Angular Momentum Calculator

Related Calculators

Angular momentum (L): Rotational momentum. In this calculator it is the amount of spin or circular-motion momentum, and bigger values are harder to change [2].

Moment of inertia (I): A measure of how mass is spread relative to the rotation axis. Mass farther from the axis gives a larger I [2].

Angular speed (omega): How fast something spins. The formulas use radians per second, so the calculator converts rpm or deg/s to rad/s before solving.

Conservation of angular momentum: If the net external torque is zero or very small, the total angular momentum before and after a change stays the same, so L1 = L2 [2].

Radius (r or R): Distance from the axis to the mass or outer edge. Do not confuse radius with diameter.

Mistake: Entering rpm or deg/s into Angular speed omega (rad/s) or Initial angular speed omega1 (rad/s) without changing Angular speed unit (for omega inputs) .
Fix: Set the unit selector first, then check Angular speed (omega) in rad/s to confirm the conversion looks right.

Mistake: Typing diameter into Radius r (m) or Shape radius R (m) .
Fix: Enter half the diameter, because radius is the distance from the center to the edge.

Mistake: Leaving Use moment of inertia helper on while also expecting the manual Moment of inertia I (kg*m^2) box to be used.
Fix: Turn the helper Off to use the manual I value, or keep it On and read Computed moment of inertia from shape (I) .

Mistake: Using a negative value for Mass m (kg) , Radius r (m) , Shape radius R (m) , or Shape length L (m) .
Fix: Enter positive sizes and mass. If you need direction, use the sign on Angular momentum L (kg*m^2/s) only when you are entering L directly.

Mistake: Mixing up conservation inputs, such as entering the final value into Initial moment of inertia I1 (kg*m^2) or the initial value into Final moment of inertia I2 (kg*m^2) .
Fix: Treat 1 as before the change and 2 as after the change, then compare Initial angular momentum (L1) and Final angular momentum (L2) .

Mistake: Choosing the wrong formula setup, like using rigid-body mode for a single object moving in a circle.
Fix: Use Point mass in circular motion when the needed inputs are Mass m (kg) , Speed v (m/s) , and Radius r (m) .

• This calculator uses magnitudes for common intro-physics cases and does not do full vector direction math.
• Conservation mode assumes the net external torque about the chosen axis is zero or small enough to ignore.
• The built-in shape helper uses standard formulas for ideal shapes and the stated axes, so real objects with uneven mass distribution can have different I values.
• Inputs labeled in meters, kilograms, seconds, and radians per second must be entered in those units, except omega fields when you intentionally change Angular speed unit (for omega inputs).
• Negative mass, radius, and length are treated as input errors because physical size values must be positive.
• If a value such as I2 is extremely close to zero, results like omega2 become very sensitive and may be shown in scientific notation.
• Rounding can make Initial angular momentum (L1) and Final angular momentum (L2) look slightly different even when the exact calculation matches.

Core formulas

The calculator uses the formula that matches your selected mode.

L = I * ω

L = m * v * r

L1 = L2

I1 * ω1 = I2 * ω2

For rigid-body motion, angular momentum is found from moment of inertia and angular speed [1]. For the shape helper, the moment of inertia formulas come from standard rigid-body results for common shapes [1].

Angular speed conversion

If you choose rpm or deg/s in Advanced options, the calculator first converts your omega input to rad/s, because rad/s is the form used in the formulas.

ω(rad/s) = ω(rpm) * (2 * pi / 60)

ω(rad/s) = ω(deg/s) * (pi / 180)

It also shows the converted value as Angular speed (omega) in rad/s so you can check the unit change before trusting the final answer.

Moment of inertia helper

When the helper is On, the computed shape value replaces the manual I input in modes that need I.

Disk or solid cylinder: I = (1/2) * m * R^2

Thin hoop or ring: I = m * R^2

Solid sphere: I = (2/5) * m * R^2

Thin spherical shell: I = (2/3) * m * R^2

Thin rod about center: I = (1/12) * m * L^2

Thin rod about one end: I = (1/3) * m * L^2

These are standard textbook moment-of-inertia formulas for common ideal shapes [1].

What the calculator solves

In rigid-body mode, it can solve for Angular momentum (L), Moment of inertia (I), or Angular speed (omega). In point-mass mode, it can solve for Angular momentum (L), Mass (m), Radius (r), or Speed (v). In conservation mode, it solves the before-and-after relation using Initial moment of inertia I1 (kg*m^2), Initial angular speed omega1 (rad/s), Final moment of inertia I2 (kg*m^2), and Final angular speed omega2 (rad/s).

Mini example

Suppose a rigid body has I = 0.25 kg*m^2 and angular speed 12 rad/s.

L = I * ω = 0.25 * 12 = 3

The result is 3 kg*m^2/s.

For conservation, if I1 = 0.40 kg*m^2, ω1 = 10 rad/s, and I2 = 0.20 kg*m^2, then:

L1 = I1 * ω1 = 0.40 * 10 = 4

ω2 = L1 / I2 = 4 / 0.20 = 20

So the final spin is faster because the final moment of inertia is smaller, which matches angular momentum conservation when outside torque is negligible [2].

Assumptions used in the math

The calculator ignores full vector direction and uses signed values only where a sign can be entered directly. Conservation results assume no important external torque acts during the change. Shape-helper results assume ideal shapes and the exact axis listed, so real equipment can differ from the computed value.