Angular Momentum Calculator – Spin, Conservation and Torque
Angular momentum is the rotational counterpart of linear momentum: the quantity that a spinning body keeps hold of until something twists it. This angular momentum calculator handles both of the definitions you meet in a mechanics course — the point-particle form L = m·v·r·sin θ and the rigid-body form L = I·ω — and then follows them through to the problems they are actually used for: conservation, angular impulse, and the signed total for a system of several bodies.
Two formulas, one quantity
For a single mass moving past a chosen axis, angular momentum is the cross product L = r × p, whose magnitude is m·v·r·sin θ. Only the component of the motion perpendicular to the position vector counts, which is why the perpendicular lever arm r·sin θ appears. A 2 kg mass travelling at 15 m/s on the end of an 0.8 m string has p = 30 kg·m/s and L = 24 kg·m²/s. Tilt the velocity to 35° from the string and the same particle carries only 13.766 kg·m²/s.
For a rigid body turning about a fixed axis, every particle shares one angular velocity, and summing m·r² over the whole body gives the moment of inertia. Angular momentum then collapses to L = I·ω. A 5 kg solid disk of radius 0.4 m has I = ½mr² = 0.4 kg·m², so at 20 rad/s it carries L = 8 kg·m²/s and stores E = ½Iω² = 80 J of rotational energy.
Why the skater speeds up
The single most useful property of angular momentum is that it is conserved whenever the net external torque is zero, giving I₁ω₁ = I₂ω₂. A skater spinning at 2 rad/s with arms out at I₁ = 4.5 kg·m² carries L = 9 kg·m²/s. Pulling the arms in to I₂ = 1.5 kg·m² cannot change that number, so the spin rate triples to 6 rad/s.
Torque is the rate of change of angular momentum
Newton's second law for rotation is τ = dL/dt, which integrated over a constant torque gives the angular impulse ΔL = τ·Δt. Applying 12 N·m for 5 s to a 3 kg·m² rotor at rest delivers ΔL = 60 kg·m²/s, so ω_f = 60 / 3 = 20 rad/s. The same answer arrives through the angular acceleration route: α = τ/I = 4 rad/s² and ω = αΔt = 20 rad/s. The rotor sweeps θ = ½αΔt² = 50 rad, and the work τθ = 600 J matches ½Iω² = 600 J exactly — a cross-check the calculator prints on every run.
Units, signs and scale
The SI unit kg·m²/s is dimensionally identical to J·s and N·m·s, so those three columns always show the same number. Imperial engineering uses lb·ft²/s and slug·ft²/s, and quantum mechanics counts angular momentum in units of the reduced Planck constant ℏ = 1.0546 × 10⁻³⁴ J·s. Direction follows the right-hand rule: counter-clockwise in the viewing plane points out of the page and is taken as positive, clockwise as negative, which is what makes the multi-particle mode a signed sum rather than a plain total.
From flywheels to planets
The same arithmetic spans an enormous range. A reaction wheel on a satellite trades a few kg·m²/swith its spacecraft to point a camera. Earth's spin, modelled as a uniform sphere with I = ⅖mR² and ω = 2π/T, works out at roughly 7.07 × 10³³ kg·m²/s, while its orbit about the Sun carries about 2.66 × 10⁴⁰ kg·m²/s — some seven orders of magnitude more. Nothing in the formulas changes; only the exponents do, which is why the calculator switches to scientific notation automatically at those magnitudes.