Moment of Inertia Designer

Which distances matter to rotational inertia? Mass farther from the chosen axis contributes more inertia: doubling a point mass’s distance makes its contribution four times larger. With the same torque, a more compact distribution accelerates faster. Shifting the axis away from the centre of mass adds the total mass times the offset squared.

Move mass pairs, change a continuous annular carrier and shift the bearing. Compare angular acceleration and rotational energy under the same torque.

Inertia about the bearing $0.326\,\mathrm{kg\,m^2}$

Angular velocity $2.66\,\mathrm{rad\,s^{-1}}$

Rotor angular momentum $0.867\,\mathrm{kg\,m^2\,s^{-1}}$

Motor work $1.15\,\mathrm{J}$

Model assumptions

A horizontal balanced rotor consists of a uniform annulus and four equal point masses on massless supports. Total mass is fixed when changing the bead fraction. The bearing is fixed in the inertial frame, with an optional body-fixed offset from the centre of mass. The rotor rotates about that bearing; the centre of mass then follows a circle. The bearing supplies constraint force and no axial torque or work. A constant pure couple drives rotation. Layout edits prepare a new rigid-body trial, not mass transfer during rotation. Compare the same mass, outer radius and torque with the ideal uniform disk and thin hoop curves. Bead size, support geometry, carrier thickness and surface markings are schematic. Friction, compliance and gravity within the horizontal plane are omitted. The displayed trial can end earlier at high acceleration; all quantities are simulated model outputs.

Focus question: How can you change angular acceleration without changing total mass or torque?
One-sentence intuition: Pick the axis first: a bead moved twice as far from it contributes four times as much, and moving the bearing adds the parallel-axis term.

Core mathematical model

Moment of inertia about the bearing

I=\int r_\perp^2\,\mathrm{d}m=I_{\mathrm{CM}}+Md^2

The rotor is balanced, so its mass centre sits at the centre of the carrier. The bearing offset is measured from that centre.

Uniform annulus and bead pairs

I_{\mathrm{CM}}=\frac{M(1-f)}2(R^2+R_{\mathrm{hole}}^2)+\frac{Mf}2(r_x^2+r_y^2)

The total bead mass is shared equally among four beads. The spokes that hold them have no mass.

Rotation under a constant couple

\dot\omega=\frac\tau I,\qquad\theta(t)=\omega_0t+\frac{\tau t^2}{2I}

During this trial only the angular speed changes; the mass distribution and the bearing stay fixed.

Work and angular impulse, checked separately

\Delta K=\tau\Delta\theta,\qquad\Delta L=\tau t,\qquad K=\frac12I\omega^2

Work and angular impulse use the same signed torque, but work builds up with the angle turned and angular impulse with the time elapsed.

Common difficulties

Same mass, different inertia

Typical misconceptionOnly the total mass determines rotational response.

Better mental modelCompare a layout with the beads close in and one with them at the rim, under the same torque. Once each mass is weighted by its squared distance from the axis, the totals differ.

A moment of inertia needs an axis

Typical misconceptionA body has one moment of inertia, whatever the axis.

Better mental modelShift the bearing and look at the parallel-axis contribution it adds.

Rearranging the beads starts a new run

Typical misconceptionDragging a bead during playback conserves angular momentum.

Better mental modelChanging the layout restarts this rigid-body trial. Reshaping a body while it spins would need a separate model for the work done by internal forces.

Run the experiment

01
Design a fast response
Move the bead pairs close to the axis and complete the torque trial.
What to observe: A smaller moment of inertia gives a larger angular acceleration.
02
Compare a rim and a disk
Use the rim preset and compare it with the disk reference curve.
What to observe: The same total mass placed farther from the axis speeds up more slowly.
03
Move the bearing
Change the bearing offset without changing the mass or the torque.
What to observe: The moment of inertia gains a parallel-axis term.
04
Brake, then reverse
Set a positive initial spin and a negative torque.
What to observe: The motor’s work first takes energy out; after the spin reverses, it puts energy back in.