CENTRE OF MASS

e centre of mass of a rigid body is an imaginary point

at a xed position with respect to the body as a whole.

e position of centre of mass of a rigid body depends

on two factors.

(i) e geometrical shape of the body.

(ii) e distribution of mass in the body.

Position of Centre of Mass (COM) of Two

Particles

Centre of mass of two particles of mass m 1 and m 2

separated by a distance r lies in between the two

particles. e distance of centre of mass from any of the

particle (r) is inversely proportional to the mass of the

particle (m)

i.e. r

m

∝

1

or r

r

m

m

1

2

2

1

=

or m 1 r 1 = m 2 r 2

or r m

mm

rr m

mm

1 2 r

21

2 1

12

=

+









=

+









and

Here, r 1 = distance of COM from m 1

and r 2 = distance of COM from m 2

From the above discussion, we see that

r 1 = r 2 = 1/2 r if m 1 = m 2 , i.e., COM lies midway between

the two particles of equal masses.

Centre of Mass of a System of ‘N’ Discrete

Particles

Consider a system of N point masses m 1 , m 2 , m 3 ,

................ mn whose position vectors from origin O

are given by rr 12 ,,r 3 ,............... rn respectively. en the

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

given by



 

r

mr mr mr

mm m

nn

n

cm =

+++

++ +

11 22

12

........

........

;





r

mr

m

ii

i

n

i

i

cm = n

=

=

∑

∑

1

1

r

M

mrii

i

n

cm =

=

∑

1

1

where M =















∑= 

mi

i

n

1

is the total mass of the system.

Centre of Mass of a Continuous Mass

Distribution

For continuous mass distribution the centre of mass can

be located by replacing summation sign with an integral

sign. Proper limits for the integral are chosen according

to the situation

x

xdm

dm

y

ydm

dm

z

zdm

cm cm cm dm

==∫ =

∫

∫

∫

∫

∫

,,

O X

C

m 2

mn

m 1

.....

.

®r 1 ®rcm

®r 2

®rn

Y

F CUS

2019

Class

XI

NEET/JEE

UNIT - 4 : SYSTEM OF PARTICLES AND ROTATIONAL MOTION

Focus more to get high rank in NEET/JEE (Main and Advanced) by reading this column. This specially designed column

is updated year after year by a panel of highly qualified teaching experts well-tuned to the requirements of these

Entrance Tests.