Class 11 Physics Rotational Motion Centre of mass

Centre of mass

• Imaginary point where the whole mass of system can be assumed to be concentrated
• The centre of mass of two bodies lies in a straight line.
(Here m1 & m2 are two bodies such that m1 is at a distance x1 from O, & m2 at a distance x2 from O. )

• The coordinates of centre of mass of a body is given by (X,Y,Z)

Here,

• M = S mi, the index i runs from 1 to n
• mi is the mass of the ith particle
• the position of the ith particle is given by (xi, yi, zi).

• If we increase the number of elements n , the element size Dmi decreases , and coordinates of COM is given by:

,  ,

Where x,y,z = coordinates of COM of small element dm

• The vector expression equivalent to these three scalar expressions is

Where

• R = position of COM of body
• r = position of COM of small element of mass dm

Consider a thin rod of length l, taking the origin to be at the geometric centre of the rod and x-axis to be along the length of the rod, we can say that on account of reflection symmetry, for every element dm of the rod at x, there is an element of the same mass dm located at –x.

The net contribution of every such pair to the integral and hence the integral x dm itself is zero. Thus the COM coincides with the geometric centre.

• The same symmetry argument will apply to homogeneous rings, discs, spheres, or even thick rods of circular or rectangular cross section; their centre of mass coincides with their geometric centre.

Example - Find COM of semicircular ring.

Solution: Let us consider a semicircular ring of mass M and radius R.

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