Forces · Model
Drawing and adding forces
A sledge is pulled one way with 40 N and the other way with 25 N. One single arrow does the job of both. How long is it, and which way does it point?
Start here
Two pulls, one rope, and the winner is a subtraction.
In a tug of war the rope does not care how many people are holding it. It moves according to one number: how much more one side is pulling than the other.
A sledge is pulled right with 40 N and left with 25 N. What single force would have exactly the same effect?
15 N to the right. The 25 N pull cancels 25 N of the 40 N pull, and what is left over is the only thing the sledge responds to. That leftover has a name: the resultant force.
An arrow is how a force gets written down. Where it starts says which object is being pushed or pulled, which way it points says the direction, and its length says the size — so two arrows drawn the same length are claiming two forces are equal.
At the bench · the sledge on ice
Set two pulls. Read one arrow.
Set a pull to begin
Commit first. You set both pulls to 30 N. What does the single arrow look like?
40 N
25 N
Pull to the right
—
Pull to the left
—
Resultant force
—
The relationship · a beam, not a triangle
Two pulls the opposite way: one cancels part of the other.
The two lower bars fill the top one exactly, because 25 N and 15 N make 40 N. That is why this relationship gets a beam and not a triangle: nothing here is being multiplied.
Along one line, opposite ways: resultant = bigger force − smaller force, pointing the way of the bigger one.
Same way along the line: add them.
Opposite ways along the line: subtract.
Every force, and the resultant, is in newtons (N).
Worked example · one step at a time
40 N right, 25 N left. Find the resultant.
Step 0 of 5
Convert
40 N stays 40 N · 25 N stays 25 N
Both pulls are already in newtons, so there is nothing to convert.
Formula
resultant = bigger pull − smaller pull
The two pulls are along one line and point opposite ways.
Insert
resultant = 40 N − 25 N
Bigger first, so the subtraction cannot come out negative.
Fine-tune
40 − 25 = 15
Subtract, and keep the direction of the bigger pull.
Answer
resultant = 15 N to the right
Size, unit and direction. Fifteen on its own is not an answer.
Worked example · one step at a time
A tug pulls right with 1.2 kN. A second pulls left with 400 N.
Step 0 of 5
Convert
1.2 kN × 1000 = 1200 N
A kilonewton is a thousand newtons, and two pulls cannot be compared until they share a unit.
Formula
resultant = bigger pull − smaller pull
Still one line, still opposite ways.
Insert
resultant = 1200 N − 400 N
The converted pull goes in. The 1.2 never does.
Fine-tune
1200 − 400 = 800
Newtons take away newtons, so the answer is in newtons.
Answer
resultant = 800 N to the right
Subtract 400 from 1.2 and the resultant comes out pointing the wrong way.
Your turn · the same five steps
Your bench: 40 N to the right, 25 N to the left.
Write all five lines before you check. The numbers are the ones your own sledge is showing.
Both ropes are slack. Set a pull above zero and the five lines come back.
The five lines, marked
Convert
40 N stays 40 N · 25 N stays 25 N
Both pulls are already in newtons, so there is nothing to convert.
Formula
resultant = bigger pull − smaller pull
Opposite ways along one line, so subtract.
Insert
resultant = 40 N − 25 N
Both pulls are in newtons already.
Fine-tune
40 − 25 = 15
Keep the direction of the bigger pull.
Answer
resultant = 15 N to the right
Size, unit and direction, all three.
The five lines give 15 N to the right, and the arrow on the bench is drawn that length.
A trolley is pushed right with 2.5 kN while a rope drags it left with 900 N.
This one needs the Convert line to do some work.
The five lines, marked
Convert
2.5 kN × 1000 = 2500 N
A kilonewton is a thousand newtons, so multiply by 1000 before comparing the two.
Formula
resultant = bigger pull − smaller pull
One line, two opposite directions.
Insert
resultant = 2500 N − 900 N
The converted push goes in. The 2.5 never does.
Fine-tune
2500 − 900 = 1600
Newtons take away newtons, so the answer is in newtons.
Answer
resultant = 1600 N to the right
Work with 2.5 instead of 2500 and the resultant comes out pointing left.
The five lines give 1600 N to the right. The whole question turned on the first one.
Key fact
An arrow's length is the size of the force and its direction is the direction of the force. Forces along one line add if they point the same way and subtract if they point opposite ways, and the single force left over — the resultant — is in newtons with a direction.
Think again
“The bigger pull wins, so the sledge goes at 40 N.”
Two different things are being run together here. The 40 N pull does win, in the sense that the sledge ends up going that way — but the sledge does not respond to 40 N, it responds to 15 N, because 25 N of that pull is being cancelled by the other rope. And a force is not a speed: 15 N does not tell you how fast the sledge goes, only how hard it is being pushed along. Change the sledge for a heavier one and the same 15 N produces less of a change, which is a hint about the next lesson.
“Arrows on a diagram should be drawn the same length so it looks tidy.”
A force arrow is a measurement, not a decoration. Two arrows of the same length are a claim that the two forces are equal, and if they are not equal you have drawn something false — the diagram now says the sledge is going nowhere. Draw the bigger force longer, always, and label both with their size in newtons so the reader is not left estimating from your drawing. This is why examiners can mark a diagram wrong even when the words beside it are right.
Mastery ladder
Not started yet.
Rungs 3 and 4 you mark yourself.
Rung 1 · Calculate
A box is pushed forwards with 90 N while friction pushes backwards with 34 N. What is the resultant force?
Rung 2 · The one that catches people
Two arrows on a diagram are drawn the same length, one pointing left and one pointing right. What does the diagram say?
Rung 3 · Draw and explain
A trolley is pulled forwards with 60 N and dragged backwards by 20 N of friction. Describe the arrow diagram you would draw for it, then give the resultant force.
Rung 4 · Take it somewhere new
A parachutist is falling. Gravity pulls down with 700 N and the parachute pushes up with 700 N. Then the parachutist pulls a cord and the parachute pushes up with 760 N instead. Work out the resultant force in each case and say what changed.
Key note
Draw a force as an arrow: it starts on the object being pushed or pulled, points the way the force acts, and its length is the size in newtons. Along a single line, forces pointing the same way add and forces pointing opposite ways subtract. What is left over is the resultant force, and it is the only force the object responds to.
Going further
Three forces along one line are no harder than two: take everything pointing right as positive, everything pointing left as negative, and add them all up. A cyclist with 120 N of pedalling, 30 N of air resistance and 15 N of friction has 120 − 30 − 15 = 75 N left over, forwards. The same trick handles four forces, or ten. What it will not handle is forces that are not along the same line — a rope pulling upwards at an angle while gravity pulls straight down. Adding those needs a method that is not subtraction, and that method is GCSE work; at this stage the honest answer is that the arrows tell you the answer is somewhere between the two, and you leave it there.
One thing the arrow says that this lesson has not used: where it starts. Two equal and opposite forces on the same object, drawn at the same point, do nothing at all. Draw the same two forces at opposite ends of a steering wheel and the wheel turns, even though the resultant is still zero. That is a second effect a force can have, and it is why the arrow's starting point is part of the diagram and not just a place to begin drawing.
Before this lesson
Connects to
At GCSE this becomes
- Free-body diagrams, scale drawings and resolving forces that are not along one line.
Where to next
Ask Mr Badmus AI
Want to check a diagram you have drawn?
The sledge on the bench is on ice: friction and air resistance are left out so that only the two pulls you set are acting along the line. The bars in the beam are drawn to one scale, so equal lengths mean equal newtons.
Lesson content © MrBadmusAI.