/plastic — stays permanently deformed.
5. The experiment
To investigate how extension varies with force:
- Hang the spring from a clamp and stand, with the stand weighted for stability
- Measure the original length with a ruler clamped vertically alongside
- Add a known mass, and calculate its weight (W = mg)
- Measure the new length and calculate the extension
- Repeat, adding masses one at a time
- Remove the masses one at a time, re-measuring, to check the spring returns to its original length
- Plot force against extension
Variables:
Independent: the force (load) applied. Dependent: the extension. Control: the same spring throughout, and the same temperature.
The load is what you CHANGE — it cannot be a control variable. Stating that the load should be kept constant was a recorded error; it is the independent variable.
Don’t list irrelevant control variables. Mentioning temperature is reasonable; padding the list with variables that don’t affect the result was flagged as a recorded error. Keep to what genuinely matters — the same spring, and not exceeding the limit.
Include a conclusion in an experiment plan — tutors noted this specifically. State what the graph shows and what it tells you about Hooke’s law.
Improving accuracy:
- Read the ruler at eye level to avoid parallax error
- Use a pointer or set square against the spring
- Repeat and average
- Add masses gently so the spring doesn’t oscillate
Read graph values carefully — misreading a graph was a recorded cause of wrong force calculations. Zoom in and use the gridlines.
6. Springs in combination
In series (end to end): each spring feels the full load, so each extends fully — the total extension is larger and the combination is less stiff.
In parallel (side by side): the load is shared, so each extends less — the combination is stiffer.
For two identical springs in parallel, each carries half the load, so the total extension is half that of one spring — giving a combined spring constant of 2k.
7. Energy stored in a spring
A stretched spring stores elastic potential energy.
On a force–extension graph, the area under the line is the energy stored.
For a spring obeying Hooke’s law, that area is a triangle: E = ½ F x.
8. Mistakes that cost marks
Using total length instead of extension.
Not converting mm or cm to metres.
Saying force and extension are inversely proportional.
Giving the wrong unit for k.
Drawing a force–extension line that misses the origin.
Confusing the limit of proportionality with the elastic limit.
Calling the load a control variable.
Listing irrelevant control variables.
Omitting a conclusion from an experiment plan.
Misreading graph values.
Frequently asked questions
What is Hooke’s law? Extension is directly proportional to the force applied, up to the limit of proportionality.
What is the equation? F = k x, where x is the extension.
What is extension? Stretched length − original length.
What is the spring constant? The force per unit extension, k = F/x, in N/m. It measures stiffness.
What does the gradient of a force–extension graph show? The spring constant.
What is the limit of proportionality? The point beyond which extension is no longer proportional to force — where the graph starts to curve.
What is the elastic limit? The point beyond which the spring will not return to its original length.
What are the variables in the experiment? Independent: force. Dependent: extension. Control: same spring, same temperature.
How do I find the energy stored? The area under the force–extension graph, ½Fx for a Hookean spring.
What happens with two springs in parallel? Each takes half the load, so the extension is halved — the combination is stiffer.
Quick revision checklist
- I can state Hooke’s law with its condition
- I know F = kx
- I calculate extension by subtracting the original length
- I convert lengths to metres
- I know k is in N/m and means stiffness
- I know force and extension are directly proportional
- I can read k from the gradient of a graph
- I know the line passes through the origin
- I can identify the limit of proportionality
- I know the difference between it and the elastic limit
- I can describe the experiment fully
- I can state the independent, dependent and control variables correctly
- I include a conclusion in a plan
- I can find the energy stored from the area
These notes cover Hooke’s law and springs in the Cambridge IGCSE Physics (0625) syllabus and are written for Grade 9–11 / Year 10–11 students. They are based on teaching patterns observed across a large set of one-to-one IGCSE Physics lessons, with particular attention to the errors students make most often and the wording examiners reward. Always check the current syllabus and formula list for your own exam series.
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