A square plate lies in the xy plane with its centre at the origin and its edges parallel to the x and y axes. Its moments of inertia about the x, y and z axes are `I_(x), I_(y)` and `I_(Z)` respectively, and about a diagonal it is `I_(D)`

Options

A. `I_(X) = I_(Y) = 1/2I_(Z)`
B. `I_(X) = I_(Y) = 2I_(x)`
C. `I_(D) = I_(x)`
D. `I_(D) = I_(Y)`

Explanation

To solve the problem, we need to find the relationships between the moments of inertia of a square plate about its axes and its diagonal. Here are the steps to derive the required relationships:

### Step 1: Understand the Geometry
The square plate lies in the xy-plane with its center at the origin. The edges of the square are parallel to the x and y axes.

### Step 2: Identify Moments of Inertia
For a square plate of uniform density:
- The moment of inertia about the x-axis, \( I_x \)
- The moment of inertia about the y-axis, \( I_y \)
- The moment of inertia about the z-axis, \( I_z \)
- The moment of inertia about a diagonal, \( I_d \)

### Step 3: Use the Perpendicular Axis Theorem
According to the perpendicular axis theorem, for a planar body lying in the xy-plane:
\[ I_z = I_x + I_y \]
Since the plate is square and symmetrical, we have:
\[ I_x = I_y \]
Let \( I_x = I_y = I \). Therefore, we can rewrite the equation as:
\[ I_z = I + I = 2I \]
From this, we can express \( I \) in terms of \( I_z \):
\[ I = \frac{1}{2} I_z \]
Thus, we have:
\[ I_x = I_y = \frac{1}{2} I_z \]

### Step 4: Find the Moment of Inertia about the Diagonal
For a square plate, the moment of inertia about a diagonal can also be derived using the symmetry of the square. The moment of inertia about the diagonal \( I_d \) can be expressed as:
\[ I_d = \frac{1}{2} I_z \]
This is because the diagonal divides the square into two equal halves, and the distribution of mass is symmetric about the diagonal.

### Step 5: Relate \( I_d \) to \( I_x \)
From the previous steps, we have:
\[ I_d = \frac{1}{2} I_z \]
And since we found that \( I = \frac{1}{2} I_z \), we can conclude:
\[ I_d = I_x \]
Thus, we have:
\[ I_d = I_x = I_y \]

### Final Relationships
From the above deductions, we can summarize the relationships:
1. \( I_x = I_y = \frac{1}{2} I_z \)
2. \( I_d = I_x = I_y \)

### Summary
- The moments of inertia about the x and y axes are equal and half of the moment of inertia about the z-axis.
- The moment of inertia about the diagonal is equal to the moments of inertia about the x and y axes.