Moment of Inertia Converter
Convert between kg·m², g·cm², slug·ft², lb·ft², and other moment of inertia units.
Moment of Inertia Converter
Enter a moment of inertia and select its unit — all other units update instantly.
Why Mass Farther From the Axis Matters More — I = Σmr²
Same mass m at different distances — all rotate together
Each particle's contribution to I:
Total I = 0.25mr² + 1mr² + 4mr² = 5.25 mr² — the particle at 2r alone contributes 76% of the total.
Common Moment of Inertia Values
How to Convert Between Kg·m², Lb·Ft², and G·cm²
Your flywheel design specifies I = 0.35 kg·m². Your US supplier quotes in lb·ft². Type 0.35, select kg·m², and see that it equals 8.31 lb·ft², 3,500,000 g·cm², and 120.4 lb·in². The interactive visualisation shows how mass distribution affects the result.
The converter handles 9 units: kg·m², g·cm², kg·cm², slug·ft², slug·in², lb·ft², lb·in², oz·in², and oz·ft². A physics student measuring a lab disk at 2,500 g·cm² sees it equals 0.00025 kg·m² and 0.00828 lb·ft². An aerospace engineer working with a satellite at 50 kg·m² sees it equals 3.74 slug·ft². The interactive demo shows three equal masses at distances r, 2r, and 3r contributing I, 4I, and 9I.
Why distance matters more than mass for flywheels
A flywheel stores rotational energy as E = 0.5 x I x ω². Doubling the moment of inertia doubles the stored energy. But I depends on mass distribution: a flywheel with mass at the rim has 2x the I of a solid disk of the same mass. This is why flywheels are designed with thick rims: maximum I for minimum mass.
A skater spinning at 3 rev/s with arms extended (I = 4.0 kg·m²) pulls arms in and I drops to 1.0 kg·m². Angular momentum L = Iω is conserved, so ω increases from 3 to 12 rev/s. The skater spins 4x faster by reducing I to 1/4. This demonstrates the I = mr² relationship: halving the radius quarters the moment of inertia.
A 4-cylinder engine has a crankshaft with I = 0.02 kg·m². Each piston contributes differently depending on its position in the firing order. The crankshaft must be balanced so that the total I is uniform through the rotation cycle. Uneven I causes vibration. Converting between lb·in² (US machining) and kg·m² (SI simulation) ensures the balance is correct.
How the Moment of Inertia Converter Handles Kg·m² and Lb·Ft²
All conversions pass through kilogram-meters squared (kg·m²). When you enter 0.35 kg·m² (a car flywheel), the tool divides by each target unit's factor: by 0.042140 for lb·ft², by 10^7 for g·cm², by 0.002926 for lb·in². The result: 0.35 kg·m² = 8.31 lb·ft² = 3,500,000 g·cm² = 120.4 lb·in².
Why g·cm² is 10 million times smaller than kg·m²
1 g·cm² = (0.001 kg) x (0.01 m)² = 0.001 x 0.0001 = 10^-7 kg·m². The centi prefix (10^-2) is squared for the distance term, and the kilo-to-gram conversion (10^-3) applies to mass. So 1 g·cm² = 10^-7 kg·m², meaning kg·m² is 10 million times larger. This is why lab apparatus outputs in g·cm² look like huge numbers (2,500,000) while the same object in kg·m² is tiny (0.25).
Shape formulas and how mass distribution affects I
Solid disk: I = 0.5 x m x r². Thin hoop: I = m x r² (twice the disk because all mass is at the rim). Solid sphere: I = 0.4 x m x r². Thin rod about centre: I = (1/12) x m x L². For a 2 kg disk at 0.1 m radius: I = 0.5 x 2 x 0.01 = 0.01 kg·m². For a 2 kg hoop at the same radius: I = 2 x 0.01 = 0.02 kg·m². The hoop has 2x the I of the disk.
The parallel axis theorem (I = I_cm + m x d²) lets you calculate I about any axis if you know I about the centre of mass. For a disk of I_cm = 0.01 kg·m² and mass 2 kg, shifted 0.05 m from centre: I = 0.01 + 2 x 0.0025 = 0.015 kg·m². This is widely used in engineering for compound shapes and assemblies.
Frequently Asked Questions
How do I convert 0.35 kg·m² to lb·ft²?
Divide by 0.042140: 0.35 / 0.042140 = 8.31 lb·ft². This is a typical car flywheel moment of inertia. For a quick estimate, multiply kg·m² by 23.73 to get lb·ft² (since 1/0.042140 = 23.73). The converter shows all units simultaneously so you can verify the relationships.
What affects moment of inertia more, mass or distance?
Distance, because I depends on r². Doubling the distance from the axis quadruples I. Doubling the mass only doubles I. A 1 kg mass at 2 m radius has I = 4 kg·m². A 4 kg mass at 1 m radius has I = 4 kg·m². Same I, but the 4 kg mass has 4x the weight. This is why flywheels put mass at the rim.
What is the moment of inertia of a solid disk vs a hoop?
Solid disk: I = 0.5 x m x r². Thin hoop: I = m x r². For the same mass and radius, the hoop has twice the moment of inertia because all its mass is at the maximum distance from the axis. A 2 kg disk at 0.1 m: I = 0.01 kg·m². A 2 kg hoop at 0.1 m: I = 0.02 kg·m². The hoop stores twice the rotational energy at the same speed.
How does a figure skater spin faster by pulling arms in?
Angular momentum L = Iω is conserved. With arms extended, I = 4.0 kg·m² and ω = 3 rev/s. Pulling arms in reduces I to 1.0 kg·m². Since L is constant, ω must increase to 12 rev/s. The skater spins 4x faster by reducing I to 1/4. This demonstrates the I = mr² relationship.
What is the parallel axis theorem?
I = I_cm + m x d², where I_cm is the moment of inertia about the centre of mass, m is total mass, and d is the distance between axes. For a 2 kg disk with I_cm = 0.01 kg·m², shifted 0.05 m: I = 0.01 + 2 x 0.0025 = 0.015 kg·m². This theorem is essential for calculating I of complex assemblies and compound shapes.
What is the moment of inertia of Earth?
Earth has I = 8.04 x 10^37 kg·m². It is approximately a solid sphere with mass 5.97 x 10^24 kg and radius 6.371 x 10^6 m. I = 0.4 x m x r² = 0.4 x 5.97e24 x (6.371e6)^2 = 8.04e37 kg·m². Earth's rotation rate changes slightly due to tidal friction, which changes its moment of inertia by about 10^-8 per century.