A body has its centre of maas at the origin. The x-coordinates of the particles

Options

A. may be all positive
B. may be all negative
C. may be all non negative
D. may be positive for some case and negative in other cases.

Answer

To solve the problem, we need to analyze the conditions under which the center of mass (COM) of a system of particles can be at the origin (x = 0).

### Step-by-step Solution:

1. **Understanding the Center of Mass Formula**:
The x-coordinate of the center of mass (COM) for a system of particles is given by the formula:
\[ x_{COM} = \frac{m_1 x_1 + m_2 x_2 + m_3 x_3 + \ldots}{m_1 + m_2 + m_3 + \ldots} \]
where \( m_i \) is the mass of the \( i^{th} \) particle and \( x_i \) is its x-coordinate.

2. **Setting the Condition for COM at the Origin**:
Since the center of mass is at the origin, we have:
\[ x_{COM} = 0 \]
This implies:
\[ m_1 x_1 + m_2 x_2 + m_3 x_3 + \ldots = 0 \]

3. **Analyzing the Implications**:
- If all \( x_i \) (x-coordinates of the particles) are positive, then \( m_1 x_1 + m_2 x_2 + m_3 x_3 + \ldots \) would be positive, which contradicts the condition that it equals zero. Therefore, not all x-coordinates can be positive.
- If all \( x_i \) are negative, then the sum would also be negative, which again contradicts the condition that it equals zero. Thus, not all x-coordinates can be negative either.
- If some \( x_i \) are positive and some are negative, it is possible for their weighted sum to equal zero, satisfying the condition for the center of mass at the origin.

4. **Conclusion**:
- It is possible for some particles to have positive x-coordinates and others to have negative x-coordinates, resulting in a net sum of zero.
- It is also possible for some particles to have zero x-coordinates, which would also contribute to the center of mass being at the origin.

### Final Answer:
The x-coordinates of the particles can be:
- Some positive and some negative (Option D).
- Some can be zero (Option C).