M051 · Rotation and mass distribution

Moment-of-Inertia Designer

Which distances matter to rotational inertia? Distances from the rotation axis you choose. Each mass adds its mass times the square of that distance, so moving a bead twice as far out makes its share four times larger. Placing the axis some distance from the center of mass adds the total mass times that distance squared.

Build a three-dimensional rotor from a ring-shaped carrier and two symmetric pairs of beads. Drag the beads or the bearing in the coordinate view to set the starting layout while the total mass stays fixed, then apply a calibrated torque. Watch the bearing, the moving mass centre and the signed vectors along the axis. The parts of the moment of inertia, the angular speed and the work update together, and a faint curve shows a centred uniform disk of the same mass and radius for comparison.

Recorded time $0\,\mathrm{s}$

Inertia about the bearing $0\,\mathrm{kg\,m^2}$

Inertia about the mass center $0\,\mathrm{kg\,m^2}$

Radius of gyration about bearing $0\,\mathrm{m}$

Angular acceleration $0\,\mathrm{rad\,s^{-2}}$

Angular speed $0\,\mathrm{rad\,s^{-1}}$

Unwrapped rotation angle $0\,\mathrm{rad}$

Angular momentum about bearing $0\,\mathrm{kg\,m^2\,s^{-1}}$

Rotational kinetic energy $0\,J$

Applied couple work $0\,J$

Work-energy balance defect $0\,J$

Angular-impulse balance defect $0\,\mathrm{kg\,m^2\,s^{-1}}$

Centered equal-mass disk reference $0\,\mathrm{kg\,m^2}$

Centered equal-mass thin hoop reference $0\,\mathrm{kg\,m^2}$

Predict the motion, then test your design

Which distances matter to rotational inertia?

Signed axial vectorsAngular momentum, angular velocity and torque arrows each have their own fixed scale.Release positionsGold wireframes mark where the mass centers started. The motion is drawn at true scale, with no magnified displacement.Total mass

M

2.00\,\mathrm{kg}

Outer radius

R

0.60\,\mathrm{m}

Carrier hole radius ratio

R_{\mathrm{hole}}/R

0.00\,

Fraction of mass in beads

f

0.60\,

Horizontal bead distance ratio

r_x/R

0.80\,

Vertical bead distance ratio

r_y/R

0.45\,

Signed bearing offset ratio

d/R

0.00\,

Applied couple

\tau

0.40\,\mathrm{N\,m}

Initial angular speed

\omega_0

0.00\,\mathrm{rad\,s^{-1}}

Inspect recorded time

t/t_{\mathrm{end}}

0.00\,\%

Drag the hollow purple beads to set how far each pair sits from the center, and drag the red bearing to choose a parallel axis. Every drag has a matching keyboard slider. Compare layouts with the same mass, radius and torque. When the rotor turns off center, its kinetic energy includes the motion of its center of mass; the bearing itself stays fixed.

The rotor is an ideal rigid, flat body turning in a horizontal plane on a fixed bearing, driven by a pure couple. Each bead carries one quarter of the total bead mass. The ring-shaped carrier is uniform and the spokes are massless. Changing the layout resets the starting state instead of reshaping a spinning rotor. Under constant torque the motion is solved exactly, and the work and angular-impulse balances are checked independently.

Start with the essentials

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
Drag the red marker 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.