About the Angular Momentum Calculator
This angular momentum calculator works out how much “rotational momentum” an object carries. For a spinning rigid body, choose a shape (solid disk, ring, solid or hollow sphere, rod about its centre or end) or enter your own moment of inertia, then give the mass, size and spin rate in rpm or rad/s; it returns the moment of inertia, the angular momentum L = Iω and the rotational kinetic energy. For a single particle moving past a pivot, it uses L = m·v·r·sin θ instead.
It is meant for physics students studying rotational dynamics and for anyone curious about flywheels, figure skaters, divers and spinning planets. Enter what percentage of the original moment of inertia remains after a change — a skater pulling in her arms, say — and the calculator applies conservation of angular momentum to give the new spin rate and kinetic energy.
Shapes are idealised and uniform, rotating about the axis named in the list. Conservation applies only when no external torque acts; the extra kinetic energy when spinning up comes from the work done pulling the mass inward.
With the default inputs, the angular momentum (l) is 1.131 kg·m²/s. Change any value above to recalculate instantly.
How to use the angular momentum calculator
- 1Choose a rotating rigid body (by shape or known I) or a moving particle.
- 2For a body, pick its shape and enter mass and radius (or length), or enter I directly.
- 3Enter the spin rate in rpm, rad/s or rev/s — or the particle’s speed and angle.
- 4Optionally set the final moment of inertia as a percentage for a conservation problem.
- 5Read L, the kinetic energy, and the new spin rate after the change.
Formula and method
Angular momentum is the rotational counterpart of linear momentum. For a rigid body spinning about a fixed axis it equals the moment of inertia times the angular velocity, L = Iω, where I depends on how the mass is distributed (for example ½mr² for a solid disk and mr² for a thin ring). For a particle, L = r × p, whose magnitude is m·v·r·sin θ, with θ the angle between the position vector and the velocity. ω in rpm is converted with ω (rad/s) = rpm × 2π ÷ 60.
With no external torque, L is conserved, so if the moment of inertia falls to a fraction f of its value the angular speed rises by 1/f. Rotational kinetic energy ½Iω² = L²/(2I) then rises by the same factor 1/f, the energy coming from the work done pulling mass toward the axis.
- L
- Angular momentum (kg·m²/s)
- I
- Moment of inertia (kg·m²)
- ω
- Angular velocity (rad/s)
- m, v, r
- Particle mass, speed and distance from pivot
- θ
- Angle between r and v
Worked examples
2 kg solid disk, 0.3 m radius, at 120 rpm
I = ½ × 2 × 0.3² = 0.09 kg·m² and ω = 120 × 2π ÷ 60 = 12.57 rad/s, so L = 1.131 kg·m²/s and KE = 7.11 J. Halving I (50%) doubles the speed to 240 rpm while L stays 1.131, and the kinetic energy doubles to 14.2 J.
0.1 kg ball on a 0.5 m string at 5 m/s
L = m·v·r = 0.1 × 5 × 0.5 = 0.25 kg·m²/s. Treated as a point mass, I = 0.1 × 0.5² = 0.025 kg·m² and ω = v/r = 10 rad/s, giving KE = ½ × 0.025 × 10² = 1.25 J.
1.5 kg, 1.2 m rod spinning about its centre at 30 rpm
I = 1.5 × 1.2² ÷ 12 = 0.18 kg·m². At 30 rpm (π rad/s), L = 0.18π = 0.566 kg·m²/s and KE = ½ × 0.18 × π² = 0.888 J.
Frequently asked questions
What is the formula for angular momentum?+
For a rigid body, L = Iω (moment of inertia times angular velocity). For a single particle, L = r × p, with magnitude m·v·r·sin θ. The SI unit is kg·m²/s, equivalent to joule-seconds.
Why does a skater spin faster when she pulls her arms in?+
With negligible friction there is no external torque, so angular momentum Iω stays constant. Pulling her arms in lowers her moment of inertia, so her angular speed must increase in proportion.
Is rotational kinetic energy conserved when I changes?+
No. KE = L²/2I, so when I falls the kinetic energy rises. The extra energy comes from the muscular work done pulling mass inward against the centrifugal tendency.
What is the difference between angular momentum and angular velocity?+
Angular velocity ω measures how fast something turns (rad/s). Angular momentum L = Iω also accounts for how mass is spread around the axis, so a heavy flywheel and a light wheel at the same ω carry very different L.
How do I convert rpm to rad/s?+
Multiply by 2π and divide by 60. For example 120 rpm × 2π ÷ 60 = 12.57 rad/s. One revolution per second (1 Hz) equals 2π ≈ 6.283 rad/s.