Angular Momentum Calculator

Angular Momentum Calculator for a point mass (L = mvr sin θ) or a rigid body (L = Iω). It also solves conservation problems and Bohr orbits, with torque, energy and US unit values.

How the Angular Momentum Calculator Works

The Angular Momentum Calculator opens with one dropdown, Formula for an Object Rotating. Choose the kind of motion you have, and only the fields that formula needs stay on screen.

Central Point is for a mass circling a point, such as a ball on a string or a planet in orbit. Known Inertia and Common Shape are for a body spinning on an axis. Conservation covers a skater or diver changing shape, and Two Bodies adds up a pair turning on one axis. Bohr Orbit covers a single electron around a nucleus.

The first two modes also run backward. Set Solve For to mass, velocity or radius, and the tool rearranges L = mvr around an angular momentum you already know. Known Inertia can return the moment of inertia or the angular velocity in place of L.

Each input has its own unit list. Mass takes kg, g or lb, speed takes m/s, km/h, ft/s or mph, and spin takes rad/s, rpm, rev/s or degrees per second. The answer comes back in kg·m²/s, with US and CGS values on the first result card.

Angular Momentum Formula

Angular momentum, L, measures how much rotation a body carries about a chosen point or axis. The formula depends on whether the body is a single mass on a path or a solid object spinning, and the answer comes out in kg·m²/s either way.

What are the two formulas for angular momentum?

Physics courses teach one formula for a moving particle and one for a spinning body. The first applies to a point mass on a circular path, where you multiply its mass, its speed and the radius of the circle.

$$L = m v r$$

The second applies to a rigid body turning about a fixed axis, such as a wheel, a flywheel or the Earth. Multiply its moment of inertia by its angular velocity, with the angular velocity in radians per second.

$$L = I \omega$$

Here m is mass in kg, v is speed in m/s and r is the distance from the axis in m. I is the moment of inertia in kg·m² and ω is the angular velocity in rad/s. The two formulas agree because a point mass has I = mr² and ω = v ÷ r, so Iω works out to mvr.

How to calculate angular momentum?

Take the default entries, a 0.2 kg ball moving at 4 m/s on a 0.75 m string. L = 0.2 × 4 × 0.75 = 0.6 kg·m²/s. The same ball has I = 0.2 × 0.75² = 0.1125 kg·m² and ω = 4 ÷ 0.75 = 5.3333 rad/s, and 0.1125 × 5.3333 gives 0.6 again.

For a spinning body, convert the spin to rad/s before you multiply. One rpm equals 2π ÷ 60 = 0.10472 rad/s, and the Angular Momentum Calculator does this for you when you pick rpm. A flywheel with I = 2.5 kg·m² turning at 12 rad/s carries L = 2.5 × 12 = 30 kg·m²/s.

Is mvr the formula of angular momentum?

Yes, but only when the velocity is at right angles to the radius, as it is in circular motion. The general form in OpenStax University Physics, section 11.2 includes the angle θ between the position and the velocity.

$$L = m v r \sin\theta$$

At θ = 90° the sine is 1 and the formula drops back to mvr. At 30° the default ball would carry half as much, 0.3 kg·m²/s, because only v sin θ = 2 m/s of its speed takes it around the center.

How is angular momentum defined?

Angular momentum is defined as the cross product L = r × p. Here r runs from a chosen origin to the particle, and p = mv is its linear momentum. L points at right angles to both, in the direction your right thumb shows when your fingers curl the way the mass turns.

The definition does not need a circle. A particle on a straight path has angular momentum about any point its path misses, equal to its momentum times the perpendicular distance to that path. This is the translational angular momentum formula, L = mv × r⊥.

OpenStax fires a 2.0 g bullet at 500 m/s into the rim of a disk with a 0.5 m radius. Before impact the bullet travels in a straight line, yet about the disk’s axle it carries L = 0.002 × 500 × 0.5 = 0.5 kg·m²/s. That value sets the spin of the disk after the hit.

Angular Momentum Calculator With Angle

Open the Angle and Stopping Time panel under the main fields to set the angle between r and v. It accepts degrees or radians, for any value above 0° and below 180°. At exactly 0° or 180° the mass moves straight along r and L is zero, so those two entries are rejected.

The OpenStax meteor example shows why the angle matters. A 15 kg meteor falls straight down at 2.0 km/s, at a point 25 km to one side of an observer and 25 km up. Its distance from the observer is 35.355 km, and the angle between r and v is 135°.

