Four particles of mass `5,3,2,4 kg` are at the points `(1,6),(-1,5),(2,-3),(-1,-4)` . Find the coordinates of their centre of mass.

Explanation

To find the coordinates of the center of mass of the four particles, we will follow these steps:

### Step 1: Identify the masses and their coordinates
We have four particles with the following masses and coordinates:
- Mass \( m_1 = 5 \, \text{kg} \) at position \( (1, 6) \)
- Mass \( m_2 = 3 \, \text{kg} \) at position \( (-1, 5) \)
- Mass \( m_3 = 2 \, \text{kg} \) at position \( (2, -3) \)
- Mass \( m_4 = 4 \, \text{kg} \) at position \( (-1, -4) \)

### Step 2: Calculate the total mass
The total mass \( M \) of the system is given by:
\[ M = m_1 + m_2 + m_3 + m_4 = 5 + 3 + 2 + 4 = 14 \, \text{kg} \]

### Step 3: Calculate the x-coordinate of the center of mass
The x-coordinate of the center of mass \( x_{cm} \) is calculated using the formula:
\[ x_{cm} = \frac{m_1 x_1 + m_2 x_2 + m_3 x_3 + m_4 x_4}{M} \]
Substituting the values:
\[ x_{cm} = \frac{5 \cdot 1 + 3 \cdot (-1) + 2 \cdot 2 + 4 \cdot (-1)}{14} \]
Calculating each term:
- \( 5 \cdot 1 = 5 \)
- \( 3 \cdot (-1) = -3 \)
- \( 2 \cdot 2 = 4 \)
- \( 4 \cdot (-1) = -4 \)

Now, summing these values:
\[ x_{cm} = \frac{5 - 3 + 4 - 4}{14} = \frac{2}{14} = \frac{1}{7} \]

### Step 4: Calculate the y-coordinate of the center of mass
The y-coordinate of the center of mass \( y_{cm} \) is calculated using the formula:
\[ y_{cm} = \frac{m_1 y_1 + m_2 y_2 + m_3 y_3 + m_4 y_4}{M} \]
Substituting the values:
\[ y_{cm} = \frac{5 \cdot 6 + 3 \cdot 5 + 2 \cdot (-3) + 4 \cdot (-4)}{14} \]
Calculating each term:
- \( 5 \cdot 6 = 30 \)
- \( 3 \cdot 5 = 15 \)
- \( 2 \cdot (-3) = -6 \)
- \( 4 \cdot (-4) = -16 \)

Now, summing these values:
\[ y_{cm} = \frac{30 + 15 - 6 - 16}{14} = \frac{23}{14} \]

### Step 5: Write the coordinates of the center of mass
The coordinates of the center of mass are:
\[ \left( x_{cm}, y_{cm} \right) = \left( \frac{1}{7}, \frac{23}{14} \right) \]

### Final Answer
The coordinates of the center of mass are:
\[ \left( \frac{1}{7}, \frac{23}{14} \right) \]