The moment of inertia is a measure of an object's tendency to resist rotational changes. This quantity depends on the mass density distribution of the object or objects in question, along with the length of the moment arm, or the vector that runs from the object's centre of mass to the major axis of rotation. The following steps will describe how to calculate the inertia tensor, or the matrix-like object that defines all inertial elements which account for the on- and off-axis combinations of inertia contributions from the centre of mass frame. This procedure, which requires some knowledge of calculus and linear algebra, is most helpful for students in an intermediate-level physics course.
Orient your object in the centre of mass frame. Its centre of mass should be located at the (0,0,0) point in the xyz plane. Identify the generalised position vector ri (r = [xi, yi, zi]'). This will represent the position of an infinitesimal piece of the object outside of the origin, which we will use to generalize the calculation. Define the variable "mi" as an infinitesimal mass within the object or system of objects; this is located at the point referenced by the position vector.
Write down the inertia tensor. This will look like a matrix of the following rows, written from top to bottom: [Ixx, Ixy, Ixz], [Iyx, Iyy, Iyz], [Izx, Izy, Izz] .
Substitute the following relationships for each element of the inertia tensor: Ixy = Iyx = -sum[mi(xi)(yi)] ; Ixz = Izx = -sum[mi(xi)(zi)] ; Iyz = Izy = -sum[mi(yi)(zi)] ; Ixx = sum[mi(y^2 + z^2)] ; Iyy = sum[mi(x^2 + z^2)] ; Izz = sum[mi(x^2 + y^2)] .
Write the relationship for the change in mass as one travels along the moment arm outward from the centre of mass for each element in the tensor. You should end up with an infinitesimal change in dimensions of length or angular units. Substitute this quantity into each element of the moment of inertia tensor for the "mi" variable. Integrate all elements of the moment of inertia tensor over this change in mass to get the specific equation for each tensor element as it pertains to the problem at hand. These axes upon which the moment arm depends should be apparent from the moment arm vector that was identified in the first step.
Substitute all known variables into the tensor elements to get the desired quantities. You should now have a tensor of inertial elements that accounts for every combination of axes, including the principle moments and products of inertia. The elements commonly called the "moments of inertia" in this tensor are the Ixx, Iyy and Izz elements.
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