In classical mechanics, the stretch rule (sometimes referred to as Routh's rule) states that the moment of inertia of a rigid object is unchanged when the object is stretched parallel to an axis of rotation that is a principal axis, provided that the distribution of mass remains unchanged except in the direction parallel to the axis.^[1] This operation leaves cylinders oriented parallel to the axis unchanged in radius.

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This rule can be applied with the parallel axis theorem and the perpendicular axis theorem to find moments of inertia for a variety of shapes.

The (scalar) moment of inertia of a rigid body around the z-axis is given by:

${\displaystyle I_{z}=\int _{V}d^{3}r\,\rho (\mathbf {r} )\,r^{2}}$

Where ${\displaystyle r}$ is the distance of a point from the z-axis. We can expand as follows, since we are dealing with stretching over the z-axis only:

${\displaystyle I_{z}=\int _{0}^{L}dz\int _{x,y}dx\,dy\,\rho (x,y,z)\,r^{2}}$

Here, ${\displaystyle L}$ is the body's height. Stretching the object by a factor of ${\displaystyle a}$ along the z-axis is equivalent to dividing the mass density by ${\displaystyle a}$ (meaning ${\displaystyle \rho '(x,y,z)=\rho (x,y,z/a)/a}$), as well as integrating over new limits ${\displaystyle 0}$ and ${\displaystyle aL}$ (the new height of the object), thus leaving the total mass unchanged. This means the new moment of inertia will be:

${\displaystyle {\begin{aligned}I_{z}'&=\int _{0}^{aL}dz\int _{x,y}dx\,dy\,\rho '(x,y,z)\,r^{2}\\[8pt]&=\int _{0}^{L}a\,dz'\int _{x,y}dx\,dy\,{\frac {\rho (x,y,z/a)}{a}}\,r^{2}\\[8pt]&=\int _{0}^{L}dz'\int _{x,y}dx\,dy\,\rho (x,y,z')\,r^{2}=I_{z}\end{aligned}}}$