Centre of Mass

Every particle is attracted towards the centre of the earth by the force of gravity and the centre of gravity of a body is the point where the resultant force of attraction of the weight of the body acts.

Centre of mass: If a system of parallel forces proportional to the masses of the various particles of a body are assumed to act on it, their resultant passes through a fixed point, irrespective of the direction of the parallel forces and that point is called the centre of mass.

In a uniform gravitational field, the centre of mass of the body and the centre of gravity coincide with each other. For small bodies (compared to the size of the earth) the centre of gravity and the centre of mass coincide.

If the two masses are equal, the centre of mass lies midway between the particles.

The position of center of mass depends on the shape of the body and distribution of mass. In symmetrical bodies in which distribution of mass is homogeneous, the centre of mass coincides with geometrical center.

Centre of mass of regular bodies:

  • Uniform rod: middle point of the rod
  • Cubical box: Point of intersection of diagonals
  • Circular ring: centre of the ring
  • circular disc: centre of the disc
  • Sphere: centre of the sphere

The velocity of centre of mass of an isolated system remains constant as long as no external force acts on the system. Its acceleration is zero.

The momentum of centre of mass of the system is equal to the sum of individual momentum of n particles of the system.

If an external force is acting on the system of particles, centre of mass behaves as if total force is acting only at that point.

Characteristics of Centre of Mass

  1. The position of centre of mass depends upon the shape of the body and the distribution of mass.
  2. If the origin is at the centre of mass, then sum of moments of mass (Σmixi) of the system about the centre of mass is zero.
  3. Centre of mass may be within the body or on or outside the material of the body.
  4. It is not necessary that mass should be present at centre of mass. In the case of uniform ring, centre of mass is outside the material where no mass is present.
  5. The location of the centre of mass is independent of the reference frame used to locate it.
  6. The centre of mass of a system of particles depends only on the masses of the particles and their relative positions.

Laws of Motion of Centre of Mass

  1. The centre of mass of a system of particles moves as though total mass of the system is concentrated at a point and external forces were applied at that point.
  2. The motion of the centre of mass of the body is called the translational motion of the body.
  3. The internal forces will not effect the motion of the centre of mass.
  4. If no external force acts on a system, the acceleration of centre of mass is zero, the velocity and momentum of the centre of mass remains constant, though velocity and momentum of individual particles vary.
  5. The motion of the centre of mass can be studied using Newton's laws of motion.