Find the momentof inertia of a uniform square plate of mass m and edge a about one of its diagonals.

Options

A. `(Ml^2)/6`
B. `(Ml^2)/12`
C. `(Ml^2)/3`
D. `(Ml^2)/4`

Answer

To find the moment of inertia of a uniform square plate of mass \( m \) and edge \( a \) about one of its diagonals, we can follow these steps:

### Step 1: Understand the Geometry
We have a square plate with mass \( m \) and side length \( a \). The diagonals of the square intersect at the center of the square and are perpendicular to each other.

### Step 2: Moment of Inertia about the Center of Mass
The moment of inertia of a square plate about an axis perpendicular to its plane and passing through its center of mass is given by the formula:
\[ I_{CM} = \frac{m a^2}{6} \]

### Step 3: Apply the Perpendicular Axis Theorem
The perpendicular axis theorem states that for a planar object, the moment of inertia about an axis perpendicular to the plane (z-axis) is equal to the sum of the moments of inertia about two perpendicular axes (x and y axes) in the plane of the object:
\[ I_z = I_x + I_y \]
In our case, the two axes in the plane of the square are the diagonals. Since the square is symmetric, we have:
\[ I_x = I_y = I_{diagonal} \]
Thus, we can rewrite the equation as:
\[ I_{CM} = I_{diagonal} + I_{diagonal} = 2 I_{diagonal} \]
This implies:
\[ I_{diagonal} = \frac{I_{CM}}{2} \]

### Step 4: Substitute the Value of \( I_{CM} \)
Now, substituting the value of \( I_{CM} \):
\[ I_{diagonal} = \frac{1}{2} \left( \frac{m a^2}{6} \right) = \frac{m a^2}{12} \]

### Conclusion
Thus, the moment of inertia of the uniform square plate about one of its diagonals is:
\[ I_{diagonal} = \frac{m a^2}{12} \]

### Final Answer
The correct option is \( \frac{m a^2}{12} \).

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