The moment of inertia of a uniform rod of mass 0.50 kg and length 1 m is 0.10 kg m^2 about a line perpendicular to the rod. Find the distance of this line from the middle point of the rod.

Explanation

To solve the problem, we will use the Parallel Axis Theorem to find the distance \( d \) from the center of the rod to the line about which the moment of inertia is given.

### Step-by-Step Solution:

1. **Identify Given Values:**
- Mass of the rod \( m = 0.50 \, \text{kg} \)
- Length of the rod \( l = 1 \, \text{m} \)
- Moment of inertia about the line \( I_{XY} = 0.10 \, \text{kg m}^2 \)

2. **Calculate Moment of Inertia about the Center of Mass:**
The moment of inertia of a uniform rod about an axis through its center of mass (perpendicular to its length) is given by the formula:
\[ I_{CM} = \frac{m l^2}{12} \]
Substituting the values:
\[ I_{CM} = \frac{0.50 \times (1)^2}{12} = \frac{0.50}{12} = 0.04167 \, \text{kg m}^2 \]

3. **Apply the Parallel Axis Theorem:**
The Parallel Axis Theorem states that the moment of inertia about any axis parallel to the axis through the center of mass is given by:
\[ I_{XY} = I_{CM} + m d^2 \]
Rearranging this to find \( d^2 \):
\[ d^2 = \frac{I_{XY} - I_{CM}}{m} \]

4. **Substitute Known Values:**
Now substituting the known values into the equation:
\[ d^2 = \frac{0.10 - 0.04167}{0.50} \]
\[ d^2 = \frac{0.05833}{0.50} = 0.11666 \]

5. **Calculate \( d \):**
Taking the square root to find \( d \):
\[ d = \sqrt{0.11666} \approx 0.342 \, \text{m} \]

### Final Answer:
The distance of the line from the middle point of the rod is approximately \( 0.342 \, \text{m} \).