Rotational Motion

ω = ω₀ + αt, θ = ω₀t + ½αt² — see disk spin with live ω(t) and θ(t) plots.

I formulas for 6 common shapes — solid/hollow sphere, cylinder, rod.

τ = r × F = rF sinθ — drag angle and watch rotation direction flip.

I = I_cm + Md² — shift the axis and see I grow quadratically with d.

KE_rot = ½Iω² — compare with translational KE for rolling objects.

L = Iω conserved — pull string shorter, ω grows as (r₀/r)².

a = g sinθ/(1 + k²) — 4 shapes race down an incline; sphere wins.

v = ωr, a_t = αr, a_c = ω²r — drag a particle on a rotating disc to see all three.

Two masses on a beam — adjust weights and arms; see torque balance live.

k = √(I/M) — see equivalent ring radius for ring, disc, rod, sphere, shell.

Independent v and ω — detect skidding (v > ωR) vs spinning (ωR > v) vs pure rolling.

Spinning top — see L precess about vertical at Ω = mgr/(Iω). Real animatronic feel.