Forces and elasticity

What you'll learn

How forces can stretch, compress, or bend objects.
The difference between elastic and inelastic deformation.
How to use $F = k e$ for springs and other elastic objects.
How to calculate elastic potential energy and analyse force-extension graphs.

Changing shape: deformation

When forces act on an object, they can change its shape. The object might be:

stretched — pulled longer, like a spring or elastic band.
compressed — squashed shorter, like foam or a spring being pushed.
bent — curved, like a ruler pressed in the middle.

Definition

Deformation

A deformation is a change in the shape of an object caused by forces.

For a stationary object to change shape, more than one force must act. If only one unbalanced force acted on the object, it would accelerate rather than just deform. With balanced forces, the resultant force can be zero, but the forces can still squash, stretch, or bend the object because they act at different places.

Diagram showing stretching, compression and bending caused by multiple balanced forces

Example

Explaining why two forces are needed

A foam block is stationary while being compressed between two hands. Each hand pushes with 12 N.

The left hand pushes the block to the right with 12 N, and the right hand pushes it to the left with 12 N.
The forces are equal and opposite, so the resultant force on the block is 0 N.
The block does not accelerate, but it is compressed because the two forces act on opposite sides and push its particles closer together.

Elastic and inelastic deformation

Some objects return to their original shape when the forces are removed. Others stay permanently changed.

Definition

Elastic deformation

Elastic deformation happens when an object returns to its original shape after the deforming forces are removed.

Definition

Inelastic deformation

Inelastic deformation happens when an object does not return completely to its original shape after the deforming forces are removed. The object has been permanently deformed.

A spring being gently stretched usually shows elastic deformation. Plasticine being squashed shows inelastic deformation.

Common Mistake

Elastic does not mean stretchy forever

An object can be elastic only over a certain range. If you stretch a spring too far, it may stop returning to its original length and become permanently deformed.

Extension and compression

For springs, we usually measure how much the length changes.

Definition

Extension

The extension, $e$ , is the increase in length of an object when it is stretched:

\text{stretched length} - \text{original length}

Extension is measured in metres, m.

If a spring is compressed instead of stretched, the same ideas apply, but $e$ means the compression — the decrease in length.

Tip

Use the change in length

In calculations, do not use the total length of the spring. Use the extension or compression: final length minus original length.

Hooke’s law: force and extension

For many elastic objects, such as a spring, the extension is directly proportional to the force applied — but only up to a certain point.

Definition

Directly proportional

Two quantities are directly proportional if doubling one doubles the other, and their graph is a straight line through the origin.

For a spring obeying Hooke’s law:

F = k e

where:

$F$ is the force in newtons, N.
$k$ is the spring constant in newtons per metre, N/m.
$e$ is the extension in metres, m.

Definition

Spring constant

The spring constant, $k$ , tells you how stiff a spring is. A larger spring constant means more force is needed for the same extension.

Example

Calculating a spring constant

A spring extends by 6.0 cm when a force of 3.0 N is applied. Calculate its spring constant.

Convert the extension into metres: 6.0 cm is 0.060 m.
Rearrange $F = k e$ to make $k$ the subject:
$\frac{F}{e}$
Substitute the values:
$N/m\begin{aligned} k &= \frac{3.0\ \text{N}}{0.060\ \text{m}} \\ k &= 50\ \text{N/m} \end{aligned}$

Common Mistake

Forgetting centimetres to metres

If the extension is given in centimetres or millimetres, convert it to metres before using $F = k e$ .

The limit of proportionality

The force-extension relationship is linear only up to the limit of proportionality.

Definition

Limit of proportionality

The limit of proportionality is the point beyond which force and extension are no longer directly proportional.

Before this limit, the graph of force against extension is a straight line through the origin. After this limit, the graph curves, so the relationship is non-linear.

Force-extension graph showing linear region, limit of proportionality and stored elastic potential energy

A linear relationship gives a straight-line graph. A non-linear relationship gives a curved graph or a line whose gradient changes.

On a force-extension graph with force on the vertical axis and extension on the horizontal axis:

gradient=Fe=k\text{gradient} = \frac{F}{e} = k

So, in the straight-line region, the gradient gives the spring constant.

