How to use this calculator
Pick the body and the situation, then enter the mass, radius and the incline angle or pulling force. Enter μs to check whether it slips (and μk for what happens if it does), and a distance for the speed reached from rest.
Formulas
No slipaG = αr
Inclinea = g sin θ/(1 + k²/r²)
FrictionF = IGa/r²
Slip checkF ≤ μsN
Energymgh = ½mv² + ½IGω²
Worked example
Solid cylinder, 20 kg, r = 0.3 m, on a 30° incline, μs = 0.25
- IG = ½mr², so k²/r² = 0.5.
- a = 9.81 sin 30°/1.5 = 3.27 m/s², α = a/r = 10.9 rad/s².
- F = mg sin θ − ma = 98.1 − 65.4 = 32.7 N and N = mg cos θ = 169.9 N, so μ needed = 0.192 < 0.25: it rolls.
- After 5 m from rest: v = √(2 × 3.27 × 5) = 5.72 m/s.
Frequently asked questions
Which rolls down fastest: a sphere, a cylinder or a hoop?
The one with the smallest k²/r²: a solid sphere (0.4), then a solid cylinder (0.5), then a hoop (1). Mass and radius cancel.
Does friction do work in rolling without slipping?
No. The contact point is momentarily at rest, so static friction does no work and energy is conserved.
What if it slips?
Then friction is kinetic, F = μkN, and the linear and angular accelerations are found separately; a ≠ αr.