Race a full toilet roll against an empty one down a ramp: the full one usually wins! Objects with more of their mass near the axis turn more easily, so more of their energy goes into moving forward.
How to use the Toilet Paper Race Calculator
- Shape
- Mass
- Inner radius r₁
- Outer radius r₂
- Ramp height H
- Ramp length L
- Gravitational acceleration g
Results update as you type, every field has its own unit menu, and everything is calculated in your browser.
How it works
Choose the shape, enter the mass, radii and the ramp dimensions.
Inputs explained
| Field | What to enter |
|---|---|
| Shape | Choose one: Solid cylinder, Thick tube (full roll), Thin-walled tube (empty roll) |
| Mass | Units: g, kg, mg, lb, oz, gr |
| Inner radius r₁ | Units: cm, mm, m, in, ft |
| Outer radius r₂ | Units: cm, mm, m, in, ft |
| Ramp height H | Units: m, cm, mm, km, in, ft … |
| Ramp length L | Units: m, cm, mm, km, in, ft … |
| Gravitational acceleration g | Units: m/s², ft/s², g, cm/s² |
What each result means
- Moment of inertia = [I÷(m·r²)]×[Mass]×[Outer radius r₂]^2
- Acceleration down the ramp = [Gravitational acceleration g]×[Ramp height H]÷[Ramp length L]÷(1 + [I÷(m·r²)])
- Time to the bottom = sqrt(2×[Ramp length L]÷([Gravitational acceleration g]×[Ramp height H]÷[Ramp length L]÷(1 + [I÷(m·r²)])))
- Speed at the bottom = sqrt(2×[Gravitational acceleration g]×[Ramp height H]÷(1 + [I÷(m·r²)]))
Example values
How the main result changes when Mass changes and everything else stays at the example values above:
| Mass (g) | Time to the bottom (s) |
|---|---|
| 60 | 1.5985 |
| 90 | 1.5985 |
| 120 | 1.5985 |
| 150 | 1.5985 |
| 180 | 1.5985 |
| 240 | 1.5985 |
Formula
Solid cylinder: I = ½mr² · Thick tube: I = ½m(r₁² + r₂²) · Thin shell: I = mr²
a = g·sinθ/(1 + I/(mr₂²)), t = √(2L/a), v = √(2gH/(1 + I/(mr₂²)))
Worked example
Inputs — Shape: Thick tube (full roll); Mass: 120 g; Inner radius r₁: 2 cm; Outer radius r₂: 5.5 cm; Ramp height H: 0.5 m; Ramp length L: 2 m; Gravitational acceleration g: 9.80665 m/s².
Results — I/(m·r²) = 0.5661; Moment of inertia = 2.055 kg·cm²; Acceleration down the ramp = 1.56544 m/s²; Time to the bottom = 1.5985 s; Speed at the bottom = 2.50235 m/s.
Key terms
- Moment of inertia
- Resistance to angular acceleration.
Frequently asked questions
Why does a full roll beat an empty one?
Its mass is closer to the axis, giving a smaller I/(mr²).
Does mass matter?
No – only the distribution of mass.
Does the ramp length matter?
It changes the time but not the order of the race.
What about slipping?
The calculation assumes rolling without slipping.