Definition

Spring constant

The force needed to produce each metre of extension in a spring, measured in newtons per metre.

Different springs need different forces to produce the same extension, because some springs are stiffer than others.
The spring constant, $k$ , is a measure of how stiff a spring is.
A spring with a large spring constant is stiff, so it needs a large force for each metre of extension.
1. A car suspension spring is very stiff, with a spring constant of tens of thousands of newtons per metre.
A spring with a small spring constant is easy to stretch, so a small force produces a large extension.
1. The small spring inside a retractable pen has a spring constant of only a few hundred newtons per metre.
The spring constant is measured in newtons per metre, $N/m\text{N/m}$ $N/m$ .
1. A spring constant of $N/m50\ \text{N/m}$ means that a force of $N50\ \text{N}$ is needed for each metre of extension, which is $N0.5\ \text{N}$ for each centimetre.

The spring equation

For a spring in the linear part of its behaviour, the force, the spring constant and the extension are linked by an equation you must recall.
In words, $constant×extension\text{force exerted by a spring}=\text{spring constant}\times\text{extension}$ .
In symbols, the equation is $F = k x$ .
1. $F$ is the force in newtons, $N\text{N}$ .
2. $k$ is the spring constant in newtons per metre, $N/m\text{N/m}$ .
3. $x$ is the extension in metres, $m\text{m}$ .
The equation applies only while the spring obeys Hooke's law, so it cannot be used beyond the limit of proportionality.
The force stretching the spring and the force the spring pulls back with are equal in size, so $F$ can stand for either.
The same equation works for a spring that is compressed, with $x$ standing for the decrease in length.

Calculating the spring constant

Rearranging $F = k x$ gives $k=Fxk=\dfrac{F}{x}$ , so the spring constant is the force divided by the extension it produces.
Rearranging for the extension gives $x=Fkx=\dfrac{F}{k}$ .
Convert the extension into metres before substituting, because the spring constant is in newtons per metre.
1. Divide a length in centimetres by $100$ to give metres, so $m4.0\ \text{cm}=0.040\ \text{m}$ .
2. Divide a length in millimetres by $1000$ to give metres, so $m25\ \text{mm}=0.025\ \text{m}$ .
If a question gives two lengths instead of an extension, subtract the original length from the stretched length first.

The spring constant from a graph

Definition

Gradient

The steepness of a line on a graph, equal to the change in the vertical quantity divided by the change in the horizontal quantity.

On a graph with force on the vertical axis and extension on the horizontal axis, the gradient of the straight section equals the spring constant.
The spring constant is found as $extensionk=\dfrac{\text{change in force}}{\text{change in extension}}$ , using two well-separated points on the straight line.
A steeper line on this graph belongs to a stiffer spring with a larger spring constant.
Many practicals plot extension on the vertical axis against force, and on that graph the gradient is $1k\dfrac{1}{k}$ , so the spring constant is the change in force divided by the change in extension, not the gradient itself.
Draw a large triangle on the straight part of the line to find the gradient, because a large triangle reduces the effect of small reading errors.

Example

Spring constant from a force and two lengths

A spring has an original length of $cm12.0\ \text{cm}$ and stretches to $cm16.0\ \text{cm}$ when a force of $N8.0\ \text{N}$ is applied.

x=16.0\ \text{cm}-12.0\ \text{cm}=4.0\ \text{cm}=0.040\ \text{m}

k=\frac{F}{x}=\frac{8.0\ \text{N}}{0.040\ \text{m}}=200\ \text{N/m}

The spring constant is $N/m200\ \text{N/m}$ .

Example

Force needed for a given extension

A spring has a spring constant of $N/m25\ \text{N/m}$ and must be stretched by $cm8.0\ \text{cm}$ .

x=8.0\ \text{cm}=0.080\ \text{m}

F=kx=25\ \text{N/m}\times0.080\ \text{m}=2.0\ \text{N}

A force of $N2.0\ \text{N}$ is needed.

Example

Extension of a trampoline spring

A trampoline spring has a spring constant of $N/m5000\ \text{N/m}$ and carries a force of $N350\ \text{N}$ when a gymnast lands.

x=\frac{F}{k}=\frac{350\ \text{N}}{5000\ \text{N/m}}=0.070\ \text{m}

The spring extends by $m0.070\ \text{m}$ , which is $cm7.0\ \text{cm}$ .

