Question

Four masses of \(1\text{ kg}, 2\text{ kg}, 3\text{ kg}\) and \(4\text{ kg}\) are kept at co-ordinates \((0, 0)\text{m, (0, 1)\text{m}\) and \((1, 0)\text{m}\) respectively. Using the co-ordinates of centre of mass its position vector is}

• A. \(0.5\hat{i} + 0.7\hat{j}\)
• B. \(0.7\hat{i} + 0.5\hat{j}\)
• C. \(0.4\hat{i} + 0.6\hat{j}\)
• D. \(\hat{i} + \hat{j}\)

Hint:

COM: Weighted average of coordinates

Answer 1

The Correct Option is C

Concept: \[ x_{cm} = \frac{\sum m_i x_i}{\sum m_i}, \quad y_{cm} = \frac{\sum m_i y_i}{\sum m_i} \]

Step 1: Assign coordinates. \[ (0,0): 1\,kg,\quad (0,1): 2\,kg,\quad (1,0): 3\,kg,\quad (1,1): 4\,kg \] Total mass = 10

Step 2: Calculate $x_{cm}$. \[ x_{cm} = \frac{(1\times0)+(2\times0)+(3\times1)+(4\times1)}{10} = \frac{7}{10} = 0.7 \]

Step 3: Calculate $y_{cm}$. \[ y_{cm} = \frac{(1\times0)+(2\times1)+(3\times0)+(4\times1)}{10} = \frac{6}{10} = 0.6 \]

Step 4: Conclusion.
Position vector = $0.4\hat{i} + 0.6\hat{j}$ Final Answer: Option (C)