Lesson 3 · 16 min
Friction and ramps
The inclined plane is not a puzzle. It is a fixture, a hopper, a road grade, a wedge.
Resolve the weight
On a ramp of angle θ, weight mg splits into mg sinθ down the plane and mg cosθ into the plane. The normal is usually N = mg cosθ (no other perpendicular forces). Kinetic friction is μk N, opposite velocity. Static friction is ≤ μs N, opposite impending slip.
a = g (sin θ − μk cos θ)
If sinθ > μs cosθ, static friction cannot hold it. That is the angle of repose: θc = arctan μs.
Worked example
Pallet on a loading dock
- Given: θ = 12°
- Given: μs = 0.30
- Given: μk = 0.22
- Given: m = 80 kg
Find: Does it sit? If not, a?
- Compare tanθ to μs. tan 12° ≈ 0.213, μs = 0.30. 0.213 < 0.30, so it sits.
- If someone greases it so μs drops to 0.18: tanθ > μs, it goes.
- Then a = g(sin12° − 0.18 cos12°) ≈ 9.81(0.208 − 0.176) ≈ 0.31 m/s²
Sits at μs = 0.30. With μs = 0.18, slides at 0.31 m/s².
Check
The critical angle at which a block starts to slide depends on
Drive the numbers. Then go back to the algebra.
Bench
Incline
θc = arctan μs
- N
- 43.3 N
- mg sinθ
- 23.0 N
- fs max
- 17.3 N
- a
- 2.01 m/s²
Angle of repose 21.8°. The block is sliding.