The inertia tensor for a solid sphere with radius $$ b $$ and mass $$ m $$ rotated about the center of mass is $$ \left[\begin{array}{ccc}\frac{2}{5} m b^{2} & 0 & 0 \ 0 & \frac{2}{5} m b^{2} & 0 \ 0 & 0 & \frac{2}{5} m b^{2}\end{array} ight] $$. What are the eigenvalues of the tensor if $$ b=5 $$ and $$ m=4 $$ ? Enter the eigenvalues from smallest to largest. $$ I_{1}= $$ Ex: 1.23 $$ I_{2}= $$ Ex: 1.23 $$ I_{3}= $$ Ex: 1.23

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The inertia tensor for a solid sphere with radius $$ b $$ and mass $$ m $$ rotated about the center of mass is $$ \left[\begin{array}{ccc}\frac{2}{5} m b^{2} & 0 & 0 \ 0 & \frac{2}{5} m b^{2} & 0 \ 0 & 0 & \frac{2}{5} m b^{2}\end{array}ight] $$. What are the eigenvalues of the tensor if $$ b=5 $$ and $$ m=4 $$ ? Enter the eigenvalues from smallest to largest. $$ I_{1}= $$ Ex: 1.23 $$ I_{2}= $$ Ex: 1.23 $$ I_{3}= $$ Ex: 1.23

UpStudy AI · Accepted Answer

We are given the inertia tensor:

$$
\mathbf{I} = \begin{bmatrix}\frac{2}{5} m b^{2} & 0 & 0 \0 & \frac{2}{5} m b^{2} & 0 \0 & 0 & \frac{2}{5} m b^{2}\end{bmatrix}
$$

Substitute $$ b = 5 $$ and $$ m = 4 $$:

$$
\frac{2}{5} m b^{2} = \frac{2}{5} \times 4 \times 5^{2} = \frac{2}{5} \times 4 \times 25
$$

Calculate the multiplication:

$$
\frac{2}{5} \times 4 = \frac{8}{5}
$$

and

$$
\frac{8}{5} \times 25 = 8 \times 5 = 40
$$

Thus, the inertia tensor becomes:

$$
\mathbf{I} = \begin{bmatrix}40 & 0 & 0 \0 & 40 & 0 \0 & 0 & 40\end{bmatrix}
$$

Since the matrix is diagonal with the same entries, the eigenvalues are simply the diagonal elements. Therefore, the eigenvalues are:

$$
I_{1} = 40, \quad I_{2} = 40, \quad I_{3} = 40
$$
The eigenvalues are all 40.

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