PHYS 1441 Secton 00 Lecture #16 Monday, Mar. 4, 008 Potental Energy Conservatve and Non-conservatve Forces Conservaton o Mechancal Energy Power Today s homework s homework #8, due 9pm, Monday, Mar. 31!! Monday, Mar. 4, 008 PHYS 1441-00, Sprng 008 1
Term exam # Announcements Ths Wednesday, March 6, n class Wll cover CH4.1 CH6.6 Mxture o multple choce problems and numerc problems Practce problems can be ound at the URL: http://www-hep.uta.edu/~yu/teachng/sprng08-1441- 00/lectures.html Readng assgnment: CH6.9
Work and Knetc Energy A meanngul work n physcs s done only when the sum o the orces exerted on an object made a moton to the object. What does ths mean? Mathematcally, the work s wrtten as the product o magntudes o the net orce vector, the magntude o the dsplacement vector and cosne o the angle between them. However much tred your arms eel, you were just holdng an object wthout movng t you have not done any physcally meanngul work to the object. W Knetc Energy s the energy assocated wth the moton and capacty to perorm work. Work causes change o KE ater the completon Work-Knetc energy theorem 1 K W mv K K ΔK F d ( ) ( F ) d cosθ NmJoule 3
Potental Energy Energy assocated wth a system o objects Stored energy whch has the potental or the possblty to work or to convert to knetc energy What does ths mean? In order to descrbe potental energy, PE, a system must be dened. The concept o potental energy can only be used under the specal class o orces called the conservatve orce whch results n the prncple o conservaton o mechancal energy. EM KE + PE KE + PE What are other orms o energes n the unverse? Mechancal Energy Chemcal Energy Bologcal Energy Electromagnetc Energy Nuclear Energy These derent types o energes are stored n the unverse n many derent orms!!! I one takes nto account ALL orms o energy, the total energy n the entre unverse s conserved. It just transorms rom one orm to another. 4
Gravtatonal Potental Energy The potental energy s gven to an object by the gravtatonal eld n the system o Earth by vrtue o the object s heght rom an arbtrary zero level h m h mg PE m When an object s allng, the gravtatonal orce, Mg, perorms the work on the object, ncreasng the object s knetc energy. So the potental energy o an object at a heght y, whch s the potental to do work s expressed as F g y The work done on the object by the gravtatonal orce as the brck drops rom h to h s: What does ths mean? F y cosθ g F g y mgh PE mgh W g PE PE mgh mgh ΔPE Work by the gravtatonal orce as the brck drops rom h to h s the negatve change o the system s potental energy Potental energy was spent n order or the gravtatonal orce to ncrease the brck s knetc energy. 5
Ex. 7 A Gymnast on a Trampolne A gymnast leaves the trampolne at an ntal heght o 1.0 m and reaches a maxmum heght o 4.80 m beore allng back down. What was the ntal speed o the gymnast? 6
W W gravty mg h ( h ) ( )( ) o ( ) v g h h o o 9.80m s 1.0 m 4.80 m 8.40m s v o Ex. 7 Contnued From the work-knetc energy theorem 1mv 1mv o Work done by the gravtatonal orce mg h ( h ) Snce at the maxmum heght, the nal speed s 0. Usng work-ke theorem, we obtan o mv 1 o 7
Example or Potental Energy A bowler drops bowlng ball o mass 7kg on hs toe. Choosng the loor level as y0, estmate the total work done on the ball by the gravtatonal orce as the ball alls on the toe. Let s assume the top o the toe s 0.03m rom the loor and the hand was 0.5m above the loor. M U W g U mgy 7 9.8 0.5 34.3J Δ U ( U ) U mgy 7 9.8 0.03.06J 3.4J 30J 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 the ball n hs hand and on hs toe. Assumng the bowler s heght s 1.8m, the ball s orgnal poston s 1.3m, and the toe s at 1.77m. U 7 9.8 1.3 89.J W g U mgy ( ) mgy ( ) Δ U ( U ) U 3.J 30J 7 9.8 1.77 11.4J 8
h m mg N Conservatve and Non-conservatve Forces The work done on an object by the gravtatonal orce does not depend on the object s path n the absence o a retardaton orce. l When drectly alls, the work done on the object by the gravtaton orce s θ When sldng down the hll o length l, the work s How about we lengthen the nclne by a actor o, keepng the heght the same?? W g g nclne Stll the same amount o work W g mg sn θ l mgh W g So the work done by the gravtatonal orce on an object s ndependent o the path o the object s movements. It only depends on the derence o the object s ntal and nal poston n the drecton o the orce. F mg l l ( snθ ) mgh mgh Forces lke gravtatonal and elastc orces are called the conservatve orce 1. I the work perormed by the orce does not depend on the path.. I the work perormed on a closed path s 0. Total mechancal energy s conserved!! EM KE + PE KE + PE 9
