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Moment of Inertia of Particles Calculator

Enter the mass and distance of each particle to find the total moment of inertia.

How it is calculated

The tool adds up the contribution of each point mass about one axis: I = Σ mi × ri2. Here mi is the mass in kilograms and ri is its perpendicular distance from the rotation axis in metres. There is room for up to four particles. A particle counts only when both its mass and its distance are filled in; if one of the two is empty the whole pair is skipped without a warning. Because the distance is squared, its sign does not matter. The result is in kg·m² to four decimals, along with the total mass and the number of particles used. Results are always labelled kg and m, so convert grams or centimetres first.

Worked example

Three particles about the same axis: 2 kg at 3 m, 5 kg at 1 m and 1.5 kg at 2 m. First particle: 2 × 32 = 18. Second: 5 × 12 = 5. Third: 1.5 × 22 = 6. Adding gives 18 + 5 + 6 = 29, shown as 29.0000 kg·m². The total mass is 2 + 5 + 1.5 = 8.5 kg and the number of particles is 3. The heaviest particle contributes the least, because the distance counts squared.

Try this example →

Common mistakes

  • Measuring the distance from the centre of mass or from another point, instead of from the axis you actually rotate about. The same masses give a different I for a different axis.
  • Entering distances in centimetres or masses in grams while the result is labelled in kg and m. The numbers are used as given, so a value in cm gives I that is 10,000 times too large.
  • Leaving the distance blank for a particle on the axis. The pair is then skipped, so its mass is missing from the total mass and the particle count. Enter 0 instead.
  • Using the straight-line distance between two particles instead of each particle's perpendicular distance to the axis.
  • Trying to enter more than four particles. Only four pairs are read, so a larger system needs to be split into several runs and added.

Sources

About this calculator

The moment of inertia of particles calculator computes the rotational inertia of a system of discrete point masses about a chosen axis using I = Σ mᵢrᵢ², summing each particle's mass mᵢ multiplied by the square of its perpendicular distance rᵢ from the axis of rotation. Unlike a continuous rigid body, where you'd need to integrate a density function over the shape, a set of point particles reduces the problem to simple addition.

Moment of inertia is the rotational analog of mass in Newton's second law — it determines how much torque is needed to achieve a given angular acceleration (τ = Iα) — and the particle-sum form is the building block used to derive the standard formulas for rods, disks, and spheres by treating the object as infinitely many point masses. Mechanical and aerospace engineering students, and engineers modeling multi-body systems like flywheels, satellites, or linkages use this calculator to get I directly from a list of masses and distances instead of deriving it from scratch.

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Comments (1)

  • Editorial team

    You can approximate a continuous body by splitting it into up to four pieces and entering each piece as a point mass at its centre. For a uniform rod about one end, four equal pieces give 0.3281 times mass times length squared against the exact one third, about 1.6% low, because each piece's own spread is ignored. What system are you modelling: a dumbbell, a rotor, a rod?

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