Resultant forces and free body diagrams
Most objects have several forces acting on them at once. What matters for how they move is the single force that would have the same effect as all of them together: the resultant force.
Part 1 of 3: Learn it
In short
- Forces in the same direction add; forces in opposite directions subtract.
- If the resultant is zero, the object stays still or keeps moving at a steady velocity.
- Two forces at right angles combine like the sides of a right-angled triangle.
Where this is in your specification
Spec points: AQA 4.5.1.4 (Combined Trilogy 6.5.1.4; resolving forces is Higher), Edexcel Topic 9, OCR P2.3
| Board | Topic: Forces, work, elasticity, moments and pressure |
|---|---|
| AQA 8463 | 4.5.1 to 4.5.5 (8463); 6.5.1 to 6.5.3 (8464) |
| Edexcel 1PH0 | Topics 8, 9 and 15 |
| OCR J249 | P2.3 |
| Eduqas C420 | 3.1 to 3.3 |
| Cambridge IGCSE 0625 | 1.3, 1.5, 1.8 |
| National 5 C857 75 | Dynamics: Newton's laws |
Forces along a line
Pick one direction as positive. Add the forces pointing that way and subtract the ones pointing the other way. The answer's sign tells you the direction of the resultant. Always give a size and a direction: "150 N forwards", not just "150".
Balanced and unbalanced
| Resultant force | What happens |
|---|---|
| Zero (balanced) | a still object stays still; a moving object keeps the same speed in the same direction |
| Not zero (unbalanced) | the object speeds up, slows down or changes direction, in the direction of the resultant |
Free body diagrams
Draw the object as a box or a dot, with one arrow for each force acting on it. Each arrow starts on the object, points the way the force acts, and is labelled with its name and size. Longer arrows show bigger forces. Only include forces acting on that object, not forces it exerts on other things.
Forces at an angle (Higher)
Two forces at right angles can be combined by drawing them tip to tail: the resultant joins the start of the first to the end of the second. Use Pythagoras for its size and trigonometry (or a scale drawing) for its direction.
Working backwards, a single force can be split (resolved) into two parts at right angles, such as a horizontal part and a vertical part.
Forces of 12 N to the left and 7 N to the right act on a box. What is the resultant?
Show the answer
5 N to the left.
Part 2 of 3: See it worked
Worked examples
Example 1
A boat's engine pushes it forwards with 600 N. Water resistance is 450 N. Find the resultant force.
- Take forwards as positive
- 600 − 450 = 150
Answer: 150 N forwards, so the boat speeds up.
Example 2
A lift has an upward tension of 9000 N in its cable and a weight of 8500 N. Find the resultant force and say what happens if the lift is at rest.
- Up is positive: 9000 − 8500 = 500
- The resultant is upwards and not zero
Answer: 500 N upwards, so the lift starts to move up.
Example 3
Two forces of 30 N east and 40 N north act on an object. Find the size and direction of the resultant. (Higher)
- Size: √(30² + 40²) = √2500 = 50 N
- Angle from east: tan⁻¹(40 ÷ 30) = 53.1°
Answer: 50 N at 53° north of east.
Common mistakes
- Adding forces that act in opposite directions.
- Giving the size of the resultant with no direction.
- Saying a moving object must have a resultant force on it.
- Drawing forces on a free body diagram that the object exerts on something else.
A car travels at a steady speed with a driving force of 2000 N. What is the total resistive force?
Show the answer
2000 N: the forces are balanced, so the resistive forces equal the driving force.
Part 3 of 3: Test yourself
Check yourself
Answer each one in your head or on paper first, then open it to check.
Forces of 12 N to the left and 7 N to the right act on a box. What is the resultant?
5 N to the left.
A car travels at a steady speed with a driving force of 2000 N. What is the total resistive force?
2000 N: the forces are balanced, so the resistive forces equal the driving force.
What happens to a stationary object when the forces on it are balanced?
It stays still.
Jobs that use this
- Structural engineer (opens a new tab)
- Aerospace engineer (opens a new tab)
- Mechanical engineer (opens a new tab)
Each link opens the job profile on the National Careers Service (England). In the rest of the UK: My World of Work (Scotland), Careers Wales, nidirect careers (Northern Ireland).
Full lessons and marked practice for this course are coming soon to Brainlag Learn. See courses