Include the ring area when deriving disk inertia

Include the ring area when deriving disk inertia

A disk with radial density cannot be treated as a line of equally weighted radii. An annulus has area 2πr dr, so its mass is that area times local surface density. Omitting the area factor changes both total mass and rotational inertia.

The hypothetical thin disk has fixed mass 4 kg and density σ₀(1 + βr²/R²). Integrating annuli gives M = πσ₀R²(1 + β/2). Multiplying each annulus by r² before integration gives I = πσ₀R⁴(1/2 + β/3). Therefore I/(MR²) = (1/2 + β/3)/(1 + β/2).

At β = 2 and R = 2 m, the ratio is 7/12 and I = 28/3 kg m². This exceeds the uniform disk value because more of the fixed mass lies farther from the center. It remains below the thin-ring ratio 1; the distribution has not become a ring.

The mass inside half radius is a separate integral: its fraction is (1/4 + β/32)/(1 + β/2). At β = 2 that fraction is 5/32, giving 0.625 kg. Neither a half-radius cut nor a half-area argument alone gives the mass of a nonuniform region. Radius scaling at fixed β changes inertia as R² while the normalized density adjusts to preserve mass.

For an isolated axial redistribution at fixed radius and mass, changing β from 0 to 2 increases inertia by a factor 7/6. With no external axial torque, angular speed becomes 6/7 of its initial value and rotational kinetic energy also becomes 6/7 at fixed angular momentum. The missing rotational energy belongs to internal work or other degrees of freedom; zero external torque is not zero internal work.

Changed-case check

Predict before reading the comparison. At β = 0 and R = 3 m, I = MR²/2 = 18 kg m² and the half-radius region contains 1 kg. Increasing R at fixed mass is not the same operation as adding more material at fixed density.

Retained investigation

Normalize the density, derive mass and inertia from separate ring integrals, and compare the uniform and graded cases. Which assumption must change if radius grows but both density and mass are claimed fixed?

Source and boundary

Primary conceptual reference checked 30 October 2026: OpenStax University Physics Volume 1. All models, packets and teaching prose are original. These are bounded mathematical investigations, not measured apparatus, physical procedures, practical certification or institutional credit.