As in (4.3) and (4.4), find the y and z components of (4.2) and the

other 6 components of the inertia tensor. Write the corresponding components

of the inertia tensor for a set of masses or an extended body as in (4.5).

Physics definitions are given.

Tensors, like scalars and vectors, are mathematical constructs that can be

used to describe physical qualities. Tensors are just a combination of scalars

and vectors, with a scalar being a zero rank tensor and a vector being a first

rank tensor.

Define angular momentum of a point mass.

$L=\varpi \left(r^2\varpi \left(r.\text{'}\right)r\right)$

Express the definition along the other axes.

$$\begin{gathered} L=m\left({\omega }^2{y}_v-\left(ii\right)\right) \\ L=m\left({\omega }^2{}_z-\left(ii\right)z\right) \end{gathered}$$

Compare the above equations with $L_i=l_{ij}{\omega }_j$.

$$\begin{gathered} l_{yx}=-myx \\ {l}_{yy}=m\left(x^2+y^2\right) \\ {l}_{yz}=-myz \end{gathered}$$

Continue the comparison.

$$\begin{gathered} l_{zx}=-mzx \\ {l}_{zy}=mzy \\ {l}_{zz}=m\left(x^2+y^2\right) \end{gathered}$$

Express the components of the inertia tensor based on the earlier equations.

$\delta =m\left(r^2{}_{ij}-ij\right)$

Extend the previous equations to a multi-particle system.

$$\begin{gathered} L=\sum _k{m}_k{r}_k\times \left(\omega \times r_k\right) \\ =\sum _k{m}_k\left(r_k^2\omega -\left(r_k.\omega \right)r_k\right) \\ {\omega }_1=\sum _k{m}_k^j\left(r{r}_k^2{}_i-{}_{k.jj{k}_i;}\right) \\ =\sum _k{\delta }_k^{}\left(r{r}_k^2{T}_{ij}-\underset{kj}{\omega }k.j\right) \end{gathered}$$

Continue the evaluation.

${\delta }_{ij}^{}=\sum _{k}m{T}_k\left(r_k^2{}_{ij}-kjk.j\right)$

Replace summation with integral to evaluate over a continuous system.

${\delta }_{ij}^{}=\int r\left(r{r}_{}^2{m}_i{}_{ij}\right)d$

Thus, the solution is${\delta }_{ij}^{}=\sum _k{m}_k\left(r_k^2{}_{ij}{-}_{k,}{}_{kj}\right)$