# Does the moment of inertia scale quintically for similar solids?

In this list

https://en.wikipedia.org/wiki/List_of_moments_of_inertia

there appears to be the pattern that the moment of inertia of a similar solid scales quintically. For example, given a sphere of radius $$r_1$$ with mass $$m_1$$, we have $$I_1 = \frac{2}{5}m_1 r_1^2.$$

If we then have another sphere of $$r_2 = c r_1$$ for some constant $$c$$,then we see that $$I_1 = \frac{2}{5}(c^3 m_1) (cr_1)^2.$$ $$= c^5\frac{2}{5}m_1 r_1^2.$$

Is this property true in general for all similar 3-dimensional solids?

## 1 Answer

Yes. If you have an arbitrary mass distribution with density $$\rho(\mathbf r)$$, and you scale up the object by $$c$$, then that corresponds to the density $$\rho_c(\mathbf r) = \rho(\mathbf r/c)$$. That distribution has total mass \begin{align} M_c & = \int \rho_c(\mathbf r)\mathrm d\mathbf r = \int \rho(\mathbf r/c)\mathrm d\mathbf r \\ & = \int \rho(\mathbf r')\mathrm d\mathbf r' = c^3\,M, \end{align} where the nontrivial step is in the change of variables to $$\mathbf r' = \mathbf r/c$$, with $$\mathrm d\mathbf r' = c^{-3} \mathrm d \mathbf r$$. Similarly, the $$(i,j)$$-th component of the moment of inertia tensor transforms as \begin{align} I_{ij}^{(c)} & = \int x_ix_j \rho_c(\mathbf r)\mathrm d\mathbf r = \int x_ix_j \rho(\mathbf r/c)\mathrm d\mathbf r \\ & = \int c^2x_i'x_j' \rho(\mathbf r')c^3\mathrm d\mathbf r' = c^5\,I_{ij}. \end{align}