A orce s conservatve when the work t does on a movng object s ndependent o the path between the object s ntal and nal postons. The work done by the gravtatonal orce s W gravty mgh o mgh ( h ) mg h o 10
A orce s conservatve when t does no work on an object movng around a closed path, startng and nshng at the same pont. The work done by the gravtatonal orce s ( ) W mg h h gravty o snce h o h W gravty 0 11
So what s the conservatve orce agan? A orce s conservatve when the work t does on a movng object s ndependent o the path between the object s ntal and nal postons. A orce s conservatve when t does no work on an object movng around a closed path, startng and nshng at the same pont. The work s done by a conservatve orce, the total mechancal energy o the system s conserved! EM KE PE + KE + PE 1
Some examples o conservatve and non-conservatve orces 13
Non-conservatve orce An example o a non-conservatve orce s the knetc rctonal orce. W ( F cos ) θ s cos180 s k o s k The work done by the knetc rctonal orce s always negatve. Thus, t s mpossble or the work t does on an object that moves around a closed path to be zero. The concept o potental energy s not dened or a nonconservatve orce. 14
Work-Energy Theorem In normal stuatons both conservatve and non-conservatve orces act smultaneously on an object, so the work done by the net external orce can be wrtten as W W + c W nc W KE KEo ΔKE W c W gravty mgho mgh PEo PE ( PE ) PEo ΔPE Δ KE ΔPE +W W nc ΔKE +ΔPE THE WORK-ENERGY THEOREM Work done by a non-conservatve orce causes changes n knetc energy as well as Monday, the potental Mar. 4, 008energy (or the PHYS total 1441-00, mechancal Sprng 008 energy) o an object n moton. 15 nc
Conservaton o Mechancal Energy Total mechancal energy s the sum o knetc and potental energes h h 1 m m mg What does ths mean? Let s consder a brck o mass m at the heght h rom the ground What happens to the energy as the brck alls to the ground? PE g Δ PE PE PE The brck gans speed By how much? v gt The lost potental energy s converted to knetc energy!! The total mechancal energy o a system remans constant n any solated system o objects that nteracts only through conservatve orces: K Prncple o mechancal energy conservaton What s the brck s potental energy? So what? The brck s knetc energy ncreased K And? mgh E E 1 0 E + PE0 K KE+ PE 1 m gt mv ( ) + 16 PE
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Ex.8: Daredevl Motorcyclst A motorcyclst s tryng to leap across the canyon by drvng horzontally o a cl 38.0 m/s. Ignorng ar resstance, nd the speed wth whch the cycle strkes the ground on the other sde. 18
Usng Mechancal Energy Conservaton mgh + mv mgh + mv 1 1 o o gh o + v 1 o gh ( ) + v g h h + v 1 o 0 v PE + KE mgh o mv 1 o mgh + mv mgh mv 1 mgh + mv o 1 o 1 1 1 o mgh + mv mgh + mv ME E E o o v Solve or v ( ) + g ho h vo ( )( ) ( ) v 9.8m s 35.0m + 38.0m s 46.m s 19
PE mgh 0 h{ Example A ball o mass m s attached to a lght cord o length L, makng up a pendulum. The ball s released rom rest when the cord makes an angle θ A wth the vertcal, and the pvotng pont P s rctonless. Fnd the speed o the ball when t s at the lowest pont, B. B m θ A L T 0 mg m KE mv / Usng Newton s nd law o moton and recallng the centrpetal acceleraton o a crcular moton Compute the potental energy at the maxmum heght, h. Remember where the 0 s. Usng the prncple o mechancal energy conservaton K + U K + U b) Determne tenson T at the pont B. v Fr T mg ma r m r v T mg+ m L v mg + L gl + gl m L v ( 1 cosθ ) h L Lcosθ A L 1 cosθ A U mgh mgl ( 1 cos θ A ) 0 + mgh mgl ( 1 cos θ ) A Cross check the result n a smple stuaton. What happens when the ntal angle θ A s 0? T 1 mv gl( 1 cos θ ) v gl( 1 cosθ ) m g + A gl v m L A ( 1 cos θ ) L T mg A ( 3 cosθ ) A ( ) 0 A mg
A 0.kg rocket n a reworks dsplay s launched rom rest and ollows an erratc lght path to reach the pont P n the gure. Pont P s 9m above the startng pont. In the process, 45J o work s done on the rocket by the nonconservatve orce generated by the burnng propellant. Ignorng ar resstance and the mass lost due to the burnng propellant, what s the nal speed o the rocket? W ME nc ( 1 mgh ) mv + ( 1 mgh ) o mvo + ME Ex. 1 Freworks 0 1
Now usng Work-Energy Theorem W nc ΔPE +ΔKE mgh mgh o + mv mv 1 1 o W nc mg h h + mv Snce v 0 0 ( ) 1 o 45 J 0.0 kg 9.80m s 9.0 m + 1 ( )( )( ) 0.0 kg v ( ) v 61m s