Centre of Mass Class 11 Notes | EduRev

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Class 11 : Centre of Mass Class 11 Notes | EduRev

The document Centre of Mass Class 11 Notes | EduRev is a part of the Class 11 Course Physics For JEE.
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Centre of Mass

  • Every physical system has associated with it a certain point whose motion characterizes 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 of the system.

Centre of Mass of a System of `N' Discrete Particles

Consider a system of N point masses m1, m2, m3, .................. mn whose position vectors from origin O are given by Centre of Mass Class 11 Notes | EduRev........... respectively. Then the position vector of the centre of mass C of the system is given by:

Centre of Mass Class 11 Notes | EduRev

Centre of Mass Class 11 Notes | EduRev ; Centre of Mass Class 11 Notes | EduRevCentre of Mass Class 11 Notes | EduRev

where, Centre of Mass Class 11 Notes | EduRev is called the moment of mass of particle with respect to origin.

Centre of Mass Class 11 Notes | EduRev is the total mass of the system.

Further, Centre of Mass Class 11 Notes | EduRev and Centre of Mass Class 11 Notes | EduRev

So, the cartesian co-ordinates of the COM will be

xcom = Centre of Mass Class 11 Notes | EduRevCentre of Mass Class 11 Notes | EduRev

or xCOM =Centre of Mass Class 11 Notes | EduRev

Similarly, yCOM =Centre of Mass Class 11 Notes | EduRev and Centre of Mass Class 11 Notes | EduRev

Note:

  • If the origin is taken at the centre of mass then Centre of Mass Class 11 Notes | EduRev = 0. hence, the COM is the point about which the sum of "mass moments" of the system is zero.
  • If we change the origin then Centre of Mass Class 11 Notes | EduRev changes. So Centre of Mass Class 11 Notes | EduRev also changes but exact location of center of mass does not change.

Position of COM of two Particles

Consider two particles of masses m1 and m2 separated by a distance l as shown in figure.

Centre of Mass Class 11 Notes | EduRev

Let us assume that m1 is placed at origin and m2 is placed at position (l, 0) and the distance of centre of mass from m1 & m2 is r1 & r2 respectively.

Centre of Mass Class 11 Notes | EduRev

So xCOM = Centre of Mass Class 11 Notes | EduRev

r1 = Centre of Mass Class 11 Notes | EduRev = Centre of Mass Class 11 Notes | EduRev ...(1)

r2 = Centre of Mass Class 11 Notes | EduRev = Centre of Mass Class 11 Notes | EduRev ...(2)

From the above discussion, we see that

r1 = r2 = Centre of Mass Class 11 Notes | EduRev if m1 = m2, i.e., COM lies midway between the two particles of equal masses.

Similarly, r1 > r2 if m1 < m2 and r1 < r2 if m2 < m1 i.e., COM is nearer to the particle having larger mass.

From equation (1) & (2),

m1r1 = m2r2 ...(3)

Centre of mass of two particle system lie on the line joining the centre of mass of two particle system.


Q.1. Two particle of mass 1 kg and 2 kg are located at x = 0 and x = 3 m. Find the position of their centre of mass. 

Ans.

Centre of Mass Class 11 Notes | EduRev

Since, both the particles lie on x-axis, the COM will also lie on x-axis. Let the COM is located at x = x, then

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

and r2 = distance of COM from the particle of mass 2 kg = (3 -x)

Using Centre of Mass Class 11 Notes | EduRev

or Centre of Mass Class 11 Notes | EduRev

or x = 2 m

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


Q.2. Two particle of mass 4 kg & 2 kg are located as shown in figure then find out the position of centre of mass. 

Centre of Mass Class 11 Notes | EduRev

Ans. First find out the position of 2 kg mass

x2kg = 5 cos 37° = 4 m

y2kg = 5 sin 37° = 3 m

So these system is like two particle system of mass 4 kg and 2kg are located (0, 0) and      (4, 3) respectively, then

xcom = Centre of Mass Class 11 Notes | EduRev = Centre of Mass Class 11 Notes | EduRev = Centre of Mass Class 11 Notes | EduRev = Centre of Mass Class 11 Notes | EduRev

ycom = Centre of Mass Class 11 Notes | EduRev = Centre of Mass Class 11 Notes | EduRev = 1 m    Centre of Mass Class 11 Notes | EduRev

So position of C.O.M is Centre of Mass Class 11 Notes | EduRev


Q.3. Two particles of mass 2 kg and 4 kg lie on the same line. If 4 kg is displaced rightwards by 5 m then by what distance 2 kg should be move for which centre of mass will remain at the same position. 