Enter 15 kg, 2,000 m/s, 35.355 km and 135°, and the result rounds to 7.5 × 10⁸ kg·m²/s, the textbook answer. The perpendicular-distance shortcut agrees: 15 × 2,000 × 25,000 m, with the 25 km sideways offset standing in for r sin θ.

At any angle other than 90°, the Angular Momentum Calculator titles its energy card Turning-Part Energy. The card shows ½m(v sin θ)², the share of the kinetic energy tied to motion around the origin. For the meteor that is 15 MJ, out of a full ½mv² of 30 MJ.

L = Iω for Common Shapes

Common Shape mode in the Angular Momentum Calculator removes the step of finding I by hand. Choose the shape and axis, then enter the mass, the radius or rod length, and the angular velocity. The tool builds the moment of inertia first and then multiplies by ω.

The table lists the seven shapes and the angular momentum each one carries. Every row uses a 2 kg body, a 0.3 m radius or rod length, and a spin of 300 rpm, which is 31.416 rad/s. The hollow cylinder has a 0.2 m inner radius.

Mass far from the axis counts more. The hoop carries twice the angular momentum of the solid disk at equal mass, radius and spin, and a rod swung about one end carries four times what it does about its center.

For an axis that does not pass through the center of the shape, open Offset and Stopping Time and fill in Axis Offset (d). The Angular Momentum Calculator applies the parallel-axis theorem, which adds md² to the moment of inertia before the multiplication by ω.

$$I = I_{cm} + m d^2$$

A 0.3 m rod about its center with d = 0.15 m gives 0.015 + 2 × 0.15² = 0.06 kg·m², the same as Rod About One End. The 2 kg disk spun about a point on its rim, d = 0.3 m, has I = 0.27 kg·m² and carries 8.4823 kg·m²/s, three times its value about the center.

Conservation of Angular Momentum Calculator

When no outside torque acts on a system, its angular momentum stays fixed. Conservation mode applies that rule to a body that changes shape while it spins, so the product of inertia and angular velocity is the same before and after.

$$I_1 \omega_1 = I_2 \omega_2$$

Enter the starting moment of inertia and spin, then either the final inertia or the final spin. The Angular Momentum Calculator returns the missing value, along with the final rotation rate, the time per turn and the change in rotational energy.

OpenStax section 11.3 works a gymnast leaving the high bar. An 80 kg gymnast modeled as a 1.8 m rod has I₁ = 21.6 kg·m² and turns at 1 rev/s, which the rev/s unit takes as given. Tucked to half that length, I₂ = 5.4 kg·m².

With those entries the final spin is 25.133 rad/s, which is 240 rpm or 4 rev/s, and L holds at 135.72 kg·m²/s. Rotational energy rises from 426.37 J to 1.7055 kJ. The extra 1.2791 kJ is work done by the gymnast’s muscles in pulling mass toward the axis.

Energy can fall as well, as in the OpenStax coupled flywheels. A wheel at 600 rpm drops onto a resting wheel with three times its inertia, so I rises fourfold and the pair turns at 150 rpm. Three quarters of the starting energy is lost to friction between the two surfaces.

The third Solve For option in the Angular Momentum Calculator, ω₂ After Adding Mass, handles a load placed on a spinning platform. It treats the load as a point mass at radius r, so the final inertia is the starting inertia plus mr².

$$\omega_2 = \frac{I_1 \omega_1}{I_1 + m r^2}$$

Take a 500 kg·m² merry-go-round turning at 2 rad/s. A 25 kg child steps on 2 m from the center, and the spin drops to 1.6667 rad/s, or 15.915 rpm. Rotational energy falls from 1 kJ to 833.33 J, the rest going into friction as the child is brought up to speed.

Total Angular Momentum

A system often has more than one rotating part, such as two wheels on a shaft or a planet that both orbits and spins. The total is one value about one axis, and the turning direction of each part decides whether it adds or subtracts.

How to calculate the total angular momentum?

To find total angular momentum, add the angular momentum of every part about the same axis, and give opposite turning directions opposite signs. For particles, sum each mv × r⊥. For rigid bodies on one axis, sum each Iω, as in the formula below.

$$L = I_1 \omega_1 \pm I_2 \omega_2$$

Two Bodies mode in the Angular Momentum Calculator does this for a pair. Enter I and ω for each body, then set Body 2 Spins to Same Way or Opposite Way. The defaults, 2.5 kg·m² at 12 rad/s and 1 kg·m² at 20 rad/s, total 50 kg·m²/s the same way and 10 kg·m²/s opposite.

The same mode shows what happens if the two lock together, as in a clutch. The pair settles at the total L divided by I₁ + I₂, which is 14.286 rad/s for the defaults. Energy falls from 380 J to 357.14 J, and the 22.857 J gap turns to heat where the surfaces rub.