Example

Interpreting force-extension data

A spring has these results: 1.0 N gives 0.020 m extension, 2.0 N gives 0.040 m, 3.0 N gives 0.060 m, but 4.0 N gives 0.090 m. Decide where the graph stops being linear.

Compare the first three ratios:
$1.00.020=502.00.040=503.00.060=50\begin{aligned} \frac{1.0}{0.020} &= 50 \\ \frac{2.0}{0.040} &= 50 \\ \frac{3.0}{0.060} &= 50 \end{aligned}$
The spring constant is the same, so this part is linear.
Check the next ratio:
$4.00.090≈44\frac{4.0}{0.090} \approx 44$
The ratio has changed, so the gradient is no longer constant.
The limit of proportionality has been exceeded somewhere between 3.0 N and 4.0 N.

Elastic potential energy

When you stretch or compress a spring, you do work on it. If the spring is not permanently deformed, this work is stored as elastic potential energy.

Definition

Elastic potential energy

Elastic potential energy, $E_e$ , is energy stored in an object when it is stretched or compressed elastically.

Up to the limit of proportionality:

Ee=12ke2E_e = \frac{1}{2} k e^2

where:

$E_e$ is elastic potential energy in joules, J.
$k$ is spring constant in N/m.
$e$ is extension in m.

This equation is also the area under the straight-line part of the force-extension graph.

Example

Calculating elastic potential energy

A spring has a spring constant of 80 N/m and is stretched by 5.0 cm. Calculate the elastic potential energy stored.

Convert the extension into metres: 5.0 cm is 0.050 m.
Substitute into $Ee=12ke2E_e = \frac{1}{2} k e^2$ :
$Ee=12×80×(0.050)2E_e = \frac{1}{2} \times 80 \times (0.050)^2$
Calculate the energy:
$J\begin{aligned} E_e &= 40 \times 0.0025 \\ E_e &= 0.10\ \text{J} \end{aligned}$

Common Mistake

Only while the spring behaves elastically

The equation $Ee=12ke2E_e = \frac{1}{2} k e^2$ is for the linear elastic region, up to the limit of proportionality.

Required practical: force and extension for a spring

You need to be able to describe how to investigate the relationship between force and extension for a spring.

Required practical setup with spring, clamp stand, ruler, masses and safety mat

The independent variable is the force applied to the spring. The dependent variable is the extension. Important control variables include using the same spring and measuring the extension in the same way each time.

A good method is:

Set up a spring hanging from a clamp stand beside a vertical ruler. Put a safety mat underneath.
Measure the original length of the spring, $L_0$ , before adding masses.
Add a mass hanger and masses one at a time.
For each mass, wait for the spring to stop moving, then measure the new length, $L$ .
Calculate the extension using $e = L - L_0$ .
Calculate the force using weight: $W = m g$ .
Plot a graph of force against extension and draw a best-fit line for the linear region.
Find the spring constant from the gradient of the straight-line section.

Tip

Mass is not force

Mass is measured in kilograms, kg. The stretching force is the weight of the masses, measured in newtons, N. At GCSE, you often use $9.8\ \text{N/kg}$ or approximately $10\ \text{N/kg}$ if the question says to.

To improve accuracy, use a pointer attached to the spring, keep the ruler close to the spring, and read the scale at eye level to avoid parallax error.

Exam technique

In the exam

Check whether the question gives total length or extension; use the change in length for $e$ .
Convert cm or mm into m before substituting into spring equations.
On graphs, use the straight-line section only to calculate $k$ from the gradient.
Mention the limit of proportionality if the graph stops being a straight line.
For practical questions, include repeat readings, a safety mat, and reading the ruler at eye level.

Self review

Check yourself

What is the difference between elastic and inelastic deformation?
A spring extends by 0.040 m when a force of 2.0 N is applied. What is its spring constant?
How would you use a force-extension graph to decide whether the limit of proportionality has been exceeded?

Forces and elasticity

What you'll learn

How forces can stretch, compress, or bend objects.
The difference between elastic and inelastic deformation.
How to use $F = k e$ for springs and other elastic objects.
How to calculate elastic potential energy and analyse force-extension graphs.

