Question

Three point particles of masses 1.0 kg, 1.5 kg and 2.5 kg are placed at three corners of a right-angle triangle of sides 4.0 cm, 3.0 cm and 5.0 cm as shown in the figure. The center of mass of the system is at a point:

• A. 0.6 cm right and 2.0 cm above 1 kg mass
• B. 1.5 cm right and 1.2 cm above 1 kg mass
• C. 2.0 cm right and 0.9 cm above 1 kg mass
• D. 0.9 cm right and 2.0 cm above 1 kg mass

Hint:

Always choose the origin at the location of one of the masses. This sets its contributions to zero, significantly simplifying the arithmetic and reducing chances of calculation errors.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The Center of Mass (COM) coordinates for a system of point masses are calculated as the mass-weighted average of their positions.

Step 2: Key Formula or Approach:
\[ X_{cm} = \frac{m_1x_1 + m_2x_2 + m_3x_3}{m_1 + m_2 + m_3} \]
\[ Y_{cm} = \frac{m_1y_1 + m_2y_2 + m_3y_3}{m_1 + m_2 + m_3} \]
Let 1 kg mass be at origin \( (0,0) \). Then 1.5 kg is at \( (3,0) \) and 2.5 kg is at \( (0,4) \).

Step 3: Detailed Explanation:
1. Assign coordinates based on the diagram:
- Mass \( m_1 = 1.0 \text{ kg} \) at \( (x_1, y_1) = (0, 0) \).
- Mass \( m_2 = 1.5 \text{ kg} \) at \( (x_2, y_2) = (3, 0) \).
- Mass \( m_3 = 2.5 \text{ kg} \) at \( (x_3, y_3) = (0, 4) \).
Total mass \( M = 1.0 + 1.5 + 2.5 = 5.0 \text{ kg} \).
2. Calculate \( X_{cm} \):
\[ X_{cm} = \frac{(1.0 \times 0) + (1.5 \times 3) + (2.5 \times 0)}{5.0} = \frac{4.5}{5.0} = 0.9 \text{ cm} \]
3. Calculate \( Y_{cm} \):
\[ Y_{cm} = \frac{(1.0 \times 0) + (1.5 \times 0) + (2.5 \times 4)}{5.0} = \frac{10}{5.0} = 2.0 \text{ cm} \]

Step 4: Final Answer:
The center of mass is 0.9 cm to the right and 2.0 cm above the 1 kg mass.