Consider a uniform square plate of side 'a' and mass 'm' The moment of inertia of heis plate about an axis perpendiucalar to its plane and passing through one of its corners is

Options

A. `(5)/(6)ma^2`
B. `(1)/(2)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 '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
First, we need to calculate the moment of inertia of the square plate about an axis passing through its center. The formula for the moment of inertia \( I_{CM} \) of a square plate about an axis perpendicular to its plane and passing through its center is given by:

\[ I_{CM} = \frac{1}{6} m a^2 \]

### Step 2: Finding the Distance to the Corner
Next, we need to find the distance from the center of the plate to the corner where the axis is located. The distance from the center of the square to a corner can be calculated as follows:

The diagonal of the square is given by:

\[ d = a\sqrt{2} \]

Since we need the distance from the center to the corner, we take half of the diagonal:

\[ \text{Distance} = \frac{d}{2} = \frac{a\sqrt{2}}{2} \]

### Step 3: Applying the Parallel Axis Theorem
Now we can apply the Parallel Axis Theorem, which states that:

\[ I = I_{CM} + m d^2 \]

where \( d \) is the distance from the center of mass to the new axis. Substituting the values we have:

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

Calculating \( \left(\frac{a\sqrt{2}}{2}\right)^2 \):

\[ \left(\frac{a\sqrt{2}}{2}\right)^2 = \frac{2a^2}{4} = \frac{a^2}{2} \]

Now substituting this back into the equation:

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

### Step 4: Simplifying the Expression
Now we combine the terms:

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

### Final Result
Thus, the moment of inertia of the square plate about the axis passing through one of its corners is:

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