Calculate the moment of inertia of a cylinder of length 2 m, radius 5 cm and density `8 xx 10^(3) kg//m^3` about (i) the axis of the cylinder and (ii) and axis passing through the centre and perpendicular to its length.

Explanation

To solve the problem of calculating the moment of inertia of a cylinder, we will follow these steps:

### Given Data:
- Length of the cylinder, \( L = 2 \, \text{m} \)
- Radius of the cylinder, \( r = 5 \, \text{cm} = 0.05 \, \text{m} \)
- Density of the cylinder, \( \rho = 8 \times 10^3 \, \text{kg/m}^3 \)

### Step 1: Calculate the Volume of the Cylinder
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
Substituting the values:
\[ V = \pi (0.05)^2 (2) = \pi (0.0025)(2) = 0.005\pi \, \text{m}^3 \]

### Step 2: Calculate the Mass of the Cylinder
The mass \( m \) can be calculated using the formula:
\[ m = \rho V \]
Substituting the values:
\[ m = 8 \times 10^3 \times 0.005\pi \approx 125.6 \, \text{kg} \]

### Step 3: Calculate the Moment of Inertia about the Axis of the Cylinder (I1)
The moment of inertia \( I_1 \) of a solid cylinder about its own axis is given by:
\[ I_1 = \frac{1}{2} m r^2 \]
Substituting the values:
\[ I_1 = \frac{1}{2} \times 125.6 \times (0.05)^2 = \frac{1}{2} \times 125.6 \times 0.0025 \]
Calculating:
\[ I_1 = \frac{1}{2} \times 125.6 \times 0.0025 \approx 0.157 \, \text{kg m}^2 \]

### Step 4: Calculate the Moment of Inertia about an Axis Perpendicular to its Length (I2)
The moment of inertia \( I_2 \) about an axis passing through the center and perpendicular to its length is given by:
\[ I_2 = \frac{1}{12} m L^2 + \frac{1}{4} m r^2 \]
Substituting the values:
\[ I_2 = \frac{1}{12} \times 125.6 \times (2)^2 + \frac{1}{4} \times 125.6 \times (0.05)^2 \]
Calculating each term:
\[ I_2 = \frac{1}{12} \times 125.6 \times 4 + \frac{1}{4} \times 125.6 \times 0.0025 \]
\[ I_2 = \frac{1}{12} \times 502.4 + \frac{1}{4} \times 0.314 \]
Calculating:
\[ I_2 = 41.8667 + 0.0785 \approx 41.94 \, \text{kg m}^2 \]

### Final Results:
1. Moment of inertia about the axis of the cylinder, \( I_1 \approx 0.157 \, \text{kg m}^2 \)
2. Moment of inertia about the axis perpendicular to its length, \( I_2 \approx 41.94 \, \text{kg m}^2 \)