Mass Moment of Inertia Calculator – Calculate I for All Shapes & Formulas

  • Adam
  • November 27, 2025

Calculate mass moment of inertia instantly for cylinders, spheres, rods, plates, and more. Free calculator with 10+ shapes, parallel axis theorem, radius of gyration, and step-by-step solutions in kg·m².

Mass Moment of Inertia Calculator - Calculate I for Common Shapes

A mass moment of inertia calculator determines the rotational inertia of objects based on their mass distribution and geometry. The mass moment of inertia measures an object's resistance to rotational acceleration about an axis, playing a crucial role in rotational dynamics, mechanical engineering, and structural analysis.

Calculate Mass Moment of Inertia

Result:

What is Mass Moment of Inertia?

Mass moment of inertia, also called rotational inertia, is a scalar quantity that quantifies an object's resistance to angular acceleration about a specified axis. It is the rotational equivalent of mass in linear motion, representing how mass is distributed relative to the rotation axis. The farther mass is distributed from the axis, the greater the moment of inertia.

Key Characteristics of Mass Moment of Inertia:

  • Scalar quantity: Has magnitude but no direction (unlike angular momentum which is a vector)
  • Units: kg·m² in SI units, or lb·ft² in imperial units
  • Axis-dependent: Value changes depending on the chosen axis of rotation
  • Mass distribution: Depends on both total mass and how mass is distributed from the axis
  • Always positive: Cannot be negative since it involves squared distances
  • Symbol: I (moment of inertia) or J (sometimes used in engineering)

Mass Moment of Inertia Formula

The mass moment of inertia is calculated differently depending on whether you're dealing with a point mass, a system of particles, or a continuous rigid body.

Fundamental Formula for Point Mass

I = mr²

Where:

  • I = mass moment of inertia (kg·m²)
  • m = mass (kg)
  • r = perpendicular distance from the rotation axis (m)

General Formula for Continuous Bodies

For a continuous distribution of mass, the moment of inertia is calculated using integration:

I = ∫ r² dm

Where the integration is performed over the entire mass of the object

Radius of Gyration

The radius of gyration (k) is a useful concept that relates moment of inertia to mass:

I = mk²

or

k = √(I/m)

k represents the distance from the axis where all mass could be concentrated to give the same moment of inertia

Moment of Inertia for Common Shapes

Different geometric shapes have specific formulas for calculating their mass moment of inertia about various axes:

Shape Axis of Rotation Moment of Inertia Formula
Point Mass Distance r from point I = mr²
Solid Cylinder/Disk Central axis (along length) I = ½mR²
Hollow Cylinder Central axis I = mR²
Solid Sphere Through center I = (2/5)mR²
Hollow Sphere (thin shell) Through center I = (2/3)mR²
Thin Rod Through center, perpendicular I = (1/12)mL²
Thin Rod Through end, perpendicular I = (1/3)mL²
Thin Ring/Hoop Through center, perpendicular to plane I = mR²
Rectangular Plate Through center, perpendicular I = (1/12)m(a² + b²)
Solid Cone About axis of symmetry I = (3/10)mR²

Parallel Axis Theorem

The parallel axis theorem allows you to calculate the moment of inertia about any axis parallel to an axis through the center of mass. This is one of the most important theorems in rotational mechanics.

I = Icm + md²

Where:

  • I = moment of inertia about the new parallel axis (kg·m²)
  • Icm = moment of inertia about axis through center of mass (kg·m²)
  • m = total mass (kg)
  • d = perpendicular distance between the two parallel axes (m)

Important Notes:

  • The parallel axis theorem only applies to parallel axes
  • One axis must pass through the center of mass
  • Always adds to the moment of inertia (I ≥ Icm)
  • The moment of inertia is minimum when the axis passes through the center of mass

Perpendicular Axis Theorem

The perpendicular axis theorem applies only to planar (flat) objects. It relates the moment of inertia about three mutually perpendicular axes:

Iz = Ix + Iy

Where:

