Question

Do Problem (4.8) in tensor notation and compare the result with your solution of (4.8).

Short answer

Answer

The result has been proven.

Step-by-step solution

Given Information

The kinetic energy is T and the tensor inertia is$l_{\omega }$

Definition of a tensor of inertia.

The tensor of inertia is a mathematical representation of how mass is distributed in a rigid body.

Prove the result.

The formula states that kinetic energy,

$T=\frac{1}{2}m\left(\omega \times r\right)·\left(\omega \times r\right)$.

The value of ${\left(\omega \times r\right)}_i$is ${\left(\omega \times r\right)}_i={\epsilon }_{ijk}{\omega }_j{r}_k$.

Substitute the above value in the equation mentioned below.

$T=\frac{1}{2}m{\epsilon }_{ijk}{\omega }_j{r}_k{\epsilon }_{ilm}{\omega }_l{r}_m$

The value of ${\epsilon }_{ijk}{\epsilon }_{ilm}$is ${\epsilon }_{ijk}{\epsilon }_{ilm}={\delta }_{ji}{\delta }_{km}-{\delta }_{jm}{\delta }_{kl}$.

Substitute the above value in the equation mentioned below.

$$\begin{gathered} T=\frac{1}{2}m{\omega }_j\left({\delta }_{ji}{\delta }_{km}-{\delta }_{jm}{\delta }_{kl}\right)r_k{r}_m{\omega }_l \\ T=\frac{1}{2}m{\omega }_j\left({\delta }_{ji}{r}_k{r}_k-r_j{r}_l\right){\omega }_l \end{gathered}$$

The tensor of inertia is $l_{ji}=\left({\delta }_{ji}{r}^2-r_j{r}_i\right)$.

Where $r^2=r_k{r}_k.$

Hence, $T=\frac{1}{2}m{\omega }_j{l}_{ji}\omega$.

The result has been proven.