CHAPTER 8 Potental Energy and Conservaton o Energy One orm o energy can be converted nto another orm o energy. Conservatve and non-conservatve orces Physcs 1 Knetc energy: Potental energy: Energy assocated wth moton Energy assocated wth poston Potental energy U: Can be thought o as stored energy that can ether do work or be converted to knetc energy. When work gets done on an object, ts potental and/or knetc energy ncreases. There are derent types o potental energy: Gravtatonal energy Elastc potental energy (energy n an stretched sprng) Others (magnetc, electrc, chemcal, ) Physcs Dr. Abdallah M. Azzeer 1
Gravtatonal Potental Energy Potental Energy (PE) Energy assocated wth poston or conguraton o a mass. Consder a problem n whch the heght o a mass above the Earth changes rom y 1 to y : y W grav =? UP W g = - mg s = - mg (y -y 1 ) s F HAND v = const a = Down W g = +mg s mg W g = - mg ( y -y 1 ) y 1 Physcs 3 mgy U g gravtatonal potental energy (PE) U -U 1 =U W g = - mg ( y -y 1 )= U 1 -U = - U g W g = - U g Changng the conguraton o an nteractng system requres work example: ltng a book The change n potental energy s equal to the negatve o the work done U But Work/Knetc Energy Theorem says: W = K g W W U K K U Physcs 4 Dr. Abdallah M. Azzeer
K U Total Mechancal Energy The change n potental energy s equal to the negatve o the work done U W But Work/Knetc Energy Theorem says: W = K W U K K U K K1 U U1 K U K1 U1 cons tant E Total mechancal energy NOTE that the ONLY orces s gravtatonal energy whch dong the work The sum o K and U or any state o the system = the sum o K and U or any other state o the system In an solated system acted upon only by conservatve orces Mechancal Energy s conserved Physcs 5 Example 8.1 A bowler drops bowlng ball o mass 7 kg on hs toe. Choosng loor level as y=, estmate the total work done on the ball by the gravtatonal orce as the ball alls. Let s assume the top o the toe s.3 m rom the loor and the hand was.5 m above the loor. U mgy 79.8.5 34.3 J U mgy 79.8.3.6 J W g U U U 3.4 J 3 y y J Physcs 6 Dr. Abdallah M. Azzeer 3
b) Perorm the same calculaton usng the top o the bowler s head as the orgn. Assumng the bowler s heght s 1.8 m What has to change? Frst we must re compute the postons o ball at the hand and o the toe. Assumng the bowler s heght s 1.8 m, the ball s orgnal poston s 1.3 m, and the toe s at 1.77 m. U U mgy 79.8 1.3 89.J mgy 79.8 1.77 11.4J Wg U U U 3.J 3J Physcs 7 F S kx Work done by Sprng dw F dx S S W x F dx x kx dx k x x Elastc Potental Energy 1 S x S x 1 U kx S W U U U S S S S Physcs 8 Dr. Abdallah M. Azzeer 4
Conservatve Forces (a) A orce s conservatve work done by that orce actng on a partcle movng between ponts s ndependent o the path the partcle takes between the two ponts (b) The total work done by a conservatve orce s zero when the partcle moves around any closed path and returns to ts ntal poston Physcs 9 Conservatve Forces To repeat the dea on the last slde: We have seen that the work done by a conservatve orce does not depend on the path taken. W W 1 = W Thereore the work done n a closed path s. W 1 W W NET = W 1 -W = W 1 -W 1 = W 1 Physcs 1 Dr. Abdallah M. Azzeer 5
Work done by gravty L:\13 Phys LECTURES SLIDES\13Phys_Sldes_T1Y3839\CH7Flash W g = F. r = mg r cos = mg h W g = mgh (Depends only on h!) W NET = W 1 + W +...+ W n r m m m g h = F r 1 + F r +... + F r n = F (r 1 + r +...+ r n ) = F r = F h W g = mg h h m r 1 r r r 3 mg Depends only on h, not on path taken! r n Physcs 11 Non-conservatve orces: A orce s non-conservatve t causes a change n mechancal energy; mechancal energy s the sum o knetc and potental energy. Example: Frctonal orce. Ths energy cannot be converted back nto other orms o energy (rreversble). Work does depend on path. For straght lne W = - d For sem-crcle path W = - ( d /) Work vares dependng on the path. Energy s dsspated The presence o a non conservatve orce reduces the ablty o a system to do work (dsspatve orce) Physcs 1 Dr. Abdallah M. Azzeer 6
Energy dsspaton: e.g. sldng rcton As the parts scrape by each other they start small-scale vbratons, whch transer energy nto atomc moton The atoms vbratons go back and orththey have energy, but no average momentum. The ncreased atomc vbratons appear to us as a rse n the temperature o the parts. The temperature o an object s related to the thermalenergythas.frctontranserssome energy nto thermal energy Physcs 13 When there s NO work done by APPLIED FORCES, the total mechancal energy s constant or CONSEVED I W a K +U = K 1 +U 1 +W a OR K + U= W nc W nc = work done by ANY other orces than gravtatonal orce sprng orces (e.g. any appled non conservatve orce or rctonal orce) Physcs 14 Dr. Abdallah M. Azzeer 7
Three dentcal balls are thrown wth the same ntal speed rom the top o a buldng. v v cosθˆ v snθjˆ Total Energy 1 E K U mv mgh g At y = 1 1 E mv mv mgh v v g h ˆ : v v c o s θ y x jˆ : v v sn θ gt 1 y h v snθ t gt v snθ v sn θ gh t g v y v sn θ gh v v v v sn θ gh v cos θ x y v g h Physcs 15 READ Quck Quz 8.7 & 8.8 A ball connected to a massless sprng suspended vertcally. What orms o potental energy are assocated wth the ball sprng Earth system when the ball s dsplaced downward? Physcs 16 Dr. Abdallah M. Azzeer 8
READ Example 8. A ball s dropped rom a heght h above the ground. Intally, the total energy o the ball Earth system s potental energy, equal to mgh relatve to the ground. At the elevaton y, the total energy s the sum o the knetc and potental energes. Physcs 17 Example 8.3 Nose crusher? A bowlng ball o mass m s suspended rom the celng by a cord o length L. The ball s released rom rest when the cord makes an angle A wth the vertcal. (a) Fnd the speed o the ball at the lowest pont B. (b) What s the tenson T B n the cord at pont B? (c) The ball swngs back. Wll t crush the operator s nose? Physcs 18 Dr. Abdallah M. Azzeer 9
Example 8.4 (a) An actor uses some clever stagng to make hs entrance. M actor = 65 kg, M bag = 13 kg, R = 3 m What s the max. value o can have beore sandbag lts o the loor? R cos R (b) Free-body dagram or actor at the bottom o the crcular path. (c) Free-body dagram or sandbag. K U K U 1 M actorv M actor gy y R R cos θ R (1 cos θ ) Physcs 19 v gr(1cos θ ) How we can obtan v???? F T M g M y actor actor T=M g M actor actor v R v R For the sandbag not to move a= T=M bag g =6 Physcs Dr. Abdallah M. Azzeer 1