The turning effect of a force, how to balance moments, and how levers and gears make hard jobs easier.
Objectives
Explain what a moment is and why the distance from the pivot matters.
Calculate a moment using moment = force × perpendicular distance, with the correct unit.
Use the principle of moments to find an unknown force or distance on a balanced object.
Explain how levers make a job easier, and how gears change speed and turning effect.
Key equations
moment of a force = force × distance (normal to the direction of the force)
M=Fd
M moment (Nm)
F force (N)
d distance from the pivot (m)
Moments, levers and gears explained
Start here
Have you ever tried to open a door by pushing close to the hinges? It is really hard. Push at the handle, on the far side, and it swings open with hardly any effort. Nothing about you changed. Only the place you pushed changed.
That difference is what this lesson is about. A force can do more than push something along. It can also make something turn. The turning effect of a force is called a moment, and the fixed point it turns about is called the pivot.
Here is a picture to hold on to. Think of a see-saw in a playground. A small child can balance a big child if the small child sits much further from the middle. Where you sit matters just as much as how heavy you are.
You use moments all the time. A spanner, a bottle opener, a wheelbarrow, a pair of scissors and the gears on a bike all depend on them. Once you know the rule, you can predict whether a see-saw will balance.
By the end of this lesson you will be able to do three things. First, work out a moment from a force and a distance. Second, find a missing force or distance when something is balanced. Third, explain in plain words why levers and gears make hard jobs easier.
Turning effects and the pivot
A pivot is a fixed point that an object can turn about. A door turns about its hinges, a see-saw about its middle and a spanner about the nut it grips. When you push on the object, it tends to rotate about the pivot.
The turning effect of a force is called its moment. A moment can turn an object clockwise, the same way as the hands of a clock, or anticlockwise, the opposite way. We always say which way it turns, because the two kinds can cancel out.
Two things decide how big a moment is. One is the size of the force, because a harder push turns the object more. The other is the distance from the pivot to where the force acts, because a push far from the pivot turns the object more.
Go back to the door. A 10 N push near the hinges has a small turning effect. The same 10 N push at the handle has a much bigger one, because the handle is much further from the hinges. The force is the same, so the distance made the difference.
This is why spanners have long handles and door handles are on the edge, not the middle. A longer handle puts your hand further from the pivot. You get a bigger moment for the same push, so the stuck nut or the heavy door gives way.
Bars showing the turning effect of the same push on a door at three places. The turning effect is smallest near the hinge and biggest at the handle.
Calculating a moment
Now let us put numbers on it. In words: moment = force × perpendicular distance from the pivot. In symbols: M = F × d. You use it whenever you are asked how big a turning effect is.
The units matter. The force F is in newtons, N, and the distance d is in metres, m. So the moment M is in newton metres, written N m. If the distance is given in centimetres, divide by 100 to change it to metres first.
Perpendicular distance means the shortest distance from the pivot to the line along which the force acts, measured at right angles to that line. On a door this is simply how far your hand is from the hinges, if you push straight at the door. If you push at a slant, the perpendicular distance is smaller.
Here is a quick one. A force of 5 N acts 0.4 m from a pivot. Write the equation, then put in the numbers: moment = 5 × 0.4 = 2 N m. Always finish with the unit.
A sense check helps. If the force or the distance gets bigger, the moment gets bigger. If your answer gets smaller when you make the force bigger, you have probably divided when you should have multiplied.
A formula triangle for moments with M at the top and F and d along the bottom. Cover the one you want to find. M equals F times d.
Balanced objects and the principle of moments
An object is balanced when it is not turning. Think of a see-saw level with the ground. Each child makes a moment about the middle, and the two moments cancel each other out.
We name the moments by direction. The child on the left pushes down and turns the see-saw anticlockwise. The child on the right turns it clockwise. When the see-saw is level, those two moments are equal.
This is the principle of moments. For a balanced object, the total clockwise moment about the pivot equals the total anticlockwise moment about the same pivot. If there are several forces on one side, work out each moment and add them up.
Look at the picture. On the left a 20 N force acts 3 m from the pivot, so the moment is 20 × 3 = 60 N m. On the right a 30 N force acts 2 m from the pivot, so the moment is 30 × 2 = 60 N m. The two are equal, so the beam balances.
Notice that the forces are different but the moments are equal. The bigger force sits closer to the pivot. The smaller force sits further out. That is how a light child can balance a heavy one.
A beam on a pivot. A 20 N force acts 3 m to the left and a 30 N force acts 2 m to the right. Both moments are 60 N m so the beam balances.
Finding a missing force or distance
The principle of moments lets you find an unknown. Work out the moment you know, set it equal to the moment on the other side, and solve. It is the same pattern every time, so learn the pattern.
Here is an example. A child weighing 150 N sits 2 m from the pivot of a see-saw. First work out the moment: 150 × 2 = 300 N m. A second child weighing 100 N sits on the other side. For balance, 100 × d must also be 300 N m.
