General Physics (PHY 2130)

Transcription

1 General Physcs (PHY 30) Lecture 8 Moentu Collsons Elastc and nelastc collsons

2 Lghtnng Reew Last lecture:. Moentu: oentu and pulse oentu conseraton Reew Proble: A cannon on a ralroad car s facng n a drecton parallel to the tracks. It fres a 98-kg shell at a speed of 05 /s (relate to the ground) at an angle of 60.0 aboe the horzontal. If the cannon plus car hae a ass of kg, what s the recol speed of the car f t was at rest before the cannon was fred?

3 A cannon on a ralroad car s facng n a drecton parallel to the tracks. It fres a 98-kg shell at a speed of 05 /s (relate to the ground) at an angle of 60.0 aboe the horzontal. If the cannon plus car hae a ass of kg, what s the recol speed of the car f t was at rest before the cannon was fred? [Hnt: A coponent of a syste s oentu along an axs s consered f the net external force actng on the syste has no coponent along that axs.] shell shell 98kg 05 / s, 60 F Ext ( horzontal) If we are only concerned wth the horzontal drecton, C Cf we hae 0 C Cf S C S Sf car 0 p p + Cf Sf kg aboe horzontal, S f Sf 0 (Horzontal) car (98kg)(05 cos kg 0? / s) 0.0 / s Car can oe only n horzontal drecton. So, the recol speed of the car s 0.0 /s 3

4 Conseraton of Moentu The prncple of conseraton of oentu states when no external forces act on a syste consstng of two objects that collde wth each other, the total oentu of the syste before the collson s equal to the total oentu of the syste after the collson

5 Conseraton of Moentu Matheatcally: + f + f Moentu s consered for the syste of objects The syste ncludes all the objects nteractng wth each other Assues only nternal forces are actng durng the collson Can be generalzed to any nuber of objects

6 Types of Collsons Moentu s consered n any collson what about knetc energy? Inelastc collsons Knetc energy s not consered KE KE f + lost Soe of the knetc energy s conerted nto other types of energy such as heat, sound, work to peranently defor an object Perfectly nelastc collsons occur when the objects stck together Not all of the KE s necessarly lost energy

7 Types of Collsons Elastc collsons Knetc energy before knetc energy after Inelastc collsons Knetc energy before knetc energy after Perfectly nelastc collsons the bodes rean stuck together after a collson Moentu s consered for all types collsons, f F ext 0

8 Exaple: Perfectly Inelastc Collsons When two objects stck together after the collson, they hae undergone a perfectly nelastc collson Suppose, for exaple, 0, 50 /s. Conseraton of oentu becoes ) + ( ( + ) f f E.g., f (000kg)(50 f 000 kg, kg kg s) + 0 (500kg) s 0 s. 500 kg f :,

9 Exaple: Perfectly Inelastc Collsons What aount of KE lost durng collson? KE KE before after + (000 kg)(50 ( + ) f (500 kg)(0 s) s) J J Δ 6 KE lost J lost n heat/ glung /sound/

10 Elastc Collsons Both oentu and knetc energy are consered Typcally hae two unknowns + + f + f f + f Sole the equatons sultaneously

11 Exaple: Elastc Collsons Suppose, for exaple, -0/s, 50 /s, 000kg, 500kg Assue elastc collson P before KE before + (000 kg)(50 s) + (500 kg)( kg s J J 5 J s) For perfectly elastc collson: kg 6 J s f + f + f f f f s s

12 Reeber: If the collson s nelastc, KE s not consered If the collson s elastc, KE s consered

13 Proble Solng for One -Densonal Collsons Set up a coordnate axs and defne the eloctes wth respect to ths axs It s conenent to ake your axs concde wth one of the ntal eloctes In your sketch, draw all the elocty ectors wth labels ncludng all the gen nforaton

14 Sketches for Collson Probles Draw before and after sketches Label each object nclude the drecton of elocty keep track of subscrpts

15 Sketches for Perfectly Inelastc Collsons The objects stck together Include all the elocty drectons The after collson cobnes the asses

16 Proble Solng for One-Densonal Collsons Wrte the expressons for the oentu of each object before and after the collson Reeber to nclude the approprate sgns Wrte an expresson for the total oentu before and after the collson Reeber the oentu of the syste s what s consered If the collson s nelastc, sole the oentu equaton for the unknown Reeber, KE s not consered If the collson s elastc, you can use the KE equaton to sole for two unknowns

