All the particles of body are situated at a distance R from the origin. The distance of the centre of mass of the body from the origin is also R. Is this statement true or false?

Explanation

To determine whether the statement "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 also R" is true or false, we can analyze the situation step by step.

### Step 1: Understanding the Position of Particles
Given that all particles of the body are situated at a distance R from the origin, we can represent this mathematically. Each particle can be represented by its position vector, which has a magnitude of R.

### Step 2: Considering the Center of Mass
The center of mass (COM) of a system of particles is given by the formula:
\[ \vec{R}_{cm} = \frac{1}{M} \sum m_i \vec{r}_i \]
where \(M\) is the total mass of the system, \(m_i\) is the mass of each particle, and \(\vec{r}_i\) is the position vector of each particle.

### Step 3: Analyzing Different Configurations
1. **Case 1**: If all particles are located at the same point on the circumference of a circle of radius R (e.g., at angles 0°, 90°, 180°, and 270°), the center of mass will also be at the origin since the contributions from all directions will cancel out. Thus, the center of mass is at the origin, which is a distance of 0 from the origin, not R.

2. **Case 2**: If all particles are located at the same point (e.g., all at (R, 0)), then the center of mass will also be at that point, which is R from the origin.

### Step 4: Conclusion
Since we can have configurations where the center of mass is at the origin (distance 0) even though all particles are at a distance R from the origin, the statement is **false**.

### Final Answer
The statement is **false**.

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