in state i at t i, Initial State E = E i

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1 Physcs 01, Lecture 1 Today s Topcs n More Energy and Work (chapters 7 & 8) n Conservatve Work and Potental Energy n Sprng Force and Sprng (Elastc) Potental Energy n Conservaton of Mechanc Energy n Exercse n Hope you ve prevewed! Revew: Energy Transfer and Energy Conservaton n state at t, Intal State E E Durng Process t t f Energy may transfer n/out from/to neghborng system Energy Conservaton: n state f at t f >t, Fnal State E E f Let E transfer net energy transfer n t t f E f - E E transfer If E transfer >0 (.e. more energy transferred n) E f > E If E transfer 0 (.e. zero net energy transferred) E f E If E transfer <0 (.e. more energy transferred out) E f < E Revew: Knetc Energy nd Work q The Energy assocated wth moton s called Knetc Energy (KE). KE ½ mv (unt: kgm /s Joule) q common way of transferrng knetc energy s by Work (W) on a segment : W F Δ x F cosθ Δx on a path: W unt of work: Nmkgm /s Joule path F d x Mechanc Energy Ø For each conservatve force, a correspondng potental energy (PE) can be defned n assocaton wth t: W conservatve - (PE f PE ) Ø Two PEs that have been ntroduced: Gravtatonal: PE g mgh W mg - (mgh f - mgh ) Sprng : PE s ½ kx W s - (½kx f - ½kx ) Ø Mechanc Energy: E ½ mv + mgh + ½ kx ( KE + PE 1 + PE + ) q Extended Work Energy Theorem: (KE + PE 1 + PE + ) f - (KE + PE 1 + PE + ) W non_conservatve q Energy Conservaton(Work-Knetc energy Theorem): ΔKE KE KE W ( W, f multple forces) all forces E f E 1

2 Summary: Two Forms of Work Energy Theorem q Work Knetc Energy Theorem ΔΚE KE f KE W total q Extended Work Energy Theorem Trval Quz q Durng a, the total work done to a system s zero. Whch of the followng statement s true? The system s mechanc energy must reman unchanged after the The system s knetc energy must reman unchanged after the ΔE E f E W non_conservatve oth of above must be true None of above s necessarly true Mechanc energy E KE + PE 1 +PE + Trval Quz q Durng a, the total non-conservatve work done to a system s zero. Whch of the followng statement s true? The system s mechanc energy must reman unchanged after the The system s knetc energy must reman unchanged after the oth of above must be true None of above s necessarly true The Roles of Conservatve and Non-Conservatve Forces q Work Energy Theorem: Wtotal Wc _ + Wnc_ cons non_ cons ΔKE KE KE q Conservatve forces store/release energy n the form of Potental Energes (ΔPE) W non_ cons Wnc_ ΔKE + ΔPE1 + ΔPE + non_ cons q Non-Conservatve forces transfer mechanc energy nto other forms of energy (heat, sound,...) : f... heat sound lght external W non _ cons non _ cons W nc _ ΔME... the energy related to heat/temperature s called nternal energy

3 Conservaton of Mechanc Energy E f - E W non-conservatve v If there s no work done by non-conservatve forces: E f E Problem Solvng Procedure Ø Identfy system (movng objects + earth + sprng ) Ø Identfy all forces (remember Free ody Dagram?) Ø For each conservatve force, dentfy/use potental energy Ø For each non conservatve force, calculate work(w). Ø Identfy mechanc energy for ntal and fnal states (E and E f ) Ø E KE +PE ½ mv + mgh + ½ kx (f any) Ø E f KE f +PE f ½ mv f + mgh f + ½ kx f (f any) Ø Use energy conservaton: W non_cons ΔE E f E ( f W0, use E f E ) Ø Solve for unknowns Exercse : Moton On Curved Track chld s sldng down an rregularly curved, frcton-less, track as shown. What s hs speed at the bottom? Soluton: Force-acceleraton approach à very complcated. Energy pproach: E KE + PE ½ mv + mgh ntal: v 0 (KE 0), PE mgh E mgh ottom: h f 0 (PE f 0) E f ½ mv f Energy Conservaton: E E f mgh ½ mv f v f (gh) ½ h Smple Exercse q car movng at 60 mph has a brakng dstance (dstance between the brakng and full stop) of 45m, what s the breakng dstance f the car s at 10mph? (assumng constant brakng force.) 90m, 180 m, other W f k Δx WΔEE f E 1 mv f 1 mv From 60 to 0 mph: v f 0, v _ 60 60mph W mv _ 60 From 10 to 0 mph: v f 0, v _10 10mph W mv _10 Δx 10 W 10 0 v _10 10 Δx 60 W 60 0 v _ Δx 4Δx 180m

