Centre of mass of a system of particles in a plane

Centre of Mass of a System of Particles in a Plane

Basic Concepts

• The centre of mass is the average location of the mass of a system of particles.
• It can be thought of as the balance point of all the mass in the system.
• For a system of particles in a plane, each particle has a position vector relative to a certain origin.

Position Vector

• The position vector of a particle is represented by an arrow from the origin to the particle’s location.
• A particle’s position vector is denoted r. The i-th particle has position vector r_i.

Mass of Particles

• Each particle in the system has a mass, denoted as m. The i-th particle has mass m_i.
• The total mass, M, is the sum of the masses of all the particles in the system: M = Σm_i.

Centre of Mass Position Vector

• The position vector of the centre of mass, R, is the mass-weighted average of all the particle’s position vectors.
• It can be calculated using the formula: R = (1/M) Σm_i * r_i, where the sum is over all particles in the system.
• This formula shows that more mass at a certain location pulls the centre of mass towards that location.

Conceptual Significance

• The centre of mass of a system of particles does not necessarily coincide with the physical centre of the system.
• For physical objects, the centre of mass is the point about which the object will naturally balance.
• If a system is free to move, it will move as if it were a single particle located at the centre of mass, subject to the total external forces acting on the system.