Inclined plane and friction
components: P∥ = mg sin θ, P⊥ = mg cos θ
The canonical mechanics problem: a body on a slope, with dry friction. The tool first settles the real question, does it hold or slide, through the criterion tan θ ≤ μs; gives the critical sliding angle, which depends neither on mass nor on gravity; decomposes the weight into driving and normal components; and serves the sliding acceleration when it exists, never when the body holds. It carefully separates the static friction actually exerted, which exactly balances the slope, from the available maximum μs·N, the most frequent confusion of the chapter.
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) = 9.8067 × (sin 30° − 0.4 × cos 30°) = 1.5062 m/s²
The body slides, with an acceleration of 1.5062 m/s² along the slope.
The critical angle arctan(μs) depends neither on the mass nor on gravity: the full crate and the empty crate slide at the same angle, on Earth as on the Moon. The mass cancelled between the weight that pushes and the weight that presses. Measuring that angle is in fact the simplest way to measure μs: tilt until it slides.
The sliding acceleration does not depend on the mass either: g(sin θ − μk cos θ), the mass cancelled again. Heavier does not rush down faster.
The model is Coulomb dry friction: constant coefficients, independent of speed and contact area, a rigid body sliding without rolling. Real coefficients depend on surface condition, humidity and temperature; table values are orders of magnitude, not constants of nature.
Scientific dossier
What the tool computes, what it assumes, where it stops being valid, and where its data comes from.
Method & formulascomponents: P∥ = mg sin θ, P⊥ = mg cos θ
components: P∥ = mg sin θ, P⊥ = mg cos θ
holds if: tan θ ≤ μs
critical angle: θc = arctan(μs)
sliding: a = g(sin θ − μk cos θ)
The whole problem lies in decomposing the weight: the parallel component pushes the body down the slope, the normal component presses it against it, and the available friction is proportional to the latter. The holding criterion compares the two, and the mass cancels out: only the angle and the coefficient remain.
- μs
- · static friction coefficient: the contact’s maximum holding capacity, before any motion.
- μk
- · kinetic coefficient, always at most μs: sliding takes less than breaking loose.
- Critical angle
- · arctan(μs), the angle beyond which the body slides. Independent of mass.
- Normal reaction
- · the support force, perpendicular to the slope: mg cos θ on a simple plane.
Validity domainThe model is Coulomb dry friction: constant coefficients, independent of speed and contact area, a rigid body that slides without rolling or tipping.
The model is 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, which this criterion does not cover; a rolling wheel is rolling, not sliding. μk above μs is refused as inconsistent, and the angle lives strictly between 0 and 90 degrees.
Reading the resultThe distinction that matters: as long as the body holds, static friction is not at its maximum: it equals exactly what balances the driving component, no more, no less.
The distinction that matters: as long as the body holds, static friction is not at its maximum: it equals exactly what balances the driving component, no more, no less. The maximum μs·N is a capacity, not a value. And two different masses behave identically: same critical angle, same sliding acceleration, the mass having cancelled everywhere. That is why measuring the sliding angle is the simplest way to measure μs.