Table of Contents
The Toilet Paper Rolling Race
Roll different cylinders down a ramp - which arrives first? The winner depends on mass distribution. Objects with mass near the center (solid cylinder, solid sphere) roll faster because less energy goes into rotation. A toilet paper roll is a hollow cylinder (I = m(R²+r²)/2) falling between a solid cylinder and thin shell.
The key parameter is I/(mR²): solid sphere = 0.4, solid cylinder = 0.5, toilet paper roll = ~0.59, thin shell = 1.0. Lower ratio means faster rolling because more kinetic energy goes into translation rather than rotation.
Formulas
Rolling Race Results (1m ramp)
| Object | I/(mR²) | Speed |
|---|---|---|
| Solid sphere | 0.400 | 3.74 m/s |
| Solid cylinder | 0.500 | 3.62 m/s |
| Toilet paper roll | ~0.59 | 3.51 m/s |
| Thin shell | 1.000 | 3.13 m/s |
FAQ
Does mass matter?
No! Mass cancels in rolling equations. A heavy and light solid cylinder of same radius finish together. Only shape (mass distribution) matters - analogous to all objects falling at the same rate regardless of mass.
Why does a full roll beat an empty tube?
A full roll has more mass near center relative to outer radius, lowering I/(mR²). An empty tube is a thin shell (ratio ~1.0). As paper is used, the roll gets relatively slower in a race.