17 Exaple: A projectle of.0 kg ass approaches a statonary body of 5.0 kg ass at 0.0 /s and, after colldng, rebounds n the reerse drecton along the sae lne wth a speed of 5.0 /s. What s the speed of the 5.0 kg ass after the collson? Gen:.0 kg 5.0 kg 0.0 /s 0 /s f -5.0 /s Idea: the collson s elastc (nothng stcks ). Use conseraton of oentu n one denson: p p f p + p p f + p f 7 Fnd: f? + 0 f + f f ( ) f.0 kg 5.0 kg 3.0/ s ( 0 /s ( 5.0 /s) )

18 Two-densonal Collsons For a general collson of two objects n threedensonal space, the conseraton of oentu prncple + f + f ples that the total oentu of the syste n each drecton s consered x + x fx + fx and y + y Use subscrpts for dentfyng the object, ntal and fnal, and coponents fy + fy

19 Two-densonal Collsons y f φ x φ F ext 0, p p f f Just lke for forces: work wth x and y coponents x coponents x + x fx + fx y coponents y + y fy + fy 9

20 Let s watch a oe!

21 Exaple: What would happen after the collson? Statonary It s also possble for two bodes to undergo scatterng

22 Exaple: Assue: and 5 /s What would happen after the collson? Statonary It s also possble for two bodes to undergo scatterng For ths proble: assue that θ φ 60

23 Exaple: Gen: asses: elocty: 5 /s 0 /s angles: θ φ 60 Fnd: f? f? Use oentu conseraton n each drecton (x and y): p p yf xf f sn 60 f sn 60 f sn 60 f f f ( 0.5) + ( 0.5) sn 60 5 s, as p f cos60 + f cos60 p 5 s f f f f y x 0 ( 5 s)

24 ConcepTest A boy stands at one end of a floatng raft that s statonary relate to the shore. He then walks to the opposte end, towards the shore. Does the raft oe (assue no frcton)?. No, t wll not oe at all. Yes, t wll oe away fro the shore 3. Yes, t wll oe towards the shore

25 ConcepTest A boy stands at one end of a floatng raft that s statonary relate to the shore. He then walks to the opposte end, towards the shore. Does the raft oe (assue no frcton)?. No, t wll not oe at all. Yes, t wll oe away fro the shore 3. Yes, t wll oe towards the shore Note: Snce oentu s consered n the boy-raft syste and nether was ong at frst, the raft ust oe n the drecton opposte to the boy s.

26 Exaple: A 00 kg car traelng ntally wth a speed of 5.0 /s n an easterly drecton crashes nto the rear end of a 9000 kg truck ong n the sae drecton at 0.0 /s. The elocty of the car rght after the collson s 8.0 /s to the east. (a) What s the elocty of the truck rght after the collson? (b) How uch echancal energy s lost n the collson? Account for ths loss n energy. 5.0 /s 0.0 /s 8.0 /s 6 Tf (a) F ext Tf 0, p p ( C C + T T T ) C f Cf C / s The truck s ong at 0.9 /s to the east C + T T (00kg 5 / s) + (9000kg 0 / s) (00kg 8) 9000kg C Cf + T Tf (b) Change n Mechancal energy ΔK K f K T Tf + Mechancal energy lost 4955 J CCf T T + CC 4955 J What happened to ths energy???

27 7 Center of Mass The center of ass (CM) s the pont representng the ean (aerage) poston of the atter n a body. Ths pont need not be located wthn the body. The center of ass (of a two body syste) s found fro: x x c + + x Ths s a weghted aerage of the postons of the partcles that copose a body. (A larger ass s ore portant.)

28 8 In 3-densons wrte r rc where total The coponents of r c are: x c x y c y z c z

29 9 Exaple: Fnd the x-coordnate of the center of ass of the coposte object shown n the fgure. The sphere, cylnder, and rectangular sold all hae a unfor coposton. Ther asses and densons are: sphere: 00 g, daeter 0 c; cylnder: 450 g, length 7 c, radus 5.0 c; rectangular sold: 35 g, length n x-drecton 6 c, heght 0 c, depth c. The x-coordnate of each three-densonal shape s dway along ts horzontal denson. To fnd the x-coponent of the center of ass of the coposte object. x CM s x s + c x c + r x r s + c + r (00 g)(5.0 c) + (450 g)(0 c +7 c) + (35 g)(0 c +7 c +6 c) 00 g g + 35 g c (notce: NOT at the center, whch s at.5 c!)

30 Moton of the Center of Mass Velocty of center of ass: CM p Moentu of an extended body: CM For an extended body: F ext CM 30