LINEAR MOMENTUM Physical quantities that we have been using to characterize the motion of a particle

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Frank Johnston
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1 LINEAR MOMENTUM Physical quantities that we have been using to chaacteize the otion of a paticle v Mass Velocity v Kinetic enegy v F Mechanical enegy + U Linea oentu of a paticle (1) is a vecto!

2 Siple exaple about collisions Wall 0.1 Kg Wall 0.1 Kg B E F O R E A F T E R What is the change in LINEAR MOMENTUM?

3 Execise B E F O R E A F T E R

4 What is the LINEAR MOMENTUM of this acoscopic object? Answe: Add up the LINEAR MOMENTUM of evey individual icoscopic constituents of the acoscopic object

5 LINEAR MOMENTUM of a syste coposed of N paticles Next task: To descibe the otion of a syste of paticles

6 d dt (3) CASE 1: No extenal foces acting on the syste (only intenal ones) Notice: 0 (4)

7 No fiction when no net extenal foces ae acting on the acoscopic object (5) Exaple 10 k/h 6 k/h 5 k/h 11 k/h M kg kg

8 Question M Kg Kg

9 CASE 2: Extenal foces (as well as intenal foces, of couse) act on the syste

10 Afte the kick

11 The Cente of Mass

12 Exaple

13 Exaple 0 c î + 0 c

14 Exaple Exaple 0.5 c î c Exaple

15 These types of pobles can be solved by syety aguents and soe judicious ticks. To that effect, let s show fist a geneal esult that states that the cente of ass of a coplicated geoety object can obtained by a) fist finding the cente of ass of sub-egions pats, and b) then finding the final cente of ass based on the knowledge of the cente of ass of those sub-egions. Conside an abitay distibution of N paticles, as shown in the figue below. The paticles have been nubeed, accodingly. In geneal, the position of the cente of ass (with espect to soe efeence) is given by,... R N N CM N N N Notice, we can beak down the nueato into an abitay nube of subsets goups of paticles (see also the othe figue below): R CM We also egoup in the denoinato 5 5 N

16 CM R A B Goup-A has a ass equal to ( ), let s call it M A Goup-B has a ass equal to ( ), let s call it M B etc Notice the nueato can be e-witten as follows B B B ) ( A A A M M M M M M CM R B B B ) ( A A A M M M M M M CM R CM of goup A CM of goup B That is, when tying to find the CM of an object whose ass is distibuted in a soewhat coplicated geoetical distibution, we can sub-divide the object into salle siple objects. Each division can be epesented by its coesponding CM.

17 Exaple

20 Motion of the Cente of Mass So it is like all the ass of the syste wee concentated at. This syste of paticles can be eplaced by 2 3 M 1

21 We want to pove the following: Fo otion analysis, the acoscopic ball can be eplaced by a point object located at the cente of ass.

22 1

23 Expession 1 also iplies d

27 Exaple: The ballistic pendulu The objective is to find the speed v of the bullet A block of wood is hanging fo two long cods. A bullet is fied into the block, coing quickly to est. The block+bullet syste then swing upwads aising a vetical distance h6.3 c v M g h M 5.4 kg 9.5 g h 6.3 c Pocedue-1 Initial kinetic enegy: (1/2) v 2 Initial potential enegy: 0 Final kinetic enegy: 0 Final potential enegy: (+M)gh The consevation of the echanical enegy iplies (1/2) v 2 (+M)gh Fo which we can solve fo v. What is wong with the pocedue above? Answe: The consevation of echanical-enegy is not valid in this case. Pat of the bullet s initial kinetic enegy is dissipated as heat

28 Is the wok / kinetic-enegy theoe K W valid in this case? Answe: Yes Pocedue-2 Step-1 Consevation of linea oentu We assue that the collision is vey bief, such that a) Duing the collision the foces on the block (gavitation and tension fo the cod) ae balanced. That is, no net extenal foce is acting on the bullet-block syste. Hence, the syste can be consideed isolated and, theefoe, its total linea oentu is conseved b) The collision is in one diension (just afte the collision the syste oves hoizontally.) Befoe the collision P i v Just afte the collision, the syste oves with velocity V (unknown yet) P f ( + M)V The consevation of the linea oentu iplies v ( + M)V V [ / ( + M) ] v (1) Step-2 Consevation of the echanical enegy Afte the collision thee ae not dissipative foces. Initial kinetic enegy: (1/2) ( +M) V 2 (1/2) ( +M) [ 2 / ( + M) 2 ] v 2 (1/2) [ 2 / ( + M) ] v 2 Initial potential enegy: 0

29 Final kinetic enegy: 0 Final potential enegy: (+M)gh The consevation of the echanical enegy iplies (1/2) [ 2 / ( + M) ] v 2 (+M)gh v 2 2[ (+M) 2 / 2 ]gh Substituting values v 630 /s

1121 T Question 1

1121 T Question 1 1121 T1 2008 Question 1 ( aks) You ae cycling, on a long staight path, at a constant speed of 6.0.s 1. Anothe cyclist passes you, tavelling on the sae path in the sae diection as you, at a constant speed