The angular momentum calculator above evaluates L = Iω and computes the moment of inertia for you from a shape library, so you do not have to look up a table of formulas first. It is worth naming the boundary with two adjacent tools before anything else, because an automated glance at the titles will not see it. The momentum calculator solves p = mv, measured in kg·m/s. The centripetal force calculator solves F = mv² / r, measured in newtons. This page solves L = Iω, measured in kg·m²/s. Three different quantities, three different units, three different conservation stories.
Arb Digital publishes them separately because they answer questions that cannot substitute for one another. A flywheel spinning about its own axis has substantial angular momentum, zero linear momentum, and no centripetal force acting on the body as a whole. Getting the right one is not a matter of taste, and the units are the fastest way to check you have.
What This Angular Momentum Calculator Does
You choose a shape, give it a mass and a size, and set how fast it is turning. The tool works out the moment of inertia about the stated axis, converts your spin rate into radians per second, multiplies the two, and reports the angular momentum in kilogram metres squared per second. If you already have a published inertia figure for a rotor or gearbox, the custom option takes it directly.
The supporting grid gives four figures. The moment of inertia is shown explicitly because it is the part most people get wrong, and because seeing it lets you sanity-check the shape you picked. The rotational kinetic energy, ½Iω², is the energy stored in the spin, which is the quantity that matters for flywheel storage and for how much damage a burst rotor does. The rim speed is ωr, the linear speed of the outermost point, which is what limits real machines through material strength. The final item is the conservation demonstration: what the spin rate becomes if the same mass is pulled in to half its radius with no external torque applied.
How to Use It
- Pick the shape and the axis together. Moment of inertia is defined about a specific axis, and the same rod has four times the inertia about its end that it has about its centre. Choosing the wrong axis is a bigger error than any rounding.
- Read the size box carefully. For discs, hoops, spheres and point masses it is a radius. For the two rod options it is the full length of the rod.
- Set the spin rate unit before typing. RPM and rad/s are direct. The third option lets you enter a rim speed in metres per second, which the tool divides by the radius to get ω.
- Use custom inertia for real hardware. Motor and gearbox datasheets publish inertia figures, usually in kg·m² or sometimes g·cm². Convert to kg·m² first — the factor is 10⁻⁷.
- Compare the rotational energy against the angular momentum. They behave differently when the radius changes, and the last grid item plus the note underneath make that difference visible.
The Formula: How Angular Momentum Is Calculated
For a rigid body rotating about a fixed axis, L = Iω, where I is the moment of inertia about that axis in kg·m² and ω is the angular velocity in rad/s. OpenStax University Physics Volume 1, section 11.2 on angular momentum, derives this from the more general vector definition l = r × p for a single particle and shows that summing over a rigid body collapses it to L = Iω.
The moment of inertia values this tool uses are the standard results: ½mr² for a solid disc or cylinder about its centre, mr² for a thin hoop or a point mass, ⅖mr² for a solid sphere about its centre, ⅔mr² for a hollow spherical shell, 1⁄12mL² for a rod about its centre and ⅓mL² for a rod about one end. Section 10.4 of the same text derives them and covers the parallel axis theorem you need when the axis is offset.
Work through the default. A solid disc of 2 kg and 0.3 m radius spinning at 1,500 RPM. The moment of inertia is ½ × 2 × 0.3² = 0.09 kg·m². The angular velocity is 1,500 × 2π ÷ 60 = 157.08 rad/s. So L = 0.09 × 157.08 = 14.14 kg·m²/s. The rotational energy is ½ × 0.09 × 157.08² = 1,110 J, and the rim is moving at 157.08 × 0.3 = 47.1 m/s.
Why L = Iω Is Not p = mv in Disguise
The analogy between linear and rotational mechanics is genuinely useful, and it is also the reason the two get conflated. Mass becomes moment of inertia, velocity becomes angular velocity, and p = mv becomes L = Iω. But an analogy between equations is not an equality between quantities.
Linear momentum has units of kg·m/s and is conserved when no external force acts. Angular momentum has units of kg·m²/s and is conserved when no external torque acts. Those are separate conservation laws with separate conditions, and a system can satisfy one while violating the other. Fire a bullet and the recoiling rifle conserves linear momentum. Spin a wheel on a frictionless axle and it conserves angular momentum while its linear momentum stays at zero the whole time, because the centre of mass never goes anywhere.
There is also a subtler difference that matters. Angular momentum is defined relative to a chosen origin, so the same particle can have different angular momentum about different points, while linear momentum has no such dependence. That is why the axis matters so much on this page and why the shape dropdown pairs a shape with an axis rather than listing shapes alone. When your question really is about a mass travelling in a straight line, the momentum calculator is the correct tool.
Why It Is Not Centripetal Force Either
The other adjacent page is the centripetal force calculator, and the distinction is arguably sharper. Centripetal force is a force, measured in newtons, and it answers a question about the present instant: how hard must something pull inward right now to keep this mass on this circle at this speed? Remove the force and the object flies off on a tangent immediately.
