Three identical particles each of mass `0.1 kg` are arranged at three corners of a square of side `sqrt(2) m`. The distance of the centre of mass from the fourth corners is

Options

A. `2//3 m`
B. `4//3 m`
C. `1m`
D. `8//3 m`

Explanation

To find the distance of the center of mass from the fourth corner of a square with three identical particles at three corners, we can follow these steps:

### Step 1: Define the Problem
We have three identical particles, each with a mass \( m = 0.1 \, \text{kg} \), located at three corners of a square with a side length of \( a = \sqrt{2} \, \text{m} \). We need to find the distance of the center of mass from the fourth corner of the square.

### Step 2: Set Up the Coordinate System
Let's place the square in the coordinate system:
- Corner 1 (0, 0)
- Corner 2 (\( \sqrt{2}, 0 \))
- Corner 3 (\( \sqrt{2}, \sqrt{2} \))
- Corner 4 (0, \( \sqrt{2} \))

We will take the origin (0, 0) as the position of the first particle.

### Step 3: Identify the Positions of the Particles
The coordinates of the three particles are:
- Particle 1: \( (0, 0) \)
- Particle 2: \( (\sqrt{2}, 0) \)
- Particle 3: \( (\sqrt{2}, \sqrt{2}) \)

### Step 4: Calculate the Center of Mass
The center of mass \( (x_{cm}, y_{cm}) \) can be calculated using the formula:
\[ x_{cm} = \frac{m_1 x_1 + m_2 x_2 + m_3 x_3}{m_1 + m_2 + m_3} \]
\[ y_{cm} = \frac{m_1 y_1 + m_2 y_2 + m_3 y_3}{m_1 + m_2 + m_3} \]

Since all masses are equal, we can simplify:
\[ x_{cm} = \frac{0.1 \cdot 0 + 0.1 \cdot \sqrt{2} + 0.1 \cdot \sqrt{2}}{0.1 + 0.1 + 0.1} = \frac{0 + 0.1\sqrt{2} + 0.1\sqrt{2}}{0.3} = \frac{0.2\sqrt{2}}{0.3} = \frac{2\sqrt{2}}{3} \]

\[ y_{cm} = \frac{0.1 \cdot 0 + 0.1 \cdot 0 + 0.1 \cdot \sqrt{2}}{0.1 + 0.1 + 0.1} = \frac{0 + 0 + 0.1\sqrt{2}}{0.3} = \frac{0.1\sqrt{2}}{0.3} = \frac{\sqrt{2}}{3} \]

Thus, the coordinates of the center of mass are:
\[ (x_{cm}, y_{cm}) = \left(\frac{2\sqrt{2}}{3}, \frac{\sqrt{2}}{3}\right) \]

### Step 5: Calculate the Distance from the Fourth Corner
The coordinates of the fourth corner are \( (0, \sqrt{2}) \). We can find the distance \( d \) between the center of mass and the fourth corner using the distance formula:
\[ d = \sqrt{(x_{cm} - x_4)^2 + (y_{cm} - y_4)^2} \]
Substituting the values:
\[ d = \sqrt{\left(\frac{2\sqrt{2}}{3} - 0\right)^2 + \left(\frac{\sqrt{2}}{3} - \sqrt{2}\right)^2} \]
\[ = \sqrt{\left(\frac{2\sqrt{2}}{3}\right)^2 + \left(\frac{\sqrt{2}}{3} - \frac{3\sqrt{2}}{3}\right)^2} \]
\[ = \sqrt{\left(\frac{2\sqrt{2}}{3}\right)^2 + \left(-\frac{2\sqrt{2}}{3}\right)^2} \]
\[ = \sqrt{\frac{8}{9} + \frac{8}{9}} = \sqrt{\frac{16}{9}} = \frac{4}{3} \, \text{m} \]

### Final Answer
The distance of the center of mass from the fourth corner is \( \frac{4}{3} \, \text{m} \).

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