Forces and springs

What you'll learn

Why stretching, bending or compressing an object needs more than one force.
The difference between elastic and inelastic distortion.
How to use $\times x$ and $\frac{1}{2} \times k \times x^2$ .
How to investigate springs using a force-extension graph.

Forces can change shape

A force is a push or pull, measured in newtons (N). Forces can change an object’s speed, direction, or shape.

A distortion is a change in shape. For springs and other elastic objects, the main distortions are:

Stretching — the object gets longer.
Compressing — the object gets shorter.
Bending — the object curves or changes shape sideways.

To stretch, compress or bend an object, you need forces acting at different points. If you only apply one force to a free object, it mostly accelerates instead of changing shape. In real situations, the “second force” might come from your other hand, a wall, a clamp, or the ground.

Key Idea

Changing shape needs force pairs

Stretching, bending and compressing require more than one force because different parts of the object must be forced in different directions.

Example

Explaining why a spring stretches

A student hangs a mass from a spring clamped to a stand. Explain why the spring stretches.

The mass pulls down on the bottom of the spring because its weight acts downwards.
The clamp pulls up on the top of the spring, holding it in place.
These forces act at different ends of the spring, so the spring is stretched rather than simply falling down.

Elastic and inelastic distortion

Definition

Elastic distortion

Elastic distortion happens when an object returns to its original shape and length after the forces are removed.

Definition

Inelastic distortion

Inelastic distortion happens when an object does not fully return to its original shape after the forces are removed. The change is permanent.

A spring used carefully usually shows elastic distortion: remove the masses and it goes back to its original length. But if you stretch it too far, it may become permanently longer. That is inelastic distortion.

Example

Deciding whether distortion is elastic

A spring is originally 12.0 cm long. It is stretched to 18.0 cm by a load. When the load is removed, its length is 12.0 cm again. Decide whether the distortion was elastic or inelastic.

Compare the final length after the load is removed with the original length.
The final length is 12.0 cm, which matches the original length of 12.0 cm.
The spring has returned to its original length, so the distortion was elastic.

Extension: measuring how much the spring stretches

The extension, symbol $x$ , is the increase in length compared with the original length.

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

Extension is measured in metres (m) when you use it in equations.

Common Mistake

Extension is not the total length

If a spring goes from 10 cm to 14 cm, the extension is 4 cm, not 14 cm.

The spring equation: force, spring constant and extension

For a spring in its linear elastic region, force and extension are linked by:

\times x

where:

$F$ is the force exerted on the spring, in newtons (N)
$k$ is the spring constant, in newtons per metre (N/m)
$x$ is the extension, in metres (m)

You are expected to recall and use this equation for Edexcel 1PH0.

Definition

Spring constant

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

Rearranged forms are:

\frac{F}{x}

\frac{F}{k}

Example

Calculating the spring constant

A spring extends by 5.0 cm when a force of 3.0 N is applied. Calculate its spring constant.

Convert the extension into metres: 5.0 cm is 0.050 m.
Use the rearranged equation $\frac{F}{x}$ .
Substitute the values:

\frac{3.0}{0.050} = 60 \text{ N/m}

So the spring constant is 60 N/m.

Tip

Quick unit check

If you are calculating $k$ , the unit should be N/m. If your extension is still in cm, your answer will be wrong by a factor of 100.

Linear and non-linear relationships

A linear relationship gives a straight-line graph. For a spring obeying $\times x$ , the force-extension graph is a straight line through the origin.

A non-linear relationship gives a curved graph. This happens after the spring passes the limit of proportionality, which is the point where force and extension stop being directly proportional.

Force-extension graph showing the linear elastic region, limit of proportionality and non-linear region

The gradient of a graph is its steepness: vertical change divided by horizontal change. On a force-extension graph, the gradient in the straight-line section is the spring constant $k$ .

\frac{\text{change in force}}{\text{change in extension}}

Common Mistake

Only use F = kx in the linear region

The equation $\times x$ only works while the spring is showing linear elastic distortion. Past the limit of proportionality, $k$ is no longer constant.

Example

Finding spring constant from a graph

A force-extension graph is straight. One point on the line is force 4.0 N at extension 0.020 m. Find $k$ .

In the straight-line region, use the graph gradient: $\frac{F}{x}$ .
Substitute the point from the graph:

\frac{4.0}{0.020} = 200 \text{ N/m}

A spring constant of 200 N/m means the spring needs 200 N for 1 m of extension, if it stayed in the linear region.

Work done in stretching a spring

When you stretch a spring, you transfer energy to it. This stored energy is called elastic potential energy.

Work done means energy transferred by a force moving through a distance. Work done and energy are both measured in joules (J).

For a spring in the linear region:

\frac{1}{2} \times k \times x^2

where:

$E$ is the energy transferred in stretching, in joules (J)
$k$ is the spring constant, in N/m
$x$ is the extension, in m

This equation is for use in this topic; you must be confident substituting values into it and using metres for extension.

Key Idea

Why there is a half

The stretching force increases from zero to its final value, so the average force during the stretch is half the final force. That is why the energy equation includes $12\frac{1}{2}$ .

