Moments, levers and gears (physics only)

What you'll learn

How forces can make objects rotate about a pivot.
How to calculate a moment using $M = F d$ .
How to use the balancing rule for clockwise and anticlockwise moments.
How simple levers and gears transmit turning effects.

This is part of the separate GCSE Physics content, so it is worth learning carefully if you study GCSE Physics.

Forces can cause rotation

You already know that a force is a push or pull. A force can change the speed, shape, or direction of motion of an object.

Sometimes, instead of just moving an object in a straight line, a force makes it rotate. For example:

pushing a door open about its hinge
using a spanner to turn a nut
pressing down on one side of a seesaw
turning bicycle pedals

Pivot

A pivot is the point about which an object turns. A door turns about its hinges, and a seesaw turns about its central support.

Definition

Pivot

A pivot is the fixed point or axis about which an object rotates.

If you push directly at the pivot, the object usually will not turn much. The further away from the pivot you apply the force, the larger the turning effect.

Moment of a force

The moment of a force is the turning effect of that force about a pivot.

Definition

Moment of a force

The moment of a force is its turning effect about a pivot. A larger moment means a stronger tendency to rotate.

A moment can be clockwise or anticlockwise, depending on which way the force tends to turn the object.

For example, pushing down on the right side of a seesaw gives a clockwise moment about the pivot. Pushing down on the left side gives an anticlockwise moment.

Distance from the pivot

The distance in the moments equation is not just any distance. It must be the perpendicular distance from the pivot to the line of action of the force.

The line of action is an imaginary straight line in the direction the force acts.

Labelled diagram showing a force acting on a beam about a pivot, with the perpendicular distance to the force's line of action and clockwise and anticlockwise moments

Definition

Perpendicular distance

The perpendicular distance is the shortest distance from the pivot to the line of action of the force, measured at right angles to that line.

Common Mistake

Using the wrong distance

Do not automatically use the length of the object. In the equation, $d$ is the perpendicular distance from the pivot to the force’s line of action.

Calculating moments

The size of the moment is found using:

M = F d

where:

$M$ is the moment of the force, measured in newton-metres, Nm
$F$ is the force, measured in newtons, N
$d$ is the perpendicular distance from the pivot to the line of action of the force, measured in metres, m

Key Idea

Bigger force or bigger distance means bigger moment

For the same pivot, increasing the force or applying the force further from the pivot increases the moment.

Example

Calculating a moment

A 15 N force is applied at right angles to a door, 0.80 m from the hinge. Calculate the moment about the hinge.

The hinge is the pivot, and the force acts at right angles to the door, so the perpendicular distance is $md=0.80\ \text{m}$ .
Use $M = F d$ and substitute the values:
$15\ \text{N} \times 0.80\ \text{m}$
Calculate the moment:
$12\ \text{Nm}$

So the moment about the hinge is 12 Nm.

Tip

Quick sense check

If either the force or the distance doubles, the moment doubles. This is a useful way to spot answers that are much too big or too small.

Balanced objects and moments

An object is balanced if it has no overall turning effect about the pivot. This means it will not start rotating.

For a balanced object:

moment\text{total clockwise moment} = \text{total anticlockwise moment}

This is very common in questions about seesaws, beams, rulers, and levers.

Key Idea

The balancing rule

If an object is balanced, the total clockwise moment about the pivot equals the total anticlockwise moment about the pivot.

When more than one force acts, calculate each moment separately. Then add the clockwise moments together and add the anticlockwise moments together.

Rearranging the moments equation

You may need to find a missing force or distance.

From:

M = F d

you can rearrange to:

\frac{M}{d}

or:

\frac{M}{F}

Example

Finding a missing force on a balanced beam

A beam is balanced on a pivot. A 6 N force acts downward 0.40 m to the left of the pivot. A second force acts downward 0.30 m to the right of the pivot. Calculate the size of the second force.

The left-hand force produces one moment:
$6\ \text{N} \times 0.40\ \text{m} = 2.4\ \text{Nm}$
The beam is balanced, so the moment on the right must also be 2.4 Nm.
Use $F=MdF=\frac{M}{d}$ for the right-hand force:
$\frac{2.4\ \text{Nm}}{0.30\ \text{m}} = 8\ \text{N}$

So the second force is 8 N.

