04 / CHAPTER 03
Moment of inertia
Understand rotational inertia as mass distribution, not just a table of numbers.
ANCHOR EQUATION
Moment of inertia
In symbols: I = ∫r²dm
Rotational inertia depends on how mass is distributed.
Moment of inertia is weighted rotational mass
Mass farther from the axis is harder to spin because the same angular speed gives it greater linear speed. Moment of inertia records this distribution, not merely the total mass.
Add the kinetic energies
A particle at radius r has speed v = rω and kinetic energy ½m(rω)² = ½(mr²)ω². For many particles, sum: K_rot = ½(Σm_i r_i²)ω². In the continuous limit I = ∫r² dm, so rotational dynamics uses τ = Iα.
A 2 kg mass at 0.5 m contributes I = 2(0.5²) = 0.50 kg·m²; moving the same mass to 1.0 m contributes 2.0 kg·m², four times larger. Radius enters squared.
The axis is part of the definition
A disk’s inertia about its center differs from its inertia about a rim axis. A table of I values is meaningful only after the axis and mass distribution are matched to the physical setup.
Work it: Approximate an impeller as 0.8 kg concentrated at 0.06 m plus a 0.2 kg hub at 0.01 m. Calculate I and rotational energy at 150 rad/s.