Toilet Paper Race
Calculate mass moment of inertia for toilet paper roll race physics demo.
Toilet Paper Roll Race Physics
The classic toilet paper race demonstrates rotational inertia. Two rolls with the same mass and outer radius can accelerate differently down a ramp if one is a hollow cylinder and the other is solid. Mass distribution determines moment of inertia and therefore linear acceleration.
Hollow Cylinder Inertia
$$I = \frac{m(R^2 + r^2)}{2}$$For a thin-walled hollow cylinder, \(R\) is the outer radius and \(r\) is the inner radius. A solid cylinder of the same mass and outer radius has \(I = \frac{1}{2}mR^2\), which is always smaller, so the solid roll wins the race.
Rolling Acceleration
$$a = \frac{g\sin\alpha}{1 + I/(mR^2)}$$On a frictionless incline at angle \(\alpha\), a rolling object accelerates more slowly when rotational inertia is larger. The calculator compares solid and hollow rolls using a 30° reference incline.
Related tools: Gravitational Force Calculator and Toilet Paper Calculator.
Frequently Asked Questions
Why does the solid roll win the race?
A solid cylinder concentrates mass closer to the rotation axis, giving lower moment of inertia. More energy goes into linear motion rather than rotation, so it accelerates faster down the ramp.
What if the hollow roll has less mass?
Mass matters independently of shape. This calculator assumes equal mass for a fair comparison. A lighter hollow roll could win if its mass advantage outweighs the inertia penalty.
Does the inner radius of toilet paper matter?
Yes. A larger cardboard tube increases inner radius \(r\), raising moment of inertia and slowing the roll. A nearly full roll behaves more like a thick-walled cylinder.
What assumptions does the acceleration formula use?
It assumes pure rolling without slipping, no air resistance, and a uniform cylinder. Real toilet paper may unroll, which changes the effective inertia during the race.
Why is a 30° incline used for comparison?
The reference angle provides concrete acceleration values in m/s². The solid-versus-hollow ranking does not depend on the incline angle because both use the same ramp.
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