Rotational Dynamics — Formula Sheet

Angular displacement (θ), angular velocity (ω = dθ/dt), and angular acceleration (α = dω/dt)

Relationship between linear and angular quantities

v = ωr, a_tangential = αr

Centripetal force = mω²r = mv²/r in magnitude

Minimum speed at top

v_min = √(rg) for string/track contact

Centripetal force F = mω²r = mv²/r is the real force directed towards the center that maintains circular motion.

Moment of inertia I = Σmᵢrᵢ² is the rotational analog of mass, representing resistance to angular acceleration.

For continuous bodies, I = ∫r²dm.

Parallel axes theorem

I₀ = Iᶜ + Mh², where h is the distance between parallel axes, one passing through the center of mass.