Common Mistake

Forgetting to compare directions

Moments only balance if clockwise moments equal anticlockwise moments. Do not just add all the forces or all the distances together.

Levers

A lever is a rigid object that turns about a pivot. Levers are used to transmit the rotational effect of a force.

Examples include:

a crowbar
scissors
a bottle opener
a spanner
a seesaw

A lever can make a job easier by allowing a smaller force to produce a larger moment. This usually happens when the force is applied far from the pivot.

Definition

Lever

A lever is a rigid object that rotates about a pivot to transmit the turning effect of a force.

A long-handled spanner is easier to use than a short-handled spanner because the force is applied further from the pivot. For the same force, the moment is larger.

Example

Using a lever to increase force

A crowbar is used to lift a heavy object. A person pushes down with 40 N at a distance of 0.75 m from the pivot. The heavy object is 0.15 m from the pivot. Estimate the upward force on the heavy object, assuming the lever is balanced.

Calculate the input moment from the person’s push:
$40\ \text{N} \times 0.75\ \text{m} = 30\ \text{Nm}$
For balance, the moment lifting the heavy object must also be 30 Nm.
Use $F=MdF=\frac{M}{d}$ for the force on the heavy object:
$\frac{30\ \text{Nm}}{0.15\ \text{m}} = 200\ \text{N}$

So the lever can produce an upward force of about 200 N on the object.

Tip

Levers do not create energy

A lever can increase the force, but the end where you apply the force usually moves a greater distance. The machine makes the force more useful, not magical.

Gears

A gear is a wheel with teeth around its edge. Gears work by meshing their teeth together, so one rotating gear forces another gear to rotate.

Definition

Gear

A gear is a toothed wheel that transmits rotation and turning effects to another gear.

The gear you turn or drive is called the input gear. The gear being driven is called the output gear.

When two gears mesh:

they turn in opposite directions
the teeth transmit forces between the gears
those forces create moments about the gears’ axles
different-sized gears can change the speed and turning effect of the rotation

You may also see the word torque. At GCSE, you can think of torque as the turning effect of a force, very similar to moment.

Labelled diagram of two meshed gears showing opposite rotation directions, with the larger gear turning slower with larger torque and the smaller gear turning faster with smaller torque

Bigger and smaller gears

If a large gear drives a smaller gear, the smaller gear turns faster. However, the smaller gear has a smaller turning effect.

If a small gear drives a larger gear, the larger gear turns more slowly. However, the larger gear has a greater turning effect.

This is why low gears on a bike help you climb hills: they give a larger turning effect at the wheel, even though the wheel turns more slowly for each pedal rotation.

Example

Explaining a simple gear system

A large input gear turns clockwise and drives a smaller output gear. Explain what happens to the output gear.

Because the gears are meshed, the output gear must rotate in the opposite direction, so it turns anticlockwise.
The output gear is smaller, so it turns faster than the larger input gear.
The faster output rotation comes with a smaller turning effect, so this gear arrangement increases speed but reduces the moment available at the output.

Common Mistake

Same direction gears

Two gears touching directly do not turn in the same direction. Meshed gears always rotate in opposite directions.

Linking it all together

Moments, levers, and gears are all about turning effects.

A force can cause rotation if it acts at a distance from a pivot. The moment depends on both the size of the force and the perpendicular distance from the pivot. Levers use this idea with a rigid bar, and gears use forces between teeth to pass rotation from one axle to another.

Exam technique

In the exam

Identify the pivot first, then decide which forces produce clockwise and anticlockwise moments.
Use the perpendicular distance from the pivot to the force’s line of action, not just the length of the object.
For balanced objects, set total clockwise moment equal to total anticlockwise moment, then rearrange $M = F d$ carefully.

Self review

Check yourself

Why does pushing a door near the handle produce a larger moment than pushing near the hinge?
A force of 20 N acts 0.50 m from a pivot. What is the moment?
If a large gear drives a smaller gear, what happens to the direction, speed, and turning effect of the smaller gear?

Moments, levers and gears (physics only)

What you'll learn

How forces can make objects rotate about a pivot.
How to calculate a moment using $M = F d$ .
How to use the balancing rule for clockwise and anticlockwise moments.
How simple levers and gears transmit turning effects.

This is part of the separate GCSE Physics content, so it is worth learning carefully if you study GCSE Physics.

Forces can cause rotation

You already know that a force is a push or pull. A force can change the speed, shape, or direction of motion of an object.

