CHAPTER 8 Potential Energy and Conservation of Energy

Transcription

1 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

2 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 w done U = W But Work/Knetc Energy Theorem says: W = K g W = U = K K + U = Physcs 4 Dr. Abdallah M. Azzeer

3 K + U = Total Mechancal Energy The change n potental energy s equal to the negatve o the work w done U = W But Work/Knetc Energy Theorem says: W = K W = U = K K + U = K K1 + U U1 = K + U = K1 + U1 = constant = 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 d 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 = = 34.3 J U = mgy = =.6 J W g = ( U U ) U = =3.4 J 3 M J Physcs 6 Dr. Abdallah M. Azzeer 3

4 b) Perorm the same calculaton usng the top o the bowler s head as the orgn. 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 = ( ) = 89.J mgy = ( ) = 11.4J W g = U = ( U U) = 3.J 3J Physcs 7 r F S r = kx Work done by Sprng r r dws = FS dx W x F r dx r x kx dx 1 = = = k x x Elastc Potental Energy ( ) ( ) S x S x 1 U = kx S ( ) W = U = U U S S S S Physcs 8 Dr. Abdallah M. Azzeer 4

5 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

6 Work done by gravty W g = F. r = mg r cos θ= mg h W g = mgh (Depends only on h!) r m θ mg W NET = W 1 + W W n m h = F r 1 + F r F r n = F ( r + r r ) n = F r = F h W g = mg h h r 1 r r r 3 m 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 = - (π 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

7 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 thermal energy t has. Frcton transers some 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

8 Three dentcal balls are thrown wth the same ntal speed rom the top o a buldng. v = v cosθˆ + v snθjˆ 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 Total Energy 1 E = K + U = mv + mgh g At y = 1 1 E = mv = mv + mgh v = v + g h 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

9 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 θ 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

10 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 θ can have beore sandbag lts o the loor? R cos θ 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(1 cos 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 θ =6 Physcs Dr. Abdallah M. Azzeer 1