Chapter 7. Potential Energy and Conservation of Energy


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1 Chapter 7 Potental Energy and Conservaton o Energy 1 Forms o Energy There are many orms o energy, but they can all be put nto two categores Knetc Knetc energy s energy o moton Potental Potental energy s energy o poston 2 1
2 Knetc Energy s energy o moton Electrcal Energy s the movement o electrcal charges. Thermal Energy, or heat, s the nternal energy n substances the vbraton and movement o the atoms and molecules wthn substances. Radant Energy s electromagnetc energy that travels n transverse waves. Moton Energy s the movement o objects and substances rom one place to another. Sound s the movement o energy through substances n longtudnal (compresson/rareacton) waves 3 Potental energy s energy o poston Gravtatonal Energy s the energy o poston or place. Chemcal Energy s energy stored n the bonds o atoms and molecules. Nuclear Energy s energy stored n the nucleus o an atom the energy that holds the nucleus together. Stored Mechancal Energy s energy stored n objects by the applcaton o a orce. 4 2
3 Part 1 Potental Energy 5 Potental Energy s energy o poston Potental energy U s energy that can be assocated wth the conguraton o a system o objects that exert orces on one another. I the conguraton o the system changes, then the potental energy o the system can also change Types o potental energy studed n Unversty Physcs courses Gravtatonal Potental Energy Elastc Potental Energy 6 3
4 4 7 Work and Potental Energy Change n potental energy W U U U = = Δ 8 Determnng Potental Energy Values 1D case 3D case Attenton: the equaton can NOT be used or rctonal orces (see later conservatve and nonconservatve orces ) = = Δ x x dx x F U U U ) ( = = Δ r r r d r F U U U r r ) (
5 Gravtatonal Potental Energy Gravtatonal potental energy s assocated wth the state o separaton between objects wth masses ΔU = y F( x) dx = ( mg) dy = mg( y y) y ΔU = mgδy Only changes n potental energy are physcally meanngul. However to smply calculatons we may select a reerence pont y where U =0, then U ( y) = mgy 9 Note: U = mgy The gravtatonal potental energy assocated wth a partcleearth system depends ONLY on the vertcal poston y (or heght) o the partcle relatve to the reerence poston (y=0), not on the horzontal poston 10 5
6 Elastc Potental Energy Gravtatonal potental energy s assocated wth the state o compresson or extenson o an elastc (sprnglke) objects. A good approxmaton or many sprngs s Hook s law then F = kx ΔU = F( x) dx = x x 1 2 ( kx) dy = k( x 2 x Choosng the reerence pont when the sprng s relaxed 2 ) 1 U ( x) = kx Conservatve orces or more mathematcs General denton o conservatve orce: Force s conservatve work that t does around a closed curve s zero. Equvalent to statement: or conservatve orce, work s ndependent o the path that connects ntal and nal ponts There s a potental energy assocated wth a conservatve orce Snce, rctonal orce does negatve work n both drectons, there s no potental energy assocated wth the 12 rctonal orce 6
7 Path Independence 13 Introducng potental 14 7
8 Potental and orce 15 Onedmensonal case 16 8
9 Zero o Potental Energy 17 Constant gravtatonal orce 18 9
10 Elastc sprng 19 Part 2 Conservaton o Mechancal Energy 20 10
11 Conservaton o Mechancal Energy The mechancal energy o a system s the sum o ts potental energy U and the knetc energy K o the objects wthn t E = K + U mech In an solated system where only conservatve orces cause energy change, the knetc energy and potental energy can change, but ther sum, the mechancal energy o the system cannot change K + U = K + U 21 lttle math 22 11
12 on practcal sde When the mechancal energy o a system s conserved (the system s solated and orces are conservatve), we can relate the sum o knetc energes and potental energes at one nstant to that at another nstant wthout consderng the ntermedate moton 23 Fndng the orce rom potental (n 1D) Potental U(x) Force F(x) du( x) F( x) = dx 24 12
13 Energy dagrams 25 Turnng ponts 26 13
14 Equlbrum ponts 27 Bungee jump example Student jumps o a brdge 52 meters above a rver wth a bungee cord ted around hs ankle. He alls 15 meters beore the bungee cord begns to stretch. Student s mass s 75 kg and the cord (sprng) constant s k=50 N/m. I we neglect ar resstance, estmate how ar below the brdge the student would all beore comng to stop
15 soluton K + U = K + U example K K U kd + U 2 = = mgd mg( L + d) 1 2 kd 2 mgl = 0 or k=50 N/m, L=15 m, m=75 kg d=40 m and L+d = 55 m the brdge s 52 meters oops 29 Part 3 Conservaton o Energy 30 15
16 Conservaton o Energy The total energy o a system can change only by amounts o energy that are transerred to or rom that system ΔE = ΔK + ΔU + ΔE thermal + ΔE nternal The total energy o an solated system cannot change ΔK + ΔU + ΔE thermal + ΔEnternal = 0 31 Conservaton o Energy (more) The law o conservaton o energy says that energy s nether created nor destroyed. When we use energy, t doesn t dsappear. We change t rom one orm o energy nto another
17 Energy Ecency Energy ecency s the amount o useul energy you get rom a system. A perect, energyecent machne would change all the energy put n t nto useul. Convertng one orm o energy nto another orm always nvolves a loss o usable energy. Example: human body s a very necent machne. Fuel s ood. (gves the energy to move, breathe, thnk). Human body s less than ve percent ecent most o the tme. The rest o the energy s lost as heat. 33 Sources o energy Energy sources are classed nto two groups renewable and nonrenewable. Energy can be converted nto secondary energy sources lke electrcty and hydrogen
18 Part 4 Frcton nvolved 35 Smultaneous Presence o Conservatve and NonConservatve Forces 36 18
19 Frcton nvolved For physcs 231 we may consder rcton as a transer to thermal energy. E = thermal k ( x) x x dx ΔK + ΔU + ΔEthermal = 0 or a constant rctonal orce E thermal = k ( x x ) K + U = K + U + k ( x x ) 37 Block on nclne example 38 19
20 Soluton example 39 20
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