Changing shape: deformation

When forces act on an object, they can change its shape. The object might be:

stretched — pulled longer, like a spring or elastic band.
compressed — squashed shorter, like foam or a spring being pushed.
bent — curved, like a ruler pressed in the middle.

Definition

Deformation

A deformation is a change in the shape of an object caused by forces.

Diagram showing stretching, compression and bending caused by multiple balanced forces

Example

Explaining why two forces are needed

A foam block is stationary while being compressed between two hands. Each hand pushes with 12 N.

The left hand pushes the block to the right with 12 N, and the right hand pushes it to the left with 12 N.
The forces are equal and opposite, so the resultant force on the block is 0 N.
The block does not accelerate, but it is compressed because the two forces act on opposite sides and push its particles closer together.

Elastic and inelastic deformation

Some objects return to their original shape when the forces are removed. Others stay permanently changed.

Definition

Elastic deformation

Elastic deformation happens when an object returns to its original shape after the deforming forces are removed.

Definition

Inelastic deformation

Inelastic deformation happens when an object does not return completely to its original shape after the deforming forces are removed. The object has been permanently deformed.

A spring being gently stretched usually shows elastic deformation. Plasticine being squashed shows inelastic deformation.

Common Mistake

Elastic does not mean stretchy forever

An object can be elastic only over a certain range. If you stretch a spring too far, it may stop returning to its original length and become permanently deformed.

Extension and compression

For springs, we usually measure how much the length changes.

Definition

Extension

The extension, $e$ , is the increase in length of an object when it is stretched:

\text{stretched length} - \text{original length}

Extension is measured in metres, m.

If a spring is compressed instead of stretched, the same ideas apply, but $e$ means the compression — the decrease in length.

Tip

Use the change in length

In calculations, do not use the total length of the spring. Use the extension or compression: final length minus original length.

Hooke’s law: force and extension

For many elastic objects, such as a spring, the extension is directly proportional to the force applied — but only up to a certain point.

Definition

Directly proportional

Two quantities are directly proportional if doubling one doubles the other, and their graph is a straight line through the origin.

For a spring obeying Hooke’s law:

F = k e

where:

$F$ is the force in newtons, N.
$k$ is the spring constant in newtons per metre, N/m.
$e$ is the extension in metres, m.

Definition

Spring constant

The spring constant, $k$ , tells you how stiff a spring is. A larger spring constant means more force is needed for the same extension.

Example

Calculating a spring constant

A spring extends by 6.0 cm when a force of 3.0 N is applied. Calculate its spring constant.

Convert the extension into metres: 6.0 cm is 0.060 m.
Rearrange $F = k e$ to make $k$ the subject:
$\frac{F}{e}$
Substitute the values:
$N/m\begin{aligned} k &= \frac{3.0\ \text{N}}{0.060\ \text{m}} \\ k &= 50\ \text{N/m} \end{aligned}$

Common Mistake

Forgetting centimetres to metres

If the extension is given in centimetres or millimetres, convert it to metres before using $F = k e$ .

The limit of proportionality

The force-extension relationship is linear only up to the limit of proportionality.

Definition

Limit of proportionality

The limit of proportionality is the point beyond which force and extension are no longer directly proportional.

Before this limit, the graph of force against extension is a straight line through the origin. After this limit, the graph curves, so the relationship is non-linear.

Force-extension graph showing linear region, limit of proportionality and stored elastic potential energy

A linear relationship gives a straight-line graph. A non-linear relationship gives a curved graph or a line whose gradient changes.

On a force-extension graph with force on the vertical axis and extension on the horizontal axis:

gradient=Fe=k\text{gradient} = \frac{F}{e} = k

So, in the straight-line region, the gradient gives the spring constant.

Example

Interpreting force-extension data

A spring has these results: 1.0 N gives 0.020 m extension, 2.0 N gives 0.040 m, 3.0 N gives 0.060 m, but 4.0 N gives 0.090 m. Decide where the graph stops being linear.

