Normalize a radial density before integrating disk inertia

Choose annuli as the differential elements

A thin annulus at radius r and width dr has area 2pi r dr. With surface density sigma(r), its mass is 2pi r sigma(r)dr. Its contribution to inertia about the perpendicular center axis adds another factor r². Omitting either the circumference factor or the inertia factor changes both units and physical weighting.

Hold total mass fixed explicitly

The family has sigma(r)=sigma 0(r/R)^n and fixed total mass M=2 kg. Integrating mass from 0 toR gives M=2pi sigma 0 R²/(n+2), so sigma 0=M(n+2)/(2piR²). A comparison at different n must recompute this normalization; holding sigma 0 fixed instead would compare different masses.

Integrate the inertia

Now I=2pi sigma 0 integral r³(r/R)^n dr=2pi sigma 0 R⁴/(n+4). Substituting the mass normalization gives I=MR²(n+2)/(n+4). At n=0 this is the uniform-disk factor 1/2. At n=2 it becomes 2/3; larger n puts a greater share of the same mass near the edge.

Check a limiting geometry

For nonnegative n, the shape factor remains below 1 and tends toward 1 as mass concentrates near the rim. A thin hoop of the same mass and radius has I=MR². The limit is a distribution argument, not an instruction to machine an increasingly concentrated material or a guarantee of mechanical strength.

Distinguish a sphere from a disk

Optional geometric extension: a thin spherical shell has inertia(2/3)r² dm about a diameter, not r²dm, because not all shell mass is the same perpendicular distance from that axis. Integrating uniform volume density over shells gives I=(2/5)MR² for a solid sphere. If spherical density instead scales with radius, normalization and shell weighting give(4/9)MR². Importing a disk formula into a sphere would omit geometry.

Worked original case

M=2,R=2,n=2 gives sigma 0=1/pi kg/m² and I=16/3 kg m². At the same mass/radius, the uniform disk gives 4 and the thin hoop 8.

Investigation

At R=2 compare n=0,2,4. Derive sigma 0 before I, retain normalized shape factors and compare the thin-rim limit. Explain which extra geometric factor appears for a spherical shell.

Public numerical transfer practice

Try a new practice case

A uniform thin disk has mass 4 kg and radius 3 m. What is its central perpendicular inertia in kg m²?

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Original teaching and numerical cases checked 30 October 2026 against the official AP Physics C Mechanics framework and OpenStax fixed-axis rotation. Supplied/generated evidence is not physical measurement, a load rating or laboratory participation. No rotating machinery, hanging weights or mechanical construction is requested. Public feedback and retained self-reports do not certify independent learner mastery.