Angular Momentum Calculator
Angular momentum measures how much rotational motion an object carries, combining how its mass is distributed with how fast it spins. Work it forward from moment of inertia and angular velocity or from a point mass on a circle, reverse solve for any single unknown, switch between RPM and rad/s and metric or imperial units, and read off the rotational kinetic energy too.
Formula
Worked example
A disk with I = 0.5 kg·m² spinning at ω = 10 rad/s: L = 0.5 × 10 = 5 kg·m²/s, with E = ½ × 0.5 × 10² = 25 J. A 2 kg ball moving at 3 m/s on a 1.5 m radius: L = 2 × 3 × 1.5 = 9 kg·m²/s. To find the spin that gives L = 5 from I = 0.5, reverse solve: ω = L ÷ I = 5 ÷ 0.5 = 10 rad/s.
How It Works
Angular momentum (L) quantifies the rotational motion of a body about an axis. For an extended object the calculator multiplies its moment of inertia I, a measure of how its mass is distributed relative to the axis, by its angular velocity ω in radians per second, giving L = Iω. For a single point mass travelling in a circle, it instead multiplies the mass, its tangential speed, and the radius of its path, since L = mvr is the special case where I = mr² and v = ωr. Both routes return the same physical quantity, expressed in kilogram metres squared per second (kg·m²/s).
RPM, imperial units and rotational energy
Real spin rates rarely come in radians per second, so the angular velocity field lets you enter revolutions per minute (RPM), revolutions per second, or degrees per second, and converts to rad/s before computing. The point-mass mode accepts mass in kilograms, grams, pounds or ounces, velocity in m/s, km/h, mph or ft/s, and radius in metres, centimetres, feet or inches, so you can mix everyday units and still get a clean SI result. Alongside L the calculator reports the rotational kinetic energy E = ½Iω². Note that L scales linearly with the spin rate while energy scales with its square, so doubling ω doubles L but quadruples the stored energy.
Reverse solving for any variable
Pick the reverse solve mode to find any single unknown when you already know L. Choose the relationship (L = Iω or L = mvr), pick which variable to solve for, and fill in the rest. The calculator rearranges the equation for you: ω = L ÷ I, I = L ÷ ω, m = L ÷ (v·r), v = L ÷ (m·r) or r = L ÷ (m·v). This is handy for conservation problems, where you know the angular momentum is fixed and want the new spin rate after the moment of inertia changes, or for designing a flywheel to a target L. If a rearrangement would divide by zero, the result is left blank rather than returning infinity.
Choosing the Right Inputs
Use the moment-of-inertia mode when you already know I for your object, such as a flywheel, wheel, or rigid body. Standard formulas give I = ½mr² for a solid disk, mr² for a hoop, and ⅖mr² for a solid sphere. Use the point-mass mode for an object small enough to treat as a single point orbiting an axis, like a ball on a string or a satellite far from its planet. When you do not know one quantity but do know L, reach for reverse solve instead of rearranging by hand.
Common Moments of Inertia (about the central axis)
| Object | Moment of inertia I |
|---|---|
| Point mass (distance r) | m·r² |
| Thin hoop / ring | m·r² |
| Solid disk / cylinder | ½·m·r² |
| Solid sphere | ⅖·m·r² |
| Hollow sphere (shell) | ⅔·m·r² |
| Thin rod (about center) | 1/12·m·L² |
Plug these I values into the L = Iω mode. m is mass, r is radius, L is length.
Frequently asked questions
What units does angular momentum use?
In SI units, angular momentum is measured in kilogram metres squared per second (kg·m²/s), which is equivalent to joule-seconds (J·s). This calculator returns its result in kg·m²/s for every mode, even when you enter the inputs in RPM, pounds, mph or other units.
When should I use L = mvr instead of L = Iω?
Use L = mvr when you can treat the object as a single point mass moving in a circle, like a ball whirled on a string. Use L = Iω for an extended rigid body whose mass is spread out, where you first need its moment of inertia I. The two agree because a point mass has I = mr² and v = ωr.
Can I enter angular velocity in RPM?
Yes. Switch the angular velocity unit to RPM, revolutions per second, or degrees per second and the calculator converts to radians per second for you. To convert RPM by hand, multiply by 2π/60 (about 0.1047); for degrees per second, multiply by π/180.
How do I solve for the spin rate or mass from a known angular momentum?
Choose the reverse solve mode, pick the relationship (L = Iω or L = mvr), and select the variable to find. The calculator rearranges the equation, for example ω = L ÷ I or m = L ÷ (v·r), and fills in your other values. This is the fastest way to handle conservation of angular momentum problems.
What is the rotational kinetic energy shown alongside L?
It is the energy stored in the spin, E = ½Iω², reported in joules. Unlike angular momentum, which scales linearly with the spin rate, kinetic energy scales with the square of it, so doubling the angular velocity quadruples the energy while only doubling L.