# Motion Of Centre Of Mass

## Center of Mass:

The center of mass is a unique point in space where the weighted relative position of the distributed mass sums up to zero. It is the point where a force applied causes the motion in the direction of motion without any rotation. The mass of the body is distributed around the center of mass.

Let us consider a system of particles with masses m1, m2, m3, …..and their distance from the centre of mass is r1, r2, r3,….

In terms of the distance of the centre of mass from the reference point, we can write,

$MR=\sum m_{i}r_{i}=m_{1}r_{1}+m_{2}r_{2}+m_{3}r_{3}+………………+m_{n}r_{n}$

Differentiating both the sides with respect to time, we get,

$M\frac{dR}{dt}=m\frac{dr_{1}}{dt}+m\frac{dr_{2}}{dt}+m\frac{dr_{3}}{dt}+……………..+m\frac{dr_{n}}{dt}$

Or,

$MV = mv_{1} + mv_{2} + mv_{3} +…………..+ mv_{n}$

Where V is the velocity of the centre of mass, v1 is the velocity of the first particle; v2 is the velocity of the second particle and so on.

From Newton’s second law of motion, the vector sum of all the forces acting on the first particle is F1 = ma1, that on the second particle F2 = ma2, that on the third particle F3 = ma3 and so on. The product of the total mass of the system and the acceleration is equal to the vector sum of all the forces acting on the system.

Mathematically, M×A = F1 + F2 + F3 + ……+ Fn

The forces acting on the particle include external forces, exerted by bodies outside the system and internal forces, exerted by the particles on each other. Since the internal forces occur in equal and opposite pairs, they cancel each other and the sum of forces is zero. Thus, only the external forces contribute to the equation.

MA = Fext, where Fext represents the sum of all the external forces acting on the system.

The centre of mass of a system of particles moves as if all the mass of the system is concentrated at the centre and all the external forces were applied at that point. In order to ease the calculations involved in the dynamics of an extended body, we treat any extended body as the system of particles, and the motion of centre of mass of the system is calculated by taking the mass of the whole system to be concentrated at the centre of mass and the external forces acting on system are said to be acting on the centre of mass.

Stay tuned with Byju’s to learn more about the motion of the center of mass of a system, Newton’s second law of motion and much more.

#### Practise This Question

Calculate the velocity of the centre of mass of the system of particles shown in figure.