How this calculator works
For rolling about the symmetry axis, a solid cylinder has I = mr^2/2, a tube has I = m(r_outer^2+r_inner^2)/2 and a thin shell has I = mr^2. Static friction couples translation and rotation on the ramp. Starting from rest, the ideal acceleration is g sin(theta)/(1+I/(mr^2)). The net force is ma; it is not the full downhill component of gravity. Adequate static friction is assumed, with no slipping or rolling resistance.
Formula
I_solid = mr^2/2; I_tube = m(ro^2+ri^2)/2; I_shell = mr^2. a = g sin(theta)/(1+I/(mr^2)); t = sqrt(2L/a); v = sqrt(2aL).
Worked example
A 1 kg solid cylinder with radius 0.1 m has I = 0.005 kg m2. At g = 10 m/s2 on a 30 degree ramp, acceleration is 3.333333 m/s2. Over 2 m it takes 1.095445 s from rest.
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Frequently asked questions
Does a heavier cylinder roll faster?
Mass cancels for the same shape and mass distribution in this ideal model.
Why does a thin shell roll more slowly?
More of its energy goes into rotation because its inertia relative to mr^2 is larger.
Can a toilet paper roll match this model?
A tightly wound rigid roll can be approximated by a tube. Unwinding, deformation and slipping change its motion.