The sum of moments of masses of all the particle in a system about the centre of mass is always

Options

A. zero
B. maximum
C. infinite
D. minimum

Answer

To solve the question regarding the sum of moments of masses of all the particles in a system about the center of mass, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Moment of Mass**:
The moment of mass for a particle is defined as the product of its mass and its distance from a reference point (in this case, the center of mass). Mathematically, it can be expressed as:
\[ \text{Moment of mass} = m_i \cdot x_i \]
where \( m_i \) is the mass of the particle and \( x_i \) is its distance from the center of mass.

2. **Defining the Center of Mass**:
The center of mass (CM) of a system of particles is the point where the total mass of the system can be considered to be concentrated. The position of the center of mass is given by:
\[ x_{CM} = \frac{\sum_{i=1}^{n} m_i x_i}{\sum_{i=1}^{n} m_i} \]
where \( n \) is the number of particles.

3. **Setting Up the Equation**:
For a system of \( n \) particles, we can express the sum of moments about the center of mass:
\[ \sum_{i=1}^{n} m_i (x_{CM} - x_i) = 0 \]
This equation states that the weighted distances of the masses from the center of mass sum to zero.

4. **Simplifying the Equation**:
Rearranging the equation gives:
\[ \sum_{i=1}^{n} m_i x_{CM} - \sum_{i=1}^{n} m_i x_i = 0 \]
Since \( x_{CM} \) is a constant for the system, we can factor it out:
\[ x_{CM} \sum_{i=1}^{n} m_i - \sum_{i=1}^{n} m_i x_i = 0 \]
This implies that:
\[ \sum_{i=1}^{n} m_i x_i = \sum_{i=1}^{n} m_i x_{CM} \]
Hence, the moments about the center of mass sum to zero.

5. **Conclusion**:
Therefore, the sum of the moments of masses of all the particles in a system about the center of mass is always zero.

### Final Answer:
The sum of moments of masses of all the particles in a system about the center of mass is always **0**.