The resultant is the single force that replaces several forces. You add forces as vectors, so direction changes the answer, and size alone is not enough.
This lesson is part of SPM Physics force and motion II. It uses the triangle from resolving forces into components.
What are the three cases?
Choose the case from the angle between the forces.
- Same line: add when the forces point the same way, subtract when they oppose.
- Right angle: use Pythagoras for the size and tan θ for the direction.
- Any other angle: use the parallelogram method or components.
For example, 12 N to the right and 5 N to the left give a resultant of 12 − 5 = 7 N to the right.
Worked example 1: a right angle
A boat is pulled by two ropes, 6.0 N east and 8.0 N north. Find the resultant.
Magnitude = √(6.0² + 8.0²) = √100 = 10 N. Direction: tan θ = 8.0 ÷ 6.0, so θ = 53.1°, which is 53° north of east.
The answer is less than 6.0 + 8.0 = 14 N, because the two forces do not point the same way.
Worked example 2: two equal forces at 60°
Two forces of 10 N act on a point, with 60° between them. Find the resultant.
Draw a parallelogram. The resultant is the diagonal, which lies between the two forces and splits the 60° angle into two 30° angles. Each force has a component along the diagonal of 10 cos 30° = 8.66 N.
The resultant = 2 × 8.66 = 17.3 N, in the direction halfway between the forces. The sum of magnitudes would be 20 N, which is too large, so 17.3 N passes the check.
The mistake to avoid
The common mistake is to add the magnitudes at any angle. The table shows the error size.
| Angle between two 10 N forces | Correct resultant | Student’s 10 + 10 |
|---|---|---|
| 0° | 20 N | 20 N |
| 60° | 17.3 N | 20 N |
| 90° | 14.1 N | 20 N |
| 180° | 0 N | 20 N |
As the angle grows, the resultant shrinks. The sum of magnitudes is correct only at 0°. Use the units and significant figure checker to keep the answers in newtons.
Check yourself
Two forces act at right angles: 9.0 N and 12 N. Find the magnitude of the resultant and its angle to the 12 N force.
Answer
Magnitude = √(9.0² + 12²) = √(81 + 144) = √225 = 15 N.
The angle to the 12 N force: tan θ = 9.0 ÷ 12 = 0.75, so θ = 36.9°, towards the 9.0 N force.
What to study next
Move on to applying equilibrium conditions, where the resultant must be zero. You can revisit diagrams in drawing force diagrams, and try the force and motion II practice set.
If you want a teacher to check your parallelograms with you, see online one-to-one Physics tuition.