What angular momentum measures
Angular momentum is the quantity of rotation of a body. For an object spinning about a fixed axis it equals
L = I × ω
where I is the moment of inertia in kg·m² and ω the angular velocity in radians per second. The moment of inertia depends on how the mass is spread around the axis: the farther the mass, the larger I. A point mass m at radius r has I = m r².
Conservation
When no external torque acts on a system, its angular momentum stays constant. If the moment of inertia changes, the spin rate must change in the opposite direction:
I₁ ω₁ = I₂ ω₂, so ω₂ = ω₁ × I₁ / I₂
That is why a figure skater spins faster when pulling the arms in, and slows down when spreading them again.
Worked example
A skater spins at 60 revolutions per minute (6.28 rad/s) with arms out, giving a moment of inertia of about 3.5 kg·m². Pulling the arms tight against the body lowers it to about 1.2 kg·m². The angular momentum is 3.5 × 6.28 = 22.0 kg·m²/s. The new spin rate is 60 × 3.5 / 1.2 = 175 rpm, nearly three times faster.
What happens to the energy
The rotational kinetic energy is E = ½ I ω². In the example it rises from about 69 J to about 202 J. The extra 133 J comes from the work the skater’s muscles do to pull the arms inward against their tendency to fly outward. In the reverse direction, spreading the arms releases that energy.
Everyday and cosmic examples
- Diving and gymnastics: athletes tuck to rotate quickly and open out to slow down before entry or landing.
- Neutron stars: when a star’s core collapses from a radius of hundreds of thousands of kilometers to about 10 km, its spin rate rises enormously, producing pulsars that rotate many times per second.
- Helicopters: the tail rotor supplies the torque that keeps the body from spinning opposite to the main rotor.
- Office chair experiment: sitting on a spinning chair with weights in your hands, pull them in and you speed up.
Units
The calculator accepts spin rates in revolutions per minute, revolutions per second or radians per second. One revolution is 2π radians, so 1 rpm equals 0.1047 rad/s. Angular momentum is in kg·m²/s, equivalent to joule-seconds.