THE ROTATION LAB / CLASSICAL MECHANICS
Moment of Inertia Lab
Same mass. Different spin. Move the mass, turn the axis, and watch an integral become something you can see.
Every piece of mass has a vote.
Distance from the axis gives it weight.
TRY AN EXPERIMENT
Start with a disk and a hoop of equal mass and radius. The hoop puts all its mass farther from the axis.
MASS DISTRIBUTION
Solid disk
The coral line is the rotation axis. Warmer dots contribute more to I.
Small contributionLarge contributionRotation axis
EXACT MOMENT OF INERTIA1.000kg·m²
THROUGH CENTER OF MASS1.000kg·m²
ADDED BY AXIS SHIFT0.000kg·m²
The hoop has twice the inertia of this disk, even though mass and radius are the same.
03
Build the integral, one piece at a time
Each dot represents a finite mass element Δm at a sample point. Add its r⊥²Δm to the total. Finer partitions approach the continuous integral.
128 / 128 pieces included
Tap a dot above or use this slider. The white segment is the shortest distance to the axis.
- Mass Δm
- Distance r⊥
- Contribution ΔI
Where does the inertia come from?
Compare each distance band’s share of the total mass and the numerical inertia.
Mass shareInertia share
Read the distance-band values
| r⊥ band (m) | Mass | Inertia |
|---|
04
From the shape to the formula
The disk lies in the xy plane. The z axis is normal to its face.
The centered shape
Orientation matters
The parallel-axis theorem
What distance is actually squared?
The axis points along n = (sin β, 0, cos β) and is shifted by d(cos β, 0, −sin β). Distance is measured perpendicular to this line, not to a single point on it. The centered x and z moments combine because these shapes are symmetric about their coordinate planes.
05
What does moment of inertia do?
Apply the same torque to two bodies, both starting at rest. A larger I gives a smaller angular acceleration: .
Solid disk
- I (kg·m²)
- α (rad/s²)
- ω (rad/s)
- Angle (rad)
Thin hoop
- I (kg·m²)
- α (rad/s²)
- ω (rad/s)
- Angle (rad)
Ready. Both bodies start at rest.
Rigid bodies on fixed, constrained axes; friction neglected. Torque is the net component about each axis. Motion uses the exact integral result. Rotating shapes have a marked point so a symmetric disk or sphere’s rotation remains visible.
Three things to take with you
Double the mass with the same geometry and axis, and you double I.
Move a mass element twice as far from the axis and its contribution becomes four times larger.
A body has different moments about different axes. Tilting and shifting are different operations.
Continue learning: OpenStax, Calculating moments of inertia and Newton’s second law for rotation.