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The spring constant formula comes from Hooke's law, F = ke, rearranged to k = F ÷ e, where F is force in newtons and e is extension in metres. Free notes.
Spring Constant and Calculations is part of Forces and elasticity in GCSE AQA Higher Physics.
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The spring constant formula comes from Hooke's law, F = ke, rearranged to k = F ÷ e, where F is force in newtons and e is extension in metres. Free notes.
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Use this lesson to stay inside Forces and elasticity while connecting the detail back to the wider Forces topic.
Read the core explanations from Spring Constant and Calculations before testing yourself from memory.
Why does a car suspension spring feel different from a pen spring? The answer lies in a property called spring constant - a number that tells us exactly how stiff or stretchy a spring is. In real-world applications, this property affects how efficiently energy is absorbed or returned by the spring, influencing both comfort and performance, and showcases its importance in engineering design.
When you pull a spring, it fights back with a force. The stiffer the spring, the harder it fights. This relationship between force and stretch follows a beautifully simple pattern that engineers use to design everything from trampolines to skyscrapers.
Robert Hooke discovered that springs behave predictably. Double the force, double the extension. Triple the force, triple the extension. This direct relationship only works up to a point - the elastic limit.
The spring constant k describes the stiffness of a spring while it is behaving elastically. For a given spring in its elastic region, k stays constant, which is why force is proportional to extension or compression there. A stiff spring has a high k value, while an easier-to-stretch spring has a lower k value.
Force-extension graphs tell the complete story of how a spring behaves. The straight-line portion shows where Hooke's Law applies - this is the elastic region where the spring will return to its original length.
The gradient of the straight-line section equals the spring constant. A steep gradient means a stiff spring (high k). A gentle gradient means a stretchy spring (low k). Beyond the elastic limit, the line curves and the spring becomes permanently damaged.
Notice how the graph shows a straight line up to about 0.08 m extension, then starts to curve. The elastic limit occurs around 16 N force. Beyond this point, the spring won't return to its original length when the force is removed.
Spring constant calculations appear frequently in exams. The key is identifying what you know and what you need to find, then applying F = ke correctly. Watch out for unit conversions - extension must be in metres, not centimetres.
Worked example: Finding Spring Constant from Data
A spring extends from 15.0 cm to 23.0 cm when a 12 N weight is attached. Calculate the spring constant. [3 marks]
Answer: 150 N/m
Worked example: Finding Force from Spring Constant
A spring with spring constant 250 N/m is compressed by 4.0 cm. Calculate the force needed. [2 marks]
Answer: 10 N
The spring constant investigation is a classic required practical. You'll hang masses from a spring and measure the extension, then plot a graph to find the spring constant from the gradient.
Common mistakes include using the total length instead of extension, forgetting to convert mass to weight using W = mg, and trying to find the gradient from the curved portion of the graph. In GCSE questions you may use g = 9.8 N/kg, or 10 N/kg if the question tells you to use the approximation.
Spring constant questions are mark-friendly if you follow the method. Examiners award marks for showing the formula, correct substitution, and the final answer with units. The biggest trap is unit conversion - always convert centimetres to metres.
Watch out for questions that give you the total length after stretching - you need to subtract the original length to find the extension. Also remember that compression works the same way as extension in calculations.
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Move to the related lessons or back to the topic page when you need the wider sequence before exam-style practice.
Find the extension, convert it to metres, then use k = F ÷ e. A spring that extends from 15.0 cm to 23.0 cm with a 12 N weight has an extension of 8.0 cm (0.080 m), so k = 12 ÷ 0.080 = 150 N/m.
The elastic limit is the maximum force that can be applied to a spring before it becomes permanently deformed and will not return to its original length. Beyond it, the force-extension line curves and the spring is permanently damaged.
The spring constant formula comes from Hooke's law, F = ke, rearranged to k = F ÷ e, where F is force in newtons and e is extension in metres. A spring extended by 0.080 m by 12 N has k = 12 ÷ 0.080 = 150 N/m.
The gradient of the straight-line section of a force-extension graph equals the spring constant. A steep gradient means a stiff spring with a high k value, and a gentle gradient means a stretchy spring with a low k value.
Use F = ke, the spring constant × the extension in metres. A spring with a spring constant of 250 N/m compressed by 4.0 cm (0.040 m) needs a force of 250 × 0.040 = 10 N; compression counts as extension for calculations.
Extension is the increase in length of a spring when a force is applied, not its total length. If a question gives the total length after stretching, subtract the original length to find the extension, then convert to metres.
Common mistakes are using the total length instead of the extension, forgetting to convert mass to weight using W = mg, and finding the gradient from the curved part of the graph. Extension must also be in metres, not centimetres.
Compressing a spring with spring constant 250 N/m by 4.0 cm needs a force of 10 N. Converting 4.0 cm to 0.040 m, F = ke = 250 × 0.040 = 10 N.
Compression works the same way as extension in Hooke's law calculations: F = ke still applies, with e as the amount the spring is compressed, in metres. For a given spring in its elastic region, k stays constant for both.
In the spring practical, each mass is converted to a force using W = mg, with g = 9.8 N/kg, or 10 N/kg if the question allows the approximation. Forgetting this conversion is a common mistake, along with using total length instead of extension.
The elastic region is the straight-line part of a force-extension graph, where Hooke's law applies and the spring returns to its original length. The spring constant k stays constant there, so force is proportional to extension.
In car suspension, the spring constant affects how efficiently energy is absorbed or returned by the spring, which influences both comfort and performance. A car suspension spring is much stiffer than a pen spring.
A force-extension graph stops being straight at the elastic limit. In one example, the line is straight up to about 0.08 m extension and the elastic limit is around 16 N; beyond this, the spring will not return to its original length.
The spring constant cannot be found from the curved part of a force-extension graph; only the gradient of the straight-line section equals the spring constant. Using the curved portion is a common mistake in the spring practical.