Three bodies of masses m, 2m and 3m are placed at the corners of a triangle having coordinates (1, 1.5), (2.5, 1.5) and (3, 3) respectively. Calculate the coordinates of the centre of mass.

Answer

To find the coordinates of the center of mass of the three bodies with masses \( m \), \( 2m \), and \( 3m \) located at the points \( (1, 1.5) \), \( (2.5, 1.5) \), and \( (3, 3) \) respectively, we can use the formula for the center of mass:

\[ \text{Center of Mass (CM)} = \frac{\sum m_i \cdot r_i}{\sum m_i} \]

where \( m_i \) is the mass of each body and \( r_i \) is the position vector of each body.

### Step 1: Identify the masses and their coordinates
- Mass \( m_1 = m \) at coordinates \( (x_1, y_1) = (1, 1.5) \)
- Mass \( m_2 = 2m \) at coordinates \( (x_2, y_2) = (2.5, 1.5) \)
- Mass \( m_3 = 3m \) at coordinates \( (x_3, y_3) = (3, 3) \)

### Step 2: Calculate the total mass
\[ \text{Total mass} = m_1 + m_2 + m_3 = m + 2m + 3m = 6m \]

### Step 3: Calculate the x-coordinate of the center of mass
Using the formula for the x-coordinate of the center of mass:
\[ x_{CM} = \frac{m_1 x_1 + m_2 x_2 + m_3 x_3}{m_1 + m_2 + m_3} \]
Substituting the values:
\[ x_{CM} = \frac{m \cdot 1 + 2m \cdot 2.5 + 3m \cdot 3}{6m} \]
\[ = \frac{m + 5m + 9m}{6m} = \frac{15m}{6m} = \frac{15}{6} = 2.5 \]

### Step 4: Calculate the y-coordinate of the center of mass
Using the formula for the y-coordinate of the center of mass:
\[ y_{CM} = \frac{m_1 y_1 + m_2 y_2 + m_3 y_3}{m_1 + m_2 + m_3} \]
Substituting the values:
\[ y_{CM} = \frac{m \cdot 1.5 + 2m \cdot 1.5 + 3m \cdot 3}{6m} \]
\[ = \frac{1.5m + 3m + 9m}{6m} = \frac{13.5m}{6m} = \frac{13.5}{6} = 2.25 \]

### Step 5: Combine the results
The coordinates of the center of mass are:
\[ \text{CM} = (x_{CM}, y_{CM}) = \left(2.5, 2.25\right) \]

### Final Answer
The coordinates of the center of mass are \( (2.5, 2.25) \).

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