Why doesn\'t mass affect the rolling race?

In the ratio c = I/(mR²), mass cancels out. Both a heavy and light solid cylinder have c = 0.5, so they accelerate identically. Only the geometry — how mass is distributed relative to the axis — matters.

What shape wins the rolling race?

A solid sphere (c = 0.4) always wins because it has the smallest fraction of energy in rotation. Ranking: solid sphere > solid cylinder > hollow sphere > thick ring > thin hollow cylinder.

Does a full or empty toilet paper roll win?

A full roll wins. When nearly full, the TP roll\'s c is close to 0.5 (solid cylinder). As paper is used, c increases toward 1.0 (thin tube), making it slower.

What if the object slides instead of rolls?

If there\'s not enough friction the object slides, and all shapes descend at the same rate (a = g sin θ). The race differences only appear with pure rolling.

Can I use this for classroom demonstrations?

Yes! The rolling race is a classic physics demo. Use a rigid board as a ramp (30° works well), release objects simultaneously from the top, and compare to the calculator\'s predictions.

What is the moment of inertia physically?

Moment of inertia is the rotational equivalent of mass — it measures resistance to angular acceleration. Objects with mass far from the rotation axis have higher I and are harder to spin up.

Toilet Paper Race & Moment of Inertia Calculator

Race different rolling objects down a ramp. Calculate moment of inertia for solid and hollow cylinders, spheres, and toilet paper rolls. See which shape wins!

Object Mass (kg)

Outer Radius (mm)

Inner Radius (mm)

Incline Angle (°)

Ramp Length (m)

Moment of Inertia (TP)

175.4500

I = ½m(R² + r²)

c = I/(mR²)

0.5800

Inertia parameter — higher = slower rolling

Rolling Acceleration

3.104

a = g·sin(30°)/(1+c)

Speed at Bottom

2.492

Pure rolling, no slip

Time Down Ramp

0.803

1.00 m ramp at 30°

Race Winner

Solid Sphere

Finishes in 0.756 s

Race Loser

Hollow Cylinder (Thin)

Finishes in 0.903 s

Winning Margin

0.148

16.3% slower

🏁 Rolling Race — Time to Finish

1. Solid Sphere

0.756s

2. Solid Cylinder

0.782s

3. Toilet Paper Roll

0.803s

4. Hollow Sphere (Thin)

0.824s

5. Hollow Cylinder (Thin)

0.903s

Rank	Object	c = I/(mR²)	Accel (m/s²)	Speed (m/s)	Time (s)
1	Solid Sphere	0.400	3.504	2.647	0.756
2	Solid Cylinder	0.500	3.270	2.557	0.782
3	Toilet Paper Roll 🧻	0.580	3.104	2.492	0.803
4	Hollow Sphere (Thin)	0.667	2.942	2.426	0.824
5	Hollow Cylinder (Thin)	1.000	2.452	2.215	0.903

Why do different shapes roll at different speeds?

When an object rolls down a ramp, gravitational potential energy converts into both translational and rotational kinetic energy. Objects with more mass concentrated far from the center (high moment of inertia) use more energy for spinning, leaving less for forward speed.

A solid sphere (c = 0.4) beats a solid cylinder (c = 0.5), which beats a hollow sphere (c = 0.667), which beats a thin hollow cylinder (c = 1.0). A toilet paper roll falls between a solid and hollow cylinder depending on how much paper remains.

Crucially, mass doesn\\\'t matter — a heavy and light object of the same shape arrive together! Only the mass distribution (the ratio c = I/mR²) determines speed.

Planning notes, formulas, and examples

About the Toilet Paper Race & Moment of Inertia Calculator

The Toilet Paper Race & Moment of Inertia Calculator lets you simulate the classic physics demonstration: which object wins a rolling race down a ramp? The answer depends not on mass but on how that mass is distributed — the moment of inertia. A solid sphere always beats a solid cylinder, which beats a hollow sphere, which beats a thin pipe. But where does a toilet paper roll fit in?

A toilet paper roll is a thick-walled hollow cylinder whose moment of inertia I = ½m(R² + r²) changes as you use up paper. A full roll (large outer radius, small inner radius) behaves almost like a solid cylinder. A nearly empty roll (outer radius approaching inner radius) mimics a thin hollow cylinder — the slowest possible roller. This calculator lets you set the outer and inner radii, choose an incline angle and ramp length, then race the toilet paper against standard shapes.