Compare the first three ratios:
$1.00.020=502.00.040=503.00.060=50\begin{aligned} \frac{1.0}{0.020} &= 50 \\ \frac{2.0}{0.040} &= 50 \\ \frac{3.0}{0.060} &= 50 \end{aligned}$
The spring constant is the same, so this part is linear.
Check the next ratio:
$4.00.090≈44\frac{4.0}{0.090} \approx 44$
The ratio has changed, so the gradient is no longer constant.
The limit of proportionality has been exceeded somewhere between 3.0 N and 4.0 N.

Elastic potential energy

When you stretch or compress a spring, you do work on it. If the spring is not permanently deformed, this work is stored as elastic potential energy.

Definition

Elastic potential energy

Elastic potential energy, $E_e$ , is energy stored in an object when it is stretched or compressed elastically.

Up to the limit of proportionality:

Ee=12ke2E_e = \frac{1}{2} k e^2

where:

$E_e$ is elastic potential energy in joules, J.
$k$ is spring constant in N/m.
$e$ is extension in m.

This equation is also the area under the straight-line part of the force-extension graph.

Example

Calculating elastic potential energy

A spring has a spring constant of 80 N/m and is stretched by 5.0 cm. Calculate the elastic potential energy stored.

Convert the extension into metres: 5.0 cm is 0.050 m.
Substitute into $Ee=12ke2E_e = \frac{1}{2} k e^2$ :
$Ee=12×80×(0.050)2E_e = \frac{1}{2} \times 80 \times (0.050)^2$
Calculate the energy:
$J\begin{aligned} E_e &= 40 \times 0.0025 \\ E_e &= 0.10\ \text{J} \end{aligned}$

Common Mistake

Only while the spring behaves elastically

The equation $Ee=12ke2E_e = \frac{1}{2} k e^2$ is for the linear elastic region, up to the limit of proportionality.

Required practical: force and extension for a spring

You need to be able to describe how to investigate the relationship between force and extension for a spring.

Required practical setup with spring, clamp stand, ruler, masses and safety mat

A good method is:

Set up a spring hanging from a clamp stand beside a vertical ruler. Put a safety mat underneath.
Measure the original length of the spring, $L_0$ , before adding masses.
Add a mass hanger and masses one at a time.
For each mass, wait for the spring to stop moving, then measure the new length, $L$ .
Calculate the extension using $e = L - L_0$ .
Calculate the force using weight: $W = m g$ .
Plot a graph of force against extension and draw a best-fit line for the linear region.
Find the spring constant from the gradient of the straight-line section.

Tip

Mass is not force

To improve accuracy, use a pointer attached to the spring, keep the ruler close to the spring, and read the scale at eye level to avoid parallax error.

Exam technique

In the exam

Check whether the question gives total length or extension; use the change in length for $e$ .
Convert cm or mm into m before substituting into spring equations.
On graphs, use the straight-line section only to calculate $k$ from the gradient.
Mention the limit of proportionality if the graph stops being a straight line.
For practical questions, include repeat readings, a safety mat, and reading the ruler at eye level.

Self review

Check yourself

What is the difference between elastic and inelastic deformation?
A spring extends by 0.040 m when a force of 2.0 N is applied. What is its spring constant?
How would you use a force-extension graph to decide whether the limit of proportionality has been exceeded?

Forces and elasticity

What you'll learn

Changing shape: deformation

Elastic and inelastic deformation

Extension and compression

Hooke’s law: force and extension

The limit of proportionality

Elastic potential energy

Required practical: force and extension for a spring

Recap questions

You've reached the end

Forces and elasticity

What you'll learn

Changing shape: deformation

Elastic and inelastic deformation

Extension and compression

Hooke’s law: force and extension

The limit of proportionality

Elastic potential energy

Required practical: force and extension for a spring

Recap questions

You've reached the end

Forces and elasticity

What you'll learn

Changing shape: deformation

Elastic and inelastic deformation

Extension and compression

Hooke’s law: force and extension

The limit of proportionality

Elastic potential energy

Required practical: force and extension for a spring

Recap questions

You've reached the end

Forces and elasticity Revision Guide

Forces and elasticity

What you'll learn

Changing shape: deformation

Elastic and inelastic deformation

Extension and compression

Hooke’s law: force and extension

The limit of proportionality

Elastic potential energy

Required practical: force and extension for a spring

Recap questions

You've reached the end

Forces and elasticity Revision Guide