Moment of a force
Key idea: A force's moment is its magnitude multiplied by the perpendicular distance from the pivot to its line of action. A force whose line of action passes through the pivot has zero moment.
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Guided steps
- Find the pivot and describe the turning effect.
- Find the perpendicular distance to the force’s line of action.
- Follow the calculation in newtons, metres and newton metres.
- Correct the distance error and calculate the moment.
- Calculate both moments using their perpendicular distances.
- Find the smallest force, with its unit.
- Find the force needed with each lever.
- Explain the force ratio and the movement trade-off.
- Calculate both moments and B’s balancing position before testing.
- Compare both model moments and explain the released motion.
- Find the distance, moment and direction about this pivot.
- Calculate the resultant moment and state its direction.
- Review distance, unit and direction, then use the topic check.
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Push a door near its handle and it swings open easily. Push just as hard near the hinges and it hardly moves. The force is the same, so something else decides how well a force turns an object: where the force acts, and in which direction. The turning effect of a force is called its moment.
The turning effect of a force
A force can make an object turn about a fixed point called the pivot. You use turning effects every day:
- pushing a door, which turns about its hinges;
- turning a nut with a spanner;
- sitting on a seesaw;
- lifting a bottle cap with a bottle opener;
- turning a steering wheel by pushing on its rim.
In each case the turning effect is larger when the force is larger, and when the force acts farther from the pivot. That is why a door handle is on the edge farthest from the hinges, and why a long spanner loosens a tight nut more easily than a short one.
Moment = force × perpendicular distance
The moment of a force about a pivot is
with F in newtons, d in metres and the moment in newton metres (N m). Convert any distance in centimetres to metres first.
The line of action of a force is the straight line along which the force acts, extended as far as you need in both directions. The perpendicular distance d is the shortest distance from the pivot to that line, measured at right angles to it.
Why the perpendicular distance, and not just the distance to where you push? Push on the edge of a door straight towards the hinges: your force acts along the door, its line of action passes through the hinges, and the door does not turn at all. Push the same edge at right angles to the door and it turns as easily as it can. What decides the turning effect is how far the force’s line of action passes from the pivot.
For a fixed force, the moment is proportional to the perpendicular distance: twice as far gives twice the turning effect.
Moment is proportional to perpendicular distance
Moment (turning effect) versus perpendicular distance from pivot for a fixed force, showing a straight line through the origin.
If the figure is wider than its frame, scroll across to read all labels.
View figure data
This table is included in print while it is open.
| Perpendicular distance from pivot, d (m) | F = 20 N |
|---|---|
| 0.0 | 0 |
| 0.1 | 2 |
| 0.2 | 4 |
| 0.3 | 6 |
| 0.4 | 8 |
Worked example 1
Calculating a moment
Problem
A mechanic pushes on a spanner with a force of 150 N. The line of action of the force is 30 cm from the centre of the nut, measured at right angles. Find the moment of the force about the nut.
Show full solution
Convert the perpendicular distance
Method
Write the distance in metres.Reason
The moment is in N m, so the distance must be in metres. This distance is already measured at right angles to the line of action.Working
Answer
d = 30 cm = 0.30 mMultiply force by perpendicular distance
Method
Apply M = Fd.Reason
The moment is the force times the perpendicular distance from the pivot to its line of action.Working
Answer
M = (150)(0.30) = 45 N m
Quick check
Spot the mistake 1
When the moment is zero
Learner response
A force acts on the end of a 0.50 m rod, but its line of action passes through the pivot. A student writes M = F(0.50) because the rod is 0.50 m long. Find and correct the first error.
Diagnose before viewing the correction
Show solution step by step
Find the first error
Method
The 0.50 m length is not the perpendicular distance.Reason
The moment uses the distance from the pivot to the line of action, not the distance to the point where the force acts.Working
Answer
The line of action passes through the pivot, so d = 0.Correct the moment
Working
Answer
M = F × 0 = 0 N m
Common mistake
Check your understanding 1 A spanner pulled at an angle
A 20 N pull acts on a 0.25 m spanner. Its line of action passes 0.15 m from the nut. Calculate its moment, then the moment if the same pull acts perpendicular to the spanner at its end.