This calculator teaches a core concept in rotational physics: when objects roll without slipping, gravitational energy splits between translational and rotational kinetic energy. The fraction going to rotation depends solely on c = I/(mR²), making mass irrelevant and shape everything.

When This Page Helps

This calculator brings a beloved physics demonstration to life with precise calculations. Students and teachers can predict race outcomes before running the experiment, then verify their theoretical understanding against real results. Engineers use moment of inertia calculations for flywheel design, rotating machinery, and robotic wheel sizing.

The visual race chart and ranking table make the abstract concept of rotational inertia immediately intuitive: you can see exactly how much each shape\'s mass distribution slows it down.

How to Use the Inputs

Enter the mass of your object in kilograms (or use a preset).
Set the outer radius (overall roll size) in millimeters.
Set the inner radius (the cardboard tube) in millimeters. Use 0 for a solid cylinder.
Choose the incline angle of your ramp in degrees.
Set the ramp length in meters.
View the race results — objects ranked fastest to slowest.
Explore the race visualization and comparison table to understand why shapes matter.

Formula used

Moment of inertia (thick ring/toilet paper): I = ½m(R² + r²), where R = outer radius, r = inner radius.
Inertia parameter: c = I / (mR²).
Rolling acceleration on incline: a = g × sin(θ) / (1 + c).
Speed at bottom of ramp: v = √(2aL), where L = ramp length.
Time to roll down: t = √(2L / a).
For comparison shapes: solid cylinder c = 0.5, solid sphere c = 0.4, hollow sphere c = 2/3, thin hollow cylinder c = 1.0.

Example Calculation

Result: I = 17.65 × 10⁻⁶ kg·m², c = 0.540, time = 0.786 s, solid sphere wins the race

A standard toilet paper roll with 55 mm outer radius and 22 mm inner radius has c ≈ 0.54, making it slightly slower than a pure solid cylinder (c = 0.5) but much faster than a thin hollow cylinder (c = 1.0). On a 1 m ramp at 30°, the solid sphere finishes first in about 0.764 s while the TP roll takes about 0.786 s.

Tips & Best Practices

For the best classroom results, use a smooth, rigid ramp at 20–30° — too steep causes sliding.
Release objects at the same time by holding them behind a ruler, then lifting the ruler.
A can of soup (solid-ish) always beats an empty can (hollow) — demonstrate with both!
Measure your toilet paper roll\'s radii with a ruler to get accurate predictions.
The inner radius of a standard toilet paper tube is about 22 mm (44 mm diameter).
Try the experiment with a nearly empty roll vs. a fresh one to see the theory in action.

The Physics of Rolling

When an object rolls without slipping, static friction at the contact point prevents sliding while redirecting gravitational potential energy into two forms: translational kinetic energy (½mv²) and rotational kinetic energy (½Iω²). The constraint v = ωR links these, giving total KE = ½mv²(1 + c) where c = I/(mR²). Higher c means more energy absorbed by spinning, less available for moving forward.

Toilet Paper as a Physics Laboratory

A toilet paper roll is an ideal teaching tool because its moment of inertia changes continuously as paper is removed. With each sheet torn off, the outer radius decreases while mass decreases proportionally, causing c to gradually shift from ~0.53 (full) toward 1.0 (empty tube). Students can chart this progression by weighing and measuring rolls at different usage stages, creating a hands-on connection between geometry and dynamics.

Beyond the Classroom

Moment of inertia calculations are critical in engineering: flywheels store energy proportional to I, robotic wheels\' acceleration depends on I, satellite spin stabilization requires precise I calculation, and automotive driveshafts must balance rotating mass to avoid vibration. The same c = I/(mR²) parameter that determines which toilet paper roll wins a race also determines how quickly an industrial roller reaches operating speed.

Sources & Methodology

Last updated: January 15, 2025

Frequently Asked Questions

In the ratio c = I/(mR²), mass cancels out. Both a heavy and light solid cylinder have c = 0.5, so they accelerate identically. Only the geometry — how mass is distributed relative to the axis — matters.