Angular Momentum Calculator

What is the angular momentum formula?

The angular momentum formula for a body turning about a fixed axis is L = Iω: moment of inertia in kg·m² times angular velocity in rad/s, giving kg·m²/s. While no net external torque acts about the axis, L is conserved, so I₁ω₁ = I₂ω₂. Halve the moment of inertia and the spin rate doubles.

Using the conservation of angular momentum calculator

Enter the moment of inertia and spin rate before the change, and one of the two after it; the calculator solves for the fourth. Spin rates go in as rad/s, rpm, rev/s or degrees per second, and the working converts them to rad/s, the unit L = Iω needs. The angular momentum readout is I₁ω₁, which is all a plain L = Iω question asks: enter I and ω as the initial values and read L off directly.

The moments of inertia usually come from the shapes involved, and the moment of inertia calculator gives them for hoops, disks, rods, spheres and point masses. When a second body joins the first on the same axis, the final moment of inertia is the sum of the two. A child of mass m who jumps onto the rim of a merry-go-round adds m r², and a disk dropped onto a turntable adds its own ½ m r².

When angular momentum is conserved

Angular momentum about an axis stays fixed when the net external torque about that axis is zero. Forces inside the system cannot change it, which is what makes the skater and the diver possible: each changes shape by moving mass toward or away from the axis, and the spin rate changes to keep Iω the same. Two disks gripping each other exert equal and opposite torques, so the pair’s total is unchanged even though each disk’s own angular momentum changes.

An outside torque breaks the rule. A motor driving a turntable, a hand pushing a merry-go-round, or friction in a bearing all change L at the rate τ = ΔL / Δt, the rotational form of Newton’s second law. Textbook problems treat the change of shape as quick enough that bearing friction can be ignored, and the torque calculator covers the cases where a torque is the point.

Why the kinetic energy changes

Rotational kinetic energy can be written in terms of angular momentum as K = L² / 2I. With L fixed, a smaller moment of inertia means more energy and a larger one means less, and the before and after readouts show both. A skater who pulls her arms in gains energy, supplied by the work her muscles do pulling them inward against the spin. When two disks lock together the energy falls instead, lost to friction while they slip against each other, which makes it the rotational version of an inelastic collision on the collision simulator. The rotational kinetic energy calculator works ½ I ω² on its own.

Angular momentum of a moving point mass

A single particle has angular momentum about a point even when it moves in a straight line: L = m v r sin θ, where r is its distance from the point and θ the angle between r and its velocity. For a mass going round a circle, θ is 90° and L = m v r, which is Iω again with I = m r² and v = ω r. A planet obeys the same rule, and the orbit simulator shows it speeding up at closest approach because its angular momentum about the star cannot change.

Common mistakes

• Leaving the spin in rpm. L = Iω gives kg·m²/s only with ω in rad/s. Multiply rpm by 2π/60, or choose rpm in the unit menu and let the calculator convert.
• Forgetting the body that joins. After a collision on one axis the final moment of inertia is the total of every body now turning together, not the first body alone.
• Assuming kinetic energy is conserved too. It rises when mass moves inward and falls when bodies lock together. Only L stays fixed.
• Ignoring an outside torque. Conservation needs zero net external torque about the axis. A merry-go-round that someone keeps pushing does not conserve its angular momentum.