Show answer
At an angle, the perpendicular distance is 0.15 m:
At right angles to the end of the spanner, the perpendicular distance is the full 0.25 m:
Pulling at right angles gives the largest moment for a given force, because it makes the perpendicular distance as large as possible.
Getting a large moment from a small force
Rearranging M = Fd gives F = M/d. To produce a given moment, a larger perpendicular distance needs a smaller force. Door handles, long spanners and crowbars all use this.
Guided practice 1
The force needed to open a door
Problem
A door needs a moment of 12 N m about its hinges to start opening. The handle is 0.80 m from the hinges. Find the smallest force on the handle that will open the door.
Rearrange before substituting
Hints
Hint 1
choose the direction of push
Which direction of push makes the perpendicular distance largest?
Hint 2
make force the subject
Divide the moment by the perpendicular distance.
Show solution step by step
Rearrange the moment equation
Method
Make F the subject of M = Fd.Reason
The moment and the distance are known; the force is the unknown.Working
Answer
F = M/dSubstitute and state the unit
Reason
A push at right angles to the door makes the perpendicular distance as large as it can be, which makes the force as small as it can be.Working
Answer
F = 12/0.80 = 15 N
Quick check
Try it yourself 1
Opening a tin lid with a coin or a spoon
New context
A moment of 0.60 N m is needed to prise open a tin lid. Used as a lever, a coin gives a perpendicular distance of 0.020 m and a spoon handle gives 0.10 m. Find the force needed with each, and use the results to explain why the spoon works better.
Try this without the worked method
Hints
Hint 1
find what stays the same
Both tools must produce the same moment on the lid.
Hint 2
rearrange for force
Use F = M/d for each tool.
Show solution step by step
Calculate the force for each tool
Reason
The lid needs a fixed moment, so a larger perpendicular distance means a smaller force.
Working
Answer
F coin = 0.60/0.020 = 30 N, F spoon = 0.60/0.10 = 6.0 NExplain the comparison
Working
Answer
The spoon gives five times the perpendicular distance, so it needs only one fifth of the force to produce the same moment.
Quick check
Check your understanding 2 Explain the lever comparison
Use your two force calculations to explain why the spoon needs less force. What happens to the distance moved by your hand for the same small lift of the lid?
Show answer
The spoon’s perpendicular distance is five times the coin’s, so the same 0.60 N m moment needs one fifth of the force: 6 N rather than 30 N. For the same small turn and lift, the hand on the longer lever moves five times as far. Compare both the force ratio and the movement trade-off.
Clockwise and anticlockwise moments
A moment has a direction: it turns the object either clockwise or anticlockwise about the pivot, as seen in the diagram. Always state the direction with the size.
When several forces act, find each moment about the same pivot. Moments in the same direction add; moments in opposite directions work against each other. If the clockwise and anticlockwise moments are equal, the object does not start to turn. That is the principle of moments.
Check your understanding 3 Predict, balance and explain
The pivot is at 50 cm, load A is 3 N at 20 cm and B is 2 N at 85 cm. The rule weighs 1 N at 50 cm. Before Let go, calculate the load moments and predict the turn. Find B’s balancing position, then test it without changing the weights or pivot.
Show the steps Optional working
Show the steps Optional working
Show the steps Optional working
Show answer
Initially, load A gives an anticlockwise moment of 3 × (50-20)/100 = 0.90 N m, while load B gives a clockwise moment of 2 × (85-50)/100 = 0.70 N m. The rule therefore starts turning anticlockwise. Its own weight has zero moment because its line of action passes through the pivot.
