# Center of Mass

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## Definition of Center of Mass

Every single physical system has associated itself with a certain point whose motion characterizes the motion of the whole system. When the system moves under an external force, then this point moves as if the entire mass of the system were concentrated at it, and also the external force was applied at it. This point is therefore called the ‘center of mass’ of the system. The motion of a system can be described in terms of the motion of its center of mass.

While studying the dynamics of the movement of the system of particles as a whole, do not be bothered about the dynamics of an individual particle of the system. Rather focus only on the dynamics of a unique point corresponding to that system. The movement of this unique point is analogous to the movement of a single particle whose mass is equivalent to the sum of all independent particles of the system and the outcome of all the forces exerted on all the particles of the system by surrounding bodies (or) action of a field of force is exerted directly to that particle. This point is known as the center of mass of the system of particles. The idea of the center of mass (COM) is functional in analyzing the complicated movement of the system of objects, especially when two and more objects collide or an object explodes into fragments.

### Center Of Mass Formula

Center of mass formula for point objects: Zcom=$\frac{\sum_{i=0}^{n}m_{i}z_{i}}{M}$

Center of mass formula for any geometric shape: Xcom= $\frac{l}{2}$

### Center of Mass of a Two-Particle System

Let us consider a system of two particles of masses m1 and m2 located at points A and B respectively.

Let r1 and r2 be the position vectors of the particles relative to a fixed origin O. Then, the position vector rcm of the center of mass C of the system is defined by

(m1+m2) rcm =m1 r1+m2 r2.

The product of the total mass of the system and the position vector of the center of mass is equal to the sum of the products of the masses of the two particles and their respective position vectors.

### Center of Mass Examples

Example 1: In the HCL molecule, the separation between the nuclei of the two atoms is 1.27 A. Find the location of the center of mass of the molecule. A chlorine atom is 35.5 times heavier than a hydrogen atom.

Solution: The center of mass of the HCL molecule will be on the line joining H and CL atoms. Let the HCL molecule be along the X-axis, the H atom being at the origin (x=0). The center of mass relative to the H atom is given by

xcm= $\frac{m_{1}x_{1}+m_{2}x_{2}}{m_{1}+m_{2}}$

Where m1 and m2are the respective masses of H and CL atoms, and x1and x2their distances relative to the H atom.

Here, m2=35.5 m1, x1=0 and x2=1.27 A

xcm= $\frac{(m_{1}\times 0 ) + (35.5 m_{1} \times 1.27 \text A)}{m_{1} + 35.5 \text m_{1}}$

= $\frac{35.5\times1.27\text A}{36.5}$ = 1.235 A

Example 2: Two bodies of masses 0.5 kg and 1 kg are lying in the X-Y plane at points (-1, 2) and (3,4) respectively. Locate the center of mass of the system.

Solution : Here, we have

m1=0.5 kg, m2=1 kg, x1= -1, y1=2, x2=3, y2=4.

By definition, the coordinates of the center of mass of the two bodies are given by:

xcm= $\frac{m_{1}x_{1}+m_{2}x_{2}}{m_{1}+m_{2}}$ =$\frac{(0.5)(-1)+(1)(4)}{(0.5)+(1)}$=  $\frac{5}{3}$

And,  ycm= $\frac{m_{1}y_{1}+m_{2}y_{2}}{m_{1}+m_{2}}$ = $\frac{(0.5)(2)+(1)(4)}{(0.5)+(1)}$ = $\frac{10}{3}$

The coordinates of C>M> are (5/3, 10/3).

Follow and practice more center of mass examples to get a better understanding.