There is a square planar object. Moment of inertia of this object about an axis passing through its centre and parallel to its edge is I. What will be moment of inertia about one of its diagonals?

Options

A. `I`
B. `I//2`
C. `2I`
D. `I//4`

Explanation

To find the moment of inertia of a square planar object about one of its diagonals, we can use the perpendicular axis theorem. Here’s a step-by-step solution:

### Step 1: Understand the Given Information
We are given that the moment of inertia of a square planar object about an axis passing through its center and parallel to its edge is \( I \).

### Step 2: Apply the Perpendicular Axis Theorem
The perpendicular axis theorem states that for a planar lamina, the moment of inertia about an axis perpendicular to the plane (let's call it \( z \)) is equal to the sum of the moments of inertia about two mutually perpendicular axes lying in the plane (let's call them \( x \) and \( y \)):

\[ I_z = I_x + I_y \]

### Step 3: Recognize Symmetry
For a square object, the moments of inertia about the \( x \) and \( y \) axes are equal due to symmetry. Thus, we can write:

\[ I_z = I + I = 2I \]

### Step 4: Identify the Diagonal Axes
Next, we need to find the moment of inertia about one of the diagonals. Let’s denote the moment of inertia about the diagonal as \( I_d \). According to the perpendicular axis theorem, we can also apply it to the diagonal axes. If we consider two diagonals, we can denote their moments of inertia as \( I_{d1} \) and \( I_{d2} \):

\[ I_z = I_{d1} + I_{d2} \]

### Step 5: Use Symmetry Again
Since the square is symmetrical, the moments of inertia about the two diagonals are equal:

\[ I_{d1} = I_{d2} = I_d \]

Thus, we can rewrite the equation as:

\[ I_z = I_d + I_d = 2I_d \]

### Step 6: Relate the Two Equations
From Step 3, we have \( I_z = 2I \). Now we can set the two expressions for \( I_z \) equal to each other:

\[ 2I = 2I_d \]

### Step 7: Solve for \( I_d \)
Dividing both sides by 2 gives us:

\[ I_d = I \]

### Conclusion
The moment of inertia of the square planar object about one of its diagonals is equal to the moment of inertia about an axis passing through its center and parallel to its edge.

\[ \text{Moment of Inertia about one diagonal} = I \]