All the particles of a body are situated at a distance R from the origin. The distance of the centre of mass of the body from the origin is

Options

A. `=R`
B. `leR`
C. `gtR`
D. `geR`

Answer

To find the distance of the center of mass of a body where all particles are situated at a distance \( R \) from the origin, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the System**:
- All particles of the body are at a distance \( R \) from the origin. This means that each particle is located on a sphere of radius \( R \).

2. **Assuming Particle Distribution**:
- For simplicity, let's assume that the particles are evenly distributed over the surface of a sphere of radius \( R \). This means that half of the particles are on the x-axis and half are on the y-axis.

3. **Calculating the Center of Mass**:
- The center of mass (COM) is calculated using the formula:
\[ \text{COM} = \frac{\sum m_i \vec{r_i}}{\sum m_i} \]
- If we assume uniform mass distribution and equal mass for each particle, the center of mass will be influenced by the symmetry of the arrangement.

4. **Finding the Coordinates**:
- If we consider two particles at positions \( (R, 0) \) and \( (0, R) \), the coordinates of the center of mass in the x and y directions can be calculated as:
\[ x_{cm} = \frac{R + 0}{2} = \frac{R}{2} \]
\[ y_{cm} = \frac{0 + R}{2} = \frac{R}{2} \]

5. **Distance from the Origin**:
- The distance \( h \) of the center of mass from the origin can be calculated using the Pythagorean theorem:
\[ h = \sqrt{x_{cm}^2 + y_{cm}^2} = \sqrt{\left(\frac{R}{2}\right)^2 + \left(\frac{R}{2}\right)^2} \]
- Simplifying this gives:
\[ h = \sqrt{\frac{R^2}{4} + \frac{R^2}{4}} = \sqrt{\frac{2R^2}{4}} = \sqrt{\frac{R^2}{2}} = \frac{R}{\sqrt{2}} \]

6. **Conclusion**:
- Therefore, the distance of the center of mass of the body from the origin is:
\[ h = \frac{R}{\sqrt{2}} \]

### Final Answer:
The distance of the center of mass of the body from the origin is \( \frac{R}{\sqrt{2}} \).