Example

Calculating energy stored in a stretched spring

A spring has spring constant 80 N/m and is stretched by 0.15 m. Calculate the energy transferred.

Choose the energy equation because the question asks for energy stored when stretching:

\frac{1}{2} \times k \times x^2

Substitute the values carefully, remembering to square the extension:

\frac{1}{2} \times 80 \times 0.15^2

Calculate:

\times 0.0225 = 0.90 \text{ J}

So the energy stored is 0.90 J.

Common Mistake

Forgetting to square the extension

In $\frac{1}{2}kx^2$ , only the extension is squared. Do not square the spring constant.

Core practical: investigating extension and work done

In the practical, you add different masses to a spring and measure how much it extends. A typical setup uses a clamp stand, spring, mass hanger and metre ruler.

Spring extension practical setup with clamp stand, spring, masses, ruler, pointer and extension labelled

Method

Measure the original length of the spring with no load.
Add a mass to the hanger and let the spring come to rest.
Record the new length and calculate the extension.
Convert the mass to force using weight = mass × gravitational field strength, with mass in kg.
Repeat for several masses, increasing the force in equal steps if possible.
Plot force on the vertical axis against extension on the horizontal axis.
Find $k$ from the gradient of the straight-line section.
Calculate work done using $\frac{1}{2}kx^2$ , or from the area under the force-extension graph.

Variables and accuracy

The independent variable is the force you apply. The dependent variable is the extension you measure. Control variables are things you keep the same, such as the spring used and how the ruler is positioned.

To improve accuracy, use a pointer, read the ruler at eye level, and wait for the spring to stop moving before taking a reading.

Tip

Graph axes

For this practical, put force on the y-axis and extension on the x-axis. Then the gradient is $k$ .

Exam technique

In the exam

Always convert extension into metres before using $F = k x$ or $\frac{1}{2}kx^2$ .
Check whether the graph is straight before assuming the spring constant is constant.
If asked about the practical, describe both the measurements and how you process them: calculate extension, plot a graph, and find gradient or work done.

Self review

Check yourself

Why does stretching a spring require forces at both ends?
A spring goes from 8 cm to 11 cm. What is its extension in metres?
What does a curved force-extension graph tell you about the spring?

Forces and springs

What you'll learn

Why stretching, bending or compressing an object needs more than one force.
The difference between elastic and inelastic distortion.
How to use $\times x$ and $\frac{1}{2} \times k \times x^2$ .
How to investigate springs using a force-extension graph.

Forces can change shape

A force is a push or pull, measured in newtons (N). Forces can change an object’s speed, direction, or shape.

A distortion is a change in shape. For springs and other elastic objects, the main distortions are:

Stretching — the object gets longer.
Compressing — the object gets shorter.
Bending — the object curves or changes shape sideways.

Key Idea

Changing shape needs force pairs

Stretching, bending and compressing require more than one force because different parts of the object must be forced in different directions.

Example

Explaining why a spring stretches

A student hangs a mass from a spring clamped to a stand. Explain why the spring stretches.

The mass pulls down on the bottom of the spring because its weight acts downwards.
The clamp pulls up on the top of the spring, holding it in place.
These forces act at different ends of the spring, so the spring is stretched rather than simply falling down.

Elastic and inelastic distortion

Definition

Elastic distortion

Elastic distortion happens when an object returns to its original shape and length after the forces are removed.

Definition

Inelastic distortion

Inelastic distortion happens when an object does not fully return to its original shape after the forces are removed. The change is permanent.

Example

Deciding whether distortion is elastic

A spring is originally 12.0 cm long. It is stretched to 18.0 cm by a load. When the load is removed, its length is 12.0 cm again. Decide whether the distortion was elastic or inelastic.

Compare the final length after the load is removed with the original length.
The final length is 12.0 cm, which matches the original length of 12.0 cm.
The spring has returned to its original length, so the distortion was elastic.

Extension: measuring how much the spring stretches

The extension, symbol $x$ , is the increase in length compared with the original length.

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

Extension is measured in metres (m) when you use it in equations.

Common Mistake

Extension is not the total length

If a spring goes from 10 cm to 14 cm, the extension is 4 cm, not 14 cm.

The spring equation: force, spring constant and extension

For a spring in its linear elastic region, force and extension are linked by:

\times x

where:

$F$ is the force exerted on the spring, in newtons (N)
$k$ is the spring constant, in newtons per metre (N/m)
$x$ is the extension, in metres (m)

You are expected to recall and use this equation for Edexcel 1PH0.

Definition

Spring constant

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

Rearranged forms are:

\frac{F}{x}

\frac{F}{k}

Example

Calculating the spring constant

A spring extends by 5.0 cm when a force of 3.0 N is applied. Calculate its spring constant.

Convert the extension into metres: 5.0 cm is 0.050 m.
Use the rearranged equation $\frac{F}{x}$ .
Substitute the values:

\frac{3.0}{0.050} = 60 \text{ N/m}

So the spring constant is 60 N/m.

Tip

Quick unit check

If you are calculating $k$ , the unit should be N/m. If your extension is still in cm, your answer will be wrong by a factor of 100.

