NEET  >  Physics Class 11  >  Centre of Mass

Centre of Mass Notes | Study Physics Class 11 - NEET

Document Description: Centre of Mass for NEET 2022 is part of Physics Class 11 preparation. The notes and questions for Centre of Mass have been prepared according to the NEET exam syllabus. Information about Centre of Mass covers topics like Centre of Mass and Centre of Mass Example, for NEET 2022 Exam. Find important definitions, questions, notes, meanings, examples, exercises and tests below for Centre of Mass.

Introduction of Centre of Mass in English is available as part of our Physics Class 11 for NEET & Centre of Mass in Hindi for Physics Class 11 course. Download more important topics related with notes, lectures and mock test series for NEET Exam by signing up for free. NEET: Centre of Mass Notes | Study Physics Class 11 - NEET
Table of contents
Centre of Mass
1 Crore+ students have signed up on EduRev. Have you?

Centre of Mass Notes | Study Physics Class 11 - NEET

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 Notes | Study Physics Class 11 - NEET........... respectively. Then the position vector of the centre of mass C of the system is given by:

Centre of Mass Notes | Study Physics Class 11 - NEET

Centre of Mass Notes | Study Physics Class 11 - NEET ; Centre of Mass Notes | Study Physics Class 11 - NEETCentre of Mass Notes | Study Physics Class 11 - NEET
where, Centre of Mass Notes | Study Physics Class 11 - NEET is called the moment of mass of particle with respect to origin.
Centre of Mass Notes | Study Physics Class 11 - NEET is the total mass of the system.
Further, Centre of Mass Notes | Study Physics Class 11 - NEET and Centre of Mass Notes | Study Physics Class 11 - NEET
So, the cartesian co-ordinates of the COM will be
xcom = Centre of Mass Notes | Study Physics Class 11 - NEETCentre of Mass Notes | Study Physics Class 11 - NEET
or xCOM =Centre of Mass Notes | Study Physics Class 11 - NEET
Similarly, yCOM =Centre of Mass Notes | Study Physics Class 11 - NEET and Centre of Mass Notes | Study Physics Class 11 - NEET

Note:

  • If the origin is taken at the centre of mass then Centre of Mass Notes | Study Physics Class 11 - NEET = 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 Notes | Study Physics Class 11 - NEET changes. So Centre of Mass Notes | Study Physics Class 11 - NEET 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 Notes | Study Physics Class 11 - NEET

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 Notes | Study Physics Class 11 - NEET

So xCOM = Centre of Mass Notes | Study Physics Class 11 - NEET
r1 = Centre of Mass Notes | Study Physics Class 11 - NEET = Centre of Mass Notes | Study Physics Class 11 - NEET ...(1)
r2 = Centre of Mass Notes | Study Physics Class 11 - NEET = Centre of Mass Notes | Study Physics Class 11 - NEET ...(2)
From the above discussion, we see that
r1 = r2 = Centre of Mass Notes | Study Physics Class 11 - NEET 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.

Question for Centre of Mass
Try yourself:Can the centre of mass of the rigid body be located outside of the body?
View Solution

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. 

Centre of Mass Notes | Study Physics Class 11 - NEET

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 Notes | Study Physics Class 11 - NEET

or Centre of Mass Notes | Study Physics Class 11 - NEET

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 Notes | Study Physics Class 11 - NEET

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 Notes | Study Physics Class 11 - NEET = Centre of Mass Notes | Study Physics Class 11 - NEET = Centre of Mass Notes | Study Physics Class 11 - NEET = Centre of Mass Notes | Study Physics Class 11 - NEET

ycom = Centre of Mass Notes | Study Physics Class 11 - NEET = Centre of Mass Notes | Study Physics Class 11 - NEET = 1 m    Centre of Mass Notes | Study Physics Class 11 - NEET

So position of C.O.M is Centre of Mass Notes | Study Physics Class 11 - NEET

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. 

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 Notes | Study Physics Class 11 - NEET

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 Notes | Study Physics Class 11 - NEET

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. 

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 Notes | Study Physics Class 11 - NEET

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 Notes | Study Physics Class 11 - NEET

⇒ 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. 

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 Notes | Study Physics Class 11 - NEET

co-ordinate of C = (1, 0)

Centre of Mass Notes | Study Physics Class 11 - NEET

Centre of Mass Notes | Study Physics Class 11 - NEET
Centre of Mass Notes | Study Physics Class 11 - NEET
Position of centre of mass Centre of Mass Notes | Study Physics Class 11 - NEET
distance of C.O.M from point Centre of Mass Notes | Study Physics Class 11 - NEET


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 Notes | Study Physics Class 11 - NEET ...(i)
Centre of Mass Notes | Study Physics Class 11 - NEET = 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 Notes | Study Physics Class 11 - NEET = Centre of Mass Notes | Study Physics Class 11 - NEETCentre of Mass Notes | Study Physics Class 11 - NEET

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 Notes | Study Physics Class 11 - NEET
  • 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 Notes | Study Physics Class 11 - NEET
    xCOM = Centre of Mass Notes | Study Physics Class 11 - NEET = Centre of Mass Notes | Study Physics Class 11 - NEET = Centre of Mass Notes | Study Physics Class 11 - NEET
  • The y-coordinate of COM is
    yCOM = Centre of Mass Notes | Study Physics Class 11 - NEET = 0
  • Similarly, zCOM = 0
    i.e., the coordinates of COM of the rod are Centre of Mass Notes | Study Physics Class 11 - NEET, i.e, it lies at the centre of the rod.


Question for Centre of Mass
Try yourself:Does the explosion of a projectile in mid-air effects the motion of the centre of mass?
View Solution

The document Centre of Mass Notes | Study Physics Class 11 - NEET is a part of the NEET Course Physics Class 11.
All you need of NEET at this link: NEET
126 videos|449 docs|210 tests
Download as PDF

How to Prepare for NEET

Read our guide to prepare for NEET which is created by Toppers & the best Teachers

Download free EduRev App

Track your progress, build streaks, highlight & save important lessons and more!

Related Searches

Centre of Mass Notes | Study Physics Class 11 - NEET

,

Semester Notes

,

ppt

,

mock tests for examination

,

Exam

,

practice quizzes

,

MCQs

,

Viva Questions

,

Centre of Mass Notes | Study Physics Class 11 - NEET

,

Sample Paper

,

past year papers

,

Free

,

video lectures

,

Extra Questions

,

Previous Year Questions with Solutions

,

Centre of Mass Notes | Study Physics Class 11 - NEET

,

study material

,

Important questions

,

shortcuts and tricks

,

Summary

,

Objective type Questions

,

pdf

;