Angular Momentum Calculator
Calculate angular momentum L from moment of inertia and angular velocity, or from point-mass data (mass, velocity, radius). Enter your values in any unit and get L in kg·m²/s (J·s).
Angular Momentum from Inertia and Angular Velocity or Point-Mass Data
Angular Momentum Formula
Where:
- \(L\) — angular momentum (kg·m²/s, equal to J·s)
- \(I\) — moment of inertia about the rotation axis (kg·m²)
- \(\omega\) — angular velocity (rad/s)
For a point mass moving in a circle of radius \(r\) with tangential speed \(v\), the same quantity is written as \(L = m v r\), because the moment of inertia of a point mass is \(I = mr^2\) and \(\omega = v/r\).
Note: the backslash-free closing tag below belongs to the shortcode.
Angular Momentum – Calculation Example
Example: a flywheel with a moment of inertia of 2 kg·m² spins at 60 rpm. First convert the angular velocity to radians per second:
\[ \omega = 60 \,\text{rpm} \times \frac{2\pi}{60} = 2\pi \approx 6.283 \,\text{rad/s} \]
\[ L = I\omega = 2 \times 6.283 = 12.566 \,\text{kg·m}^2/\text{s} \]
Result: the angular momentum is approximately 12.57 kg·m²/s (equivalently 12.57 J·s). This matches the calculator's default values, so you can reproduce it directly.
For comparison, in point-mass mode a 1 kg object moving at 5 m/s on a circle of radius 0.8 m carries \(L = mvr = 1 \times 5 \times 0.8 = 4\) kg·m²/s.
How to Use the Angular Momentum Calculator
- Choose the Object type. Select "Rotating body (moment of inertia)" if you know \(I\) and \(\omega\), or "Point mass (mass, velocity, radius)" if you know \(m\), \(v\) and \(r\). The input fields switch automatically.
- Rigid-body mode: enter the Moment of inertia (I) and pick its unit (kg·m², g·cm², lb·ft², lb·in²), then enter the Angular velocity (ω) in rad/s, rpm, °/s, or rev/s (Hz).
- Point-mass mode: enter the Mass (m), the Tangential velocity (v), and the Radius (r) from the rotation axis, each with its own unit selector.
- Read the Angular momentum result in kg·m²/s (J·s), together with a table showing all quantities converted to SI and other common units, and the exact formulas applied to your values.
What Is Angular Momentum?
Angular momentum is the rotational analogue of linear momentum. While linear momentum \(p = mv\) describes how much straight-line motion a body carries, angular momentum \(L\) describes how much rotational motion it carries about an axis. It depends on both how massive the object is and how its mass is distributed: a spinning flywheel with most of its mass far from the axis stores much more angular momentum than a compact object spinning at the same rate.
The two input modes of this calculator are physically equivalent. A rigid body is treated as a collection of mass elements, and its resistance to spinning is summarized by the moment of inertia \(I\). A point mass on a circular path is the simplest special case, where \(I = mr^2\) and \(\omega = v/r\), so \(L = I\omega = mvr\). If you need \(I\) for a common shape, remember: a solid disk is \(\tfrac{1}{2}MR^2\), a thin hoop is \(MR^2\), and a solid sphere is \(\tfrac{2}{5}MR^2\).
The SI unit of angular momentum is kg·m²/s, which is identical to the joule-second (J·s) — the same unit used for Planck's constant. The calculator also compares your result with the reduced Planck constant \(\hbar = 1.055 \times 10^{-34}\) J·s, the natural quantum unit of angular momentum. Everyday rotating objects have angular momenta that are astronomically larger than \(\hbar\), which is why quantum effects are invisible at macroscopic scales.
Interpreting the result: a larger \(L\) means more stored rotational motion, so more effort (a bigger torque acting over a longer time) is needed to stop or reverse the rotation. This is directly related to the angular analogue of impulse — torque times time equals the change in angular momentum.
Conservation of Angular Momentum
The most important property of angular momentum is that, in the absence of an external torque, it is conserved. A spinning skater who pulls in her arms decreases her moment of inertia, so her angular velocity must increase to keep \(L = I\omega\) constant — the classic demonstration of this principle. The same physics governs collapsing stellar cores spinning up into neutron stars, gyroscopes, and satellites reorienting with reaction wheels.
Conservation of angular momentum is the rotational counterpart of the conservation of linear momentum, which you can explore with the linear momentum calculator, and it complements the conservation of energy principle in analyzing rotational collisions and spinning systems. For the circular motion underlying the point-mass mode, the centripetal force calculator is a useful companion.
