Question

If \(P\) is the midpoint of median, find distance of COM from \(P\).

Answer 1

Correct Answer: 1.32

Concept: Centre of mass coordinates: \[ x_{cm} = \frac{\sum m_i x_i}{\sum m_i} \] \[ y_{cm} = \frac{\sum m_i y_i}{\sum m_i} \] Total mass \[ M = 2 + 15 + 3 = 20\,kg \]

Step 1: Assign coordinates.} Left mass \(2kg\) \[ (0,0) \] Right mass \(3kg\) \[ (10\sqrt{3},0) \] Top mass \(15kg\) \[ (5\sqrt{3},5) \]
Step 2: Find \(x_{cm}\).} \[ x_{cm} = \frac{2(0) + 15(5\sqrt{3}) + 3(10\sqrt{3})}{20} \] \[ x_{cm} = \frac{75\sqrt{3} + 30\sqrt{3}}{20} \] \[ x_{cm} = \frac{21\sqrt{3}}{4} \]
Step 3: Find \(y_{cm}\).} \[ y_{cm} = \frac{2(0) + 15(5) + 3(0)}{20} \] \[ y_{cm} = \frac{75}{20} \] \[ y_{cm} = \frac{15}{4} \]
Step 4: Coordinates of midpoint of median \(P\).} \[ P = (5\sqrt{3}, 2.5) \]
Step 5: Find distance between COM and \(P\).} \[ d = \sqrt{(x_{cm}-x_P)^2 + (y_{cm}-y_P)^2} \] \[ d = \sqrt{(5.25\sqrt{3} - 5\sqrt{3})^2 + (3.75 - 2.5)^2} \] \[ d = \sqrt{(0.25\sqrt{3})^2 + (1.25)^2} \] \[ d = \sqrt{1.75} \] \[ d \approx 1.32 \] Final Result \[ d = 1.32 \]