Find the moment of inertia of a uniform square plate of mass `M` and edge a about one of its diagonals.

Explanation

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 \( a \). We want to find the moment of inertia about a diagonal of the square.

### Step 2: Moment of Inertia About an Axis Through the Center
First, we need to find the moment of inertia of the square plate about an axis that is perpendicular to the plane of the square and passes through its center. According to the formula for a square plate, the moment of inertia \( I_O \) about this axis is given by:
\[ I_O = \frac{1}{6} M a^2 \]

### Step 3: Moment of Inertia About the Axes Along the Sides
Next, we calculate the moment of inertia about an axis along one of the sides of the square. The moment of inertia \( I_{AB} \) about this axis is:
\[ I_{AB} = \frac{1}{3} M a^2 \]

Since the square is symmetric, the moment of inertia about the other side (let's call it \( I_{A'B'} \)) is the same:
\[ I_{A'B'} = \frac{1}{3} M a^2 \]

### Step 4: Apply the Perpendicular Axis Theorem
The perpendicular axis theorem states that for a planar body, the moment of inertia about an axis perpendicular to the plane (in this case, the axis through the center \( O \)) is equal to the sum of the moments of inertia about two perpendicular axes in the plane (the axes along the sides of the square):
\[ I_O = I_{AB} + I_{A'B'} \]
Substituting the values we have:
\[ I_O = \frac{1}{3} M a^2 + \frac{1}{3} M a^2 = \frac{2}{3} M a^2 \]

### Step 5: Moment of Inertia About the Diagonal
Now, we need to find the moment of inertia about the diagonal. We will denote the moment of inertia about the diagonal as \( I_D \). By the perpendicular axis theorem, we can relate the moment of inertia about the diagonal to the moment of inertia about the axes along the sides:
\[ I_O = I_D + I_D \]
This simplifies to:
\[ I_O = 2 I_D \]
Substituting our earlier result for \( I_O \):
\[ \frac{2}{3} M a^2 = 2 I_D \]
Dividing both sides by 2 gives:
\[ I_D = \frac{1}{3} M a^2 \]

### Conclusion
Thus, the moment of inertia of a uniform square plate of mass \( M \) and edge \( a \) about one of its diagonals is:
\[ I_D = \frac{1}{3} M a^2 \]