The radius of gyration of a uniform square plate of mass M and sida about its diagonal is

Options

A. `a/sqrt(6)`
B. `a/sqrt(12)`
C. `a/2 sqrt(2)`
D. `a/4`

Answer

To find the radius of gyration of a uniform square plate of mass \( M \) and side \( A \) about its diagonal, we can follow these steps:

### Step 1: Understand the Moment of Inertia
The moment of inertia \( I \) of a body about an axis is related to its mass and the radius of gyration \( k \) by the formula:
\[ I = M k^2 \]
We need to find the moment of inertia of the square plate about its diagonal.

### Step 2: Use the Perpendicular Axis Theorem
For a flat plate, the perpendicular axis theorem states that:
\[ I_z = I_x + I_y \]
where \( I_z \) is the moment of inertia about an axis perpendicular to the plane of the plate, and \( I_x \) and \( I_y \) are the moments of inertia about two perpendicular axes in the plane of the plate.

### Step 3: Calculate the Moment of Inertia for the Square Plate
For a square plate of side \( A \):
- The moment of inertia about an axis through the center and parallel to one side (let's say \( I_x \)) is given by:
\[ I_x = \frac{M A^2}{12} \]
- Due to symmetry, the moment of inertia about the other axis \( I_y \) is the same:
\[ I_y = \frac{M A^2}{12} \]

### Step 4: Apply the Perpendicular Axis Theorem
Now, applying the perpendicular axis theorem:
\[ I_z = I_x + I_y = \frac{M A^2}{12} + \frac{M A^2}{12} = \frac{M A^2}{6} \]

### Step 5: Find the Moment of Inertia About the Diagonal
Since the square plate is symmetric, the moment of inertia about either diagonal (let's call it \( I_d \)) will be:
\[ I_d = \frac{I_z}{2} = \frac{M A^2}{12} \]

### Step 6: Relate Moment of Inertia to Radius of Gyration
Now, using the relation \( I = M k^2 \):
\[ \frac{M A^2}{12} = M k^2 \]
Dividing both sides by \( M \):
\[ k^2 = \frac{A^2}{12} \]
Taking the square root:
\[ k = \frac{A}{2\sqrt{3}} \]

### Step 7: Rationalize the Result
To rationalize \( k \):
\[ k = \frac{A \sqrt{3}}{6} \]

### Final Answer
Thus, the radius of gyration of the uniform square plate about its diagonal is:
\[ k = \frac{A \sqrt{3}}{6} \]

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