Physcs or Scentsts and Engneers Chapter 0 Energy Sprng, 008 Ho Jung Pak
Introducton to Energy Energy s one o the ost portant concepts n scence although t s not easly dened Eery physcal process that occurs n the Unerse noles energy and energy transers The energy approach to descrbng oton s partcularly useul when the orce s not constant An approach wll nole Conseraton o Energy Ths could be extended to bologcal organss, technologcal systes and engneerng stuatons 3-Mar-08 Pak p.
Energy o Fallng Object Kneatc eq. Dene : knetc energy g Then, K U : Conseraton o Energy U K g a g y ( y Graty s the only orce actng Multplyng & puttng a y g, g( y y gy gy K U gy 3-Mar-08 Pak p. 3 y : gratatonal potental energy
Knetc Energy Knetc energy s the energy assocated wth the oton o an object Knetc energy s always gen by K ½, where s the speed o the object Knetc energy s a scalar, ndependent o drecton o Knetc energy cannot be negate Unts o energy: kg /s Joule Energy has densons o [M L T ] 3-Mar-08 Pak p. 4
Potental Energy Potental energy s the energy assocated wth the relate poston o objects that exert orces on each other A potental energy can only be assocated wth specc types o orces,.e. conserate orces There are any ors o potental energy: gratatonal, electroagnetc, checal, nuclear Gratatonal potental energy s assocated wth the dstance aboe Earth s surace: U g gy 3-Mar-08 Pak p. 5
Total Mechancal Energy The su o knetc and potental energes s the echancal energy o the syste: E K U When conserate orces act n an solated syste, knetc energy ganed (or lost by the syste as ts ebers change ther relate postons s balanced by an equal loss (or gan n potental energy Ths s the Conseraton o Mechancal Energy I there s rcton or ar drag, the total echancal energy s not consered,.e. constant 3-Mar-08 Pak p. 6
Gratatonal Potental Energy The syste conssts o Earth and a book Intally, the book s at rest (K 0 at y y b (U g gy b When the book alls and reaches y a, K a gy a 0 gy b K a g (y b - y a >0 The knetc energy that was conerted ro potental energy depends only on the relate dsplaceent y b - y a It doesn t atter where we take y 0 3-Mar-08 Pak p. 7
Exaple : Launchng a Pebble Bob uses a slngshot to shoot a 0.0 g pebble straght up wth a speed o 5.0 /s. How hgh does a pebble go? Beore: K b Ater : K 0, U a Usng energy conseraton, 0 096. y a 6.5 J, Knetc energy was conerted nto gratatonal potental energy by 6.5 J. ga 6. 5 0, gy 0.96 or K U y a a gb U 0 y ga a 39. K b U gb 3-Mar-08 Pak p. 8
along a Ths s Energy Conseraton n -D We wll now analyze - D oton ds Fs as, dt ds ds g sn θ ds dt Multplyng both sdes by ds, But snθds rctonless nclne. dy. ds ds g sn θds Integratng ro ntal to nal gy gy the sae equaton as n a - D oton. s postons, 3-Mar-08 Pak p. 9 s d s
Exaple : Downhll Sleddng Chrstne runs orward wth her sled at.0 /s. She hops onto the sled at the top o a 5.0 hgh, ery slppery slope. What s her speed at the botto? Usng the conseraton o 0 0 gy 0 g( y 0 y energy, gy 0 /s Notce that ( the ass cancelled out and ( only Δy s needed. We could hae taken the botto o the hll as y 0.0 and the top as y 5.0, and gotten the sae answer. 3-Mar-08 Pak p. 0
Energy Conseraton: Pendulu As the pendulu swngs, there s a contnuous exchange between potental and knetc energes At A, all the energy s potental At B, all o the potental energy at A s transored nto knetc energy Let zero potental energy be at B At C, the knetc energy has been transored back nto potental energy 3-Mar-08 Pak p.
Exaple 3: Ballstc Pendulu A 0 g bullet s red nto a 00 g wood block hangng ro a 50 c long strng. The bullet ebeds tsel nto the block, and the block then swngs out to an angle o 40. What was the speed o the bullet? 3-Mar-08 Pak p.