4 Exercse: Roller Coaster q all rdng underneath What s the mnmum heght so the ball won t fall down at? (assumng no frcton) Ø Soluton: at, N+mgmv / R à mv mn Rmg v mn Rg Energy pproach: E KE + PE ½ mv + mgh Intal: KE 0, PE mgh mn, E mgh mn : KE ½ mv mn ½ Rmg, PE mgr, E ½ Rmg +mgr E E h mn 5/ R.5 R Exercse : all Thrown From Table q ball s thrown at an ntal speed v wth an angle of θ (above horzon) from a table of h hgh. ssumng earth gravty s the only force on the ball durng ts flght. : What s the maxmum heght the ball can reach? : Show that when the ball returns to the same heght as the table, ts speed v v. C: What s the ball s speed when t hts the ground? v θ h C Soluton : Usng Energy Conservaton v θ Quck Quz q Two cannon balls are launched from the ground at the same ntal speed. all s launched at a hgher angle than ball. What can be sad about the ball s speed when they land back to the ground? h C : all wll land back to the ground wth hgher speed Intal State: KE ½ mv, PE mgh E KE + PE ½ mv + mgh q State (maxmum heght): KE ½ mv x, PE mgh E mgh +½ mv x è Energy conservaton E E ½ mv x + mgh ½ mv + mgh h -h ½ v y / g (v y v snθ) q State (Same Level, h h ): KE ½ mv, PE mgh mgh E E ½ mv + mgh ½ mv + mgh v v q State C (ground, h C 0): KE C ½ mv C, PE C mgh C 0 E C E ½ mv C + 0 ½ mv + mgh ½ mv C ½ mv +mgh : all wll land back to the ground wth lower speed C: oth wll land back to the ground at the same speed. D: Not enough nformaton to determne as t depends on the launch angle. E: None of above s necessarly true. 4

5 Exercse: lock-sprng Collson q block movng on the frctonless surface s colldng wth a sprng wth ntal speed v. What s the maxmum compresson? Show that when the block returns, t has the same speed as orgnal. Ø Soluton: E KE + PE ½ mv + ½ kx Intal: E ½ mv state C (max-compressng): v C 0 E C 0 + ½ kx max x max v m/k State D: sprng released x 0 E D ½ mv D + 0 v D v Power q Power (P) s the rate at whch energy s transferred Energy Transfer P q Power has a unt J/s Watts (W) One non-standard unt for power s horse power: 1 hp W q For mechanc energy transfer P Energy Transfer Work de dt or nstantenously: P(t) F(t) v(t) F Δx F v Exercse: Power q sports car accelerates from zero to 30 mph n 1.5 s. How long does t take for t to accelerate from zero to 60 mph? (assumng the power output of the engne to be constant and neglectng energy loss due to frcton) Ø 3.0s, 4.5s, 6.0s, other Thnk after class: What f the car s clmbng up an nclne of 30 o? W P WΔKE 1 mv f 1 mv From 0 to 30 mph: v 0, v f _1 30mph W 1 1 mv f _1 From 0 to 60 mph: v 0, v f _ 60mph W 1 mv f _ v f _ s v f _1 5