Rotational inertia in terms of a cylindrical 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 space coordinates.

Notes:

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

Conditions:

1. The \(z\)-axis is the rotational axis of the body.

Links:

1. Wikipedia, derivable from fourth equation.

rotational_inertia

rotational_inertia of the body.

Symbol:

I

Latex:

\(I\)

Dimension:

length**2*mass

radius

radius, or distance to the rotational axis.

Symbol:

r

Latex:

\(r\)

Dimension:

length

radius_start

Initial radius.

Symbol:

r_0

Latex:

\(r_{0}\)

Dimension:

length

radius_end

Final radius.

Symbol:

r_1

Latex:

\(r_{1}\)

Dimension:

length

polar_angle

Polar angle.

Symbol:

phi

Latex:

\(\varphi\)

Dimension:

angle

polar_angle_start

Initial polar angle.

Symbol:

phi_0

Latex:

\(\varphi_{0}\)

Dimension:

angle

polar_angle_end

Final polar angle.

Symbol:

phi_1

Latex:

\(\varphi_{1}\)

Dimension:

angle

height

height.

Symbol:

h

Latex:

\(h\)

Dimension:

length

height_start

Initial height.

Symbol:

h_0

Latex:

\(h_{0}\)

Dimension:

length

height_end

Final height.

Symbol:

h_1

Latex:

\(h_{1}\)

Dimension:

length

density

density as a function of radius, polar_angle, and height.

Symbol:

rho(r, phi, h)

Latex:

\(\rho{\left(r,\varphi,h \right)}\)

Dimension:

mass/volume

law

I = Integral(rho(r, phi, h) * r^3, (r, r_0, r_1), (phi, phi_0, phi_1), (h, h_0, h_1))

Latex:

\[I = \int\limits_{h_{0}}^{h_{1}}\int\limits_{\varphi_{0}}^{\varphi_{1}}\int\limits_{r_{0}}^{r_{1}} \rho{\left(r,\varphi,h \right)} r^{3}\, dr\, d\varphi\, dh\]