Example

Spring constant from a graph

A graph has force on the vertical axis and extension on the horizontal axis, and its straight line passes through the origin and the point $N)(0.030\ \text{m},\ 6.0\ \text{N})$ .

k=\frac{\text{change in force}}{\text{change in extension}}=\frac{6.0\ \text{N}-0\ \text{N}}{0.030\ \text{m}-0\ \text{m}}=200\ \text{N/m}

The gradient, and so the spring constant, is $N/m200\ \text{N/m}$ .

Common Mistake

Do not use the stretched length as $x$ , because $x$ is the extension.
Do not substitute an extension in centimetres when the answer must be in newtons per metre, because the result will be $100$ times too small.
Do not give the spring constant in newtons, because its unit is newtons per metre.
Do not use $F = k x$ for a spring stretched beyond its limit of proportionality.

Exam technique

Spring constant calculations

Write $F = k x$ from memory before you substitute, because recalling the equation is part of what is being tested.
Show the conversion of the extension into metres on its own line, so that working is credited even if a later step goes wrong.
Give the final answer with the unit $N/m\text{N/m}$ .
In a graph question, check which quantity is on each axis before using the gradient, and take readings only from the straight part of the line.

Self review

State the equation that links force, spring constant and extension.
State the unit of spring constant.
Calculate the spring constant of a spring that extends by $cm5.0\ \text{cm}$ when a force of $N3.0\ \text{N}$ is applied.
Explain what a larger spring constant tells you about a spring.
Describe how to find the spring constant from a graph of force against extension.

Definition

Spring constant

The force needed to produce each metre of extension in a spring, measured in newtons per metre.

Different springs need different forces to produce the same extension, because some springs are stiffer than others.
The spring constant, $k$ , is a measure of how stiff a spring is.
A spring with a large spring constant is stiff, so it needs a large force for each metre of extension.
1. A car suspension spring is very stiff, with a spring constant of tens of thousands of newtons per metre.
A spring with a small spring constant is easy to stretch, so a small force produces a large extension.
1. The small spring inside a retractable pen has a spring constant of only a few hundred newtons per metre.
The spring constant is measured in newtons per metre, $N/m\text{N/m}$ $N/m$ .
1. A spring constant of $N/m50\ \text{N/m}$ means that a force of $N50\ \text{N}$ is needed for each metre of extension, which is $N0.5\ \text{N}$ for each centimetre.

The spring equation

For a spring in the linear part of its behaviour, the force, the spring constant and the extension are linked by an equation you must recall.
In words, $constant×extension\text{force exerted by a spring}=\text{spring constant}\times\text{extension}$ .
In symbols, the equation is $F = k x$ .
1. $F$ is the force in newtons, $N\text{N}$ .
2. $k$ is the spring constant in newtons per metre, $N/m\text{N/m}$ .
3. $x$ is the extension in metres, $m\text{m}$ .
The equation applies only while the spring obeys Hooke's law, so it cannot be used beyond the limit of proportionality.
The force stretching the spring and the force the spring pulls back with are equal in size, so $F$ can stand for either.
The same equation works for a spring that is compressed, with $x$ standing for the decrease in length.

Calculating the spring constant

Rearranging $F = k x$ gives $k=Fxk=\dfrac{F}{x}$ , so the spring constant is the force divided by the extension it produces.
Rearranging for the extension gives $x=Fkx=\dfrac{F}{k}$ .
Convert the extension into metres before substituting, because the spring constant is in newtons per metre.
1. Divide a length in centimetres by $100$ to give metres, so $m4.0\ \text{cm}=0.040\ \text{m}$ .
2. Divide a length in millimetres by $1000$ to give metres, so $m25\ \text{mm}=0.025\ \text{m}$ .
If a question gives two lengths instead of an extension, subtract the original length from the stretched length first.

The spring constant from a graph

Definition

Gradient

The steepness of a line on a graph, equal to the change in the vertical quantity divided by the change in the horizontal quantity.

On a graph with force on the vertical axis and extension on the horizontal axis, the gradient of the straight section equals the spring constant.
The spring constant is found as $extensionk=\dfrac{\text{change in force}}{\text{change in extension}}$ , using two well-separated points on the straight line.
A steeper line on this graph belongs to a stiffer spring with a larger spring constant.
Many practicals plot extension on the vertical axis against force, and on that graph the gradient is $1k\dfrac{1}{k}$ , so the spring constant is the change in force divided by the change in extension, not the gradient itself.
Draw a large triangle on the straight part of the line to find the gradient, because a large triangle reduces the effect of small reading errors.

