The Angular Momentum of a Rigid Body Rotating About a Fixed Axis — Question 4

Question 4

Annular layers contribute angular momentum in proportion to squared radius

A uniform annular plate has mass $M$, inner radius $a$ and outer radius $b$. It rotates about its central normal axis at angular speed $\omega$.

1. Integrate the ring contributions to find $L_z$.

2. For $a=b/2$, find the ratio to a solid disk of the same mass, outer radius and angular speed.

Question 4 - Solution

Annular layers contribute angular momentum in proportion to squared radius

(a) Derivation and calculation

The uniform surface density and differential ring mass are $$\sigma=\frac{M}{\pi(b^2-a^2)},\qquad dm=2\pi\sigma r\,dr.$$ Integrating, $$I_z=\int_a^b r^2\,dm=2\pi\sigma\int_a^b r^3\,dr =\frac{\pi\sigma}{2}(b^4-a^4),$$ $$\boxed{L_z=I_z\omega=\tfrac12M(a^2+b^2)\omega}.$$

(b) Result and interpretation

For $a=b/2$, $$I_{\mathrm{annulus}}=\tfrac12M(b^2/4+b^2)=\frac58Mb^2.$$ At equal mass, outer radius and angular speed, $$\boxed{\frac{L_{\mathrm{annulus}}}{L_{\mathrm{disk}}}=\frac{(5/8)Mb^2}{(1/2)Mb^2}=\frac54}.$$ Removing the central region while retaining the same mass concentrates more mass at larger radii.

Physical check

The formula approaches the disk value at $a=0$ and the thin-hoop value as $a\to b$.