Ans. Let us assume that C.O.M. lie at point C and the distance of C from 2 kg and 4 kg particles are r1 & r2 respectively. Then from relation

Centre of Mass Class 11 Notes | EduRev

m1r1 = m2r2

2r1 = 4r2                   ...(i)

Now 4 kg is displaced rightwards by 5 m then assume 2 kg is displaced leftwards by x distance to keep the C.O.M. at rest.

from relation m1r1 = m2r2'
⇒  m1(r1 + x) = m2 (r2 + y)
2(r1 + x) = 4(r2 + 5)    ...(ii)
2x = 20
x = 10 m

Centre of Mass Class 11 Notes | EduRev

To keep the C.O.M at rest 2 kg displaced 10 m left wards

Alter: If centre of mass is at rest then we can write

m1x = m2y

2 × x = 4 × 5

x = 10 m.


Q.4. Two particles of mass 1 kg and 2 kg lie on the same line. If 2 kg is displaced 10 m rightwards then by what distance 1 kg should displaced so that centre of mass will displaced 2m right wards. 

Ans. Initially let us assume that C.O.M is at point C which is r1 & r2 distance apart from mass m1 & m2 respectively as shown in figure.

Centre of Mass Class 11 Notes | EduRev

from relation m1 r1 = m2 r2

⇒ (1) r1 = 2r2

Now 2kg is displaced 10 m rightwards then we assume that 1 kg is displaced x m leftward to move the C.O.M 2m rightwards.

So from relation m1r1' = m2r2'

Centre of Mass Class 11 Notes | EduRev

⇒ 1 (x + r1 + 2) = 2 (10 + r2 - 2)
⇒ x + r1 + 2 = 20 + 2r2 - 4    ...(ii)

from eq. (i) & (ii) x = 14 m (leftwards).


Q.5. Three particles of mass 1 kg, 2 kg, and 3 kg are placed at the corners A, B and C respectively of an equilateral triangle ABC of edge 1 m. Find the distance of their centre of mass from A. 

Ans. Assume that 1 kg mass is placed at origin as shown in figure.

co-ordinate of A = (0, 0)

co-ordinate of B = (1 cos60°,1 sin60°) =Centre of Mass Class 11 Notes | EduRev

co-ordinate of C = (1, 0)

Centre of Mass Class 11 Notes | EduRev

Centre of Mass Class 11 Notes | EduRev
Centre of Mass Class 11 Notes | EduRev
Position of centre of mass Centre of Mass Class 11 Notes | EduRev
distance of C.O.M from point Centre of Mass Class 11 Notes | EduRev


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

Centre of Mass Class 11 Notes | EduRev ...(i)

Centre of Mass Class 11 Notes | EduRev = M (mass of the body)

here x,y,z in the numerator of the eq. (i) is the coordinate of the centre of mass of the dm mass.

Centre of Mass Class 11 Notes | EduRev = Centre of Mass Class 11 Notes | EduRevCentre of Mass Class 11 Notes | EduRev

Note : 

  • If an object has symmetric mass distribution about x axis then y coordinate of COM is zero and vice-versa.

Centre of Mass of a Uniform Rod

  • Suppose a rod of mass M and length L is lying along the x-axis with its one end at x = 0 and the other at x = L. Mass per unit length of the rod l = Centre of Mass Class 11 Notes | EduRev
  • Hence, dm, (the mass of the element dx situated at x = x is) = l dx
  • The coordinates of the element dx are (x, 0, 0). Therefore, x-coordinate of COM of the rod will be
    Centre of Mass Class 11 Notes | EduRev
    xCOM = Centre of Mass Class 11 Notes | EduRev = Centre of Mass Class 11 Notes | EduRev = Centre of Mass Class 11 Notes | EduRev
  • The y-coordinate of COM is
    yCOM = Centre of Mass Class 11 Notes | EduRev = 0
  • Similarly, zCOM = 0
    i.e., the coordinates of COM of the rod are Centre of Mass Class 11 Notes | EduRev, i.e, it lies at the centre of the rod.
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