Rotational inertia in terms of Cartesian integral

In case of a rigid body with a continuously distributed mass, its rotational inertia is expressed as a volume integral over the entire body, i.e. a triple integral over \(x, y, z\) in Cartesian coordinates.

Notes:

1. The integration is carried out over the entire body as to include every volume element.

Links:

1. Wikipedia, derivable from fourth equation.

rotational_inertia

rotational_inertia.

Symbol:

I

Latex:

\(I\)

Dimension:

length**2*mass

x

position on the \(x\) axis.

Symbol:

x

Latex:

\(x\)

Dimension:

length

x_start

Initial position on the \(x\) axis.

Symbol:

x_0

Latex:

\(x_{0}\)

Dimension:

length

x_end

Final position on the \(x\) axis.

Symbol:

x_1

Latex:

\(x_{1}\)

Dimension:

length

y

position on the \(y\) axis.

Symbol:

y

Latex:

\(y\)

Dimension:

length

y_start

Initial position on the \(y\) axis.

Symbol:

y_0

Latex:

\(y_{0}\)

Dimension:

length

y_end

Final position on the \(y\) axis.

Symbol:

y_1

Latex:

\(y_{1}\)

Dimension:

length

z

position on the \(z\) axis.

Symbol:

z

Latex:

\(z\)

Dimension:

length

z_start

Initial position on the \(z\) axis.

Symbol:

z_0

Latex:

\(z_{0}\)

Dimension:

length

z_end

Final position on the \(z\) axis.

Symbol:

z_1

Latex:

\(z_{1}\)

Dimension:

length

density

Mass-specific density as a function of x, y, z.

Symbol:

rho(x, y, z)

Latex:

\(\rho{\left(x,y,z \right)}\)

Dimension:

mass/volume

distance_to_axis

distance_to_axis as a function of x, y, z.

Symbol:

r(x, y, z)

Latex:

\(r{\left(x,y,z \right)}\)

Dimension:

length

law

I = Integral(rho(x, y, z) * r(x, y, z)^2, (x, x_0, x_1), (y, y_0, y_1), (z, z_0, z_1))

Latex:

\[I = \int\limits_{z_{0}}^{z_{1}}\int\limits_{y_{0}}^{y_{1}}\int\limits_{x_{0}}^{x_{1}} \rho{\left(x,y,z \right)} r^{2}{\left(x,y,z \right)}\, dx\, dy\, dz\]