Four particles each of mass `m` are placed at the corners of a square of side length `l`. The radius of gyration of the system about an axis perpendicular to the plane of square and passing through its centre is

Options

A. `l/(sqrt(2))`
B. `l/2`
C. `l`
D. `sqrt(2)l`

Explanation

To find the radius of gyration of a system of four particles each of mass `m` placed at the corners of a square of side length `l`, we can follow these steps:

### Step 1: Understand the Configuration
We have four particles located at the corners of a square. The coordinates of these particles can be defined as follows:
- Particle 1: (l/2, l/2)
- Particle 2: (l/2, -l/2)
- Particle 3: (-l/2, -l/2)
- Particle 4: (-l/2, l/2)

### Step 2: Calculate the Total Mass
The total mass \( M \) of the system is the sum of the masses of all four particles:
\[ M = m + m + m + m = 4m \]

### Step 3: Determine the Distance of Each Particle from the Axis
The radius of gyration \( k \) is defined as:
\[ k = \sqrt{\frac{I}{M}} \]
where \( I \) is the moment of inertia of the system about the given axis.

### Step 4: Calculate the Moment of Inertia
The moment of inertia \( I \) for each particle about the axis perpendicular to the plane of the square and passing through its center can be calculated using the formula:
\[ I = \sum m_i r_i^2 \]
where \( r_i \) is the distance of each particle from the axis.

The distance of each particle from the center (0, 0) is given by:
- For Particle 1: \( r_1 = \sqrt{(l/2)^2 + (l/2)^2} = \frac{l}{\sqrt{2}} \)
- For Particle 2: \( r_2 = \frac{l}{\sqrt{2}} \)
- For Particle 3: \( r_3 = \frac{l}{\sqrt{2}} \)
- For Particle 4: \( r_4 = \frac{l}{\sqrt{2}} \)

Now, substituting these into the moment of inertia formula:
\[ I = m \left(\frac{l}{\sqrt{2}}\right)^2 + m \left(\frac{l}{\sqrt{2}}\right)^2 + m \left(\frac{l}{\sqrt{2}}\right)^2 + m \left(\frac{l}{\sqrt{2}}\right)^2 \]
\[ I = 4m \left(\frac{l^2}{2}\right) = 2ml^2 \]

### Step 5: Calculate the Radius of Gyration
Now, substituting \( I \) and \( M \) into the radius of gyration formula:
\[ k = \sqrt{\frac{I}{M}} = \sqrt{\frac{2ml^2}{4m}} = \sqrt{\frac{l^2}{2}} = \frac{l}{\sqrt{2}} \]

### Final Answer
The radius of gyration of the system about the specified axis is:
\[ k = \frac{l}{\sqrt{2}} \]