Calculate the moment of inertia of a.unifprm disc of mass 0.4 kg and radius 0.1m about an axis passing through its edge and perpendicular to the plane of die disc ?

Explanation

To calculate the moment of inertia of a uniform disc of mass 0.4 kg and radius 0.1 m about an axis passing through its edge and perpendicular to the plane of the disc, we will follow these steps:

### Step 1: Identify the Moment of Inertia about the Center
The moment of inertia \( I_c \) of a uniform disc about an axis passing through its center and perpendicular to its plane is given by the formula:
\[ I_c = \frac{1}{2} m r^2 \]
Where:
- \( m \) is the mass of the disc (0.4 kg)
- \( r \) is the radius of the disc (0.1 m)

### Step 2: Substitute the Values
Substituting the values into the formula:
\[ I_c = \frac{1}{2} \times 0.4 \, \text{kg} \times (0.1 \, \text{m})^2 \]
Calculating:
\[ I_c = \frac{1}{2} \times 0.4 \times 0.01 = 0.002 \, \text{kg m}^2 \]

### Step 3: Apply the Parallel Axis Theorem
To find the moment of inertia \( I' \) about an axis passing through the edge of the disc, we use the parallel axis theorem:
\[ I' = I_c + m d^2 \]
Where:
- \( d \) is the distance from the center of mass to the new axis. For a disc, this distance is equal to the radius \( r \).

### Step 4: Substitute the Values into the Parallel Axis Theorem
Substituting the values:
\[ I' = 0.002 \, \text{kg m}^2 + 0.4 \, \text{kg} \times (0.1 \, \text{m})^2 \]
Calculating:
\[ I' = 0.002 + 0.4 \times 0.01 = 0.002 + 0.004 = 0.006 \, \text{kg m}^2 \]

### Final Result
Thus, the moment of inertia of the uniform disc about an axis passing through its edge and perpendicular to the plane of the disc is:
\[ I' = 0.006 \, \text{kg m}^2 \]

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