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System of Particles and Collision
(1) Introduction
- Untill now we have focused on describing motion of a single particle in one, two or three dimensions. By particle we mean to say that it has a size negligible in comparison to the path ravelled by it.
- When we studied Law's of motion we have applied them even to the bodies having finite size imagining that motion of such bodies can be described in terms of motion of particles.
- While doing so we have ignored the the internal structure of such bodies. Any real body we encounter in our daily life has a finite size and idealised model of particle is inadequat when we deal with motion of real bodies of finite size .
- Real bodies of finite size can also be regarded as the system of particles. While studying system of particles we will not concentrate on each and every particle of the system instead we will consider the motion of system as a whole.
- Large number of problens involving extended bodies or real bodies of finite size can be solve dby considering them as Rigid Bodies. We define rigid body as a body having definite and unchanging shape.
- A rigid body is a rigid assembly of particles with fixed inter-particle distances. In actual bodies deformations do occur but we neglect them for the sake of simplicity.
- In this chapter we will study about centre of mass of system of particles, motion of centre of mass and about collisions.
(2) Centre of mass
- Consider a body consisting of large number of particles whose mass is equal to the total mass of all the particles. When such a body undergoes a translational motion the displacement is produced in each and every particle of the body with respect to their original position.
- If this body is executing motion under the effect of some external forces acting on it then it has been found that there is a point in the system , where if whole mass of the system is supposed to be concentrated and the nature the motion executed by the system remains unaltered when force acting on the system is directly applied to this point. Such a point of the system is called centre of mass of the system.
- Hence for any system Centre of mass is the point where whole mass of the system can be supposed to be concentrated and motion of the system can be defined in terms of the centre of mass.
- Consider a stationary frame of refrance where a body of mass M is situated. This body is made up of n number of particles. Let mi be the mass and ri be the pisition vector of i'th particle of the body.
- Let C be any point in the body whose position vector with respect to origin O of the frame of refrance is Rc and position vector of point C w.r.t. i'th particle is rci as shown below in the figure.
- From triangle OCP
ri=Rc+rci (1)
multiplying both sides of equation 1 bt mi we get
miri=miRc+mirci
taking summation of above equation for n particles we get
If for a body
then point C is known as the centre of mass of the body.
- Hence a point in a body w.r.t. which the sum of the product of mass of the particle and their position vector is equal to zero is equal to zero is known as centre of mass of the body.
(3) Position of centre of mass
(i) Two particle system
- Consider a system made up of two particles whose mass are m1 and m2 and their respective position vectors w.r.t. origin O be r1 and r2 and Rcm be the position vector of centre of mass of the system as shown below in the figure. So from equation 2
- If M=m1+m2=total mass of the system , then
(ii) Many particle system
- Consider a many particle system made up of number of particles as shown below in the figure. Let m1 , m2 , m3 , . . . . . . . . . . . . . , mn be the masses of the particles of system and their respective position vectors w.r.t. origin are r1 , r2 , r3 , . . . . . . . . . . . . . . . . . , rn.
Also position vector of centre of mass of the system w.r.t. origin of the refance frame be Rcm then from equation 2
- Because of the definition of centre of mass
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