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Inclined plane and friction calculator

Whether a body on a slope holds or slides: critical angle, force decomposition and acceleration, with the actual static friction kept apart from the maximum.

Parameters

Between 0 (a table) and 90 excluded (a wall).
Wood on wood ≈ 0.4; steel on steel ≈ 0.6; rubber on dry asphalt ≈ 0.9.
At most μs.
Earth: 9.80665. Moon: 1.62. Mars: 3.72.

Result

Equilibrium

The body slides

Calculation
holds if tan θ ≤ μs: tan θ = 0.57735, μs = 0.5
Sliding acceleration
a = g(sin θ − μk cos θ) = 9.8066 × (sin 30° − 0.4 × cos 30°) = 1.5062 m/s²
30°26.6°PP cos θP sin θf
Critical sliding angle
26.565 °
Driving component (parallel to the slope)
9.8066 N
Normal component (support reaction)
16.986 N
Maximum available static friction
8.4928 N
Kinetic friction
6.7942 N
Sliding acceleration
1.5062 m/s²
Notes (2)

The body slides, with an acceleration of 1.5062 m/s² along the slope.

The sliding acceleration, g(sin θ − μk cos θ), does not depend on the mass.

Calculation, assumptions and sources

OverviewA body on a slope with dry friction: holding or sliding by the criterion tan θ ≤ μs, critical sliding angle, decomposition of the weight into driving and normal components, sliding acceleration.

A body on a slope with dry friction: holding or sliding by the criterion tan θ ≤ μs, critical sliding angle, decomposition of the weight into driving and normal components, sliding acceleration. The static friction exerted is kept apart from the available maximum μs·N.

Method & formulascomponents: P∥ = mg sin θ, P⊥ = mg cos θ
components: P∥ = mg sin θ, P⊥ = mg cos θholds if: tan θ ≤ μscritical angle: θc = arctan(μs)sliding: a = g(sin θ − μk cos θ)

The component of the weight parallel to the slope drives the body; the normal component presses it against the support, and the available friction is proportional to it. The mass cancels in the holding criterion, which depends only on the angle and μs.

μs
static friction coefficient, which sets the maximum friction before any motion.
μk
kinetic friction coefficient, at most μs.
Critical angle
arctan(μs), the angle beyond which the body slides, independent of mass and gravity. Tilting the plane until it slides measures μs.
Normal reaction
the support force, perpendicular to the slope: mg cos θ on a simple plane.
Validity domainCoulomb dry friction: constant coefficients, independent of speed and contact area; a rigid body that slides without rolling or tipping.

Coulomb dry friction: constant coefficients, independent of speed and contact area; a rigid body that slides without rolling or tipping. Real coefficients depend on surface condition, humidity and temperature; table values are orders of magnitude. A tall narrow object may tip before it slides. The angle lies strictly between 0 and 90 degrees, and μk does not exceed μs.