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 find the distance of the line from the midpoint of the rod, we can use the parallel axis theorem. The parallel axis theorem states that the moment of inertia \( I \) about any axis parallel to an axis through the center of mass is given by:

\[ I = I_{cm} + Md^2 \]

Where:
- \( I \) is the moment of inertia about the new axis,
- \( I_{cm} \) is the moment of inertia about the center of mass,
- \( M \) is the mass of the object,
- \( d \) is the distance between the two axes.

### Step 1: Identify the known values
We know:
- The moment of inertia about the new axis \( I = 0.10 \, \text{kg m}^2 \)
- The mass of the rod \( M = 0.50 \, \text{kg} \)
- The moment of inertia about the center of mass for a uniform rod \( I_{cm} = \frac{1}{12} M L^2 \), where \( L \) is the length of the rod.

### Step 2: Calculate \( I_{cm} \)
Substituting the values into the formula for \( I_{cm} \):

\[ I_{cm} = \frac{1}{12} \times 0.50 \, \text{kg} \times (1 \, \text{m})^2 = \frac{0.50}{12} = 0.04167 \, \text{kg m}^2 \]

### Step 3: Use the parallel axis theorem
Now, we can substitute \( I \) and \( I_{cm} \) into the parallel axis theorem equation:

\[ 0.10 = 0.04167 + 0.50 \times d^2 \]

### Step 4: Solve for \( d^2 \)
Rearranging the equation to isolate \( d^2 \):

\[ 0.50 \times d^2 = 0.10 - 0.04167 \]

Calculating the right side:

\[ 0.50 \times d^2 = 0.05833 \]

Now, divide both sides by 0.50:

\[ d^2 = \frac{0.05833}{0.50} = 0.11666 \]

### Step 5: Find \( d \)
Taking the square root of both sides to find \( d \):

\[ d = \sqrt{0.11666} \approx 0.342 \, \text{m} \]

### Conclusion
The distance of the line from the midpoint of the rod is approximately \( 0.342 \, \text{m} \).

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