Forces in action

What you'll learn

Why stretching, bending or compressing an object needs forces acting in different places.
How springs behave, including elastic deformation, plastic deformation and Hooke’s law.
How to calculate spring constant and work done in stretching.
How gravitational fields cause weight, free fall and gravitational potential energy.

Forces can change shape

A force is a push or pull on an object. Forces can change an object’s motion, but they can also change its shape.

To stretch, bend or compress an object, you need more than one force acting on it. A single unbalanced force mainly makes the whole object accelerate. Deformation happens when different parts of the object are pushed or pulled in different directions.

Definition

Deformation

A deformation is a change in shape. Stretching makes an object longer, compression makes it shorter or squashed, and bending changes its shape without simply making it longer or shorter.

For example, an elastic band stretches because your two hands pull opposite ends. A sponge is compressed because your hand pushes down while the table pushes up. A ruler bends because one part is held while another part is pushed.

A spring being stretched by opposite forces, showing original length, stretched length and extension

Example

Explaining why a spring stretches

A mass is hung from a spring fixed to a clamp stand. Explain why the spring is deformed.

The load pulls downward on the lower end of the spring, while the clamp pulls upward on the upper end.
These forces act in opposite directions on different parts of the spring, so the coils are pulled apart and the spring stretches.
If the clamp were removed, the spring and mass would accelerate downward instead of being held in a stretched position.

Elastic and plastic deformation

When a force is removed, objects do not all behave in the same way.

Definition

Elastic and plastic deformation

Elastic deformation means the object returns to its original shape when the force is removed.
Plastic deformation means the object does not return fully to its original shape, so the deformation is permanent.

Springs, rubber bands and bungee cords are designed to deform elastically over a useful range. Plasticine, bent paperclips and car crumple zones show plastic deformation: the shape change remains after the force is removed.

The elastic limit is the point beyond which an object will not return to its original shape. In real life, useful safety designs depend on this. A car crumple zone is meant to deform plastically to transfer energy away from the passengers, while a mattress spring should deform elastically so it returns to shape.

Extension: the key measurement for springs

For springs, you usually measure extension, not just length.

Definition

Extension

The extension, $x$ , is the increase in length of an object:

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

Extension is measured in metres (m).

Common Mistake

Using length instead of extension

If a spring is originally 12 cm long and becomes 17 cm long, the extension is 5 cm, not 17 cm. Always subtract the original length first, then convert to metres if needed.

Hooke’s law

For many springs, as long as they are not stretched too far, the force is directly proportional to the extension. This is Hooke’s law.

Key Idea

Hooke’s law

For a spring in its linear region:

F = k x

You need to recall and apply this equation. $F$ is force in newtons (N), $k$ is spring constant in newtons per metre (N/m), and $x$ is extension in metres (m).

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

Example

Calculating spring constant

A spring is 10.0 cm long at first. A 2.5 N force stretches it to 15.0 cm. Calculate the spring constant.

Calculate the extension: 15.0 cm - 10.0 cm = 5.0 cm, which is 0.050 m.
Rearrange Hooke’s law: from $F = k x$ , so $\frac{F}{x}$ .
Substitute and calculate:

\frac{2.5\ \text{N}}{0.050\ \text{m}} = 50\ \text{N/m}

Tip

Unit check

Spring constant is in N/m, so extension must be in metres. Convert centimetres to metres before substituting into $F = k x$ .

Force-extension graphs

A force-extension graph shows how the force changes as the extension changes.

A linear relationship gives a straight line. If the straight line goes through the origin, force is directly proportional to extension. A non-linear relationship gives a curve, so the gradient changes as the object stretches.

Force-extension graph showing Hooke's law region, limit of proportionality and work done as area under the graph

For a spring, the first straight-line part is the Hooke’s law region. The limit of proportionality is where the graph stops being a straight line. After this point, $F = k x$ no longer works because the spring constant is not constant.

Elastic bands often have non-linear force-extension graphs. Their loading and unloading curves may be different, meaning some energy is transferred to the surroundings, usually as thermal energy.

Tip

Gradient of the graph

If force is on the vertical axis and extension is on the horizontal axis, the gradient of the straight-line section is the spring constant, $k$ . If the axes are swapped, the gradient is not $k$ .

In the Hooke’s law practical, you add masses, convert their mass to weight, measure the spring’s length, calculate extension, and plot a graph. Adding the masses gradually helps you avoid passing the elastic limit too soon.

Work done in stretching

When you stretch a spring or elastic band, a force moves through a distance. This means work is done, so energy is transferred. If the deformation is elastic, much of this energy is stored as elastic potential energy.