Exaple 3, cont Ths proble has two parts: (a the bullet and the block ake a perectly nelastc collson, n whch oentu s consered, and (b the total ass as a syste undergoes a pendulu oton, n whch energy s consered. (a Moentu conseraton : w Snce w0 0, b0 b (b Energy conseraton : Snce Thereore, 0 and we can choose b0 w b b.0 kg 0.00 kg ( b tot gl w y b tot 0, gy ( cosθ gy gl ( cosθ ( 9.80 /s (.50 ( 0.766 30 /s 3-Mar-08 Pak p. 3 w w0 tot b b0 tot gy
Exaple 4: Roller Coaster A roller-coaster car s released ro rest at the top o the rst rse and then oes reely wth neglgble rcton. The roller coaster has a crcular loop o radus R n a ertcal plane. (a Suppose rst that the car barely akes t around the loop: at the top o the loop the rders are upsde down and eel weghtless,.e. n 0. Fnd the heght o the release pont aboe the botto o the loop, n ters o R so that n 0. 3-Mar-08 Pak p. 4
(a At the top o K gh Exaple 4, cont are n ree all. g gr R Energy s consered between the release and the top o U g 0 gh the loop, the car and rders K the loop : U gr gr Fro Newton's g g(r h.5r nd law, 3-Mar-08 Pak p. 5
Exaple 4, cont (b Show that the noral orce on the car at the botto o the loop exceeds the noral orce at the top o the loop by sx tes the weght o the car. (b Energy conseraton :(botto Newton's nd law :(botto Substtutng b and t, n n b b gh b gh b g R g ( h / R n g( 5 h / R The noral orce on each rder ollows the sae rule. Such a large noral orce s ery uncoortable or the rders. Roller coasters are thereore bult helcal so that soe o the gratatonal energy goes nto axal oton. 3-Mar-08 Pak p. 6 b n b (top (top n t gh n t 6g t t g R t t gh 4gR g(r
Hooke s Law Force exerted by a sprng s always drected opposte to the dsplaceent ro equlbru Hooke s Law: F s kδs Δs s the dsplaceent ro the equlbru poston k s the sprng constant and easures the stness o the sprng F s called the restorng orce I t s released, t wll oscllate back and orth 3-Mar-08 Pak p. 7
Measurng the Force Constant At equlbru, a y F y 0, F Thereore, g k Δs Measure s g kδs ( N/ and g Δs a y 0 to deterne k. 3-Mar-08 Pak p. 8
Energy o Object on a Sprng Use the chan rule, ds ds k( s s e ds dt or k( s s ds d Dene u Integratng, s s F s k( s s s d a s e s, e k( s s s s e ds Δs, Δs Δs e s s kudu ds dt hence du s ds ds ku ds. Δs Δs ( Δs k( Δs ( Δs k( Δs 3-Mar-08 Pak p. 9 k k
Elastc Potental Energy Potental energy o an object attached to a sprng s U s ½ k(δs The elastc potental energy depends on the square o the dsplaceent ro the sprng s equlbru poston The sae aount o potental energy can be conerted to knetc energy the sprng s extended or contracted by the sae dsplaceent 3-Mar-08 Pak p. 0
Exaple 5: Sprng Gun Proble A sprng-loaded toy gun launches a 0 g plastc ball. The sprng, wth sprng constant 0 N/, s copressed by 0 c as the ball s pushed nto the barrel. When the trgger s pulled, the sprng s released and shoots the ball out. What s the ball s speed as t leaes the barrel? Assue rcton s neglgble. k ( Δx k( Δx, 0 k( x x e 0 k( x x e (0 N/( 0.0 0 0.00 kg 3.6 /s 3-Mar-08 Pak p.
Exaple 6: Pushng Apart A sprng wth sprng constant 000 N/ s sandwched between a.0 kg block and a.0 kg block on a rctonless table. The blocks are pushed together to copress the sprng by 0 c, then released. What are the eloctes o the blocks as they ly apart? 3-Mar-08 Pak p.
0 0 Exaple 6, cont Fro energy conseraton, k( Δx Fro oentu conseraton, Substtutng ths nto the energy equaton,,,,,,, k( Δx ( k(δx,,,,, 0, k( Δ x.8 /s,, 0,,, k(δx 3.6 /s 3-Mar-08 Pak p. 3,,,,, k( Δx,
Exaple 7: Sprng and Graty A 0 kg box sldes 4.0 down the rctonless rap shown n the gure, then colldes wth a sprng whose sprng constant s 50 N/. (a What s the axu copresson o the sprng? (b At what copresson o the sprng does the box hae ts axu elocty? 3-Mar-08 Pak p. 4
Exaple 7, cont (a Choose the orgn o the coordnate syste at the pont o axu copresson. Take the coordnate along the rap to be s. K 0 s ( Δs ( 5 N/( Δs ( 49 kg /s Δs U k U.46, g 0 K U s U g 0 0 g(4.0 Δssn 30.07 (unphyscal Δs 96 kg /s o 0 3-Mar-08 Pak p. 5
Exaple 7, cont (b For ths part o the proble, t s easest to take the orgn at the pont where the sprng s at ts equlbru poston. K k U ( Δs o o ( Δs ( g sn 30 ( Δs g( 4.0 sn 30 To nd s U ax g gy kδs g sn 30 s wrt Δs, take the derate o o k K U d dδs U g 0 gy 0, Δs 0 ths equaton and put g sn 30 k 0.96 3-Mar-08 Pak p. 6 o 0 d dδs 0.
