The moment of inertia when rotating around the axis through and in direction :

If the vector is omitted, you get a moment of inertia matrix about that point:

Using this matrix and a normalized vector, the moment of inertia around any axis can be found:

If the point is omitted, you get a moment of inertia matrix around the center of mass:

This is equivalent to specifying the center point by RegionCentroid:

Compute the moment of inertia for a region with symbolic parameters:

MomentOfInertia gives the moment of inertia for a body with constant mass density 1. Compute the moment of inertia for a Cuboid with length , width , height , and a constant mass density :

The mass of a body with a constant mass density is given by m=ρ Volume[body]. Express the moment of inertia in terms of mass of the body:

Moment of inertia of a 2D region embedded in 3D:

The top-left 2×2 submatrix is equal to moment of inertia of the 2D region:

The only other nonzero entry of the matrix is on the diagonal. It is the polar moment of area about the axis and, by the perpendicular axis theorem, it is equal to the sum of the other two entries on the diagonal:

Visualize the region:

The principal axes for a body are given by the eigenvectors of the moment of inertia matrix:

Find the principal axes:

When a moment of inertia matrix is diagonal, then the principal axes coincide with the coordinate axes:

In this case, the principal axes coincide with the coordinate axes:

A moment of inertia matrix can be thought as a matrix defining an Ellipsoid:

The inertia ellipsoid is defined as , where is the centroid:

Use Eigensystem to find the principal axes:

Show inertial ellipsoid together with its principal axes:

The angular momentum for a rotating rigid body is given by , where is the moment of inertia and is the angular velocity. Suppose a ball of radius 1 and uniform density 1 rotates around the axis on a string with adjustable length . When , the angular velocity is . Find the angular velocity as a function of :

Compute knowing that, according to the angular momentum conservation law, if no torque is applied to a body, its angular momentum remains constant:

Plot as a function of :

The kinetic energy for a rotating rigid body is given by , where is the moment of inertia and is the angular velocity. Compute the kinetic energy of a ball with radius and uniform density , rotating with angular velocity around an axis passing through its center:

A kinetic energy recovery system (KERS) stores kinetic energy for a car when braking. Suppose the energy is stored in a rotating steel cylinder with radius and length . How fast does the cylinder need to rotate to store of energy, assuming the density of steel is :

The moment of inertia for the steel cylinder:

Find the required angular velocity for storage:

In terms of revolutions per minute:

The kinetic energy for a rigid body is given by , where is an angular velocity vector and is the moment of inertia matrix. Find the rotational energy for a cuboid with sides 2, 4, and 6 when rotating around different axes. In all cases, the mass of the rotating body is the same, but how the mass is distributed relative to the rotation axis differs and is represented by moment of inertia:

Rotating around the axis:

Rotating around the axis:

Rotating around the axis:

For a rotating rigid body, the relation between the torque and the angular acceleration is given by , where is the moment of inertia. Compute the torque required to give a cube with one-meter sides of lead and angular acceleration of :

Compute the moment of inertia, assuming the density of lead is :

Compute the torque:

Convert to SI units:

The parallel axis theorem gives a relation between the moment of inertia around an arbitrary axis and the moment of inertia around the parallel axis passing through the center of mass, where is the total mass and is the distance between the axes:

Verifying the parallel axis theorem :

If the point is omitted, the RegionCentroid is taken as a default:

For a given point , one will get the same result by translating the body by and calculating the moment of inertia around the origin:

The moment of inertia is the integral of square distance to the axis of rotation:

The moment of inertia matrix is defined by an integral over the body:

Use Integrate to compute moment of inertia for a body varying mass density :

The moment of inertia matrix is symmetric:

The moment of inertia matrix is positive semidefinite:

Moment of inertia matrix in 2D can be computed using RegionMoment:

Moment of inertia matrix in 3D can be computed using RegionMoment:

The moment of inertia of a Point is the square of the distance of the point to the axis:

For a set of points, it is the sum of moments of inertia:

The moment of inertia for disjoint bodies is the sum of moments of inertia: