Dimensional Formula of Angular Momentum — [ML²T⁻¹]

Method 1: From L = mvr Angular momentum L = Linear momentum × Perpendicular distance L = p × r = m × v × r

Dimensions: • m (mass) = [M] • v (velocity) = [LT⁻¹] • r (radius/distance) = [L]

Therefore: [L] = [M] × [LT⁻¹] × [L] = [M] × [L²T⁻¹] = [ML²T⁻¹]

Method 2: From L = Iω • I (moment of inertia) = mr² → [ML²] • ω (angular velocity) = [T⁻¹] (radians per second)

[L] = [ML²] × [T⁻¹] = [ML²T⁻¹] ✓

Method 3: From L = Torque × Time • Torque (τ) = r × F → [L] × [MLT⁻²] = [ML²T⁻²] • L = τ × t → [ML²T⁻²] × [T] = [ML²T⁻¹] ✓

All three methods give: [ML²T⁻¹]

SI unit: kg·m²·s⁻¹ (kilogram metre squared per second) Also written as: J·s (joule-second = same dimensions as Planck's constant h)

Angular momentum (L):

For a particle: • L = r × p = r × mv (cross product) • Magnitude: L = mvr sinθ (θ = angle between r and v) • For circular motion (θ = 90°): L = mvr

For a rigid body: • L = Iω • I = moment of inertia, ω = angular velocity

Conservation of Angular Momentum: • If no external torque acts on a system, angular momentum is conserved • L = Iω = constant • When I decreases, ω increases (and vice versa)

Examples of conservation:

1. Figure skater spinning: • Arms in: small I → high ω (fast spin) • Arms out: large I → small ω (slow spin)
2. Diver doing somersaults: • Tucked position: small I → faster rotation
3. Planetary motion: • Planets conserve L around the Sun (Kepler's 2nd Law)

Relation to Planck's constant: • Planck's constant h has the same dimensions as angular momentum: [ML²T⁻¹] • This is why angular momentum is quantised in atomic systems (multiples of ℏ = h/2π)

Quantities with same dimensions as Angular Momentum [ML²T⁻¹]:

1. Planck's constant (h): E = hf → h = E/f = [ML²T⁻²] / [T⁻¹] = [ML²T⁻¹] ✓

2. Action (in classical mechanics): [ML²T⁻¹]

3. Impulse × distance: [MLT⁻¹] × [L] = [ML²T⁻¹]

Quickly compare other dimensional formulas: Quantity | Dimensional Formula Angular Momentum | [ML²T⁻¹] Planck's constant | [ML²T⁻¹] Moment of Inertia | [ML²] Torque | [ML²T⁻²] Kinetic Energy | [ML²T⁻²] Linear Momentum | [MLT⁻¹] Force | [MLT⁻²] Pressure | [ML⁻¹T⁻²]