Moment of Inertia Converter
Convert kg·m², g·cm², lb·ft², slug·ft², and other units of rotational inertia.
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Relative Value
*Diagram shows values relative to the selected base unit (kg·m²).
Unit Information
What is Kilogram square meter (kg·m²)?
Kilogram square meter (kg·m²) is the SI unit of moment of inertia. It represents the moment of inertia of a point mass of one kilogram rotating at a radius of one meter from the axis of rotation. For a composite object, it depends on the total mass and how that mass is distributed relative to the axis of rotation.
What is Kilogram square centimeter (kg·cm²)?
This unit measures the moment of inertia in kilograms for a mass distribution measured in square centimeters. 1 kg·m² = 10,000 kg·cm². It's used for smaller components where measuring dimensions in centimeters is more practical.
What is Kilogram square millimeter (kg·mm²)?
This is an even smaller metric unit, where 1 kg·m² = 1,000,000 kg·mm². It's useful for very small or lightweight rotating parts in precision engineering.
What is Gram square centimeter (g·cm²)?
This is the standard CGS (centimeter-gram-second) unit for moment of inertia. It's frequently used in scientific literature and for specifying properties of small mechanical or electronic components. 1 kg·m² = 10,000,000 g·cm².
What is Gram square millimeter (g·mm²)?
A smaller CGS unit, useful for very fine mechanics and instrumentation. 1 g·cm² = 100 g·mm².
What is Pound square foot (lb·ft²)?
Pound square foot (lb·ft²) is an imperial/US customary unit of moment of inertia, where 'pound' refers to the unit of mass (avoirdupois pound). It signifies the moment of inertia related to a mass in pounds distributed at distances measured in feet from the axis of rotation. This unit is often used in engineering applications within regions that use the imperial system.
What is Pound square inch (lb·in²)?
This unit uses pounds for mass and inches for distance, commonly used for smaller components in the US system. Since there are 144 square inches in a square foot, 1 lb·ft² = 144 lb·in².
What is Ounce square inch (oz·in²)?
This unit is used for very small or lightweight components in the imperial system, like those found in small motors or servos. Since 1 pound = 16 ounces, 1 lb·in² = 16 oz·in².
What is Slug square foot (slug·ft²)?
This is the unit of moment of inertia in the English Engineering system. A 'slug' is a unit of mass defined such that it simplifies the equation τ = Iα (Torque = Moment of Inertia × Angular Acceleration) in the imperial system, avoiding the need for a gravitational constant. 1 slug is approx 32.2 lbs. This unit is used in some US aerospace and mechanical engineering fields.
Formulas
τ = I × α
Newton's Second Law for Rotation: Torque (τ) equals Moment of Inertia (I) times Angular Acceleration (α).
L = I × ω
Angular Momentum (L) is the product of Moment of Inertia (I) and Angular Velocity (ω).
K_rot = ½ × I × ω²
Rotational Kinetic Energy (K_rot) is one-half the Moment of Inertia (I) times the square of the Angular Velocity (ω).
I = Σ(mᵢrᵢ²)
The general formula for moment of inertia is the sum (Σ) of each mass particle (mᵢ) multiplied by the square of its perpendicular distance (rᵢ) from the axis of rotation.
I = I_cm + md²
The Parallel Axis Theorem, used to find the moment of inertia about an axis parallel to one through the center of mass (I_cm).
1 slug·ft² ≈ 1.356 kg·m²
Conversion between the English Engineering unit and the SI unit.
1 kg·m² ≈ 23.73 lb·ft²
Conversion between the SI unit and a common imperial unit.
Key Reference Points
- A thin rod of mass 'm' and length 'L' rotating about its center: I = (1/12)mL².
- A solid disk of mass 'm' and radius 'r' rotating about its center axis: I = ½mr².
- A hollow hoop of mass 'm' and radius 'r': I = mr² (has a larger moment of inertia than a solid disk of same mass/radius).
- A solid sphere of mass 'm' and radius 'r': I = (2/5)mr².
- Concentrating mass further from the axis of rotation significantly increases the moment of inertia (due to the r² term).
- A car wheel and tire assembly has a moment of inertia that affects acceleration and braking.
- A helicopter's main rotor has a very large moment of inertia, storing energy that allows for autorotation in case of engine failure.
- A yo-yo's moment of inertia determines how long it can 'sleep' (spin at the bottom of the string).
- The moment of inertia of a planet affects its rotational period and precession (wobble).
- In a centrifuge, the moment of inertia of the rotor is a key design parameter for achieving high rotational speeds safely.
Did You Know?
A figure skater can change their spin speed dramatically by altering their moment of inertia. Pulling their arms and legs in reduces their moment of inertia, causing them to spin faster (due to conservation of angular momentum).
The moment of inertia of an object is not a single value; it depends on the chosen axis of rotation. An object will have different moments of inertia about different axes.
A hollow object (like a hoop) has a greater moment of inertia than a solid object (like a disk) of the same mass and radius. This is because more of its mass is distributed farther from the axis of rotation, making it harder to start or stop rotating.
The Earth has a massive moment of inertia (approx. 8 x 10³⁷ kg·m²), which keeps its rotation incredibly stable. This stability is responsible for the consistent length of a day.
For a planar object (a flat, 2D shape), the moment of inertia about an axis perpendicular to the plane is the sum of the moments of inertia about two perpendicular axes in the plane that intersect at the same point.
Cats can famously twist their bodies mid-air to land on their feet. They do this by changing their moment of inertia, pulling in their front legs while extending their back legs, allowing them to rotate their front and back halves separately to reorient themselves.
For a 3D object, the moment of inertia is actually a 'tensor,' a more complex mathematical object represented by a 3x3 matrix. This accounts for the fact that an applied torque might cause rotation around a completely different axis if the object is asymmetrical.
Satellites and spacecraft use reaction wheels or control moment gyroscopes to change their orientation (attitude). These devices work by changing the angular momentum of spinning flywheels, which, by conservation of angular momentum, causes the spacecraft itself to rotate.
The gyroscopic effect of the spinning wheels on a bicycle contributes to its stability. The wheels have a moment of inertia, and their angular momentum makes them resist changes in their orientation, helping the rider stay upright.
A tightrope walker carries a long pole to increase their moment of inertia. This makes them more stable and resistant to tipping over, as a larger torque is required to cause any significant rotation.
The radius of gyration (k) is a distance from the axis of rotation at which the entire mass of the body could be concentrated without changing its moment of inertia (I = mk²).