Question

Two masses 2 kg and 6 kg lie on the x-axis at distances of 3 m and 6 m, respectively, from the origin. The distances of the centre of mass of the system from the origin and from the 2 kg mass are in the ratio

• A. 3 : 7
• B. 1 : 4
• C. 2 : 1
• D. 7 : 3
• E. 3 : 1

Hint:

To simplify the ratio \(5.25 / 2.25\), think in terms of 0.75 units. \(5.25 = 7 \times 0.75\) and \(2.25 = 3 \times 0.75\). This gives the ratio \(7:3\) immediately.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
The centre of mass (COM) of a system of particles is the weighted average position of the particles, where the weights are their masses.

Step 2: Key Formula or Approach:
Position of COM: \(x_{cm} = \frac{m_1x_1 + m_2x_2}{m_1 + m_2}\).

Step 3: Detailed Explanation:

Step 1: Find the position of the centre of mass from the origin.
\[ x_{cm} = \frac{(2 \times 3) + (6 \times 6)}{2 + 6} \]
\[ x_{cm} = \frac{6 + 36}{8} = \frac{42}{8} = 5.25 \text{ m} \]
Distance from the origin is 5.25 m.

Step 2: Find the distance of the COM from the 2 kg mass.
The 2 kg mass is at \(x = 3\) m.
\[ \text{Distance} = |5.25 - 3| = 2.25 \text{ m} \]

Step 3: Calculate the required ratio.
\[ \text{Ratio} = \frac{5.25}{2.25} \]
Multiplying both by 100 to remove decimals:
\[ \frac{525}{225} = \frac{105}{45} = \frac{21}{9} = \frac{7}{3} \]

Step 4: Final Answer:
The ratio is 7 : 3.