Toilet paper race mass moment of inertia

A toilet-paper race is the classroom demo in which two rolls compete down a ramp, or drop from a height while one unrolls and the other stays a rigid body. The number that decides who wins is not mass alone: it is the mass moment of inertia of the roll about its cardboard axis, because rolling without slip couples translation to rotation through I. This calculator models a toilet-paper roll as a uniform hollow cylinder, takes the measured mass and the inner and outer radii of the remaining paper, and returns I = (1/2) m (ri² + ro²) in kilogram square metres.

The hollow-cylinder formula a toilet-paper race actually uses

A toilet paper roll is not a thin hoop, a solid disc, or a sphere. After you peel away the remaining paper, what is left is a thick-walled tube: a cardboard core of inner radius ri (often the outer radius of the empty tube) and an outer radius ro of the last wrap of paper, with mass m distributed through the annulus. For a uniform-density hollow cylinder rotating about its central longitudinal axis, the mass moment of inertia is the closed form I = (1/2) m (ri² + ro²).

That identity is the integral of r² dm over the annulus; the length of the roll cancels, so you never need the width of the paper, only mass and the two radii. When ri is much smaller than ro the roll behaves like a solid disc of inertia (1/2) m ro²; when the paper is almost gone and ri approaches ro, I approaches m ro², the thin-hoop limit.

Those two extremes are why a nearly empty roll and a full jumbo roll do not race the same way even if you scale mass out. In a ramp race with rolling without slip the linear acceleration is a = g sinθ / (1 + I / (m R²)), and the contact radius R is the outer radius ro. The dimensionless factor k = I / (m ro²) therefore equals (ri² + ro²) / (2 ro²), which always sits between one half and one for a physical roll.

A smaller k means a larger share of the lost potential energy goes into translation rather than spin, so that roll reaches the bottom first. This calculator returns both I and that inertia factor so a lab notebook can plug k straight into the race formula without a second rearrangement.

It does not integrate an unrolling stream of paper, model kinetic friction on the cardboard, or convert the result into a predicted finish time: those steps need the ramp angle, the length, and a statement of whether the paper is free to unwind. What it does is give the rigid-roll I that every such model starts from, with SI units and a deterministic round so two groups measuring the same roll get the same number.

Searchers who want a generic ring, disc, sphere, or rod table are on a different landing; this page only answers the toilet-paper-race hollow cylinder from mass and the two radii.

Input contract, SI units, and why inner radius must be smaller

Optional leading plus signs and scientific notation on numeric strings are allowed when they parse as finite decimals; embedded spaces, hex prefixes, and non-numeric tokens are rejected with a message that names the field. Mass must be strictly positive: a zero or negative mass is not a roll and is not billed as a successful zero inertia. Inner radius must be non-negative; outer radius must be strictly positive.

Equal radii would be a mathematical thin hoop, but a physical toilet-paper roll always has a finite wall, and the race landing refuses to silently switch formulas. Units are SI and are not converted. A typical household roll has an outer radius near five centimetres, which you must send as 0.05 m, and a cardboard core near two centimetres, sent as 0.02 m; sending millimetres as if they were metres inflates I by a factor of one million squared and will not be silently corrected.

There is no Monte Carlo sampling of ply density and no random rounding: the engine either evaluates (1/2) m (ri² + ro²) inside the supported magnitude range or rejects the input. A very small positive outer radius is valid and yields a correspondingly small I; it is not treated as zero. Keep requests inside the documented absolute range.

Where a toilet-paper-race inertia number earns its keep

Introductory mechanics labs stage toilet-paper races because the apparatus is cheap, the geometry is visible, and the result depends on rotational inertia rather than on who has the heavier roll. Students caliper the remaining paper, weigh the roll, compute I, then predict whether a full roll or a nearly spent one should win a no-slip ramp race from the factor k. Lecturers can pre-generate answer keys for arbitrary measured radii without trusting opaque calculator apps that silently switch millimetres and metres or apply a solid-disc formula to a hollow core.

Developers call the same contract from scripts to seed deterministic test vectors for rolling-without-slip demos, to normalize user-entered roll specs before a UI plots acceleration versus remaining radius, or to satisfy programming-challenge statements that demand the hollow-cylinder I rather than a hand-waved sketch. Contest practice often needs a fast, offline source of I and k so solutions can be checked without graphing calculators under time pressure.

Product and education tools use the closed form when a learner types mass and two radii and the UI must show I, k, and a clean formula reminder. Pair it with a generic moment-of-inertia calculator when the search is a ring, disc, sphere, or rod, and with a polar-moment calculator when the search is shaft torsion in metres to the fourth, not mass inertia.

Do not use this page for unrolling-drop times, yo-yo descent, or variable-mass paper leaving the roll: those problems change the Lagrangian and would silently mis-rank a race if forced through a rigid hollow cylinder. Maintenance and reliability scripts can batch measured rolls from a lab period, reject rows with ri not less than ro, and attach I and k to each specimen before the race.