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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
Fig. 1
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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