Sometimes, instead of just moving an object in a straight line, a force makes it rotate. For example:

pushing a door open about its hinge
using a spanner to turn a nut
pressing down on one side of a seesaw
turning bicycle pedals

Pivot

A pivot is the point about which an object turns. A door turns about its hinges, and a seesaw turns about its central support.

Definition

Pivot

A pivot is the fixed point or axis about which an object rotates.

If you push directly at the pivot, the object usually will not turn much. The further away from the pivot you apply the force, the larger the turning effect.

Moment of a force

The moment of a force is the turning effect of that force about a pivot.

Definition

Moment of a force

The moment of a force is its turning effect about a pivot. A larger moment means a stronger tendency to rotate.

A moment can be clockwise or anticlockwise, depending on which way the force tends to turn the object.

For example, pushing down on the right side of a seesaw gives a clockwise moment about the pivot. Pushing down on the left side gives an anticlockwise moment.

Distance from the pivot

The distance in the moments equation is not just any distance. It must be the perpendicular distance from the pivot to the line of action of the force.

The line of action is an imaginary straight line in the direction the force acts.

Labelled diagram showing a force acting on a beam about a pivot, with the perpendicular distance to the force's line of action and clockwise and anticlockwise moments

Definition

Perpendicular distance

The perpendicular distance is the shortest distance from the pivot to the line of action of the force, measured at right angles to that line.

Common Mistake

Using the wrong distance

Do not automatically use the length of the object. In the equation, $d$ is the perpendicular distance from the pivot to the force’s line of action.

Calculating moments

The size of the moment is found using:

M = F d

where:

$M$ is the moment of the force, measured in newton-metres, Nm
$F$ is the force, measured in newtons, N
$d$ is the perpendicular distance from the pivot to the line of action of the force, measured in metres, m

Key Idea

Bigger force or bigger distance means bigger moment

For the same pivot, increasing the force or applying the force further from the pivot increases the moment.

Example

Calculating a moment

A 15 N force is applied at right angles to a door, 0.80 m from the hinge. Calculate the moment about the hinge.

The hinge is the pivot, and the force acts at right angles to the door, so the perpendicular distance is $md=0.80\ \text{m}$ .
Use $M = F d$ and substitute the values:
$15\ \text{N} \times 0.80\ \text{m}$
Calculate the moment:
$12\ \text{Nm}$

So the moment about the hinge is 12 Nm.

Tip

Quick sense check

If either the force or the distance doubles, the moment doubles. This is a useful way to spot answers that are much too big or too small.

Balanced objects and moments

An object is balanced if it has no overall turning effect about the pivot. This means it will not start rotating.

For a balanced object:

moment\text{total clockwise moment} = \text{total anticlockwise moment}

This is very common in questions about seesaws, beams, rulers, and levers.

Key Idea

The balancing rule

If an object is balanced, the total clockwise moment about the pivot equals the total anticlockwise moment about the pivot.

When more than one force acts, calculate each moment separately. Then add the clockwise moments together and add the anticlockwise moments together.

Rearranging the moments equation

You may need to find a missing force or distance.

From:

M = F d

you can rearrange to:

\frac{M}{d}

or:

\frac{M}{F}

Example

Finding a missing force on a balanced beam

A beam is balanced on a pivot. A 6 N force acts downward 0.40 m to the left of the pivot. A second force acts downward 0.30 m to the right of the pivot. Calculate the size of the second force.

The left-hand force produces one moment:
$6\ \text{N} \times 0.40\ \text{m} = 2.4\ \text{Nm}$
The beam is balanced, so the moment on the right must also be 2.4 Nm.
Use $F=MdF=\frac{M}{d}$ for the right-hand force:
$\frac{2.4\ \text{Nm}}{0.30\ \text{m}} = 8\ \text{N}$

So the second force is 8 N.

Common Mistake

Forgetting to compare directions

Moments only balance if clockwise moments equal anticlockwise moments. Do not just add all the forces or all the distances together.

Levers

A lever is a rigid object that turns about a pivot. Levers are used to transmit the rotational effect of a force.

Examples include:

a crowbar
scissors
a bottle opener
a spanner
a seesaw

A lever can make a job easier by allowing a smaller force to produce a larger moment. This usually happens when the force is applied far from the pivot.

Definition

Lever

A lever is a rigid object that rotates about a pivot to transmit the turning effect of a force.