Now divide to find the distance: d = 300 ÷ 100 = 3 m. The lighter child has to sit 3 m from the pivot. Check the answer makes sense. The lighter child sits further out, which is just what we expect.
Rearranging is the part that trips people up. If moment = force × distance, then distance = moment ÷ force, and force = moment ÷ distance. Use the formula triangle if you find that easier, and always divide the moment by the number you already know.
Check the units before you start. If a distance is in centimetres, change it to metres. And ask yourself at the end: is the bigger force closer to the pivot? If not, look for a slip.
Four boxes in order. Work out the moment you know with force times distance. Set it equal to the moment on the other side. Divide by the number you know to find the unknown. Check that the bigger force is closer to the pivot.
Levers
A lever is a rod that turns about a pivot. You push or pull on it with an effort, and it moves a load. The effort is the force you apply, and the load is the force the lever has to overcome, such as the weight of the thing you are lifting.
Think of a crowbar lifting a heavy paving slab. The slab is the load, close to the pivot. Your hand is the effort, much further from the pivot. Because your hand is far out, a small effort can balance a large load.
This works because of moments. When the lever is just balanced, effort × distance of the effort = load × distance of the load. If the effort is five times further from the pivot than the load, the effort only needs to be one fifth of the load.
That is why levers are called force multipliers. A spanner, a bottle opener, a wheelbarrow and a pair of scissors all work this way. Each lets a small push do a big job.
There is always a price. The effort moves a much longer distance than the load does. You gain in force but you lose in distance, so you cannot get something for nothing.
A lever balanced on a pivot. A 100 N load is 1 m on one side and a 20 N effort is 5 m on the other. Both moments are 100 N m.
Gears
Gears are wheels with teeth that mesh together, which means the teeth fit between each other. When one gear turns, its teeth push the teeth of the next gear. Gears pass on the turning effect of a force from one place to another.
Meshing gears turn in opposite directions. If the first gear turns clockwise, the second turns anticlockwise. A chain or belt between two gears can make both turn the same way, but two gears touching always turn opposite ways.
The number of teeth decides the speed. The gear that is turned by the motor or by your pedals is called the driving gear. The gear it turns is the driven gear. If the driven gear has more teeth, it turns slower.
Work it out by counting teeth. A gear with 20 teeth drives a gear with 40 teeth. The driven gear has twice as many teeth, so it turns at half the speed. Ignoring friction, it has twice the turning effect, which means twice the moment.
A bike is a good example. Going uphill you choose a low gear, with a big gear on the back wheel. The wheel turns more slowly for each pedal turn, but you get a bigger turning effect, which helps you climb. On the flat you choose the opposite and go faster.
Recap
A force can turn an object about a pivot. The turning effect is the moment. The moment is bigger when the force is bigger or when it acts further from the pivot.
The equation is moment = force × perpendicular distance from the pivot, M = F × d, in newton metres, N m. Change centimetres to metres before you multiply.
When an object is balanced, the clockwise moments equal the anticlockwise moments. Use this to find a missing force or distance, and expect the smaller force to sit further from the pivot.
A lever lets a small effort far from the pivot balance a large load close to it, but the effort moves further. Meshing gears turn in opposite directions. A small gear driving a large gear gives a slower, stronger turn.
How this is asked in an exam: a calculation wants the equation, the numbers and the answer with the unit N m. An explain question wants the link: further from the pivot, so a bigger moment. A gear question wants a count of teeth and a clear comparison of speed and turning effect.
Key points
A moment is the turning effect of a force about a pivot.
moment = force × perpendicular distance from the pivot, in N m.
A bigger force or a bigger distance gives a bigger moment.
When balanced, clockwise moments equal anticlockwise moments.
A lever lets a small effort far from the pivot balance a large load near it.
A larger driven gear turns slower and has a larger moment.
Worked examples
Example 1
Calculate the moment of a force of 15 N at a perpendicular distance of 0.40 m from a pivot.
Write the equation: moment = force × perpendicular distance.
Substitute the numbers: moment = 15 × 0.40.
Work it out and add the unit: moment = 6 N m.
Answer: 6 N m
Common slips
Slip: moment = force ÷ distance. A moment is force multiplied by distance, not divided.
Slip: moment = 15 + 0.40 = 15.4. Adding does not give a moment. Multiply the force by the distance.
Slip: moment = 6 N. The unit of a moment is the newton metre, N m, not N.
Example 2
A force of 40 N acts 25 cm from a pivot. Calculate the moment in N m.
Change the distance to metres: 25 cm ÷ 100 = 0.25 m.
Write the equation and substitute: moment = 40 × 0.25.
Work it out and add the unit: moment = 10 N m.
Answer: 10 N m
Common slips
Slip: Use 25 as it is, because it is already a number. The unit N m needs metres. Change 25 cm to 0.25 m first.