A body at rest goes in as ω₂ = 0. Drop a 1 kg·m² disk at rest onto a 2.5 kg·m² disk turning at 12 rad/s, and L stays at 30 kg·m²/s. The pair turns at 8.5714 rad/s, and the energy falls from 180 J to 128.57 J, a loss of 51.429 J.

With three or more parts, sum by hand. OpenStax adds three particles with values of −16, −20 and +6 kg·m²/s and gets −30 kg·m²/s, not 42. Run each particle in Central Point mode, note which way it turns, and combine the signed results.

A planet has an orbital part and a spin part. With figures from the NASA Earth Fact Sheet, the orbital part is 5.9722 × 10²⁴ kg × 29,780 m/s × 1.49598 × 10¹¹ m, which is 2.66 × 10⁴⁰ kg·m²/s in Central Point mode.

The spin part is Iω. NASA lists Earth’s moment of inertia as 0.3308 MR², with R = 6,371 km and one turn every 23.9345 hours. That gives I = 8.019 × 10³⁷ kg·m², ω = 7.2921 × 10⁻⁵ rad/s and a spin of 5.85 × 10³³ kg·m²/s.

To add the two in Two Bodies mode, enter the orbit as I = mr² = 1.3366 × 10⁴⁷ kg·m² with ω = v ÷ r = 1.9907 × 10⁻⁷ rad/s. Share from body 1 then reads Over 99.99%, because the orbit holds about 4.5 million times more angular momentum than the spin.

Solid Sphere in Common Shape mode returns 7.07 × 10³³ kg·m²/s for the same spin, about 21% high. The shape formulas assume uniform density, and Earth’s factor of 0.3308 sits below the 0.4 of a uniform sphere because its core is denser than its crust.

How to calculate specific angular momentum?

Specific angular momentum is L divided by mass, h = L ÷ m = r × v × sin θ, in m²/s. Orbital mechanics uses it because it does not depend on the mass of the satellite, and it keeps one value all the way around an orbit.

The Angular Momentum Calculator lists it on the energy card of Central Point mode as Specific L. The default ball gives 0.6 ÷ 0.2 = 3 m²/s. For Earth’s orbit around the Sun it is 29,780 m/s × 1.49598 × 10¹¹ m = 4.455 × 10¹⁵ m²/s.

Angular Momentum vs Momentum

Momentum on its own means linear momentum, the product of mass and velocity along a line. Angular momentum is its counterpart for rotation, built from the same mass and speed plus a distance from an axis.

What is angular momentum vs momentum?

Linear momentum measures motion along a line, and angular momentum measures motion around a point. A force changes the first and a torque changes the second. Each one is conserved when nothing outside the system supplies that force or torque.

A body can have one without the other. A flywheel bolted to a bench has angular momentum and no linear momentum, while a puck sliding straight at you has linear momentum and no angular momentum about your position.

What is the formula for calculating momentum?

Linear momentum is p = mv, mass times velocity, and it points the same way as the velocity. That is the real formula for momentum at everyday speeds. Close to the speed of light it becomes p = γmv, where γ = 1 ÷ √(1 − v²/c²).

To calculate momentum, multiply kg by m/s. The default 0.2 kg ball at 4 m/s has p = 0.8 kg·m/s, shown in Central Point mode as Linear momentum (mv). The Momentum Calculator works with p = mv directly, and the Conservation of Momentum Calculator applies it to collisions, where the total momentum is the vector sum of each body’s mv.

What unit is momentum calculated in?

Linear momentum is in kg·m/s, which equals one newton-second. Angular momentum carries one more length, so its unit is kg·m²/s, with dimensions ML²T⁻¹. That unit is the same as N·m·s and J·s, the unit of the Planck constant.

The first result card of the Angular Momentum Calculator converts L to slug·ft²/s, lb·ft²/s and g·cm²/s. One slug·ft²/s is 1.3558 kg·m²/s, and one lb·ft²/s is 0.04214 kg·m²/s. The default 0.6 kg·m²/s reads 0.44254 slug·ft²/s, 14.238 lb·ft²/s or 6,000,000 g·cm²/s.

What is the symbol for angular momentum, and why is L used?

The symbol is L, set in bold or with an arrow when direction matters. OpenStax writes a lowercase l for a single particle and a capital L for a system or a rigid body. Atomic physics adds S for spin and J for the total of orbital and spin.

Physics references give no agreed origin for the letter. It is a convention and not an abbreviation, and it stays clear of p for linear momentum, I for moment of inertia and ω for angular velocity. Chemistry reuses it as the orbital quantum number l.

