Calculating Moments of Inertia: Formulas and Examples

Calculate a body’s mass moment of inertia by summing or integrating each bit of mass times the square of its distance from a chosen axis: I = ∫r² dm. Enter the object’s mass and dimensions, then select a shape and axis to estimate its moment of inertia in kg·m²; the result describes resistance to angular acceleration about that axis.

Formula

The general definition is I = ∫r² dm, where I is mass moment of inertia in kg·m², r is the perpendicular distance from the rotation axis to a small mass element in metres, and dm is that element’s mass in kilograms. For separate particles or components, use I = Σmᵢrᵢ².

For the calculator’s common shapes, the formulas are:

• Uniform solid disk or cylinder about its central symmetry axis: I = ½MR².
• Thin hoop about its central symmetry axis: I = MR².
• Uniform solid sphere about a diameter: I = ⅖MR².
• Thin uniform rod about its midpoint, with the axis perpendicular to its length: I = ¹⁄₁₂ML².
• The same rod about an axis through one end and perpendicular to its length: I = ⅓ML².

M is total mass in kilograms; R is radius in metres; and L is rod length in metres. In the calculator, “radius or length” supplies R for the disk, hoop, and sphere, or L for either rod. The shape selector chooses the formula. The displayed result is rounded to three decimal places.

Worked Example

With the defaults, the calculator uses a 2 kg solid disk of radius 0.5 m, rotating about its central axis. Apply I = ½MR²:

1. Square the radius: (0.5 m)² = 0.25 m².
2. Multiply by the mass: 2 kg × 0.25 m² = 0.5 kg·m².
3. Take half: I = ½ × 0.5 = 0.250 kg·m².

This is the disk’s mass moment of inertia about the specified axis. Changing the axis or the mass distribution changes the answer, even if the object’s total mass and outer dimensions stay the same.

What Moment of Inertia Means

Moment of inertia is the rotational counterpart of mass in straight-line motion. Newton’s rotational equation for a rigid body rotating about a fixed principal axis is τ = Iα: torque τ, measured in newton-metres, produces angular acceleration α, measured in radians per second squared. For a given torque, a larger I means less angular acceleration.

Distance matters strongly because it is squared. Moving mass twice as far from the axis contributes four times as much to I. This is why a hoop has a larger moment of inertia than a solid disk with the same mass and radius: more of the hoop’s mass lies near the outer edge. It also explains why a rotating performer can change spin rate by moving their arms inward or outward, provided external torque is small.

Do not confuse mass moment of inertia, measured in kg·m² (or slug·ft²), with area moment of inertia, also called the second moment of area. Area moment of inertia has units of length to the fourth power, such as m⁴ or in⁴, and is used in beam-bending calculations. The two quantities are related by the idea of distance squared, but they describe different physical properties and cannot be substituted for one another.

Choosing the Right Axis and Shape

A moment of inertia is always relative to an axis. Before using a standard formula, identify where the axis passes and which direction it points. A disk about its center axis is not equivalent to the same disk rotating about a line through its rim. Likewise, a rod’s midpoint and end formulas differ.

Standard formulas assume idealized, uniform shapes and specific axes. A “thin hoop” treats the mass as concentrated at one radius; a solid disk assumes mass fills its circular area uniformly. Real objects with holes, spokes, varying thickness, attached parts, or nonuniform materials may need a more detailed calculation.

• For an object assembled from parts, calculate each part’s I about the required axis and add the results.
• For a part whose known formula uses a parallel axis through its own centre of mass, use the parallel-axis theorem: I = I_cm + Md². Here d is the perpendicular separation between the parallel axes.
• For an irregular object with known density, divide it into small elements and evaluate I = ∫r² dm, often by integration or numerical methods.

The parallel-axis theorem applies to axes that are parallel; it does not allow an arbitrary change in axis direction. For a general three-dimensional body, inertia can depend on orientation and is represented by an inertia tensor rather than one single value.

Units, Uses, and Limits

Use consistent units throughout. If mass is in kilograms and dimensions in metres, I is in kg·m². In US customary engineering units, mass may be expressed in slugs and dimensions in feet, giving slug·ft². If a source gives weight in pounds-force rather than mass, convert it to mass before applying a mass-inertia formula; weight and mass are not interchangeable. Convert centimetres to metres or inches to feet before squaring lengths.

Moment of inertia is used to estimate rotational acceleration, energy, and response in mechanisms such as wheels, flywheels, doors, and rotating machinery. Rotational kinetic energy about a fixed axis is K = ½Iω², where angular speed ω is in radians per second and energy is in joules. For an object that changes shape or rotates about a moving axis, the basic fixed-axis formulas may not by themselves describe its full motion.

The calculator is a shape-based estimate, not a measurement of every real object. Its result depends on the selected ideal shape, correct dimensions, uniformity assumptions, and specified axis. A three-decimal display does not imply that the input measurements are accurate to that precision; round the final value to match the reliability of the data.

Common Calculation Mistakes

• Using the wrong axis: verify the line and direction of rotation before choosing a formula.
• Using diameter instead of radius: formulas for disks, hoops, and spheres generally require radius. If only diameter D is known, use R = D/2.
• Forgetting to square the distance: both the formula and units include length squared.
• Mixing units: convert all dimensions and mass to a consistent system before calculating.
• Assuming all objects with the same mass and size have the same I: distribution of mass relative to the axis matters.

Frequently Asked Questions

How do you calculate moment of inertia from mass and radius?
Choose a shape and axis first. For a uniform solid disk about its central axis, use I = ½MR²; for a thin hoop about that axis, use I = MR². Other shapes have different formulas.

Why is distance squared in the formula?
Rotational effects depend on how far mass lies from the axis, and the contribution of each mass element is proportional to r². Doubling that distance therefore quadruples its contribution.

Can moment of inertia be negative?
No. For ordinary mass distributions, each contribution is dm times a squared distance, so the moment of inertia about an axis is zero or positive. A zero value is possible only when all mass lies on the axis in the ideal mathematical model.

What is the difference between mass and area moment of inertia?
Mass moment of inertia describes resistance to angular acceleration and has units such as kg·m². Area moment of inertia describes a cross-section’s geometric contribution to bending and has units such as m⁴.

How do I calculate the moment of inertia of a composite object?
Split it into simpler parts, calculate each part about the same target axis, and add the values. If a part’s formula is about a parallel axis through its centre of mass, shift it with I = I_cm + Md² before summing.

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in mass | Mass | kg | 2 | 0
in radius | Radius or length | m | 0.5 | 0
sel shape | Shape and axis | Solid disk about center=1, Thin hoop about center=2, Solid sphere about center=3, Thin rod about center=4, Thin rod about end=5
out inertia | Mass moment of inertia | kg·m^2 | mass * radius^2 * 0.5 | 3