Elastc Collsons Both oentu and knetc energy are consered (U 0 Typcally, there are two unknowns ( and, so you need both equatons: p p and K K The knetc energy equaton can be dcult to use Wth soe algebrac anpulaton, the two equatons can be used to sole or the two unknowns 3-Mar-08 Pak p. 7
3-Mar-08 Pak p. 8 -D Elastc Collsons p p K K, and substtutng nto the oentu equaton, or Sole or ( ( Cobne the two equatons, ( ( : ( ( ( ( :
-D Elastc Collsons, cont ( ( (3 :, ( Partcles exchange eloctes. (0 << ( ( 0 : (0 (00 contnues to oe at,, bounces back whle >>, 0 :, ( ( (0 whle (0 ( 0 (0 0 reans statonary oes at 3-Mar-08 Pak p. 9
-D Elastc Collsons Energy conseraton: ½ ½ ½ Moentu conseraton: x: cosθ cosφ y: 0 snθ snφ Note that we hae 3 equatons and 4 unknowns We cannot sole or the unknowns unless we hae one nal angle or nal elocty gen 3-Mar-08 Pak p. 30
Exaple 8: Bllard Ball A player wshes to snk target ball n the corner pocket. The angle to the corner pocket s 35. At what angle s the cue ball delected? Energy conseraton : Moentu conseraton : 0 cosθ snθ cos 35 sn 35 3-Mar-08 Pak p. 3
Exaple 8, cont Snce,, sn 35 Fro the thrd equaton,. snθ Substtutng ths nto the rst and second equatons, sn 35 sn θ cosθ sn 35 Fro these, cosθ cos35 snθ Ths leads to cotθ tan35 or θ, cos 35, sn 35 cosθ cos35 snθ 55. 0 sn 35 sn θ. snθ sn 35 3-Mar-08 Pak p. 3
Internal Energy So ar we hae consdered stuatons where there s no rcton In the absence o rcton, the echancal energy s consered: E K U constant Where rcton s present, soe energy s conerted to nternal energy, usually heat Here K E constant 3-Mar-08 Pak p. 33
Exaple 9: Bullet Fred nto Block A 5.00-g bullet ong wth an ntal speed o 400 /s s red nto and passes through a.00- kg block. The block, ntally at rest on a rctonless, horzontal surace, s connected to a sprng o orce constant 900 N/. The block oes 5.00 c to the rght ater pact. Fnd (a the speed at whch the bullet eerges ro the block and (b the echancal energy conerted nto nternal energy n the collson. 3-Mar-08 Pak p. 34
Exaple 9, cont (a Energy conseraton or the sprng: (b.50 /s MV Moentu conseraton n the collson : MV 3 MV (5.00 0 kg(400 /s (.00 kg(.5 /s 3 5.00 0 kg ΔE V ΔK kx M ΔU (900 N/(0.05.00 kg The lost energy s due to the rcton between the bullet and block. The block heats up a lttle. 3-Mar-08 Pak p. 35 kx 3 ( 5.00 0 kg( 00 /s ( 400 /s [ ] [ ] 374 J ( 900 N/( 5.00 0 ( 0 00 /s
Energy Dagras Partcle under a ree all 3-Mar-08 Pak p. 36
Energy Dagras, cont Partcle attached to a sprng 3-Mar-08 Pak p. 37
General Energy Dagra Equlbra are at ponts where du/dx 0. At equlbra, F 0. Away ro equlbra, du/dx 0 and F 0. 3-Mar-08 Pak p. 38
Molecular Bondng Potental energy assocated wth the orce between two neutral atos n a olecule can be odeled by the Lennard-Jones uncton: 3-Mar-08 Pak p. 39
Force Actng n a Molecule The orce s repulse (poste at sall separatons The orce s zero at the pont o stable equlbru Ths s the ost lkely separaton between the atos n the olecule (.9 x 0-0 at nu energy The orce s attracte (negate at large separatons At great dstances, the orce approaches zero 3-Mar-08 Pak p. 40
Exaple 0: Hesphercal Hll A sled starts ro rest at the top o the rctonless, hesphercal, snowcoered hll shown n the gure. (a Fnd an expresson or the sled's speed when t s at angle φ. (b Use Newton's laws to nd the axu speed the sled can hae at angle φ wthout leang the surace. (c At what angle φ ax does the sled "ly o" the hll? 3-Mar-08 Pak p. 4
3-Mar-08 Pak p. 4 Exaple 0, cont ( o 48. 3 cos 3 cos, cos cos ( two, the Equatng. and or an arbtrary angle hae We (c cos 0 hll. the leaes sled the 0, When ncreases. as decreases cos cos, (b cos cos (a ax ax ax ax ax ax ax 0 0 0 0 0 φ φ φ φ φ φ φ φ φ gr gr gr n n n R g n R g n a F gr gr gr gy gy U K U K r r