A long-handled spanner is easier to use than a short-handled spanner because the force is applied further from the pivot. For the same force, the moment is larger.

Example

Using a lever to increase force

Calculate the input moment from the person’s push:
$40\ \text{N} \times 0.75\ \text{m} = 30\ \text{Nm}$
For balance, the moment lifting the heavy object must also be 30 Nm.
Use $F=MdF=\frac{M}{d}$ for the force on the heavy object:
$\frac{30\ \text{Nm}}{0.15\ \text{m}} = 200\ \text{N}$

So the lever can produce an upward force of about 200 N on the object.

Tip

Levers do not create energy

A lever can increase the force, but the end where you apply the force usually moves a greater distance. The machine makes the force more useful, not magical.

Gears

A gear is a wheel with teeth around its edge. Gears work by meshing their teeth together, so one rotating gear forces another gear to rotate.

Definition

Gear

A gear is a toothed wheel that transmits rotation and turning effects to another gear.

The gear you turn or drive is called the input gear. The gear being driven is called the output gear.

When two gears mesh:

they turn in opposite directions
the teeth transmit forces between the gears
those forces create moments about the gears’ axles
different-sized gears can change the speed and turning effect of the rotation

You may also see the word torque. At GCSE, you can think of torque as the turning effect of a force, very similar to moment.

Labelled diagram of two meshed gears showing opposite rotation directions, with the larger gear turning slower with larger torque and the smaller gear turning faster with smaller torque

Bigger and smaller gears

If a large gear drives a smaller gear, the smaller gear turns faster. However, the smaller gear has a smaller turning effect.

If a small gear drives a larger gear, the larger gear turns more slowly. However, the larger gear has a greater turning effect.

This is why low gears on a bike help you climb hills: they give a larger turning effect at the wheel, even though the wheel turns more slowly for each pedal rotation.

Example

Explaining a simple gear system

A large input gear turns clockwise and drives a smaller output gear. Explain what happens to the output gear.

Because the gears are meshed, the output gear must rotate in the opposite direction, so it turns anticlockwise.
The output gear is smaller, so it turns faster than the larger input gear.
The faster output rotation comes with a smaller turning effect, so this gear arrangement increases speed but reduces the moment available at the output.

Common Mistake

Same direction gears

Two gears touching directly do not turn in the same direction. Meshed gears always rotate in opposite directions.

Linking it all together

Moments, levers, and gears are all about turning effects.

Exam technique

In the exam

Identify the pivot first, then decide which forces produce clockwise and anticlockwise moments.
Use the perpendicular distance from the pivot to the force’s line of action, not just the length of the object.
For balanced objects, set total clockwise moment equal to total anticlockwise moment, then rearrange $M = F d$ carefully.

Self review

Check yourself

Why does pushing a door near the handle produce a larger moment than pushing near the hinge?
A force of 20 N acts 0.50 m from a pivot. What is the moment?
If a large gear drives a smaller gear, what happens to the direction, speed, and turning effect of the smaller gear?

Moments, levers and gears (physics only)

What you'll learn

Forces can cause rotation

Pivot

Moment of a force

Distance from the pivot

Calculating moments

Balanced objects and moments

Rearranging the moments equation

Levers

Gears

Bigger and smaller gears

Linking it all together

Recap questions

You've reached the end

Moments, levers and gears (physics only)

What you'll learn

Forces can cause rotation

Pivot

Moment of a force

Distance from the pivot

Calculating moments

Balanced objects and moments

Rearranging the moments equation

Levers

Gears

Bigger and smaller gears

Linking it all together

Recap questions

You've reached the end

Moments, levers and gears (physics only)

What you'll learn

Forces can cause rotation

Pivot

Moment of a force

Distance from the pivot

Calculating moments

Balanced objects and moments

Rearranging the moments equation

Levers

Gears

Bigger and smaller gears

Linking it all together

Recap questions

You've reached the end

Moments, levers and gears (physics only) Revision Guide

Moments, levers and gears (physics only)

What you'll learn

Forces can cause rotation

Pivot

Moment of a force

Distance from the pivot

Calculating moments

Balanced objects and moments

Rearranging the moments equation

Levers

Gears

Bigger and smaller gears

Linking it all together

Recap questions

You've reached the end

Moments, levers and gears (physics only) Revision Guide