Rotational dynamics Formulas

Angular Displacement

θ = s / r, where θ = Angular displacement (rad), s = Arc length, r = Radius.

Angular Velocity

ω = θ / t; also, ω = 2π / T and ω = 2πf, where T = Time period and f = Frequency.

Angular Acceleration

α = Δω / Δt.

Relation Between Linear & Angular Quantities

v = rω; aₜ = rα (Tangential acceleration); aᵣ = v²/r = rω² (Radial/Centripetal acceleration); Total acceleration: a = √(aₜ² + aᵣ²).

Rotational Equations of Motion

ω = ω₀ + αt; θ = ω₀t + ½αt²; ω² = ω₀² + 2αθ; θ = [(ω + ω₀)/2] × t.

Torque

τ = rF sinθ; Special case: τ = rF (θ = 90°); τ = Iα.

Moment of Inertia (Definition)

I = mr²; For many particles: I = Σmr².

Radius of Gyration

I = Mk²; k = √(I/M).

Parallel Axis Theorem

I = Icm + Md².

Perpendicular Axis Theorem

Iz = Ix + Iy (Only for plane lamina).

Angular Momentum

L = Iω; also, L = r × p; Magnitude: L = rp sinθ.

Rotational Kinetic Energy

KE = ½Iω².

Total Kinetic Energy (Rolling)

KE = ½Mv² + ½Iω².

Power in Rotation

P = τω.

Work Done

W = τθ.

Rolling Motion

v = rω; Pure Rolling Condition: v = rω.

Conservation of Angular Momentum

I₁ω₁ = I₂ω₂.

Centripetal Force

Fc = mv²/r; also, Fc = mrω².

Centripetal Acceleration

ac = v²/r; ac = rω².

Angular Impulse

τt = ΔL.

Relation Between Linear & Angular Momentum

L = Iω; L = mvr.

Rotational Equilibrium

Στ = 0.

Translational Equilibrium

ΣF = 0.

Rolling Down an Inclined Plane

Acceleration: a = g sinθ / (1 + I/MR²).

Time Period

T = 2π/ω.

Frequency

f = 1/T.

Standard Moments of Inertia (Must Memorize)

Thin Ring: I(center) = MR², I(diameter) = ½MR²; Hollow Cylinder: I = MR²; Solid Disc: I(center) = ½MR², I(diameter) = ¼MR²; Solid Cylinder: I = ½MR²; Solid Sphere: I = 2MR²/5; Hollow Sphere: I = 2MR²/3; Thin Rod: About Centre: I = ML²/12, About End: I = ML²/3; Rectangular Plate: About Centre: I = M(L² + B²)/12.