The inertia tensor for an ellipsoid with semi-axes $$ a, b $$, and $$ c $$, and mass $$ m $$ rotated about the center of mass is $$ \left[\begin{array}{lc}\frac{1}{5} m\left(b^{2}+c^{2} ight) \ 0 & \frac{1}{5} m\left(a^{2}+c^{2} ight) \ 0 & 0 \ ext { axes if } a=4, b=7, c=8 ext {, and } m=1 ext { ? } \ ext { Enter the moments of inertia from smallest to largest. } \ ext { First moment }= ext { Ex: } 1.23 \ ext { Second moment }= ext { Ex: 1.23 } \ ext { Third moment }= ext { Ex: 1.23 }\end{array} ight. $$

Question

The inertia tensor for an ellipsoid with semi-axes $$ a, b $$, and $$ c $$, and mass $$ m $$ rotated about the center of mass is $$ \left[\begin{array}{lc}\frac{1}{5} m\left(b^{2}+c^{2}ight) \ 0 & \frac{1}{5} m\left(a^{2}+c^{2}ight) \ 0 & 0 \ 	ext { axes if } a=4, b=7, c=8 	ext {, and } m=1 	ext { ? } \ 	ext { Enter the moments of inertia from smallest to largest. } \ 	ext { First moment }=	ext { Ex: } 1.23 \ 	ext { Second moment }=	ext { Ex: 1.23 } \ 	ext { Third moment }=	ext { Ex: 1.23 }\end{array}ight. $$

UpStudy AI · Accepted Answer

To find the moments of inertia for the ellipsoid with semi-axes $$ a = 4 $$, $$ b = 7 $$, $$ c = 8 $$, and mass $$ m = 1 $$, we will use the formula for the inertia tensor:

$$
I = \begin{bmatrix}
\frac{1}{5} m (b^2 + c^2) & 0 & 0 \
0 & \frac{1}{5} m (a^2 + c^2) & 0 \
0 & 0 & \frac{1}{5} m (a^2 + b^2)
\end{bmatrix}
$$

Now, we will calculate each moment of inertia step by step.

1. **Calculate $$ I_{xx} = \frac{1}{5} m (b^2 + c^2) $$**:
   $$
   I_{xx} = \frac{1}{5} \cdot 1 \cdot (7^2 + 8^2)
   $$

2. **Calculate $$ I_{yy} = \frac{1}{5} m (a^2 + c^2) $$**:
   $$
   I_{yy} = \frac{1}{5} \cdot 1 \cdot (4^2 + 8^2)
   $$

3. **Calculate $$ I_{zz} = \frac{1}{5} m (a^2 + b^2) $$**:
   $$
   I_{zz} = \frac{1}{5} \cdot 1 \cdot (4^2 + 7^2)
   $$

Now, let's perform these calculations.

### Step 1: Calculate $$ I_{xx} $$

$$
I_{xx} = \frac{1}{5} \cdot (49 + 64) = \frac{1}{5} \cdot 113
$$

### Step 2: Calculate $$ I_{yy} $$

$$
I_{yy} = \frac{1}{5} \cdot (16 + 64) = \frac{1}{5} \cdot 80
$$

### Step 3: Calculate $$ I_{zz} $$

$$
I_{zz} = \frac{1}{5} \cdot (16 + 49) = \frac{1}{5} \cdot 65
$$

Now, let's compute these values.

1. $$ I_{xx} = \frac{113}{5} = 22.6 $$
2. $$ I_{yy} = \frac{80}{5} = 16 $$
3. $$ I_{zz} = \frac{65}{5} = 13 $$

Now we have the moments of inertia:

- $$ I_{xx} = 22.6 $$
- $$ I_{yy} = 16 $$
- $$ I_{zz} = 13 $$

Finally, we will sort these values from smallest to largest:

- $$ I_{zz} = 13 $$
- $$ I_{yy} = 16 $$
- $$ I_{xx} = 22.6 $$

Thus, the moments of inertia from smallest to largest are:

1. **First moment**: $$ 13 $$
2. **Second moment**: $$ 16 $$
3. **Third moment**: $$ 22.6 $$

So the final answer is:

- First moment = 13
- Second moment = 16
- Third moment = 22.6
First moment = 13 Second moment = 16 Third moment = 22.6

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