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# Linear Momentum Notes PHYS 111

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This 8 page Class Notes was uploaded by Wilson on Tuesday October 27, 2015. The Class Notes belongs to PHYS 111 at Indiana University of Pennsylvania taught by Dr. Haija in Summer 2015. Since its upload, it has received 32 views. For similar materials see Physics I Lecture in Physics 2 at Indiana University of Pennsylvania.

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Date Created: 10/27/15

LINEAR MOMENTUM 81 INTRODUCTION The definition of linear momentum for a single particle was introduced in unit 4 In this chapter the linear momentum for a system of particles will be studied and its relevance to events like collision between two or more particles will be discussed and explained When a collision between two or more objects occurs forces of contact act on the colliding objects It is this force of contact on each object that is called the force of collision In general the duration of the forces of collision is rather small and it is during this short period of time changes in the objects velocities in magnitude direction or both occurs The force of collision on an object is so large compared to external forces acting during collision so that external forces acting on objects are neglected 82 LINEAR MOMENTUM OF A SYSTEM OF PARTICLES The sum of the linear momenta of a collection of particles in a system along the X and y axes are PX PM P2 P3 P P1y P2yP3y y Since from Newton s second law the rate of change of momentum of a particle or a system of particles is equal to the external forces acing on it then A P A Py Fext x X Fext A t y A t If the net external force on the system is zero the total momentum of the system is constant Thus PX constant Py constant 82 V 83 COLLISIONS AND CHANGE IN LINEAR MOMENTUM Consider a collision between two objects During the collision each acts on the other by a force FC that will be the only force considered acting on each of the objects An example of such an event is the collision between a tennis ball and a racket or between a baseball and a bat or simply a collision between two balls two blocks or two figure skaters accidentally bumping into each other The force of collision rises from zero at the initial instant of contact to a maXimum value which then declines to zero at the instant of detachment at t t0 Fig 81 Or consider a ball of mass m that has been hit by a bat The exact dependence of the force of collision on time is not trivial to determine E It is practical to consider an average value of the collision force that acts on the ball from the very initial instant to the very nal instant of collision such that the area under the E average force versus time graph would be equal to the area under the exact force acting from O to t tc From Newton39s law the force acting on the ball is mAv mv39 v At At Ml ma 83 where the prime refers to the velocity of the object after collision while the unprimed quantity is its velocity prior to collision The product of mass m times the velocity v de nes a new quantity called the linear momentum of the object The change in the momentum AP of such an object is APP P mv39 v Fig 82 84 Note that P and v are vectors Thus in collision problems the proper sign must be af xed to them Back to Eq 83 which from Newton39s second law becomes AP Fc At 85 That is E AP At 86 A schematic of a collision between two balls of masses m1 and m2 in a head on collision is demonstrated in Fig 82 below b m2 a a 1 Since external forces in a collision are ignored the system as a whole is under no external force Therefore the force with which m1 acts on m2 during collision call it Fcz is simultaneously associated with an equal but opposite force of reaction Fcl that m2 exerts on m1 That is AP2 At then Accordingly 87 That is 88 Thus 89 AP1 At AP1 AP2 0 AP 0 P constant I Thus the momenta just prior to and right after collision by P and may be written as PP 810 Eq 89 re ects an important principle called conservation of linear momentum which applies for all colliding objects For a system consisting of two colliding objects Eq 810 can be written as I I m1V1m2V2 m1V1m2V2 811 84 IMPULSE F C The change in momentum Ap of an object ie the product At de nes another quantity known as the impulse For any object AP Fe At 812 In the above discussion the impulse imparted to masses m1 and m2 are Fcl I1 At 813 814 As can be seen from Eq 87 l1 l2 0 implying that I1 l2 815 As the impulse of an object is equal to the change in its momentum which is a vector the impulse is then a vector From the above it can be concluded that the total impulse of the system is zero That is 816 8 5 COLLISIONS IN TWO DIMENSIONS The conclusion arrived at in Eqs 925 and 926 can be generalized to two dimensional coisions Fig 83 That is the linear momentum for a system of two objects is conserved Since Eq 811 is a vector equation it can be resolved into its components Taking the plane of collision to be the xy plane I I PX PX Py Py and That is I I m1V1 m2v2x m1V1 m2V2x 817 mlv1y mzv2y mlv1y mzv2y 818 v1 I 86 TYPES OF COLLISION The kinetic energy of a system of colliding objects before and after a collision is employed as a criterion to categorize the type of collision they encounter In this regard there are two types of collision I ELASTIC COLLISION In addition to the conservation of linear momentum that applies to all collisions the conservation of kinetic energy of the colliding objects just before and just after collision also holds for elastic collisions Therefore for an elastic collision KE1 KE2 K13 KE 2 In terms of the masses and velocities of the colliding objects the above relation becomes 1 2 1 2 1 12 1 12 mv mv mv mv 2 11 2 2 2 2 11 2 2 2 823 It can be shown that for a one dimensional collision of two objects the above condition combined with the conservation of linear momentum leads to the following result II INELASTIC COLLISION In this kind of collision the kinetic energy of a system just before and just after collision is A KE 7t 0 system not conserved Thus for a system of two colliding objects KB KE 2 KE1KE2 0 325 This difference in the kinetic energy of the system appears as heat or deformation of the colliding objects

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