Inclined Plane Visualization | Friction: Does the Block Stay or Slide?

The idea: split gravity along the ramp

A block of mass m sits on a ramp tilted at an angle θ. Gravity pulls it straight down with its weight mg, but the block can only move along the ramp. So split mg into two parts: one along the slope and one straight into the ramp. The part into the ramp is cancelled by the ramp; the part along the slope is fought by friction. Who wins decides everything.

Free-body diagram of a block on a ramp at angle θ: the weight mg points straight down and splits into mg·sinθ down the slope and mg·cosθ into the ramp; the normal force N = mg·cosθ points out of the ramp and friction f points up the slope; the angle between mg and mg·cosθ is θ, the same as the ramp angle — The angle between mg and the line into the ramp is the ramp angle θ, so the parts are mg·sinθ along the slope and mg·cosθ into it.

Down the slope: mg·sinθ. Zero on flat ground, all of mg on a vertical wall.
Into the ramp: mg·cosθ. All of mg on flat ground, zero on a wall.

The normal force N = mg·cosθ

The ramp pushes back on the block, at right angles to its surface. That push is the normal force N ("normal" means perpendicular). The block does not sink into the ramp or jump off it, so the forces across the ramp cancel: N = mg·cosθ. On a steeper ramp N gets smaller, and so does the friction the ramp can give, because friction is proportional to N.

Static friction: just enough, up to μs·N

A block at rest feels static friction. It is not a fixed force: it gives exactly what is needed to hold the block, f = mg·sinθ, but never more than μs·N, where μs is the coefficient of static friction. So the block stays when

mg·sinθ ≤ μs·mg·cosθ      ⇔      tanθ ≤ μs

The mass cancels, and the test is about the angle only. The angle where tanθ = μs is the slip angle (or angle of repose): with μs = 0.5 it is arctan 0.5 ≈ 26.6°. At 25° the block holds with 8.29 N of a possible 8.89 N (Demo 1); at 27° it slips (Demo 2). Tilting a board until a coin starts to slide is how μs is measured.

Graph of force per unit weight against ramp angle from 0 to 60 degrees: the pull sinθ rises, the static limit 0.5·cosθ falls, they cross at the slip angle 26.6 degrees; the kinetic friction 0.3·cosθ is a lower curve; left of the crossing the block stays, right of it the block slides — The pull down the slope (mg·sinθ) grows with θ while the most friction can give (μs·mg·cosθ) shrinks; they cross at the slip angle.

Sliding: kinetic friction and a = g(sinθ − μk·cosθ)

Once the block moves, friction is kinetic: a fixed f = μk·N up the slope, with μk usually smaller than μs. The net force down the slope is mg·sinθ − μk·mg·cosθ, and Newton's second law F = m·a gives

a = g·(sinθ − μk·cosθ)

That a is constant, so from rest the distance along the ramp is x = ½·a·t² and the speed is v = a·t. The time to slide d = 4 m is t = √(2d/a). With the page's start values (θ = 30°, μk = 0.3): a = 9.81·(0.500 − 0.3·0.866) = 2.36 m/s², t = √(8/2.36) = 1.84 s, v = 4.34 m/s. Because kinetic friction is smaller than the static limit, a block that just breaks free jumps to a clear acceleration instead of creeping (Demo 2: 1.83 m/s² at 27°).

Special cases

case	what happens	on the page
tanθ ≤ μs, at rest	static friction holds it: f = mg·sinθ, a = 0	Demo 1: 25°, 8.29 N ≤ 8.89 N
no friction (μ = 0)	a = g·sinθ, the largest possible	Demo 3: 4.91 m/s², 1.28 s
any mass	every force is ∝ m, so a does not depend on m	Demo 4: 20 kg, still 2.36 m/s²
tanθ = μk, pushed	net force 0: constant speed	Demo 5: 45°, μk = 1, 4 m in 4 s
steep ramp	N is small, so friction matters less	Demo 6: 60°, 7.02 m/s², 1.07 s
tanθ < μk, pushed	a < 0: it slows and stops, then static friction holds it	Demo 7: stops at 1.51 m after 3.01 s

A pushed block uses x = v₀t + ½at² with v₀ = 1 m/s. It stops after t = v₀/|a| at x = v₀²/(2|a|) when a is negative and that comes before the 4 m mark.

Typical friction coefficients

surfaces	μs	μk
rubber on dry concrete	1.0	0.7–0.8
wood on wood	0.25–0.5	0.2
steel on steel (dry)	0.6	0.4
ice on ice	0.1	0.03

These are rough values; real μ depends on how clean, wet and smooth the surfaces are. The page refuses μk > μs, because kinetic friction is (almost) never stronger than static friction.

What the page leaves out

Air drag, a block that rolls or tips over instead of sliding, friction that changes with speed or temperature, stick–slip juddering, a ramp that moves, pushing the block up the slope (friction then points down), and the energy view (mg·h turns into ½mv² plus heat f·d). The block is treated as a point for the forces, so every arrow starts at its centre; the slide is always 4 m along the ramp, and g = 9.81 m/s².