Circular Motion

Angular kinematics

For uniform circular motion of radius r and linear speed v, angular speed ω = v/r, period T = 2π/ω, and frequency f = 1/T. Angular displacement is in radians (1 rev = 2π rad). The angular kinematic equations mirror the linear ones: ω = ω₀ + αt, θ = ω₀t + ½αt², ω² = ω₀² + 2αθ. A wheel accelerating from rest at α = 2 rad/s² for 5 s reaches ω = 10 rad/s and turns θ = 25 rad ≈ 4 revolutions.

Centripetal acceleration and force

A body in circular motion has centripetal acceleration a_c = v²/r = ω²r directed toward the centre. The net inward force is F_c = mv²/r = mω²r. A 1000 kg car taking a 50 m radius bend at 20 m/s needs F_c = 1000·400/50 = 8000 N. On a banked road of angle θ with no friction, tan θ = v²/(rg) gives the design speed; for a 50 m bend at 20 m/s with g = 10, tan θ = 400/500 = 0.8, so θ ≈ 38.7°.

Rotational dynamics

Torque τ = r×F = rF sin θ; angular acceleration α = τ/I, where I is the moment of inertia. For a point mass, I = mr². For a uniform disk about its centre, I = ½MR²; solid sphere I = (2/5)MR²; hollow sphere I = (2/3)MR²; thin rod about centre I = ML²/12, about end ML²/3. Rotational KE = ½Iω². A solid sphere rolling without slipping has total KE = ½mv² + ½Iω² = ½mv²(1 + 2/5) = (7/10)mv².

Angular momentum and conservation

L = Iω = mvr (for a particle with v perpendicular to r). When the net external torque is zero, L is conserved — a skater pulling arms in spins faster because I decreases and ω increases proportionally. A planet in elliptical orbit moves faster at perihelion than aphelion, the basis of Kepler's second law; L = mvr remains constant.

Artificial satellites and orbital speed

For a satellite in circular orbit at radius r from Earth's centre, gravitational pull supplies centripetal force: GMm/r² = mv²/r, so v = √(GM/r). At Earth's surface r = R, v ≈ 7.9 km/s (first cosmic speed). Geostationary orbit has T = 86 400 s, giving r ≈ 4.22×10⁷ m or about 36 000 km altitude. The escape velocity v_e = √(2GM/R) ≈ 11.2 km/s — exactly √2 times the orbital speed at the surface, a relation HRW emphasises in Chapter 13.