Slip: moment = 40 ÷ 0.25 = 160. Multiply the force by the distance. Do not divide.
Slip: moment = 40 × 25 = 1000 N m. That used 25 instead of 0.25. Divide the centimetres by 100 first.
Example 3
A 60 N weight is 0.5 m to the left of a pivot. A 40 N weight is on the right. How far from the pivot must the 40 N weight be for the beam to balance?
Work out the anticlockwise moment on the left: 60 × 0.5 = 30 N m.
For balance the clockwise moment is also 30 N m, so 40 × d = 30.
Divide to find the distance: d = 30 ÷ 40 = 0.75 m.
Answer: 0.75 m
Common slips
Slip: Anticlockwise moment = 60 + 0.5 = 60.5. Multiply force by distance. Do not add.
Slip: d = 40 ÷ 30 = 1.33 m. Divide the moment by the force: 30 ÷ 40.
Slip: d = 0.5 m, the same distance as the other side. The forces are different, so the distances have to be different too.
Questions and answers
Moments, levers and gears practice questions for GCSE Combined Science Physics. Try each one, then open the answer. Easy questions come first.
Question 1 Easy
Calculate the moment of a force of 20 N at a perpendicular distance of 0.5 m from a pivot.
Show answer
Answer: 10 N m
moment = force × distance.
20 × 0.5 = 10 N m.
Question 2 Easy
Calculate the moment of a force of 8 N at a perpendicular distance of 0.25 m from a pivot.
Show answer
Answer: 2 N m
moment = force × distance.
8 × 0.25 = 2 N m.
Question 3 Easy
State the unit of a moment.
Show answer
Answer: newton metre
A moment is force × distance.
The unit is the newton metre.
Question 4 Easy
Two gears mesh. One turns clockwise. Which way does the other turn?
Show answer
Answer: anticlockwise
Meshing gears turn in opposite directions.
So the other turns anticlockwise.
Question 5 Medium
A force of 12 N produces a moment of 3 N m. Work out the perpendicular distance from the pivot.
Show answer
Answer: 0.25 m
distance = moment ÷ force.
3 ÷ 12 = 0.25 m.
Question 6 Medium
A 4 N force acts 0.6 m from a pivot and turns clockwise. What anticlockwise moment balances it?
Show answer
Answer: 2.4 N m
Clockwise moment = 4 × 0.6 = 2.4 N m.
A balancing anticlockwise moment is 2.4 N m.
Question 7 Medium
A child weighing 300 N sits 2.0 m from the pivot of a seesaw. A child weighing 400 N sits on the other side. How far from the pivot must the second child sit to balance?
Show answer
Answer: 1.5 m
300 × 2.0 = 600 N m.
400 × d = 600, so d = 1.5 m.
Question 8 Medium
Why does a longer spanner make a tight nut easier to turn?
Show answer
Answer: the force is further from the pivot so the moment is bigger
The force acts further from the pivot.
So the moment is bigger for the same force.
Exam-style questions
Longer questions in the style of a written paper, with the marking points. These are Yumio practice questions.
Exam-style question 14 marks
A seesaw is balanced. A child weighing 250 N sits 1.6 m from the pivot. (a) Calculate the moment of this child. [2 marks] (b) A second child weighs 320 N. Calculate how far from the pivot this child sits. [2 marks]
Show answer and marking points
Answer: (a) 400 N m (b) 1.25 m
(a) moment = force × distance, 250 × 1.6 (1)
(a) 400 N m (1)
(b) 320 × d = 400 (1)
(b) 1.25 m (1)
Exam-style question 23 marks
A gear with 10 teeth drives a gear with 30 teeth. Compare the larger gear with the smaller gear. [3 marks]
Show answer and marking points
Answer: The larger gear turns at one third of the speed, has three times the moment, and turns in the opposite direction.
turns more slowly, at one third of the speed (1)
larger moment, three times as much (1)
turns in the opposite direction (1)
Common mistakes
Using the distance along the force instead of the perpendicular distance
The distance must be measured at right angles to the line of the force.
Writing the unit of a moment as N or J
The unit of a moment is the newton metre, N m.
Using centimetres in the equation without changing to metres
Divide centimetres by 100 to get metres before you multiply.
Saying the heavier child sits further from the pivot on a balanced see-saw
The smaller force has to be further from the pivot so the moments are equal.
Dividing when the question needs you to multiply, such as moment = force ÷ distance
Moment = force × distance. Only divide when you rearrange to find a force or a distance.
Questions students ask
Is Moments, levers and gears on the Foundation or Higher tier?
Moments, levers and gears is on both Foundation and Higher papers for GCSE Combined Science Physics, aimed at around grade 6.
What is a common mistake on Moments, levers and gears?
Using the distance along the force instead of the perpendicular distance. The distance must be measured at right angles to the line of the force.
Where are the answers to the Moments, levers and gears questions?
Open Show answer under each question on this page. The questions are Yumio practice questions, not past exam questions.