Formula and Method for a Toilet Paper Roll's Moment of Inertia
A toilet paper roll is, geometrically, a hollow cylinder (a "thick-walled tube" or annular cylinder): a solid ring of paper wound between an inner radius r₁ (the cardboard tube) and an outer radius r₂ (the outside of the roll), rotating about its own central axis. For any hollow cylinder of mass m, the mass moment of inertia about that central axis is I = ½m(r₁² + r₂²). This calculator also reports the radius of gyration, the dimensionless rolling-inertia ratio I/(mR²), and the linear acceleration the roll would have rolling down an incline — the physics behind a "toilet paper race."
How the calculation works
Enter the roll's mass and its inner and outer radii (with units), and the calculator converts everything to SI units (kilograms and meters) before applying I = ½m(r₁² + r₂²). It then derives the radius of gyration k = √(I/m) — the distance from the axis where a thin ring of the same mass would have identical inertia — and the ratio I/(mR²) using the outer radius R = r₂ as the rolling (contact) radius. Finally, for a roll released on an incline of angle θ and rolling without slipping, Newton's second law along the incline combined with the rotational equation of motion gives the linear acceleration a = g·sinθ / (1 + I/(mR²)), where g = 9.80665 m/s². A lower I/(mR²) ratio means less of the roll's gravitational potential energy is diverted into spin, so it accelerates faster — which is exactly why a full roll tends to out-roll a nearly empty one of the same outer size.
Common mistakes and edge cases
- Mixing units: enter both radii in the same unit selector; do not enter the outer radius in centimeters and the inner radius in inches.
- Inner radius must be smaller than outer radius: r₁ = r₂ describes a thin hoop (I = mr₂²), not a roll with paper on it — if they are equal there is no wound paper left.
- Setting r₁ = 0: this reduces the formula to a solid disk or cylinder, I = ½mr₂² — useful for modeling a bare spindle or a fully solid roll with no core hole.
- Rolling without slipping assumption: the downhill acceleration formula assumes the roll rolls (not slides) and ignores air resistance and rolling friction losses, so a real race may be slightly slower than the ideal result.