M051 · Rotation / mass distribution

Moment-of-Inertia Designer

Build a balanced rotor from an annular carrier and two symmetric bead pairs. Drag the hollow release beads or the bearing, keep total mass fixed, then apply a calibrated torque. The inertia budget, angular speed and energy ledger update together; a faint curve compares a centered uniform disk of equal mass and radius.

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 $0\,\mathrm{kg\,m^2}$

DESIGN / PREDICT / VERIFY

Which distances matter to rotational inertia?

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\,\%

Hollow purple beads set pair distances; the red bearing sets a parallel axis. All dragging has equivalent keyboard sliders. Compare layouts at equal mass, radius and torque. The energy of an off-center rotor includes its moving mass center; the bearing stays fixed.

Ideal rigid planar rotor in a horizontal plane; a fixed bearing and a pure applied couple. Each bead has one quarter of the total bead mass. The annular carrier is uniform; spokes are massless. Changing the layout resets the release rather than deforming a spinning rotor. Constant torque has an exact solution, with independent work and angular-impulse balances.

Start with the essentials

Focus question: How can you change angular acceleration without changing total mass or torque?
One-sentence intuition: Specify the axis first. Moving a bead twice as far from that axis multiplies its contribution by four; moving the bearing adds the parallel-axis term.

Core mathematical model

Sum about the bearing

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

The balanced rotor has its mass center at the carrier center. The bearing offset is measured from that center.

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 split equally across four beads. The supporting spokes carry no mass.

Constant applied couple

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

This trial changes angular speed while keeping the mass distribution and bearing fixed.

Independent balances

\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 different integrated motions.

Common difficulties

Same mass, different inertia

Typical misconceptionOnly the total mass determines rotational response.

Better mental modelCompare concentrated and rim layouts at the same torque. Their distance-weighted mass budgets differ.

The axis belongs in the specification

Typical misconceptionOne body has one universal inertia.

Better mental modelShift the bearing and inspect the added parallel-axis contribution.

Editing is a new release

Typical misconceptionDragging a bead while replaying conserves angular momentum.

Better mental modelLayout editing resets this rigid-body trial. Deforming a spinning body requires a separate internal-work model.

Run the experiment

01
Design a fast response
Concentrate the bead pairs and complete the torque trial.
What to observe: Lower inertia produces larger angular acceleration.
02
Compare a rim and a disk
Use the rim preset and compare the disk reference curve.
What to observe: The same total mass farther from the axis accelerates more slowly.
03
Move the bearing
Drag the red marker without changing mass or torque.
What to observe: The inertia budget gains a parallel-axis term.
04
Brake, then reverse
Set positive initial spin and negative torque.
What to observe: Motor work first removes energy and later adds it after reversal.