Forces are vectors
A force has magnitude (in newtons, N) and direction, so forces combine as vectors.
Resultant force
The resultant force is the vector sum of all forces.
Example: Two ropes pull a crate: 120 N east and 90 N north.
- Resultant = [120, 90]
- Magnitude: √(14,400 + 8,100) = 150 N
- Direction: tan⁻¹(90/120) ≈ 36.9° north of east
Using components
- Resolve each force into components: F cos θ and F sin θ.
- Add all x-components and all y-components.
- Find the magnitude and direction of the result.
Example: 50 N at 30° and 80 N at 120°
| Force | x | y |
|---|---|---|
| 50 N at 30° | 43.30 | 25.00 |
| 80 N at 120° | −40.00 | 69.28 |
| Sum | 3.30 | 94.28 |
Resultant ≈ √(3.30² + 94.28²) ≈ 94.3 N at about 88.0°.
Equilibrium
An object is in equilibrium when the net force is zero:
F₁ + F₂ + F₃ + … = 0
The force that balances a resultant is the equilibrant: equal in size and opposite in direction.
Example: A 20 kg sign hangs from two cables at 30° above the horizontal on each side.
- Weight: 20 × 9.8 = 196 N downward
- By symmetry, each cable has tension T.
- Vertical balance: 2T sin 30° = 196 → T = 196 N
Inclined planes
On a ramp at angle θ, a weight mg splits into:
- Along the ramp: mg sin θ
- Perpendicular to the ramp: mg cos θ
Example: 30 kg box on a 25° ramp → force down the slope ≈ 30 × 9.8 × sin 25° ≈ 124 N.
Relative velocity: navigation
Relative velocity combines an object’s velocity through a medium with the medium’s velocity.
Example: A boat heads straight across a river at 4 m/s; the current flows at 3 m/s.
- Ground velocity: [3, 4] → 5 m/s at tan⁻¹(3/4) ≈ 36.9° downstream from straight across.
Aiming to go straight across: The boat must aim upstream so its upstream component cancels the current: sin θ = 3/4 → θ ≈ 48.6° upstream. Crossing speed = √(16 − 9) ≈ 2.65 m/s.
Aircraft and wind
A plane flies at 400 km/h on a bearing of 090° (east); wind blows from the south at 50 km/h (toward the north).
- Ground velocity: [400, 50] → ≈ 403 km/h, bearing ≈ 083°.
Work
Work done by a force over a displacement: W = F · d (dot product).
Worked examples
Example 1: Forces 30 N east and 40 N west. Resultant? 10 N west.
Example 2: Equilibrant of a 100 N force at 45°? 100 N at 225°.
Example 3: A 60 N force at 60° above the horizontal. Horizontal component? 30 N.
Common mistakes
- Adding magnitudes instead of vectors.
- Wrong signs on components.
- Confusing the resultant and the equilibrant.
Try it yourself
- Resultant of 5 N north and 12 N east?
- Equilibrant of a 25 N force pointing south?
- A boat at 6 km/h straight across a 2 km/h current. Ground speed?
- Force down a 30° ramp for a 50 kg mass (g = 9.8)?
Answers: 1) 13 N, about 67.4° east of north 2) 25 N north 3) ≈ 6.32 km/h 4) 245 N
Quick recap
- Forces are vectors: find the resultant force with components.
- Equilibrium means the net force is zero.
- Vectors solve ramp, relative velocity and navigation problems.