Splitting one force into two
A single force acting at an angle can be replaced by two perpendicular components. This is the opposite of finding a resultant. Resolving forces makes problems on slopes, ropes and pulls much easier to solve because we can treat the horizontal and vertical directions separately.
Key formula / 关键公式
For a force F at angle θ to the horizontal:
Horizontal component: Fx = F cos θ
Vertical component: Fy = F sin θ
Choosing the angle carefully
The component along the direction of the angle uses cos θ, while the component across it uses sin θ. On an inclined plane at angle θ, the weight W resolves into a component W sin θ parallel to the slope (pulling the object down the slope) and W cos θ perpendicular to the slope (pressing into it).
Worked example / 例题
A rope pulls a sledge with a force of 100 N at 30° above the horizontal. Find the horizontal and vertical components.
Fx = F cos θ = 100 × cos 30° = 100 × 0.866 = 86.6 N
Fy = F sin θ = 100 × sin 30° = 100 × 0.5 = 50 N
Worked example / 例题
A 20 kg box rests on a slope inclined at 30°. Find the component of weight down the slope (g = 10 m s⁻²).
Weight W = mg = 20 × 10 = 200 N.
Component down slope = W sin θ = 200 × sin 30° = 200 × 0.5 = 100 N.
Remember / 记住
- Along the angle → cos θ; across the angle → sin θ.
- On a slope: parallel = W sin θ, perpendicular = W cos θ.
- The two components are always at 90° to each other.