  • Iz = moment of inertia about axis perpendicular to the plane (kg·m²)
  • Ix = moment of inertia about x-axis in the plane (kg·m²)
  • Iy = moment of inertia about y-axis in the plane (kg·m²)

Requirements for Perpendicular Axis Theorem:

  • Only applicable to planar (2D) laminar objects
  • All three axes must intersect at a single point
  • Two axes must lie in the plane of the object
  • The third axis must be perpendicular to the plane

How to Calculate Mass Moment of Inertia

Calculating mass moment of inertia requires understanding the object's geometry, mass distribution, and rotation axis. Follow these systematic steps:

  1. Identify the object geometry: Determine the shape (cylinder, sphere, rod, etc.) and relevant dimensions.
  2. Specify the rotation axis: Clearly define which axis the object rotates about, as moment of inertia is axis-dependent.
  3. Select the appropriate formula: Choose the formula that matches your shape and axis from the standard formulas table.
  4. Gather required parameters: Measure or determine mass (m), radius (R), length (L), or other relevant dimensions.
  5. Apply the formula: Substitute values into the formula and calculate the moment of inertia.
  6. Use theorems if needed: Apply parallel axis or perpendicular axis theorem if the axis is not standard.
  7. Verify units: Ensure the result is in kg·m² (or appropriate units).

Calculation Example 1: Solid Cylinder

Problem: Calculate the moment of inertia of a solid cylinder with mass 10 kg and radius 0.5 m about its central axis.

Solution:

Given: m = 10 kg, R = 0.5 m

For solid cylinder about central axis: I = ½mR²

I = ½ × 10 × (0.5)²

I = 5 × 0.25

I = 1.25 kg·m²

Calculation Example 2: Using Parallel Axis Theorem

Problem: A thin rod of mass 2 kg and length 1 m rotates about an axis perpendicular to the rod and 0.2 m from the center. Find the moment of inertia.

Solution:

Step 1: Calculate Icm for rod about center

Icm = (1/12)mL² = (1/12) × 2 × (1)² = 0.1667 kg·m²

Step 2: Apply parallel axis theorem

I = Icm + md²

I = 0.1667 + 2 × (0.2)²

I = 0.1667 + 2 × 0.04

I = 0.1667 + 0.08

I = 0.2467 kg·m²

Calculation Example 3: Compound Object

Problem: Calculate the total moment of inertia for a system consisting of three point masses: 2 kg at 1 m, 3 kg at 2 m, and 1 kg at 3 m from the rotation axis.

Solution:

For multiple point masses: Itotal = Σ(miri²)

I₁ = m₁r₁² = 2 × (1)² = 2 kg·m²

I₂ = m₂r₂² = 3 × (2)² = 12 kg·m²

I₃ = m₃r₃² = 1 × (3)² = 9 kg·m²

Itotal = 2 + 12 + 9

Itotal = 23 kg·m²

Applications of Mass Moment of Inertia

Understanding and calculating mass moment of inertia is essential in numerous engineering and scientific applications:

  • Mechanical Design: Designing flywheels, rotating shafts, gears, and machinery requiring precise rotational dynamics control
  • Automotive Engineering: Analyzing engine crankshafts, wheels, drive trains, and vehicle dynamics for performance optimization
  • Aerospace: Calculating spacecraft attitude control, satellite stabilization, and aircraft rotational stability
  • Robotics: Determining torque requirements for robotic joints and manipulator arms
  • Structural Engineering: Analyzing beam bending, column buckling, and structural member stress distributions
  • Sports Equipment: Optimizing golf clubs, baseball bats, tennis rackets, and figure skating techniques
  • Manufacturing: Balancing rotating equipment, designing centrifuges, and optimizing machining operations
  • Physics Research: Studying rotational motion, angular momentum conservation, and rigid body dynamics