For any force-extension graph:

Key Idea

Work from a force-extension graph

The work done in stretching is the area under the force-extension graph.

For a spring obeying Hooke’s law, the graph is a straight line through the origin, so the area is a triangle. This gives:

\frac{1}{2}kx^2

OCR lists this as an equation to apply, so it is supplied on the equation sheet, but you must know how to use it correctly. $E$ is energy transferred in joules (J), $k$ is spring constant in N/m, and $x$ is extension in m.

Common Mistake

When the spring formula works

The equation $\frac{1}{2}kx^2$ is for a linear spring starting from zero extension. For a non-linear graph, use the area under the graph instead.

Example

Calculating energy stored in a stretched spring

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

The extension is already in metres, and the spring is being treated as a linear spring, so use $\frac{1}{2}kx^2$ .
Substitute the values:

\frac{1}{2} \times 80\ \text{N/m} \times (0.15\ \text{m})^2

Calculate the squared extension and final energy:

0.90\ \text{J}

Common Mistake

Forgetting to square the extension

In $\frac{1}{2}kx^2$ , only the extension is squared. Doubling the extension makes the stored energy four times larger, as long as the spring stays in the linear region.

Gravitational fields

A field is a region where an object can experience a non-contact force. A gravitational field is the region around a mass where another mass experiences gravitational attraction.

All matter has a gravitational field, but the field is much stronger for very massive objects such as planets and stars. Near Earth, gravity pulls objects towards the centre of Earth.

Earth's gravitational field, showing field lines towards the centre and weight W = mg

Definition

Gravitational field strength

Gravitational field strength, $g$ , is the force per kilogram on a mass in a gravitational field. Near Earth’s surface, $10\ \text{N/kg}$ .

Mass and weight

Mass is the amount of matter in an object. It is measured in kilograms (kg) and does not change when you move to another planet.

Weight is the gravitational force acting on an object. It is measured in newtons (N) using a newtonmeter.

Key Idea

Weight equation

Weight is calculated using:

W = m g

You need to recall and apply this equation. $W$ is weight in N, $m$ is mass in kg, and $g$ is gravitational field strength in N/kg.

Common Mistake

Mass is not weight

In everyday language people say “weighing” when they often mean measuring mass. In physics, mass is in kg and weight is in N. Your mass would be the same on the Moon, but your weight would be smaller because the Moon’s gravitational field strength is smaller.

Example

Calculating weight on another planet

A student has a mass of 60 kg. Calculate their weight on Earth, where $10\ \text{N/kg}$ , and on Mars, where $3.7\ \text{N/kg}$ .

On Earth, use $W = m g$ :

60\ \text{kg} \times 10\ \text{N/kg} = 600\ \text{N}

On Mars, use the different gravitational field strength:

60\ \text{kg} \times 3.7\ \text{N/kg} = 222\ \text{N}

The mass stays 60 kg in both places, but the weight changes because $g$ changes.

Free fall

An object is in free fall if gravity is the only force acting on it. Near Earth’s surface, the acceleration in free fall is 10 m/s² downwards.

This value is closely linked to gravitational field strength: 10 N/kg means each kilogram experiences 10 N of gravitational force, and 10 m/s² means the object’s velocity changes by 10 metres per second every second when only gravity acts.

If air resistance is significant, the object is not in pure free fall because there is another force acting.

Gravitational potential energy

When you lift an object, you do work against gravity. Energy is transferred to the object’s gravitational potential energy store.

Key Idea

Gravitational potential energy equation

Gravitational potential energy is calculated using:

E_p = mgh

You need to recall and apply this equation. $E_p$ is energy in J, $m$ is mass in kg, $g$ is gravitational field strength in N/kg, and $h$ is vertical height in m.

Only the vertical height change matters. Lifting a bag 1 m straight up and carrying it up a ramp to end 1 m higher both increase its gravitational potential energy by the same amount, ignoring friction and other losses.

Example

Calculating gravitational potential energy

A 2.5 kg box is lifted onto a shelf 1.2 m above the floor. Calculate the increase in gravitational potential energy. Use $10\ \text{N/kg}$ .

Identify the vertical height change as 1.2 m, because gravitational potential energy depends on height gained.
Substitute into $E_p = mgh$ :

mE_p = 2.5\ \text{kg} \times 10\ \text{N/kg} \times 1.2\ \text{m}

Calculate the energy transferred:

JE_p = 30\ \text{J}

Exam technique

In the exam

For spring questions, use extension, not total length, and convert cm or mm into m before using equations.
For graphs, decide whether the relationship is linear or non-linear; use gradient for spring constant only when force is on the vertical axis and extension is on the horizontal axis.
For gravity questions, keep mass and weight separate: kg for mass, N for weight, and use $10\ \text{N/kg}$ near Earth unless another value is given.

