The M.I. of a thin rod of length 1 about the perpendicular axis through its centre is I. The M.I. of square structure made by four such rods a perpendicular axis to the plane and through the centre will be:

Options

A. 4I
B. 8I
C. 12I
D. 16I

Answer

To solve the problem of finding the moment of inertia (M.I.) of a square structure made by four thin rods about a perpendicular axis through its center, we can follow these steps:

### Step 1: Understand the Moment of Inertia of a Single Rod
The moment of inertia (I) of a thin rod of length 1 about an axis perpendicular to it and passing through its center is given as I.

### Step 2: Construct the Square Structure
The square structure is formed by four such rods. Each rod will be positioned at the sides of the square.

### Step 3: Apply the Perpendicular Axis Theorem
According to the perpendicular axis theorem, the moment of inertia of a planar object about an axis perpendicular to its plane is the sum of the moments of inertia about two perpendicular axes lying in the plane of the object and intersecting at a point on the perpendicular axis.

### Step 4: Calculate the Moment of Inertia of One Rod
For one rod, the moment of inertia about its center is I. When considering the square structure, we need to find the contribution of each rod to the total moment of inertia about the center of the square.

### Step 5: Use the Parallel Axis Theorem
For each rod, we can use the parallel axis theorem to find the moment of inertia about the center of the square. The parallel axis theorem states:
\[ I' = I + Md^2 \]
where \( I' \) is the moment of inertia about the new axis, \( I \) is the moment of inertia about the center of mass, \( M \) is the mass of the rod, and \( d \) is the distance from the center of mass to the new axis.

### Step 6: Calculate the Moment of Inertia for Each Rod
For each rod positioned at the sides of the square:
- The distance \( d \) from the center of the square to the center of each rod is \( \frac{1}{2} \) (since the length of each rod is 1).
- The moment of inertia of each rod about the center of the square is:
\[ I' = I + M \left(\frac{1}{2}\right)^2 \]
Since \( I = \frac{m l^2}{12} \) for a single rod of length \( l = 1 \), we can express \( M \) in terms of \( I \).

### Step 7: Total Moment of Inertia for the Square Structure
Since there are four rods, the total moment of inertia \( I_{total} \) about the center of the square is:
\[ I_{total} = 4 \left( I + M \left(\frac{1}{2}\right)^2 \right) \]

### Step 8: Substitute and Simplify
Substituting \( M \) with \( \frac{12I}{l^2} \) (where \( l = 1 \)):
\[ I_{total} = 4 \left( I + \frac{12I}{1} \cdot \frac{1}{4} \right) \]
\[ = 4 \left( I + 3I \right) \]
\[ = 4 \cdot 4I = 16I \]

### Conclusion
Thus, the moment of inertia of the square structure made by four thin rods about the perpendicular axis through its center is \( 16I \).