Example

Spring constant from a force and two lengths

A spring has an original length of $cm12.0\ \text{cm}$ and stretches to $cm16.0\ \text{cm}$ when a force of $N8.0\ \text{N}$ is applied.

x=16.0\ \text{cm}-12.0\ \text{cm}=4.0\ \text{cm}=0.040\ \text{m}

k=\frac{F}{x}=\frac{8.0\ \text{N}}{0.040\ \text{m}}=200\ \text{N/m}

The spring constant is $N/m200\ \text{N/m}$ .

Example

Force needed for a given extension

A spring has a spring constant of $N/m25\ \text{N/m}$ and must be stretched by $cm8.0\ \text{cm}$ .

x=8.0\ \text{cm}=0.080\ \text{m}

F=kx=25\ \text{N/m}\times0.080\ \text{m}=2.0\ \text{N}

A force of $N2.0\ \text{N}$ is needed.

Example

Extension of a trampoline spring

A trampoline spring has a spring constant of $N/m5000\ \text{N/m}$ and carries a force of $N350\ \text{N}$ when a gymnast lands.

x=\frac{F}{k}=\frac{350\ \text{N}}{5000\ \text{N/m}}=0.070\ \text{m}

The spring extends by $m0.070\ \text{m}$ , which is $cm7.0\ \text{cm}$ .

Example

Spring constant from a graph

A graph has force on the vertical axis and extension on the horizontal axis, and its straight line passes through the origin and the point $N)(0.030\ \text{m},\ 6.0\ \text{N})$ .

k=\frac{\text{change in force}}{\text{change in extension}}=\frac{6.0\ \text{N}-0\ \text{N}}{0.030\ \text{m}-0\ \text{m}}=200\ \text{N/m}

The gradient, and so the spring constant, is $N/m200\ \text{N/m}$ .

Common Mistake

Do not use the stretched length as $x$ , because $x$ is the extension.
Do not substitute an extension in centimetres when the answer must be in newtons per metre, because the result will be $100$ times too small.
Do not give the spring constant in newtons, because its unit is newtons per metre.
Do not use $F = k x$ for a spring stretched beyond its limit of proportionality.

Exam technique

Spring constant calculations

Write $F = k x$ from memory before you substitute, because recalling the equation is part of what is being tested.
Show the conversion of the extension into metres on its own line, so that working is credited even if a later step goes wrong.
Give the final answer with the unit $N/m\text{N/m}$ .
In a graph question, check which quantity is on each axis before using the gradient, and take readings only from the straight part of the line.

Self review

State the equation that links force, spring constant and extension.
State the unit of spring constant.
Calculate the spring constant of a spring that extends by $cm5.0\ \text{cm}$ when a force of $N3.0\ \text{N}$ is applied.
Explain what a larger spring constant tells you about a spring.
Describe how to find the spring constant from a graph of force against extension.

2.3.5 The spring constant

The spring equation

Calculating the spring constant

The spring constant from a graph

Spring constant from a force and two lengths

Force needed for a given extension

Extension of a trampoline spring

Spring constant from a graph

Spring constant calculations

Review 2.3.5 The spring constant by teaching Genie

2.3.5 The spring constant

The spring equation

Calculating the spring constant

The spring constant from a graph

Spring constant from a force and two lengths

Force needed for a given extension

Extension of a trampoline spring

Spring constant from a graph

Spring constant calculations

Review 2.3.5 The spring constant by teaching Genie

2.3.5 The spring constant

The spring equation

Calculating the spring constant

The spring constant from a graph

Spring constant from a force and two lengths

Force needed for a given extension

Extension of a trampoline spring

Spring constant from a graph

Spring constant calculations

Review 2.3.5 The spring constant by teaching Genie

2.3.5 The spring constant Revision Guide

2.3.5 The spring constant

The spring equation

Calculating the spring constant

The spring constant from a graph

Spring constant from a force and two lengths

Force needed for a given extension

Extension of a trampoline spring

Spring constant from a graph

Spring constant calculations

Review 2.3.5 The spring constant by teaching Genie

2.3.5 The spring constant Revision Guide