Variable-Density Flywheel Rotational Inertia (Aerospace & Mechanical Engineering)

An aerospace guidance, navigation, and control (GNC) engineer at the UNLV Howard R. Hughes College of Engineering is designing an energy-storage reaction flywheel for a low-Earth-orbit (LEO) small satellite.

The flywheel is constructed as a flat circular disk of outer radius $R = 20\text{ cm}$. To maximize rotational kinetic energy while minimizing center-of-mass balancing jitter, the disk is fabricated using a radially graded carbon nanotube matrix whose planar mass density increases quadratically toward the outer rim. In Cartesian coordinates, the area mass density distribution is modeled as:

The satellite's attitude control agility and kinetic storage capacity depend directly on the flywheel's mass moment of inertia ($I_z$) about its central perpendicular spin axis:

1. Express the circular boundary of the flywheel disk in polar coordinates $r$ and $\theta$.
2. Transform the moment of inertia integrand into polar form, incorporating the mass density function $\rho(r)$ and the Jacobian differential area element $dA = r \, dr \, d\theta$.
3. Use Fubini's product separability theorem to evaluate the double integral, computing the exact moment of inertia in $\text{g}\cdot\text{cm}^2$ and converting the result to standard SI units ($\text{kg}\cdot\text{m}^2$).
4. Explain how concentrating mass toward the outer perimeter of a reaction flywheel optimizes angular momentum storage per unit total mass in space satellite attitude control systems.