On a ramp, weight splits into mg sin θ along the slope and mg cos θ into it. Friction decides whether the block slides — the maths, with a simulation.
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How does an inclined plane work?
Gravity always pulls straight down, but a ramp only lets a block move along its surface. So weight is split in two: a component along the slope, mgsinθ, that pulls the block downhill, and a component into the slope, mgcosθ, that the surface pushes back on (the normal force). Friction depends on that normal force, which is why the angle decides whether the block stays put or slides.
Key fact
A block on a ramp slides only when tanθ>μs. At the angle of repose, θ=arctanμs, gravity along the slope exactly equals the maximum static friction — and mass cancels out.
How do you split weight into components?
Tilt your axes to match the ramp: one along the slope, one perpendicular to it. The angle between the weight and the perpendicular axis equals the ramp angle θ, so:
At θ=0 (flat) all the weight presses on the surface and nothing pulls sideways. At θ=90° (vertical) the surface carries no weight and the block is in free fall. Every ramp lies between those extremes. Perpendicular to the slope the block does not move, so N cancels the weight's perpendicular component exactly.
Example: a 5 kg block on a 30° ramp
F∥=5×9.81×sin30°=24.5 N and N=5×9.81×cos30°=42.5 N. With μs=0.5 the maximum static friction is 0.5×42.5=21.2 N, which is less than 24.5 N — so the block slides.
When does the block start to slide?
Static friction can match the pull down the slope only up to μsN. Setting mgsinθ=μsmgcosθ and cancelling mg:
Set static friction to 0.5 and raise the ramp slowly. The block stays put until the angle passes about 26.6°, then starts to slide. Change the mass — the angle does not move.
How fast does it accelerate once it slides?
Once moving, friction drops to the kinetic value μkN. Along the slope, Newton's second law gives:
ma=mgsinθ−μkmgcosθ⟹a=g(sinθ−μkcosθ)
Mass cancels again. On a frictionless ramp a=gsinθ — the ramp simply slows down free fall. With θ=30° and μk=0.2, a=9.81(0.5−0.2×0.866)=3.21m/s2, so a block slides 2 m in about 1.1 s and reaches 3.6 m/s.
An applied force F at an angle adds components along and perpendicular to the ramp. Pulling up the slope reduces the net force downhill; pulling partly away from the surface also reduces N, and with it friction. Resolve every force along the two tilted axes and sum each axis separately — the method never changes.
Frequently asked questions
Why is the normal force mgcosθ and not mg?
The surface only has to cancel the part of the weight pushing into it. On a slope that is mgcosθ; the rest of the weight, mgsinθ, is along the surface and is what pulls the block downhill.
Does mass affect whether a block slides down a ramp?
No. Both the pull down the slope and the maximum friction are proportional to m, so it cancels: the block slides when tanθ>μs, whatever its mass.
What is the acceleration on a frictionless incline?
a=gsinθ, directed down the slope. At 30° that is half of g, about 4.9m/s2.
What is the difference between static and kinetic friction?
Static friction acts on a block that is not moving and adjusts up to a maximum of μsN. Kinetic friction acts on a sliding block and is roughly constant at μkN, usually smaller than the static maximum — which is why sliding starts with a small jerk.
Why does a ramp make lifting easier?
A ramp trades distance for force: the force needed to push a load up is about mgsinθ instead of mg, but you move it a longer distance. The work against gravity is the same.