Angular Momentum in Chemistry

Atomic values of angular momentum are tiny, near 10⁻³⁴ J·s, so chemistry writes them as multiples of ħ instead of kg·m²/s. The quantum numbers n, l and mₗ in an electron configuration are what set them.

How to find angular momentum in chemistry?

Chemistry asks for the angular momentum of an electron in an atom, and that value is quantized. Two rules are taught, the older Bohr rule and the quantum mechanical one. Bohr Orbit mode in the Angular Momentum Calculator takes the orbit number n, the orbital quantum number l and the atomic number Z, and it shows both results.

What is the formula of angular momentum in Bohr’s model?

Bohr allowed only those orbits whose angular momentum is a whole multiple of the reduced Planck constant. The orbit number n counts up from 1, and each step adds one more unit of ħ to the electron.

$$L = n \hbar = \frac{n h}{2 \pi}$$

The Angular Momentum Calculator uses ħ = 1.054571817 × 10⁻³⁴ J·s, the exact value in the NIST CODATA 2022 constants. OpenStax section 6.4 writes the same condition as mₑvₙrₙ = nħ, which is L = mvr for the electron.

The table is for hydrogen, Z = 1. The radius grows as n²a₀, with a₀ = 52.918 pm, and the speed falls as αc ÷ n. Multiply the electron mass by the speed and radius in any row and you get nħ back. The last card adds the de Broglie wavelength, 664.98 pm at n = 2, and exactly n of them fit around the orbit.

Atomic Number (Z) extends the mode to one-electron ions such as He⁺ and Li²⁺. The radius becomes n²a₀ ÷ Z and the speed Zαc ÷ n, while L stays at nħ. For He⁺ in n = 1 the results are 26.459 pm, 4,375,400 m/s and −54.423 eV.

What is the angular momentum of a 2p orbital?

A 2p electron has n = 2 and orbital quantum number l = 1. Quantum mechanics sets the size of its orbital angular momentum from l alone and not from n, as given in OpenStax section 8.1.

$$L = \sqrt{l(l+1)} \, \hbar = \sqrt{2} \, \hbar$$

That is 1.4142ħ, or 1.4914 × 10⁻³⁴ J·s. Bohr’s rule gives 2ħ for n = 2, which is why the hero value and the second card differ. Bohr Orbit mode starts at n = 2 and l = 1, so the 2p values are the first ones you see.

The z-component can only equal mₗħ, with mₗ = −1, 0 or +1 for a p orbital. Every s orbital has l = 0 and no orbital angular momentum at all, a result the Bohr model cannot produce since its lowest value is ħ.

With electron spin included, the Total J row gives √(j(j + 1))ħ for j = l + ½. For 2p that is j = 3/2 and 1.9365ħ, or 2.0422 × 10⁻³⁴ J·s. The other allowed value, j = ½, works out to 0.86603ħ and is not on the card.

Torque, Energy and Spin Results

The second card doubles as an angular velocity calculator. It shows the angular velocity, the moment of inertia, the rotation rate in rpm or the time for one turn, leaving out whichever one you typed in. If your spin is given as a frequency or a period, the Angular Frequency Calculator converts it to rad/s. Set Solve For to Angular Velocity in Known Inertia mode and ω = L ÷ I becomes the main answer.

The third card gives rotational energy, ½Iω², which equals L² ÷ 2I. The default ball stores 1.6 J. Known Inertia and Common Shape modes add the same energy in ft·lbf and in watt-hours, the unit used to rate flywheel storage.

The fourth card answers how hard it is to stop the motion. Torque is the rate of change of angular momentum, so a steady torque that removes all of L in the stopping time Δt has the size shown below.

$$\tau = \frac{L}{\Delta t}$$

Stopping the default ball in 2 s takes 0.3 N·m, at an average power of 0.8 W, and it completes 0.84883 turns on the way. That is a steady slowing of 2.6667 rad/s² over 5.3333 rad, which you can check in the Angular Acceleration Calculator and the Angular Displacement Calculator.

Torque times time is the rotational form of impulse, the quantity the Impulse and Momentum Calculator handles for straight-line motion. Stopping time never changes L, I or ω, and it feeds this one card only.

Under the cards, the Same L at Other Radii table shows conservation for a point mass. Halve the radius of the default ball to 0.375 m and its speed doubles to 8 m/s, while ω and the energy both rise fourfold, to 21.333 rad/s and 6.4 J.

The Angular Momentum Calculator also writes out each step of the arithmetic and draws the right-hand rule for both formulas. More tools for spin, torque and inertia are listed under Rotational Motion Calculators.