Moment of Inertia

Moment of inertia I is the rotational analog of mass — measures resistance to angular acceleration.

For a point mass at distance r from axis: I = mr².

For a collection of particles: I = Σ m_i r_i². For continuous bodies: I = ∫r²·dm.

I depends on (i) mass distribution, (ii) chosen rotation AXIS — different axes give different I.

Common bodies (about symmetry axes): rod (about center, length L): ML²/12. Disk (about center): MR²/2. Solid sphere: 2MR²/5. Hollow sphere: 2MR²/3. Cylinder (about axis): MR²/2. Ring: MR².

Newton's 2nd law for rotation: τ = Iα.

Rotational KE: K = ½Iω². Higher I ⇒ more KE for same ω.

Engineering: flywheels store rotational energy via large I.