Angular momentum is not a force and answers a different kind of question, about what persists. It is the rotational quantity that stays fixed in the absence of external torque, which is why it explains behaviour over time — why a skater speeds up when they pull their arms in, why a gyroscope resists being tipped, why planetary orbits sweep equal areas in equal times. Centripetal force explains why the motion is circular. Angular momentum explains what the rotation will do next when something changes. Neither substitutes for the other, and the torque calculator covers the third piece, the quantity that changes angular momentum in the first place.
The Conservation Result Everyone Half-Remembers
The last grid item shows what happens when the same mass is pulled in to half its radius with no external torque. Because L is conserved and I scales with the square of the radius for every shape in the library, halving the radius quarters the inertia and therefore quadruples the angular velocity. The skater's arms are the famous example, and the numbers are less gentle than the demonstration looks: a fourfold speed increase from a single halving.
The energy does not behave the same way, and this is the part that gets skipped. Rotational energy is ½Iω², so quartering I while quadrupling ω multiplies the energy by four, not by one. Energy is not conserved in that manoeuvre. The extra energy comes from the skater's muscles doing work against the outward pull as they haul their arms in, and it is real physical work, not a bookkeeping artefact. Anyone who has tried the office-chair version knows how much effort it takes. If you want the linear analogue of that energy figure, the kinetic energy calculator handles ½mv².
Where the Moment of Inertia Goes Wrong
Almost every wrong angular momentum figure traces back to the moment of inertia rather than to the arithmetic. Three failure modes account for most of it. The first is using the wrong axis: a rod about its end has four times the inertia it has about its centre, and no other input error is that large. The second is treating a real object as an idealised shape — a car wheel is neither a solid disc nor a thin hoop but something between the two, with mass concentrated at the rim, so the disc formula understates it noticeably. The third is a unit slip, usually a diameter entered where a radius was wanted, which quadruples the inertia in one keystroke.
When the axis is offset from the centre of mass, the parallel axis theorem applies: I = Icm + md², where d is the distance between the two axes. That is not built into this tool's shape list, so compute it separately and enter the result through the custom option. For compound assemblies, moments of inertia about a common axis simply add, so you can build up a rotor from parts and enter the total.
Arb Digital builds free tools like this one because useful pages earn attention. If you want tools, calculators or content built for your own audience, we can help.
Browse All Free Tools Talk to Arb DigitalCommon Mistakes to Avoid
- Using kg·m/s as the unit — that is linear momentum. Angular momentum is kg·m²/s, and the extra metre comes from the lever arm.
- Entering a diameter where a radius is wanted — inertia scales with the square of the radius, so this single slip multiplies the answer by four.
- Picking a shape without checking the axis — a rod about one end has four times the inertia of the same rod about its centre.
- Assuming energy is conserved along with angular momentum — pulling a spinning mass inward conserves L but increases the rotational energy, because work is done against the outward pull.
- Feeding RPM straight into L = Iω — the equation needs rad/s. The conversion factor is 2π/60, and this tool applies it when you set the unit dropdown.
Related Free Tools From Arb Digital
For the straight-line quantity, use the momentum calculator; for the inward force holding an object on a circle, the centripetal force calculator. The torque calculator gives the quantity that changes angular momentum, and the angular acceleration calculator gives the rate at which ω itself changes. Convert spin rates with the angular velocity calculator, and compare the rotational energy here against the linear form in the kinetic energy calculator. Everything Arb Digital publishes is listed on the free online tools hub.
Frequently Asked Questions
Kilogram metres squared per second, written kg·m²/s. It is equivalent to a joule second. Note the squared metre: linear momentum is kg·m/s, and the extra length factor comes from the lever arm between the mass and the axis.
Linear momentum is mass times velocity and is conserved when no external force acts. Angular momentum is moment of inertia times angular velocity and is conserved when no external torque acts. They are separate quantities with separate units and separate conservation conditions. A wheel spinning on a fixed axle has large angular momentum and zero linear momentum.
Centripetal force is a force in newtons that describes what must pull inward to keep an object on a circular path at this instant. Angular momentum is a conserved rotational quantity in kg·m²/s that describes what the rotation will do when conditions change. One explains why the path is circular, the other explains why a skater speeds up when their arms come in.
Because angular momentum is conserved when no external torque acts. Moment of inertia falls with the square of the radius, so bringing the mass closer to the axis reduces the inertia and the angular velocity must rise to keep the product constant. Halving the effective radius quadruples the spin rate.
No. Rotational energy is half the moment of inertia times the angular velocity squared, so the same manoeuvre that conserves angular momentum increases the energy. The extra energy comes from the work done pulling the mass inward against the outward pull, which is why it takes real effort.
Somewhere between the solid disc value of half mass times radius squared and the thin hoop value of mass times radius squared, since a wheel carries much of its mass at the rim. If a manufacturer publishes an inertia figure, use the custom option and enter it directly rather than approximating with a shape.
Apply the parallel axis theorem: add the mass multiplied by the square of the distance between the two axes to the centre-of-mass inertia. This tool does not do that automatically, so compute the total and enter it through the custom moment of inertia box.
This tool is provided for educational and study use. It applies idealised rigid-body formulas about a fixed axis and does not model flexible bodies, offset axes or three-dimensional precession, so treat its output as a physics result rather than a substitute for a mechanical design calculation.