Frequently Asked Questions

What is the formula for mass moment of inertia?
The fundamental formula for mass moment of inertia is I = mr² for a point mass, where m is mass and r is perpendicular distance from the rotation axis. For continuous bodies, I = ∫r²dm integrated over the entire mass. Common shapes have specific formulas: solid cylinder I = ½mR², solid sphere I = (2/5)mR², thin rod about center I = (1/12)mL². The formula depends on both the object's geometry and the axis of rotation.
How do you calculate moment of inertia for different shapes?
To calculate moment of inertia for different shapes, identify the geometry and rotation axis, then apply the appropriate formula. For a solid cylinder about its central axis, use I = ½mR². For a solid sphere, use I = (2/5)mR². For a thin rod about its center, use I = (1/12)mL². Each shape has derived formulas based on integrating I = ∫r²dm over the mass distribution. Always ensure you're using the formula for the correct axis of rotation.
What is the difference between mass moment of inertia and area moment of inertia?
Mass moment of inertia (I) measures rotational inertia with units kg·m² and appears in rotational dynamics equations (τ = Iα). It involves mass distribution: I = ∫r²dm. Area moment of inertia (second moment of area) measures beam bending resistance with units m⁴ and appears in stress analysis. It involves area distribution: I = ∫r²dA. Despite similar names and symbols, they are fundamentally different quantities used in different branches of mechanics.
What is the parallel axis theorem?
The parallel axis theorem states that the moment of inertia about any axis equals the moment of inertia about a parallel axis through the center of mass plus the product of mass and the square of the distance between axes: I = Icm + md². This theorem allows you to calculate moment of inertia about any parallel axis once you know Icm. It only applies to parallel axes and always increases the moment of inertia compared to the center of mass axis.
What is the perpendicular axis theorem?
The perpendicular axis theorem applies only to planar (flat) objects and states that Iz = Ix + Iy, where Iz is moment of inertia about an axis perpendicular to the plane, and Ix, Iy are moments about two perpendicular axes in the plane. All three axes must intersect at one point. This theorem is useful for calculating perpendicular-axis moments for thin plates, disks, and other planar shapes when in-plane moments are known.
Why does moment of inertia depend on the axis of rotation?
Moment of inertia depends on the axis because it measures how mass is distributed relative to that specific axis (I = ∫r²dm, where r is distance from the axis). The same object has different moments of inertia about different axes because mass elements are at different distances from each axis. For example, a rod has I = (1/12)mL² about its center but I = (1/3)mL² about its end—four times larger because mass is farther from the end axis on average.
Can moment of inertia be negative?
No, moment of inertia cannot be negative because it's calculated as I = ∫r²dm or I = mr², where r² is always positive (squared distance), and mass is always positive. The minimum value is zero, which occurs only for a point mass on the rotation axis (r = 0). All real objects with mass distributed away from the axis have positive moment of inertia. This is analogous to mass in linear motion—always positive and representing inertia.
What is radius of gyration?
Radius of gyration (k) is the distance from the rotation axis where all mass could be concentrated to produce the same moment of inertia: I = mk² or k = √(I/m). It's a characteristic length that simplifies moment of inertia calculations. For example, if a disk has k = 0.5R where R is its radius, all mass could be placed at distance 0.5R from the axis to give the same rotational inertia. Radius of gyration is always less than or equal to the maximum distance of mass from the axis.
How does mass distribution affect moment of inertia?
Mass distribution dramatically affects moment of inertia because I depends on r² (distance squared from axis). Mass farther from the axis contributes much more to inertia than mass close to the axis. For example, a hollow cylinder (all mass at radius R) has I = mR², while a solid cylinder (mass distributed from 0 to R) has I = ½mR²—half as much despite the same total mass and radius. This is why figure skaters spin faster when pulling arms inward (reducing mass distribution).
What are the units of mass moment of inertia?
The SI unit of mass moment of inertia is kilogram meter squared (kg·m²). This comes from the formula I = mr², where mass is in kg and distance is in m. In imperial units, it's slug·ft² or lb·ft². In engineering, you might see g·cm² for small objects. Dimensionally, moment of inertia has units of [mass][length]². It's important not to confuse this with area moment of inertia, which has units of length to the fourth power (m⁴).

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