How this instrument works
Moment of inertia tells you how much torque it takes to change a system's spin rate, playing the part in rotation that ordinary mass plays in a straight-line push: τ = Iα is the rotational twin of F = ma. What makes it interesting is that I is not fixed by mass alone — it depends on where that mass sits relative to the axis, so two systems built from identical masses can behave completely differently depending on layout. This sheet handles the case of two separate masses sharing one axis, adding a m₁r₁² term and a m₂r₂² term into a single combined figure.
Addition is the whole trick here, and it works because moment of inertia is a scalar with no direction to fight against — unlike momentum or velocity, contributions from separate masses on the same axis simply pile on top of each other with no vector bookkeeping required. That differs from moving one mass further out on a single body, which the parallel-axis theorem handles; here the two masses are already independent objects at their own radii, and the total is nothing more elaborate than reading each term off and summing it. Extend the same idea to three masses, thirty, or a continuous shell of material and the sum becomes the integral ∫r² dm that defines moment of inertia in general.
Two limits are worth holding onto. Zero out either mass and its whole m·r² term drops away, leaving the total exactly equal to what a single remaining mass alone would give — the two-mass sum genuinely collapses to the one-mass case rather than merely approximating it. Zero out a distance instead and that mass's lever arm vanishes: a point mass parked precisely at the pivot never sweeps out any circle as the system spins, so it never opposes the spin either, no matter how heavy it is.
- Enter the first mass into Mass 1 and how far it sits from the axis into Distance 1 from axis.
- Enter the second mass into Mass 2 and its own distance into Distance 2 from axis; keep every field in one consistent system of units.
- Read Moment of inertia, I — the combined total, m₁r₁² + m₂r₂², recalculated the moment either field changes.
- To see the split between the two contributions, compute m₁r₁² and m₂r₂² by hand and confirm they add to the figure shown — a quick way to spot which mass actually dominates the total.
- Set either mass or either distance to zero to watch the system collapse toward the single-mass case, and confirm the total matches what that leftover term alone would give.
Worked example — two masses on one arm
Picture a light rod free to spin about one end, carrying a 2 unit mass fixed 3 units out and a 5 unit mass fixed 4 units out along the same rod. Square each distance first — 3² = 9 and 4² = 16 — then weight by its own mass: 2 × 9 = 18 for the inner mass and 5 × 16 = 80 for the outer one. Adding the two contributions gives I = 18 + 80 = 98, and entering m1 = 2, r1 = 3, m2 = 5, r2 = 4 into the fields returns that same 98.0.
The split between the two terms is lopsided on purpose: the outer mass is only two and a half times heavier than the inner one, yet its term is more than four times larger, 80 against 18, entirely because it sits farther from the axis. Slide that same 5 unit mass in to 3 units instead of 4 and its term falls to 5 × 9 = 45, dropping the combined total from 98 to 63 — more than a third smaller — even though not one unit of mass left the system.
Questions
Why do the two masses just add together?
Because moment of inertia is a scalar with no direction to cancel, every mass on a shared axis contributes its own m·r² term completely independently of the others. There's no interaction term to account for — the total is exactly m₁r₁² + m₂r₂², the same way total mass is just m₁ + m₂ when rotation isn't involved at all.
What happens if one mass sits much closer to the axis than the other?
Its term falls away fast, because the exponent lands on distance, not on mass. A mass at half the radius of an otherwise identical mass contributes only a quarter as much to the total — halving r divides its r² term by four — so trimming a system's inertia is usually achieved by moving mass inward rather than by cutting its weight.
What does this add beyond simply adding the two masses together?
Plain mass addition, m₁ + m₂, ignores position entirely and gives the same number whether the masses sit right at the axis or far out on an arm. Moment of inertia deliberately does not do that: it weights each mass by the square of its own distance first, so identical masses at different radii add up to very different totals — location, not just quantity, decides the answer.
Can this formula be extended to three or more masses?
Yes — the pattern simply repeats: add another mᵢrᵢ² term for every extra mass sharing the axis. Run this sheet for pairs and add the running totals, or work from the general identity I = Σ mᵢrᵢ², which is exactly this formula extended to as many discrete masses as the system actually has.
Why does pulling the arms in make a figure skater spin faster?
Extending the arms moves mass further from the spin axis, which raises I sharply since every added centimetre of radius counts squared. With no external torque acting mid-spin, angular momentum L = Iω has to stay constant, so a smaller I forces a larger ω — the same arithmetic behind this two-mass sum, just applied to a body instead of two isolated points.
Does it matter which mass I label '1' and which I label '2'?
No. Addition doesn't care about order, so m1 = 2, r1 = 3, m2 = 5, r2 = 4 gives the same 98 as m1 = 5, r1 = 4, m2 = 2, r2 = 3. Assign the labels however matches your own diagram or the order you measured things in.