How this instrument works
Angular momentum is rotation's bookkeeping quantity — the moment of inertia about a chosen axis, multiplied by the rate of turn about that same axis. Its unit, the kilogram metre squared per second, is dimensionally identical to the joule second, which is the unit of action and the unit of Planck's constant. That coincidence is not decorative. It is why the same quantity survives at every scale, down to an electron, whose intrinsic spin is fixed forever at ħ⁄2 ≈ 5.27 × 10⁻³⁵ kg·m²/s.
Kepler wrote the conservation law down in 1609 without a name for it: a planet sweeps equal areas in equal times, which is exactly the statement that its orbital value never changes. Newton derived that result from central forces in the Principia, Euler supplied the moment of inertia eighty years after him, and in 1918 Emmy Noether explained why such a rule had to exist at all — space has no preferred direction, and this conserved quantity is precisely what that symmetry buys you. Tidal friction is the everyday audit: Earth's spin slows by roughly two milliseconds per century while the Moon climbs away about 3.8 cm a year, and the ledger balances.
L = Iω is the single-axis form, and it hides two assumptions. First, I belongs not to the object but to the object plus the axis you picked — a door swung on its hinges and the same door spun about its centre give different figures. Second, for a body turning about anything other than a principal axis, inertia is a tensor rather than a number, and the true vector points somewhere off the spin axis. That mismatch is why an unbalanced wheel chews through bearings, and why a phone flipped about its middle axis tumbles instead of spinning cleanly.
- Put your figure in Moment of inertia (kg·m²) — the value about the axis you actually care about, not the object's mass.
- Type the spin rate into Angular velocity and pick its unit. Motor plates, turbines and flywheels nearly always quote rpm.
- Read Angular momentum (kg·m²/s) to four decimal places, with the multiplication written out in the working block.
- For a conservation problem, run the sheet twice — once before the shape change, once after. Matching results confirm no torque acted.
Worked example — the lecture-hall turntable
A student sits on a low-friction turntable with a dumbbell in each outstretched hand. Measured about the vertical axis, the whole arrangement has a Moment of inertia of 2 kg·m². A colleague gives the platform a shove and it settles at an Angular velocity of 5 rad/s — 0.80 turns a second, or 47.7 rpm on a tachometer. The sheet returns L = 2 × 5 = 10 kg·m²/s.
Now the student pulls both weights to the chest. Nothing outside the system twists the platform, so those 10 kg·m²/s survive intact and only their split between I and ω shifts. Inertia falls to roughly 0.8 kg·m², so the turn rate must rise to 10 ⁄ 0.8 = 12.5 rad/s, a shade under two revolutions a second. Rotational energy meanwhile climbs from 25 J to 62.5 J, and the extra 37.5 J came out of the student's arms hauling those masses inward. One quantity is conserved here; energy is not the one.
Questions
Why does the calculator insist on radians per second?
Because L = Iω is only numerically correct in radians — the radian is the dimensionless ratio that makes rotational formulas line up with their linear counterparts. Feed the raw product 300 deg/s and the answer lands off by a factor of 57.3. The unit menu takes care of it: rpm is scaled by 2π⁄60 = 0.10472 and degrees per second by π⁄180 = 0.017453 before anything is multiplied. A 3,000 rpm flywheel is turning at 314.16 rad/s.
Is moment of inertia just another name for mass?
No, and treating it as one is the commonest mistake on this sheet. Mass measures resistance to being pushed; moment of inertia measures resistance to being twisted, and it weights every scrap of material by the square of its distance from the axis. A 10 kg hoop of 0.5 m radius has I = 2.5 kg·m², while the same 10 kg recast as a solid disc of identical radius has only 1.25 kg·m². Move the axis and the figure changes again.
Can the result be negative?
As a vector it carries a direction, set by the right-hand rule: curl your fingers with the rotation and your thumb points along L. Call that thumb direction positive and anything turning the other way counts as negative, which is how counter-rotating parts cancel — the reason a helicopter needs a tail rotor. This instrument reports a magnitude, so enter a positive Angular velocity and attach the sign yourself when adding two spins together.
How does this differ from linear momentum?
They are two separate conserved quantities, not two views of one. Linear momentum p = mv holds steady when no net force acts; the rotational version holds steady when no net torque acts, and a body can carry one without the other. A wheel spinning on the spot has zero p and plenty of L; a brick sliding flat across ice has p and no spin at all. Torque stands to L exactly as force stands to p — the rate at which it changes.
When does L = Iω stop being true?
The moment the rotation axis is not a principal axis of the body. In general an inertia tensor maps ω to L, the two vectors need not be parallel, and no single number can describe the situation: the axis wanders and the body wobbles. The formula also assumes I holds still while you read it. If the shape is changing, L stays put while I and ω trade against each other, which is the entire point of the skater demonstration. At relativistic speeds a different framework takes over.
What magnitudes should I expect?
The range is enormous. A bicycle wheel at commuting speed sits near 1 kg·m²/s — enough to make the bike feel planted, nowhere near enough to hold it upright. A car flywheel runs in the tens. The International Space Station steers on four control moment gyroscopes storing thousands, burning no propellant until they saturate and have to be unloaded. Earth's own rotation carries about 7 × 10³³ kg·m²/s. Between a wheel and a planet lie thirty-three orders of magnitude, all obeying the same product.