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)`

Explanation

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 use the parallel axis theorem. Here’s the step-by-step solution:

### Step 1: Identify the 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 (CM) is given by the formula:
\[ I_{CM} = \frac{1}{6} m A^2 \]

### Step 2: Use the Parallel Axis Theorem
The parallel axis theorem states that if you know the moment of inertia about an axis through the center of mass, you can find the moment of inertia about any parallel axis by adding the term \( md^2 \), where \( d \) is the distance between the two axes.

In this case, the distance \( d \) from the center of mass to the corner of the square plate is:
\[ d = \frac{A}{\sqrt{2}} \]
This is because the center of mass of the square plate is located at a distance of \( \frac{A}{2} \) from each side, and the diagonal distance to the corner can be calculated using the Pythagorean theorem.

### Step 3: Calculate the Moment of Inertia about the Corner
Now we can calculate the moment of inertia about the corner using the parallel axis theorem:
\[ I = I_{CM} + md^2 \]
Substituting the values we have:
\[ I = \frac{1}{6} m A^2 + m \left(\frac{A}{\sqrt{2}}\right)^2 \]
\[ = \frac{1}{6} m A^2 + m \left(\frac{A^2}{2}\right) \]
\[ = \frac{1}{6} m A^2 + \frac{1}{2} m A^2 \]
To add these fractions, we need a common denominator:
\[ = \frac{1}{6} m A^2 + \frac{3}{6} m A^2 \]
\[ = \frac{4}{6} m A^2 = \frac{2}{3} m A^2 \]

### Final Answer
Thus, the moment of inertia of the uniform square plate about an axis perpendicular to its plane and passing through one of its corners is:
\[ I = \frac{2}{3} m A^2 \]