Forces on a slope
An object resting on a slope feels its weight, m × g, pulling straight down. It helps to split that weight into two parts:
- along the slope: F∥ = m × g × sin θ, which tends to make the object slide;
- into the slope: F⊥ = m × g × cos θ, balanced by the normal force of the surface.
Friction opposes sliding, up to a maximum of μ × N, where μ is the coefficient of friction and N the normal force.
Does it slide?
The object stays put as long as friction can balance the pull along the slope, which happens while tan θ ≤ μs, the static coefficient. The critical angle is therefore arctan(μs). Once it slides, friction drops to the kinetic value and the acceleration is:
a = g × (sin θ − μk × cos θ)
From rest, after sliding a distance d, its speed is v = √(2 × a × d).
Worked example
A 10 kg crate on a 30° ramp with a friction coefficient of 0.2, sliding 2 m:
| Result | Value |
|---|---|
| Weight along the slope | 49.03 N |
| Normal force | 84.93 N |
| Maximum friction | 16.99 N |
| Acceleration | 3.20 m/s² |
| Speed after 2 m | 3.58 m/s |
| Force to push it up at constant speed | 66.02 N |
| Angle at which it starts to slide | 11.3° |
Without friction, the acceleration would be half of g, 4.90 m/s².
Typical friction coefficients
| Surfaces | Static μ (approx.) |
|---|---|
| Rubber on dry concrete | 0.6 to 0.85 |
| Wood on wood | 0.25 to 0.5 |
| Steel on steel, dry | about 0.5 to 0.8 |
| Steel on ice | about 0.03 |
| Teflon on Teflon | about 0.04 |
The kinetic coefficient is usually somewhat lower than the static one, which is why it takes more force to start something moving than to keep it moving.
The mechanical advantage of a ramp
Without friction, pushing an object up a ramp takes a force of m × g × sin θ instead of m × g to lift it straight up. On a 30° ramp, that is half the force, over twice the distance: the work is the same, but the effort is smaller. Friction adds to the force needed going up.
Tips
- Mass cancels out of the acceleration: heavy and light objects slide alike, apart from air resistance.
- Angles are in degrees. A 1-in-12 wheelchair ramp is about 4.8°.