The resultant force is the single force that has the same effect as all the forces on an object combined. For forces along one straight line, you choose a positive direction, add the forces pointing that way, subtract the forces pointing the other way, and state the direction of the answer.
This lesson follows drawing a force diagram and feeds directly into acceleration and resultant force in forces and momentum.
How do you add forces along a line?
The size of each force is always positive. The direction decides whether you add or subtract.
- Choose a positive direction, for example right or up, and say so.
- Add forces pointing that way.
- Subtract forces pointing the opposite way.
- Read the sign. A positive answer points in your chosen direction, a negative answer points the other way.
- Write the answer with size, unit and direction. The unit is the newton (N).
If the answer is zero, the forces are balanced. The next lessons return to what that means.
Worked example
Invented situation: a sledge is pulled to the right by two ropes with forces of 30 N and 15 N. Friction of 12 N acts to the left, and air resistance of 3 N also acts to the left. Find the resultant force.
Step 1, positive direction: right is positive.
Step 2, forces to the right: 30 + 15 = 45 N.
Step 3, forces to the left: 12 + 3 = 15 N.
Step 4, resultant: 45 − 15 = 30 N.
Step 5, answer: 30 N to the right.
Check: if you add 30 N (resultant) to the 15 N that opposes it, you get 45 N, the total forward force. The two values agree.
What if the forces are at right angles?
Two forces at right angles, such as 6.0 N to the east and 8.0 N to the north, form two sides of a right-angled triangle. The resultant is the hypotenuse.
Resultant = √(6.0² + 8.0²) = √(36 + 64) = √100 = 10 N.
The direction is found with the angle whose tangent is 8.0/6.0, which is about 53° north of east. Whether your syllabus year needs this, or only the size, is on the current Cambridge page, so check it before relying on it.
The mistake to watch for
A common slip is to add all the forces as if they pointed the same way.
Mistaken working: a lift cable pulls up with 4500 N and the weight is 4000 N. Resultant = 4500 + 4000 = 8500 N.
The student added the numbers and ignored that the forces act in opposite directions.
The correction is to write the positive direction first. Up is positive: 4500 − 4000 = 500 N upwards.
A plausibility test helps: two opposing forces give a resultant smaller than the larger one. 8500 N is bigger than both, which cannot be right.
Check yourself
Try these without a calculator, then open each answer.
1. Forces of 8 N and 5 N act to the east. A force of 11 N acts to the west. Find the resultant.
Show answer
East is positive: 8 + 5 − 11 = 2. 2 N to the east.
2. A crate has a weight of 60 N. A rope pulls it up with 45 N. The ground is not involved. Find the resultant force.
Show answer
Up is positive: 45 − 60 = −15. 15 N downwards. The crate is not lifted off the ground, so in a real situation the floor would add an upward normal force, but with only these two forces the resultant is 15 N down.
3. Two forces of 9.0 N and 12 N act at right angles. Find the size of the resultant.
Show answer
√(9.0² + 12²) = √(81 + 144) = √225 = 15 N.
Where this leads next
A resultant force is what causes acceleration, so continue with relating acceleration to resultant force. The triangle and bearings reasoning board is a good place to practise combining perpendicular forces.
If multi-force questions still produce different answers each time you try them, our teachers can work through them one at a time in online one-to-one Physics tuition.