A cylinder and a sphere each have a mass of 5 kg and a radius of 0.2 m. If the cylinder rotates about an axis through its center and both of its circular faces, and the sphere rotates about an axis through its center, what is the moment of inertia of each object?



Answer :

Certainly! Let's find the moment of inertia for each object given the provided conditions: a cylinder rotating about an axis through its center and through both of its circular faces, and a sphere rotating about an axis through its center.

### Cylinder
The moment of inertia [tex]\(I\)[/tex] for a rotating solid cylinder about its center through both of its faces is given by the formula:

[tex]\[ I_{\text{cylinder}} = \frac{1}{2} m r^2 \][/tex]

where:
- [tex]\(m\)[/tex] is the mass of the cylinder
- [tex]\(r\)[/tex] is the radius of the cylinder

Given:
- Mass, [tex]\(m = 5 \, \text{kg}\)[/tex]
- Radius, [tex]\(r = 0.2 \, \text{m}\)[/tex]

Substituting the values into the formula:

[tex]\[ I_{\text{cylinder}} = \frac{1}{2} \times 5 \, \text{kg} \times (0.2 \, \text{m})^2 \][/tex]

We can evaluate:

[tex]\[ I_{\text{cylinder}} = \frac{1}{2} \times 5 \times 0.04 \][/tex]

[tex]\[ I_{\text{cylinder}} = \frac{1}{2} \times 0.2 \][/tex]

[tex]\[ I_{\text{cylinder}} = 0.1 \, \text{kg}\cdot\text{m}^2 \][/tex]

### Sphere
The moment of inertia [tex]\(I\)[/tex] for a rotating solid sphere about its center is given by the formula:

[tex]\[ I_{\text{sphere}} = \frac{2}{5} m r^2 \][/tex]

where:
- [tex]\(m\)[/tex] is the mass of the sphere
- [tex]\(r\)[/tex] is the radius of the sphere

Given:
- Mass, [tex]\(m = 5 \, \text{kg}\)[/tex]
- Radius, [tex]\(r = 0.2 \, \text{m}\)[/tex]

Substituting the values into the formula:

[tex]\[ I_{\text{sphere}} = \frac{2}{5} \times 5 \, \text{kg} \times (0.2 \, \text{m})^2 \][/tex]

We can evaluate:

[tex]\[ I_{\text{sphere}} = \frac{2}{5} \times 5 \times 0.04 \][/tex]

[tex]\[ I_{\text{sphere}} = \frac{2}{5} \times 0.2 \][/tex]

[tex]\[ I_{\text{sphere}} = 0.08 \, \text{kg}\cdot\text{m}^2 \][/tex]

### Summary
The moments of inertia for the given objects are:
- Cylinder: [tex]\(0.1 \, \text{kg}\cdot\text{m}^2\)[/tex]
- Sphere: [tex]\(0.08 \, \text{kg}\cdot\text{m}^2\)[/tex]