Question

In the diagram shown below, $ m_1 $ and $ m_2 $ are the masses of two particles and $ x_1 $ and $ x_2 $ are their respective distances from the origin $ O $. The centre of mass of the system is

• A. \( \frac{m_1 x_1 - m_2 x_2}{m_1 + m_2} \)
• B. \( \frac{m_1 x_2 + m_2 x_1}{m_1 + m_2} \)
• C. \( \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \)
• D. \( \frac{m_1 x_2 - m_2 x_1}{m_1 + m_2} \)

Hint:

The centre of mass is always found using a mass-weighted average of the positions of all particles in the system.

Answer 1

The Correct Option is C

To determine the center of mass of the system, we use the formula for the center of mass (\(X_{cm}\)) of a two-particle system:

X_cm = \( \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \)

Where:

• m₁ and m₂ are the masses of the particles.
• x₁ and x₂ are their respective distances from the origin.

Let's evaluate the given answer choices:

1. \(< \frac{m_1 x_1 - m_2 x_2}{m_1 + m_2} \)
2. \(< \frac{m_1 x_2 + m_2 x_1}{m_1 + m_2} \)
3. \(< \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \)
4. \(< \frac{m_1 x_2 - m_2 x_1}{m_1 + m_2} \)

The formula for the center of mass is correctly represented by option (C), \(\frac{m_1 x_1 + m_2 x_2}{m_1 + m_2}\).

Answer 2

The centre of mass \( x_{\text{cm}} \) for a system of two particles is given by the formula: \[ x_{\text{cm}} = \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \] where: 
- \( m_1 \) and \( m_2 \) are the masses of the particles, 
- \( x_1 \) and \( x_2 \) are the distances of the particles from the origin. This formula gives the position of the centre of mass of the system. It is a mass-weighted average of the positions of the two particles. 
Thus, the correct answer is: \[ \text{(C) } \frac{m_1 x_1 + m_2 x_2}{m_1 + m_2} \]