What is the moment of inertia?
The moment of inertia is the rotational analog of mass: it measures how hard it is to angularly accelerate a body about a chosen axis. It is I = Σ mᵢ rᵢ² for a set of particles, or I = ∫ r² dm for a continuous body, where r is each mass element’s distance from the axis. Unlike mass, it depends on both the mass distribution and the axis, so the same object has different moments of inertia about different axes.
Standard shapes and the parallel-axis theorem
For common shapes the integral is tabulated: a solid cylinder or disk about its central axis is ½ M R², a solid sphere ⅗ M R², a thin rod about its center 1⁄12 M L², and a thin hoop M R². To shift the axis away from the center of mass, use the parallel-axis theorem I = I_cm + M d², where d is the distance between the axes.
How to use the calculator
Select the shape and axis, enter the mass and dimensions, and the tool returns I in kg·m² along with the radius of gyration k = √(I/M) — the distance at which a point mass would have the same moment of inertia.
Note: the tabulated formulas assume uniform density and idealized geometry. Real parts with holes, fillets or non-uniform material need composite methods (sum or subtract the inertias of simple pieces, each shifted with the parallel-axis theorem).
Frequently asked questions
What is moment of inertia?
Moment of inertia is a body's resistance to angular acceleration about an axis, the rotational analog of mass. It equals the sum of each mass element times the square of its distance from the axis, I = sum(m*r^2).
What is the moment of inertia of a solid cylinder?
A solid cylinder or disk of mass M and radius R about its central axis has I = (1/2) M R^2. A solid sphere is (2/5) M R^2, a thin hoop is M R^2, and a thin rod about its center is (1/12) M L^2.
What is the parallel-axis theorem?
It gives the moment of inertia about any axis parallel to one through the center of mass: I = I_cm + M d^2, where d is the distance between the two axes and M is the total mass.
What are the units of moment of inertia?
In SI, moment of inertia is measured in kilogram square metres (kg*m^2). The related radius of gyration k = sqrt(I/M) has units of metres.
References
- D. Halliday, R. Resnick & J. Walker, Fundamentals of Physics (Wiley) — rotation, moment of inertia and the parallel-axis theorem.
- H. Goldstein, C. Poole & J. Safko, Classical Mechanics, 3rd ed. (Pearson) — rigid-body dynamics and the inertia tensor.
- Standard rigid-body moment-of-inertia tables (uniform solids about principal axes).