CENTRE OF MASS

 By Eeshan Baheti

In this new physics blog, I will be going to show you about the topic of centre of mass

in physics. In this blog, I will also show some example tutorials to 

make the topic understand more further. So let's begin into the topic of centre of mass.

History of Centre of Mass

The concept of centre of mass in physics in the form of centre of gravity was first

introduced by the great Greek physicist, mathematician and engineer 

Archimedes of Syracuse. 

He worked with simplified assumptions about gravity that amount to a uniform

field which thus, made the arrival of he mathematical properties

that is known today as centre of mass.

Archimedes showed that the torque exerted on the lever by weights resting at

various points along the lever is the same as what it would be if all of the weights

were moved to a single point which is the centre of mass. 

In work of floating bodies, he demonstrated that the orientation of a 

floating object is the one that makes its centre of mass as low as possible.

He developed mathematical techniques to find out the centre of mass of the objects 

of uniform density of various well defined shapes.

The later mathematicians who developed the theory of centre of mass

include Pappus of Alexandria, Guido Ubaldi, Francesco Maurolico and others

were responsible for the giving the centre of mass 
topic of physics. Other fact is that the newton's second law was reformulated 

with respect to the centre of mass in Euler's first law. 

Definition of Centre of Mass

Every physical system has associated with it a certain point whose motion

characterises the motion of the whole system. When the system moves under some

external forces, then this point moves as if the entire mass of the system

is concentrated at this point and also the external force is applied at this point for

translational motion. This point is called the centre of mass.

System of 'n' discrete particles

Now, we will see the topic of centre of mass in a more illustrated way

by the system of 'n' discrete particles. The figure of 'n' discrete particles are given below.

 

So, consider a system of n point masses m1, m2, m3, ..., mn whose position

vectors from origin O are given by r1 vector, r2 vector, r3 vector, ..., rn vector respectively.

Then the position vector of the centre of mass C of the system is given 

by the mathematical equations given below.

In the above equations, mi ri vector is called the momentum of mass of the

particle w.r.t O. Thus, the total mass of the system is

A very important point is to be noted is that if the origin is taken

 at the centre centre of the mass then

Hence, the COM is the point about which the sum 

of the "mass moments" of the system is zero.

Position of COM of two particles

Now, we will see the illustration about the position of COM 

of two particles. Its figure is given below.

Centre of mass of two particles of masses m1 and m2 separated by the r lies

in between the two particles. The distance of centre of mass from any of the particle 

that is r is inversely proportional to the mass of the particle which is m.

Here, r1 = distance of COM from m1 and 

r2 = distance of COM from m2 .

From the above discussion, we see that r1 = r2 = 1/2 

if m1 = m2, which is that COM lies midway between the two particles of 

equal masses. Similarly, r1 > r2 if m1 < m2 and r1 < r2 if m2 < m1,

which is that COM is nearer to the particle having larger mass.

 Tutorial questions of Centre of Mass

Now, I will show you the tutorials of questions of the topic of centre of mass. 

Now, lets begin the tutorial by our first question.

Question 1- Two particles of mass 1kg and 2kg are located at x = 0 and x = 3m.

Find the position of their centre of mass.

Solution 1- In this question it is given that both the particles lie on the x axis. Thus, 

the COM will also lie on the x axis. 

So, the figure according to this question can be given as

 

So, r1 = distance of COM from the particle of mass 1kg = x.

and r2 = distance of COM from the particle of mass 3kg = (3-x) 

In this you will be able to solve this by the equation and procedure by using

Now the procedure is very simple to follow, which is that we have to just 

substitute the values of r1, r2, m1 and m2 and solve this equation as following

Thus, there we have find out that the position of their centre of mass which is x = 2m.

Thus, the COM of the two particles is located at x = 2m.

ANS = 2m

Now, let's go to the next question of centre of mass.

Question 2- The position vector of three particles of masses m1 = 1kg, m2 = 2kg and

m3 = 3kg are r1 vector = (ihat + 4 jhat + khat)m, r2 vector = (2 ihat-jhat-2 khat)m respectively.

Find the position vector of their centre of mass.

Solution 2- In this question, we have to find the value of rcom vector 

which will be our answer to this question.

In this you will able to find the rcom vector by the equation and procedure by using

Now, again from here the procedure seems to be very simple that is we have to just 

substitute the values of m1, r1 vector, m2, r2 vector, m3 and r3 vector.

Thus, we have find out that the answer is 1/2(3 ihat + jhat - khat).

ANS = 1/2 (3 ihat + jhat - khat)

Now, lets go to the next question of centre of mass.

Question 3- Four particles of mass 1kg, 2kg, 3kg and 4kg are placed at the four vertices

A, B, C and D of a square of side 1m. Find the position of centre of mass of the particles.

Solution 3- Now, this question is a bit lengthy. So, please pay attention to the preceding of answer.

In this question, it is given that D is origin, DC is x axis and DY is y axis.

So, we can find the x and y axis origins which are

So, the coordinates of their COM are

Now, again we have to just substitute the values of

m1, m2, m3, m4, x1, x2, x3 and x4.

Similarly the equation of ycom is given as

Now, again we have to just substitute the values of

m1, m2, m3, m4, y1, y2, y3 and y4.

Now, we can easily find the answer which is (xcom,ycom) which is

Thus, we found the answer which is (0.5m, 0.3m).

And also the position of COM of the four particles is shown in figure below.

Ans = (0.5m, 0.3m)

So, by this question tutorial I come to the end of this blog.

If you have any doubts regarding my blog you can either mail me directly 

or join my telegram channel and ask doubt in discuss forum.

In the next blogs of physics, I would be explaining and doing more advanced 

questions on the topic of centre of mass.

So, stay tuned to my blogs.

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