Question

A large number of particles are placed around the origin, each at a distance R from the origin. Find the distance of the centre of mass from the origin.
A) Equal to R.
B) Less than or equal to R.
C) Greater than R.
D) Greater than or equal to R.

Answer 1

The centre of mass of the system of particles depends on the individual masses of the particles and the distance of each particle from the origin. Since all the particles are at an equal distance from the origin the position of the centre of mass will shift depending on the masses of the particles. In a system of two particles, the centre of mass will be nearer to the more massive particle.

Complete step by step answer.
Step 1: Sketch a rough figure of the arrangement of the particles around the origin.
Since the particles are all placed around the origin O at an equal distance of R from O, the locus of all the particles must form a circle. The figure shown below depicts this circle.

Based on the concept of centre of mass, we can predict the distance at which the centre of mass of the system of particles will lie.
The centre of mass of a system of particles depends on the masses of the particles and the distance of the particles from the origin. Here the distance of the particles from the origin remains the same as R. The masses of the particles can be different from each other or can be equal.
The centre of mass must therefore lie on the circle or inside it.
So, the distance at which the centre of mass lies from the origin is less than or equal to R.

Hence the correct option is B.

Note: The distance between each particle is not mentioned so we can assume that this distance is not constant. This distance, however, does not affect the centre of mass of the system. The position of the centre of mass will never lie outside the circle described in the sketched figure.