Consider a uniform square plate of of side and mass `m`. The moment of inertia of this plate about an axis perpendicular to its plane and passing through one of its corners is -

Options

A. `(5)/(6) ma^(2)`
B. `(1)/(12) ma^(2)`
C. `(7)/(12) ma^(2)`
D. `(2)/(3)ma^(2)`

Answer

To find the moment of inertia of a uniform square plate of side length \( A \) and mass \( m \) about an axis perpendicular to its plane and passing through one of its corners, we can follow these steps:

### Step 1: Moment of Inertia about the Center of Mass
The moment of inertia \( I_{cm} \) of a uniform square plate about an axis perpendicular to its plane and passing through its center of mass is given by the formula:

\[ I_{cm} = \frac{1}{6} m A^2 \]

### Step 2: Identify the Distance to the New Axis
Next, we need to find the distance \( d \) from the center of mass to the new axis that passes through one of the corners. The distance from the center of the square to a corner can be calculated as follows:

1. The diagonal of the square is \( A\sqrt{2} \).
2. The distance from the center to a corner is half of the diagonal:

\[ d = \frac{A\sqrt{2}}{2} = \frac{A}{\sqrt{2}} \]

### Step 3: Apply the Parallel Axis Theorem
The parallel axis theorem states that:

\[ I = I_{cm} + md^2 \]

Where:
- \( I \) is the moment of inertia about the new axis.
- \( I_{cm} \) is the moment of inertia about the center of mass.
- \( m \) is the mass of the plate.
- \( d \) is the distance from the center of mass to the new axis.

Substituting the values we have:

\[ I = \frac{1}{6} m A^2 + m \left(\frac{A}{\sqrt{2}}\right)^2 \]

### Step 4: Calculate \( md^2 \)
Now, calculate \( md^2 \):

\[ md^2 = m \left(\frac{A}{\sqrt{2}}\right)^2 = m \cdot \frac{A^2}{2} \]

### Step 5: Combine the Terms
Now substitute \( md^2 \) back into the equation for \( I \):

\[ I = \frac{1}{6} m A^2 + \frac{1}{2} m A^2 \]

To combine these fractions, we need a common denominator:

\[ \frac{1}{2} m A^2 = \frac{3}{6} m A^2 \]

So we have:

\[ I = \frac{1}{6} m A^2 + \frac{3}{6} m A^2 = \frac{4}{6} m A^2 = \frac{2}{3} m A^2 \]

### Final Answer
Thus, the moment of inertia of the square plate about the given axis is:

\[ I = \frac{2}{3} m A^2 \]