Question

What do you mean by center of mass? Calculate the position coordinates of the center of mass of two particles in a system.

Answer 1

Center of Mass
Introduction of Center of Mass

• Untill now we have focused on describing motion of a single particle in one, two or three dimensions. By particle we mean to say that it has a size negligible in comparison to the path travelled by it.
• When we study laws of motion, we apply them’ even to the bodies having finite size imagining that motion of such bodies can be described in terms of motion of particles.
• While doing so we have ignored the internal structure of such bodies. Any real body we encounter in our daily life has a finite size and idealized model of particle is inadequate when we deal with motion of real bodies of finite size.
• Real bodies of finite size can also be regarded as the system of particles. While studying system of particles, we will not concentrate on each and every particle of the system, instead we will consider the motion of system as a whole.
• Large number of problems involving extended bodies or real bodies of finite size can be solved by considering them as rigid bodies. We define rigid body as a body having definite and unchanging shapes.
• A rigid body is rigid assembly of particles with fixed inter particle distances. In actual bodies deformation do occur but we neglect them for the sake of simplicity.

Center of Mass of Two Particle System

(i) Position of center of mass for two particle system : Consider a system of two point masses (or particles) M₁ and M₂, whose position vectors at time t with reference to the origin O of the inertial frame are \(\vec{r}_{1}\) and \(\vec{r}_{2}\) respectively as shown in the figure.
The total force \(\vec{F}_{1}\) that is acting on point mass M₁ consists of two parts :
(i) A force (\(\vec{F}_{1}\))_ext which appears because of some agency acting on the system.
(ii) A force \(\vec{F}_{12}\) which appears because of point mass M₂. This force is internal force of the two. particles system

Fig.: Position of center of mass for a two particle system

So, total force is
\(\vec{F}_{1}=\vec{F}_{1} \cdot_{e x t}+\vec{F}_{12}\) ……………… (1)
Similarly the force on the other particle (for point mass M₂)

We know that all external forces act at the center of mass. If total mass of the system is
M = M₁ + M₂, then from equation (7)

The point whose position is defined by \(\vec{r}_{\mathrm{cm}}\) is called the center of mass of a two particles system. So equation (11) gives the center of mass formula for a particle system at any time ‘t’. It is a point at which the total external force is supposed to be acting.