M.I. of a thin uniform rod about the axis passing through its centre and perpendicular to its length is `ML^(2)//12` . The rod is cut transversely into two halves, which are then riveted end to end.M.I. of the composite rod about the axis passing through its centre and perpendicular to its length will be

Explanation

To solve the problem step by step, we need to calculate the moment of inertia (M.I.) of the composite rod formed by cutting a thin uniform rod into two halves and then riveting them end to end.

### Step 1: Understand the initial moment of inertia of the rod
The moment of inertia (I) of a thin uniform rod about an axis passing through its center and perpendicular to its length is given by the formula:

\[ I = \frac{ML^2}{12} \]

where:
- \( M \) is the mass of the rod,
- \( L \) is the length of the rod.

### Step 2: Cut the rod into two halves
When the rod is cut transversely into two equal halves, each half will have:
- Mass of each half: \( \frac{M}{2} \)
- Length of each half: \( \frac{L}{2} \)

### Step 3: Calculate the moment of inertia of each half
The moment of inertia of each half about its own center (which is at the midpoint of the half) is:

\[ I_{\text{half}} = \frac{\left(\frac{M}{2}\right) \left(\frac{L}{2}\right)^2}{12} = \frac{M}{2} \cdot \frac{L^2}{4} \cdot \frac{1}{12} = \frac{ML^2}{96} \]

### Step 4: Use the parallel axis theorem
When the two halves are riveted end to end, we need to find the moment of inertia about the new axis passing through the center of the composite rod.

The distance from the center of each half to the new axis (which is at the center of the composite rod) is \( \frac{L}{4} \) (since each half is \( \frac{L}{2} \) long).

Using the parallel axis theorem, the moment of inertia of each half about the new axis is:

\[ I_{\text{new}} = I_{\text{half}} + \left(\frac{M}{2}\right) \left(\frac{L}{4}\right)^2 \]

Calculating this gives:

\[ I_{\text{new}} = \frac{ML^2}{96} + \frac{M}{2} \cdot \frac{L^2}{16} = \frac{ML^2}{96} + \frac{ML^2}{32} \]

### Step 5: Finding a common denominator
To add these fractions, we need a common denominator. The least common multiple of 96 and 32 is 96.

Converting \( \frac{ML^2}{32} \) to have a denominator of 96:

\[ \frac{ML^2}{32} = \frac{3ML^2}{96} \]

### Step 6: Add the moments of inertia
Now we can add the two moments of inertia:

\[ I_{\text{total}} = \frac{ML^2}{96} + \frac{3ML^2}{96} = \frac{4ML^2}{96} = \frac{ML^2}{24} \]

### Final Result
Thus, the moment of inertia of the composite rod about the axis passing through its center and perpendicular to its length is:

\[ I = \frac{ML^2}{24} \]