Centre of Mass of a Distributed System

 

A body may be considered to be made up of an indefinitely large number of particles, each of which is attracted towards the centre of the earth by the force of gravity. These force constitute a system of like parallel forces. The resultant of these parallel forces known as the weight of the body always acts through a point, which a fixed relative to the body, whatever be the position of the body. This fixed point is called the centre of gravity of the body.

The centre of gravity of a body is the point at which the resultant of the weights of all the particles of the body acts, whatever may be the orientation or position of the body provided that its size and shape remain unaltered.

Note

1. The position of centre of mass is independent of the co-ordinate system chosen.
2. The position of centre of mass depends upon the shape of the body and distribution of mass.
3. Position of centre of mass for different bodies

S. No.	Body	Position of centre of mass
(a)	Uniform hollow sphere	Centre of sphere
(b)	Uniform solid sphere	Centre of sphere
(c)	Uniform circular ring	Centre of ring
(d)	Uniform circular disc	Centre of disc
(e)	Uniform rod	Centre of rod
(f)	A plane lamina (Square, Rectangle, Parallelogram)	Point of inter section of diagonals
(g)	Triangular plane lamina	Point of inter section of medians
(h)	Rectangular or cubical block	Points of inter section of diagonals
(i)	Hollow cylinder	Middle point of the axis of cylinder
(j)	Solid cylinder	Middle point of the axis of cylinder
(k)	Cone or pyramid	On the axis of the cone at point distance (3h/4) from the vertex where h is the height of cone

Note:

If the origin is at the centre of mass, then the sum of the moments of the masses of the system about

the centre of mass is zero i.e.

If a system of particles of masses m₁, m₂, m₃, ……  move with velocities v₁, v₂, v₃,…….. then the velocity of centre of mass v_cm = (Σm_iv_i)/ Σm_i.

(viii) If a system of particles of masses m₁, m₂, m₃, ……. move with accelerations a₁, a₂, a₃,…….

then the acceleration of centre of mass A_cm = (Σm_ia_i)/ Σm_i.

(ix) If  is a position vector of centre of mass of a system

then velocity of centre of mass

(x) Acceleration of centre of mass 

(xi) Force on a rigid body 

(xii) For an isolated system external force on the body is zero

==>

i.e., centre of mass of an isolated system moves with uniform velocity along a straight-line path.