Moment of inertia of a uniform solid cylinder
= Moment of inertia of a uniform solid cylinder
{title2=$I=ma^2/2$}
For a uniform solid cylinder of radius $a$, length $L$ and <mass density> $\rho$, a coaxial shell carries <mass> $dm=2\pi\rho Lr\,dr$. The axial <moment of inertia> is $I=\int_0^a r^2dm=\pi\rho La^4/2=ma^2/2$. The axial <moment of inertia> is independent of the length once <mass> and radius are fixed. A thin cylindrical shell instead has $I=ma^2$, demonstrating why the uniform solid assumption matters.