The steepest a loose heap will stand. It is the critical angle, so a material’s repose angle tells you its effective friction directly: μ = tan(repose).
Gravity, the normal force and friction. Resolved along the slope: a = g(sinθ − μcosθ) once sliding. Velocity and position are integrated frame by frame.
It slides if tan θ exceeds the static friction coefficient; otherwise it sits however long you wait. Mass is on both sides and cancels.
A rolling body spins up as well as moves: a = g sinθ ÷ (1 + k), with k = 2/5 for a solid sphere and 1 for a hoop.
The steepest a loose heap will stand: μ = tan(repose), about 34° for dry sand, 45° for gravel.
Whether a block slides depends on tan(angle) against the friction coefficient: mass appears on both sides and cancels. A tonne and a matchbox let go on the same slope start moving at the same angle and slide with the same acceleration: the most counter-intuitive thing about slopes. Mass still matters for the forces involved, not the acceleration.
Rolling needs friction to supply the torque: μ ≥ tanθ × k/(1+k), 0.17 for a solid sphere and 0.29 for a hoop at 30°. Below it the body slips, stops spending energy on spin, and reaches the bottom faster: friction speeding something up. The order never changes with angle: solid sphere, solid cylinder, hollow sphere, hoop.
It divides the force by up to 1/sinθ but multiplies the distance by the same factor: the work is unchanged. With friction a ramp costs energy and the gentler it is the worse: pushing up a 1:12 accessibility ramp at μ = 0.2 puts under 30% of your effort into height. It is still the right design, because the constraint was the force a person can apply, not the energy they spend.
A percentage gradient is the tangent: ten degrees is 17.6%. Builders use 1:N, the reciprocal of the same tangent, so 1:12 is 8.3% is 4.76°. Mixing them up is the commonest practical error with gradients; the page shows all three. A 45° gravel batter is stable and a 50° one is not.
Static friction (μₛ) resists the start of motion; kinetic (μₖ) acts once sliding and is almost always slightly lower: why a stuck object needs a firm push but slides more easily once moving. Right at tanθ ≈ μₛ the equilibrium is unstable: the smallest vibration tips it into sliding, so real designs build in a margin.
Because a rolling body has to spin up as well as move forward, and both come out of the same gravity. The split depends only on how far the mass sits from the axis: acceleration is g·sinθ ÷ (1 + k), where k is 2/5 for a solid sphere and 1 for a hoop. So a hoop gets exactly half the acceleration a frictionless block would, and a sphere keeps five sevenths. Mass and radius cancel out of that formula completely — a marble and a cannonball of the same shape finish together, and always have.
Yes, and it then arrives sooner. Rolling without slipping needs the surface to supply the torque, which takes μ ≥ tanθ × k/(1+k). At 30° a solid sphere needs only 0.17 but a hoop needs 0.29. Below that it slides, stops spending energy on spin, and speeds up — one of the few situations where less friction makes something arrive later, not earlier.
The steepest a loose heap will stand: about 34° for dry sand, 45° for gravel. It is the critical angle for that material, so it measures friction directly — μ = tan(repose). This is why a pile of sand always forms the same cone whatever you do to it, and why a 45° gravel batter is stable and a 50° one is not.
No. It divides the force by up to 1/sinθ, but multiplies the distance by exactly the same factor, so the work is unchanged in the ideal case. With friction a ramp costs energy, and the gentler the slope the worse it gets: pushing something up a 1:12 accessibility ramp at μ 0.2 puts under 30% of your effort into height. That is still the correct design, because the constraint was the force a person can apply, not the energy they spend.
Because a percentage gradient is the tangent, not the angle: rise over run, times 100. Ten degrees is 17.6%. Builders use a third notation, 1:N, which is the reciprocal of the same tangent — so 1:12 is 8.3% is 4.76°. All three describe one slope, and this page shows all three at once because mixing them up is the most common practical error with gradients.
Static friction (μₛ) resists the start of motion — it's what holds a stationary block in place. Kinetic friction (μₖ) acts once something is already sliding, and is almost always slightly lower. That's why a stuck object needs a firm push to get going, but slides more easily once moving.
tanθ is the ratio of the gravity component pulling the block down the slope to the component pressing it into the slope. μₛ is the maximum friction force available, as a fraction of that same pressing force. If gravity's pull exceeds friction's maximum grip, the block moves — otherwise it doesn't.
Both the gravity component along the slope and the normal force scale with mass in exactly the same way, so mass cancels out of the acceleration entirely — a heavier block doesn't slide down faster or slower than a lighter one with the same friction coefficients. (Mass still matters for the actual forces involved, just not for the resulting acceleration.)
In theory it's perfectly balanced and stays put; in practice this is an unstable equilibrium — the smallest disturbance (a vibration, a slightly uneven surface) tips it into sliding. Real designs build in a margin rather than running right at the limit.
Releases in which this page changed, newest last. Derived from the archived copy of every release, not from notes written afterwards — so it reflects what actually shipped. Site-wide passes are left out; they are in the full changelog.