Assumptions and Limitations
- Fixed rotation axis: the calculator assumes rotation about a single, fixed axis. For tumbling objects or rotation about a free axis (like a thrown wrench), angular momentum is a vector and \(I\) may change with orientation, which requires vector treatment.
- Tangential velocity: in point-mass mode, the velocity must be perpendicular to the radius. The general vector formula is \(L = mvr\sin\theta\); the calculator uses \(\sin\theta = 1\), the case of pure circular motion.
- Nonnegative inputs: magnitudes only. If you need to distinguish spin directions (clockwise vs counterclockwise), assign the sign yourself: reversing the spin direction flips the sign of \(L\) but not its magnitude.
- Unit conversions: standard factors are applied automatically — 1 rpm = 0.10472 rad/s, 1 lb·ft² = 0.04214 kg·m², 1 mph = 0.44704 m/s — so you can mix unit systems freely, but results are always reported in SI.
- Axis choice matters: the moment of inertia must be the one about the actual rotation axis. Using \(I\) about the wrong axis (e.g., about the center vs. about a pivot point) gives a wrong answer.
Frequently Asked Questions
How do I calculate angular momentum by hand, step by step?
First convert everything to SI units: moment of inertia to kg·m² and angular velocity to rad/s. Then multiply: L = I × ω. For example, a body with \(I = 0.5\) kg·m² spinning at 30 rpm gives \(\omega = 30 \times 0.10472 \approx 3.14\) rad/s, so \(L = 0.5 \times 3.14 = 1.57\) kg·m²/s. For a point mass on a circular path, simply multiply mass × tangential speed × radius.
How do I convert rpm to rad/s myself?
One revolution is \(2\pi\) radians and one minute is 60 seconds, so multiply rpm by \(2\pi/60 \approx 0.10472\). Quick reference: 60 rpm = 1 rev/s = \(2\pi \approx 6.283\) rad/s; 120 rpm ≈ 12.57 rad/s. The reverse conversion: divide rad/s by 0.10472 to get rpm. Never use rpm or degrees per second directly in \(L = I\omega\) — the formula only works with radians, since radians are dimensionless.
Is angular momentum always positive? Can it be negative?
The magnitude of angular momentum is always nonnegative, and this calculator reports magnitudes only. However, angular momentum is really a vector, and its sign (or direction) depends on the spin direction: by convention, counterclockwise rotation viewed from above the axis is positive and clockwise is negative. Reversing the spin direction flips the sign but keeps the magnitude the same. Assign the sign yourself if you need to track direction, for example when checking conservation in a collision.
What is a typical angular momentum for everyday objects?
It depends strongly on size and speed. A bicycle wheel spinning at walking pace carries roughly 1–10 kg·m²/s; a car wheel at highway speed, tens of kg·m²/s; a figure skater spinning fast, about 10–50 kg·m²/s; the Earth's rotation, a colossal \(7 \times 10^{33}\) kg·m²/s. In contrast, quantum systems like electron orbits carry angular momentum on the order of \(10^{-34}\) J·s, which is why the calculator compares your result with the reduced Planck constant.
What is the difference between angular momentum and rotational kinetic energy?
They describe different things. Angular momentum \(L = I\omega\) measures the amount of rotational motion and is conserved when no external torque acts. Rotational kinetic energy \(E = \tfrac{1}{2}I\omega^2\) measures the energy stored in the spin and can be converted to heat or work. They scale differently with speed: doubling \(\omega\) doubles \(L\) but quadruples the energy. A skater pulling in her arms keeps \(L\) constant while her kinetic energy actually increases, supplied by the work of pulling.
I know the shape of my object but not its moment of inertia. How do I find I?
Use the standard formula for the shape and axis. A solid disk or cylinder about its center is \(\tfrac{1}{2}MR^2\), a thin hoop is \(MR^2\), a solid sphere about its center is \(\tfrac{2}{5}MR^2\), and a rod about its center is \(\tfrac{1}{12}ML^2\). If the axis is parallel to but offset from the center of mass by distance \(d\), apply the parallel axis theorem: \(I = I_{cm} + Md^2\). Then plug \(I\) into the calculator with your angular velocity.
What are the most common mistakes when calculating angular momentum?
The biggest one is using angular velocity in rpm or degrees per second instead of rad/s — this inflates the result by a factor of about 10 or 57. Others: using the moment of inertia about the wrong axis (center vs. pivot), using the diameter instead of the radius, and in point-mass mode using a velocity that is not perpendicular to the radius (then you need \(L = mvr\sin\theta\)). Also, don't confuse mass in pounds with mass in kilograms — pounds must be converted (1 lb = 0.4536 kg).