Centre of mass |Basics of centre of mass | what is center of mass?

Centre of mass is very interesting topic in classical physics.it is easy to understand (actually it depends upon the tutor and curiosity of student).here we will be discussing basics of centre of mass.

so lets get started.

INTRODUCTION TO CENTRE OF MASS:

Here is a quick example. If you toss a ball into the air without much spin on the ball , its motion is simple—it follows a parabolic path. and the ball can be treated as a particle.

If, instead, you flip a baseball bat into the air, its motion is more complicated. Because every part of the bat moves differently, along paths of many different shapes, you cannot represent the bat as a particle.

 Instead, it is a system of particles each of which follows its own path through the air. However, the bat has one special point—the centre of mass—that does move in a simple parabolic path. 

The Other parts of the bat move around the center of mass. (To locate the center of mass, balance the bat on an outstretched finger; the point is above your finger, on the bat’s central axis.) 

You cannot make a career of flipping baseball bats into the air, but you can make a career of advising long-jumpers or dancers on how to leap properly into the air while either moving their arms and legs or rotating their torso. 

Your starting point would be to determine the person’s center of mass because of its simple motion.

WHAT IS CENTER OF MASS?

when we consider the motion of a system of particles , there is one point in it which behaves as though the entire mass of the system (i.e. the sum of the masses of all the individual particles) is concentrated there and its motion is the same as would ensue if the resultant of all the forces acting on all the particles were applied directly to it. this point is called the centre of mass (COM) of the system. 

the concept of com is very useful in solving many problems, in particular , those concentrated with collision of particles.

POSITION OF CENTRE OF MASS

First of all we find the position of COM of a system of particles. Just to make the subject easy we

classify a system of particles in three groups:

1. System of two particles
2. System of a large number of particles and
3. Continuous bodies. 

Now, let us take them separately.

Position of COM of two particles

Centre of mass of two particles of mass m1 and m2 separated by a distance of d lies in between the two particles. The distance of centre of mass from any of the particle (r) is inversely proportional to the mass of the particle (m).

Here, r₁ = distance of COM from m₁

and r₂ = distance of COM from  m₂

From the above discussion, we see that 

i.e. COM lies midway between the two particles of equal masses.

Position of COM of a Large Number of Particles

if we have a system consisting of n particles, of mass m_{1 ,}m_{2, .....,}m_n with r_{1 ,}r_{2 ,.....,}r_n as their position vectors at a given instant of time. The position vector r_COM of the COM of the system at that instant is given by:

Position of COM of Continuous Bodies

If we consider the body to have continuous distribution of matter the summation in the formula of COM is replaced by integration. Suppose x, y and z are the co-ordinates of a small element of mass dm, we write the co-ordinates of COM as

that's it for today, if you want to read more about the center of mass, like how to derive center of mass of different things. So let me know  in the comments, I will write a post on it. Thanks for reading hve a nice day / night.