Linear and non-linear relationships

A linear relationship gives a straight-line graph. For a spring obeying $\times x$ , the force-extension graph is a straight line through the origin.

Force-extension graph showing the linear elastic region, limit of proportionality and non-linear region

The gradient of a graph is its steepness: vertical change divided by horizontal change. On a force-extension graph, the gradient in the straight-line section is the spring constant $k$ .

\frac{\text{change in force}}{\text{change in extension}}

Common Mistake

Only use F = kx in the linear region

The equation $\times x$ only works while the spring is showing linear elastic distortion. Past the limit of proportionality, $k$ is no longer constant.

Example

Finding spring constant from a graph

A force-extension graph is straight. One point on the line is force 4.0 N at extension 0.020 m. Find $k$ .

In the straight-line region, use the graph gradient: $\frac{F}{x}$ .
Substitute the point from the graph:

\frac{4.0}{0.020} = 200 \text{ N/m}

A spring constant of 200 N/m means the spring needs 200 N for 1 m of extension, if it stayed in the linear region.

Work done in stretching a spring

When you stretch a spring, you transfer energy to it. This stored energy is called elastic potential energy.

Work done means energy transferred by a force moving through a distance. Work done and energy are both measured in joules (J).

For a spring in the linear region:

\frac{1}{2} \times k \times x^2

where:

$E$ is the energy transferred in stretching, in joules (J)
$k$ is the spring constant, in N/m
$x$ is the extension, in m

This equation is for use in this topic; you must be confident substituting values into it and using metres for extension.

Key Idea

Why there is a half

The stretching force increases from zero to its final value, so the average force during the stretch is half the final force. That is why the energy equation includes $12\frac{1}{2}$ .

Example

Calculating energy stored in a stretched spring

A spring has spring constant 80 N/m and is stretched by 0.15 m. Calculate the energy transferred.

Choose the energy equation because the question asks for energy stored when stretching:

\frac{1}{2} \times k \times x^2

Substitute the values carefully, remembering to square the extension:

\frac{1}{2} \times 80 \times 0.15^2

Calculate:

\times 0.0225 = 0.90 \text{ J}

So the energy stored is 0.90 J.

Common Mistake

Forgetting to square the extension

In $\frac{1}{2}kx^2$ , only the extension is squared. Do not square the spring constant.

Core practical: investigating extension and work done

In the practical, you add different masses to a spring and measure how much it extends. A typical setup uses a clamp stand, spring, mass hanger and metre ruler.

Spring extension practical setup with clamp stand, spring, masses, ruler, pointer and extension labelled

Method

Measure the original length of the spring with no load.
Add a mass to the hanger and let the spring come to rest.
Record the new length and calculate the extension.
Convert the mass to force using weight = mass × gravitational field strength, with mass in kg.
Repeat for several masses, increasing the force in equal steps if possible.
Plot force on the vertical axis against extension on the horizontal axis.
Find $k$ from the gradient of the straight-line section.
Calculate work done using $\frac{1}{2}kx^2$ , or from the area under the force-extension graph.

Variables and accuracy

To improve accuracy, use a pointer, read the ruler at eye level, and wait for the spring to stop moving before taking a reading.

Tip

Graph axes

For this practical, put force on the y-axis and extension on the x-axis. Then the gradient is $k$ .

Exam technique

In the exam

Always convert extension into metres before using $F = k x$ or $\frac{1}{2}kx^2$ .
Check whether the graph is straight before assuming the spring constant is constant.
If asked about the practical, describe both the measurements and how you process them: calculate extension, plot a graph, and find gradient or work done.

Self review

Check yourself

Why does stretching a spring require forces at both ends?
A spring goes from 8 cm to 11 cm. What is its extension in metres?
What does a curved force-extension graph tell you about the spring?

Forces and springs

What you'll learn

Forces can change shape

Elastic and inelastic distortion

Extension: measuring how much the spring stretches

The spring equation: force, spring constant and extension

Linear and non-linear relationships

Work done in stretching a spring

Core practical: investigating extension and work done

Method

Variables and accuracy

Recap questions

Forces and springs

What you'll learn

Forces can change shape

Elastic and inelastic distortion

Extension: measuring how much the spring stretches

The spring equation: force, spring constant and extension

Linear and non-linear relationships

Work done in stretching a spring

Core practical: investigating extension and work done

Method

Variables and accuracy

Recap questions

Forces and springs

What you'll learn

Forces can change shape

Elastic and inelastic distortion

Extension: measuring how much the spring stretches

The spring equation: force, spring constant and extension

Linear and non-linear relationships

Work done in stretching a spring

Core practical: investigating extension and work done

Method

Variables and accuracy

Recap questions

Forces and springs Revision Guide

Forces and springs

What you'll learn

Forces can change shape

Elastic and inelastic distortion

Extension: measuring how much the spring stretches

The spring equation: force, spring constant and extension

Linear and non-linear relationships

Work done in stretching a spring

Core practical: investigating extension and work done

Method

Variables and accuracy

Recap questions

Forces and springs Revision Guide