Self review

Check yourself

A spring changes length from 8.0 cm to 11.5 cm when a 1.4 N force is applied. What is its spring constant?
Why does compressing a sponge on a table involve two forces?
How are mass, weight and gravitational field strength linked?

What you'll learn

Why stretching, bending or compressing an object needs forces acting in different places.
How springs behave, including elastic deformation, plastic deformation and Hooke’s law.
How to calculate spring constant and work done in stretching.
How gravitational fields cause weight, free fall and gravitational potential energy.

Forces can change shape

A force is a push or pull on an object. Forces can change an object’s motion, but they can also change its shape.

Definition

Deformation

A deformation is a change in shape. Stretching makes an object longer, compression makes it shorter or squashed, and bending changes its shape without simply making it longer or shorter.

A spring being stretched by opposite forces, showing original length, stretched length and extension

Example

Explaining why a spring stretches

A mass is hung from a spring fixed to a clamp stand. Explain why the spring is deformed.

The load pulls downward on the lower end of the spring, while the clamp pulls upward on the upper end.
These forces act in opposite directions on different parts of the spring, so the coils are pulled apart and the spring stretches.
If the clamp were removed, the spring and mass would accelerate downward instead of being held in a stretched position.

Elastic and plastic deformation

When a force is removed, objects do not all behave in the same way.

Definition

Elastic and plastic deformation

Elastic deformation means the object returns to its original shape when the force is removed.
Plastic deformation means the object does not return fully to its original shape, so the deformation is permanent.

Extension: the key measurement for springs

For springs, you usually measure extension, not just length.

Definition

Extension

The extension, $x$ , is the increase in length of an object:

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

Extension is measured in metres (m).

Common Mistake

Using length instead of extension

If a spring is originally 12 cm long and becomes 17 cm long, the extension is 5 cm, not 17 cm. Always subtract the original length first, then convert to metres if needed.

Hooke’s law

For many springs, as long as they are not stretched too far, the force is directly proportional to the extension. This is Hooke’s law.

Key Idea

Hooke’s law

For a spring in its linear region:

F = k x

You need to recall and apply this equation. $F$ is force in newtons (N), $k$ is spring constant in newtons per metre (N/m), and $x$ is extension in metres (m).

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

Example

Calculating spring constant

A spring is 10.0 cm long at first. A 2.5 N force stretches it to 15.0 cm. Calculate the spring constant.

Calculate the extension: 15.0 cm - 10.0 cm = 5.0 cm, which is 0.050 m.
Rearrange Hooke’s law: from $F = k x$ , so $\frac{F}{x}$ .
Substitute and calculate:

\frac{2.5\ \text{N}}{0.050\ \text{m}} = 50\ \text{N/m}

Tip

Unit check

Spring constant is in N/m, so extension must be in metres. Convert centimetres to metres before substituting into $F = k x$ .

Force-extension graphs

A force-extension graph shows how the force changes as the extension changes.

Force-extension graph showing Hooke's law region, limit of proportionality and work done as area under the graph

Elastic bands often have non-linear force-extension graphs. Their loading and unloading curves may be different, meaning some energy is transferred to the surroundings, usually as thermal energy.

Tip

Gradient of the graph

If force is on the vertical axis and extension is on the horizontal axis, the gradient of the straight-line section is the spring constant, $k$ . If the axes are swapped, the gradient is not $k$ .

Work done in stretching

For any force-extension graph:

Key Idea

Work from a force-extension graph

The work done in stretching is the area under the force-extension graph.

For a spring obeying Hooke’s law, the graph is a straight line through the origin, so the area is a triangle. This gives:

\frac{1}{2}kx^2

Common Mistake

When the spring formula works

The equation $\frac{1}{2}kx^2$ is for a linear spring starting from zero extension. For a non-linear graph, use the area under the graph instead.

Example

Calculating energy stored in a stretched spring

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

The extension is already in metres, and the spring is being treated as a linear spring, so use $\frac{1}{2}kx^2$ .
Substitute the values:

\frac{1}{2} \times 80\ \text{N/m} \times (0.15\ \text{m})^2

Calculate the squared extension and final energy:

0.90\ \text{J}

Common Mistake

Forgetting to square the extension

In $\frac{1}{2}kx^2$ , only the extension is squared. Doubling the extension makes the stored energy four times larger, as long as the spring stays in the linear region.

