# 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**If a system consists of

*n*particles system*n*particles of masses

*m*

_{1},

*m*

_{2},

*m*

_{3}, ...,

*m*,

_{n}*as shown in Fig. 1, whose positions vectors are 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., .
- If a system of particles of masses
*m*_{1},*m*_{2},*m*_{3}, â€¦ move with velocities*v*_{1},*v*_{2},*v*_{3}, â€¦. then the velocity of center of mass, . - If a system of particles of masses
*m*_{1},*m*_{2},*m*_{3}, â€¦. move with accelerations*a*_{1},*a*_{2},*a*_{3}, â€¦. then the acceleration of center of mass, . - Force on a rigid body, .
- For an isolated system, external force on the body is zero.