Question

In$\mathbf{(}\mathbf{7}\mathbf{.9}\mathbf{)}$we have written the first row of elements in the inertia matrix. Write the formulas for the other6elements and compare with Section 4.

Short answer

the inertia matrix is mentioned below.

$I=\left(\begin{array}{ccc}m(r^2-{x_1}^2)& -m{x}_1{x}_2& -m{x}_1{x}_3\\ -m{x}_2{x}_1& m(r^2-{x_2}^2)& -m{x}_2{x}_3\\ -m{x}_3{x}_1& -m{x}_3{x}_2& m(r^2-{x_3}^2)\end{array}\right)$

Step-by-step solution

Given Information

The elements of the first row of the inertia matrix are mentioned below.

$\begin{array}{c}{I}_{11}=m(r^2-{x_1}^2)\\ I_{12}=-m{x}_1{x}_2\\ I_{13}=-m{x}_1{x}_3\end{array}$

Definition of a tensor of inertia.

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

Find inertia matrix.

The elements of the first row of the inertia matrix are mentioned below.

$\begin{array}{c}{I}_{11}=m(r^2-{x_1}^2)\\ I_{12}=-m{x}_1{x}_2\\ I_{13}=-m{x}_1{x}_3\end{array}$

The Formula states that $I_{nj}=m({\delta }_{nj}{r}^2-x_n{x}_j)$.

Hence, the inertia matrix is mentioned below.

$I=\left(\begin{array}{ccc}m(r^2-{x_1}^2)& -m{x}_1{x}_2& -m{x}_1{x}_3\\ -m{x}_2{x}_1& m(r^2-{x_2}^2)& -m{x}_2{x}_3\\ -m{x}_3{x}_1& -m{x}_3{x}_2& m(r^2-{x_3}^2)\end{array}\right)$