# Toilet Paper Race Calculator (Rolling Down a Ramp)

URL: https://speedycalculator.com/calculators/physics/toilet-paper-race-mass-moment-inertia/
Last updated: 2026-10-30

> The toilet paper race calculator compares how fast different rolls – full, empty tube or solid cylinder – roll down a ramp, from their mass, inner and outer radii and the ramp height and length, using the moment of inertia.

## Inputs

- Shape
- Mass (g, kg, mg, lb, oz, gr)
- Inner radius r₁ (cm, mm, m, in, ft)
- Outer radius r₂ (cm, mm, m, in, ft)
- Ramp height H (m, cm, mm, km, in, ft, yd, mi)
- Ramp length L (m, cm, mm, km, in, ft, yd, mi)
- Gravitational acceleration g (m/s², ft/s², g, cm/s²)

## Results

- I/(m·r²)
- Moment of inertia
- Acceleration down the ramp
- Time to the bottom
- Speed at the bottom

## 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)))
```

## Explanation

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

FieldWhat to enterShapeChoose one: Solid cylinder, Thick tube (full roll), Thin-walled tube (empty roll)MassUnits: g, kg, mg, lb, oz, grInner radius r₁Units: cm, mm, m, in, ftOuter radius r₂Units: cm, mm, m, in, ftRamp height HUnits: m, cm, mm, km, in, ft …Ramp length LUnits: m, cm, mm, km, in, ft …Gravitational acceleration gUnits: 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)601.5985901.59851201.59851501.59851801.59852401.5985

## 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.

## FAQ

**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.


## Sources

- [OpenStax – University Physics Volume 1](https://openstax.org/details/books/university-physics-volume-1)
- [HyperPhysics (Georgia State University)](http://hyperphysics.phy-astr.gsu.edu/hbase/hframe.html)