In one line
Resolution splits one force into perpendicular parts; composition combines several forces into one equivalent force called the resultant.
Explained simply
When you pull a suitcase by its handle at an angle, part of your pull moves it forward and part lifts it. Splitting the pull this way is resolution. Going the other way, two friends pulling a cart in different directions can be replaced by one imaginary pull doing the same job: that is composition.
Technically, a force F at angle \( \theta \) to the x-axis has rectangular components \( F_x = F\cos\theta \) and \( F_y = F\sin\theta \). The resultant R of a system is the single force producing the same external effect.
How it works
- Measure each force's angle from a fixed reference axis (usually +x).
- Resolve: \( F_x = F\cos\theta \), \( F_y = F\sin\theta \), with signs by quadrant.
- For two forces at an angle \( \alpha \) to each other you can also use the parallelogram law directly.
- Find the direction of R from the components or from the law's angle formula.
Key formula
Parallelogram law: \( R = \sqrt{P^2 + Q^2 + 2PQ\cos\alpha} \), \( \tan\beta = \dfrac{Q\sin\alpha}{P + Q\cos\alpha} \), where P, Q are the two forces (N), \( \alpha \) the angle between them, R the resultant (N) and \( \beta \) the angle R makes with P.
Components: \( F_x = F\cos\theta \), \( F_y = F\sin\theta \), \( F = \sqrt{F_x^2+F_y^2} \).
Quick example
P = 30 N and Q = 40 N act at 90° to each other. \( R = \sqrt{900 + 1600 + 0} = 50 \) N, and \( \tan\beta = 40/30 \), so \( \beta = 53.13^\circ \) from P. Check by components: taking P along x gives (30, 40), magnitude 50 N.
Common mistakes
- Using the sine component for the horizontal part when the angle is measured from the vertical; always check which axis the angle is measured from.
- Dropping signs in the 2nd, 3rd and 4th quadrants.
- Using \( \alpha \) as the angle of Q from the x-axis in the parallelogram formula instead of the angle between P and Q.
How MU asks it
Appears inside numerical problems rather than alone, and as a short 3 to 4-mark part: 'Two forces of ... N act at ... degrees; find the resultant'. Show the sketch, the formula, the substitution and give both magnitude and direction with a final diagram. Remember the prerequisite laws themselves are not examined separately, but you must use them correctly.