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Center of Mass

The center of mass of a system (body) is a point that moves as though all the masses were concentrated there and all external forces were applied there.
 
Position vector of center of mass for n particles system If a system consists of n particles of masses m1, m2, m3, ..., mn, as shown in Fig. 1, whose positions vectors are 43917.png respectively, then position vector of the center of mass will be
 
43911.png
Fig. 1
 
43905.png
 
Hence, the center of mass of n particles is a weighted average of the position vectors of n particles making up the system.

Important points about center of mass

  • The position of center of mass is independent of the co-ordinate system chosen.
  • The position of center of mass depends upon the shape of the body and distribution of mass.
  • In symmetrical bodies in which the distribution of mass is homogenous, the center of mass coincides with the geometrical center or center of symmetry of the body.
  • The center of mass changes its position only under the translatory motion. There is no effect of rotatory motion on the center of mass of the body.
  • If the origin is at the center of mass, then sum of the moments of masses of the system about the center of mass is zero, i.e., 43899.png.
  • If a system of particles of masses m1m2m3, … move with velocities v1v2v3, …. then the velocity of center of mass, 43892.png.
  • If a system of particles of masses m1m2m3, …. move with accelerations a1a2a3, …. then the acceleration of center of mass, 43886.png.
  • Force on a rigid body, 43880.png.
  • For an isolated system, external force on the body is zero.
     
    or 43874.png ⇒ 43868.png.
     
    i.e., center of mass of an isolated system moves with uniform velocity along a straight-line path.




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