Gravitational fields

A field is a region where an object can experience a non-contact force. A gravitational field is the region around a mass where another mass experiences gravitational attraction.

All matter has a gravitational field, but the field is much stronger for very massive objects such as planets and stars. Near Earth, gravity pulls objects towards the centre of Earth.

Earth's gravitational field, showing field lines towards the centre and weight W = mg

Definition

Gravitational field strength

Gravitational field strength, $g$ , is the force per kilogram on a mass in a gravitational field. Near Earth’s surface, $10\ \text{N/kg}$ .

Mass and weight

Mass is the amount of matter in an object. It is measured in kilograms (kg) and does not change when you move to another planet.

Weight is the gravitational force acting on an object. It is measured in newtons (N) using a newtonmeter.

Key Idea

Weight equation

Weight is calculated using:

W = m g

You need to recall and apply this equation. $W$ is weight in N, $m$ is mass in kg, and $g$ is gravitational field strength in N/kg.

Common Mistake

Mass is not weight

Example

Calculating weight on another planet

A student has a mass of 60 kg. Calculate their weight on Earth, where $10\ \text{N/kg}$ , and on Mars, where $3.7\ \text{N/kg}$ .

On Earth, use $W = m g$ :

60\ \text{kg} \times 10\ \text{N/kg} = 600\ \text{N}

On Mars, use the different gravitational field strength:

60\ \text{kg} \times 3.7\ \text{N/kg} = 222\ \text{N}

The mass stays 60 kg in both places, but the weight changes because $g$ changes.

Free fall

An object is in free fall if gravity is the only force acting on it. Near Earth’s surface, the acceleration in free fall is 10 m/s² downwards.

If air resistance is significant, the object is not in pure free fall because there is another force acting.

Gravitational potential energy

When you lift an object, you do work against gravity. Energy is transferred to the object’s gravitational potential energy store.

Key Idea

Gravitational potential energy equation

Gravitational potential energy is calculated using:

E_p = mgh

You need to recall and apply this equation. $E_p$ is energy in J, $m$ is mass in kg, $g$ is gravitational field strength in N/kg, and $h$ is vertical height in m.

Example

Calculating gravitational potential energy

A 2.5 kg box is lifted onto a shelf 1.2 m above the floor. Calculate the increase in gravitational potential energy. Use $10\ \text{N/kg}$ .

Identify the vertical height change as 1.2 m, because gravitational potential energy depends on height gained.
Substitute into $E_p = mgh$ :

mE_p = 2.5\ \text{kg} \times 10\ \text{N/kg} \times 1.2\ \text{m}

Calculate the energy transferred:

JE_p = 30\ \text{J}

Exam technique

In the exam

For spring questions, use extension, not total length, and convert cm or mm into m before using equations.
For graphs, decide whether the relationship is linear or non-linear; use gradient for spring constant only when force is on the vertical axis and extension is on the horizontal axis.
For gravity questions, keep mass and weight separate: kg for mass, N for weight, and use $10\ \text{N/kg}$ near Earth unless another value is given.

Self review

Check yourself

A spring changes length from 8.0 cm to 11.5 cm when a 1.4 N force is applied. What is its spring constant?
Why does compressing a sponge on a table involve two forces?
How are mass, weight and gravitational field strength linked?

What you'll learn

Forces can change shape

Elastic and plastic deformation

Extension: the key measurement for springs

Hooke’s law

Force-extension graphs

Work done in stretching

Gravitational fields

Mass and weight

Free fall

Gravitational potential energy

Review Forces in action by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Forces in action

What you'll learn

Forces can change shape

Elastic and plastic deformation

Extension: the key measurement for springs

Hooke’s law

Force-extension graphs

Work done in stretching

Gravitational fields

Mass and weight

Free fall

Gravitational potential energy

Review Forces in action by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Forces in action

What you'll learn

Forces can change shape

Elastic and plastic deformation

Extension: the key measurement for springs

Hooke’s law

Force-extension graphs

Work done in stretching

Gravitational fields

Mass and weight

Free fall

Gravitational potential energy

Review Forces in action by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Forces in action Revision Guide

Forces in action

What you'll learn

Forces can change shape

Elastic and plastic deformation

Extension: the key measurement for springs

Hooke’s law

Force-extension graphs

Work done in stretching

Gravitational fields

Mass and weight

Free fall

Gravitational potential energy

Review Forces in action by teaching Genie

Recap your knowledge with an interactive lesson

Remember key concepts with flashcards

Forces in action Revision Guide