For balance, load B needs a perpendicular distance of 0.90/2 = 0.45 m to the right of the pivot, placing it at 95 cm. Both moments are then 0.90 N m. The smaller load balances the larger one by acting farther from the pivot: equal moments, rather than equal forces or equal distances, prevent the initially stationary rule from starting to turn.
Metre rule pivoted at 50 cm with loads of 3.0 N at 20 cm, 2.0 N at 85 cm. Clockwise moments 0.70 newton metres, anticlockwise moments 0.90 newton metres. The rule is held level.
- Sum of clockwise moments
- 0.70 N m
- Sum of anticlockwise moments
- 0.90 N m
- Resultant moment
- 0.20 N m anticlockwise
- Tilt
- 0 °
- Line of action of the weight
- inside the base
Try this
0 of 4 doneBalance the rule with unequal loads, then let go. (not done yet)
The rule stays level when the sum of clockwise moments equals the sum of anticlockwise moments: a smaller load needs a longer distance.
Balance the rule with the pivot away from the 50 cm mark. (not done yet)
The rule's own weight acts at its centre of gravity, so it has a moment about any pivot that is not at the centre.
Tilt the block so that it only just topples when you let go. (not done yet)
The block topples once the line of action of its weight passes outside its base, beyond the edge it pivots on.
Change the block so that it falls back upright after a 30° tilt. (not done yet)
A lower centre of gravity and a wider base make the block more stable: it must tilt further before its weight acts outside the base.
You’ve tried every task.
Check your understanding 4 Compare the model with your prediction
Open More settings and use Position of load B to set B at 95 cm, keeping the weights and pivot unchanged. Compare the displayed clockwise and anticlockwise moments, then select Let go. Explain the result and why the rule’s weight contributes no moment here.
Show answer
At 95 cm, both displayed moments are 0.90 N m. The initially level rule stays level when released because the opposing moments balance. B is only 2 N, compared with A’s 3 N, but its 0.45 m arm is longer than A’s 0.30 m arm. The rule’s 1 N weight acts at the 50 cm pivot, giving zero perpendicular distance and zero moment. Compare both readouts, the released motion and these two explanations.
Check your understanding 5 A metre rule with an off-centre pivot
A horizontal metre rule is pivoted at 20 cm. A downward 4.0 N force acts at 80 cm. Calculate its perpendicular distance from the pivot and its moment. State the turning direction.
Show the steps Optional working
Show the steps Optional working
Show answer
The force is vertical and the rule is horizontal, so the perpendicular distance is the distance along the rule:
The force acts downwards to the right of the pivot, so the moment is clockwise.
Exam-style question 1
Two moments in opposite directions
Examination question
Two forces act on a door. Force A produces a clockwise moment of 6.0 N m about the hinges, and force B produces an anticlockwise moment of 2.5 N m. Calculate the resultant moment and state its direction. [3 marks]
Combine opposing moments
Show solution step by step
Combine the opposing moments
1 markMethod
Take clockwise as positive.Reason
Moments in opposite directions work against each other, so they need opposite signs.Working
Answer
M clockwise = +6.0 N m and M anticlockwise = -2.5 N m.Calculate the size
1 markReason
Opposite moments subtract.Working
Answer
M resultant = 6.0-2.5 = 3.5 N mState the direction
1 markWorking
Answer
The resultant moment is 3.5 N m clockwise, so the resultant tends to turn the door clockwise.
Self-mark with the mark scheme
Tick each mark point that your answer includes.
Self-mark the method, the size and the direction with its unit separately.
Common mistake
Adding the moments gives 8.5 N m and loses the first two marks: opposing moments are subtracted.
Common mistakes
- Using the distance to where the force acts, or the length of the bar, instead of the perpendicular distance to the line of action.
- Calculating with centimetres and giving the answer in N m.
- Using a mass in kg instead of a weight in N for a force due to gravity.
- Taking moments about a point other than the pivot the question gives.
- Giving the size of a moment without saying whether it is clockwise or anticlockwise.
- Writing the unit of moment as N/m. It is N m: newtons multiplied by metres.
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