General Physics (PHY 2130)

Transcription

1 General Physcs (PHY 130) Lecture 15 Energy Knetc and potental energy Conservatve and non-conservatve orces

2 Lghtnng Revew Last lecture: 1. Work and energy: work: connecton between orces and energy knetc energy Revew Problem:.

3 Potental Energy Potental energy s assocated wth the poston o the object wthn some system Potental energy s a property o the system, not the object A system s a collecton o objects or partcles nteractng va orces or processes that are nternal to the system Unts o Potental Energy are the same as those o Work and Knetc Energy

4 Gravtatonal Potental Energy Gravtatonal Potental Energy s the energy assocated wth the relatve poston o an object n space near the Earth s surace Objects nteract wth the earth through the gravtatonal orce Actually the potental energy o the earth-object system

5 Potental Energy: example

6 Work and Gravtatonal Potental Energy Consder block o mass m at ntal heght y Work done by the gravtatonal orce W grav Thus : W ( F cosθ ) s grav y y mg s ( mg cosθ ) s, but :, cosθ 1, ( y y ) mgy mgy. Ths quantty s called potental energy: PE mgy Note: W gravty PE PE Important: work s related to the derence n PE s!

7 Reerence Levels or Gravtatonal Potental Energy A locaton where the gravtatonal potental energy s zero must be chosen or each problem The choce s arbtrary snce the change n the potental energy s the mportant quantty Choose a convenent locaton or the zero reerence heght oten the Earth s surace may be some other pont suggested by the problem

8 Reerence Levels or Gravtatonal Potental Energy A locaton where the gravtatonal potental energy s zero must be chosen or each problem The choce s arbtrary snce the change n the potental energy gves the work done W W W W grav1 grav grav3 grav1 mgy W mgy mgy 1 3 grav mgy mgy mgy W 1 3, grav3,..

9 Example: What s the change n gravtatonal potental energy o the box t s placed on the table? The table s 1.0 m tall and the mass o the box s 1.0 kg. 9 Frst: Choose the reerence level at the loor. U 0 here. ΔU g mgδy mg ( y y ) ( )( 1.0 kg 9.8 m/s )( 1.0 m 0 m) J

10 10 Example contnued: Now take the reerence level (U 0) to be on top o the table so that y -1.0 m and y 0.0 m. ΔU g mgδy mg ( y y ) ( )( 1kg 9.8 m/s ) 0.0m ( 1.0 m) ( ) J The results or the energy derence do not depend on the locaton o U 0!

11 ConcepTest At the bowlng alley, the ball-eeder mechansm must exert a orce to push the bowlng balls up a 1.0-m long ramp. The ramp leads the balls to a chute 0.5 m above the base o the ramp. Approxmately how much orce must be exerted on a 5.0-kg bowlng ball? N. 50 N 3. 5 N N 5. mpossble to determne

12 ConcepTest At the bowlng alley, the ball-eeder mechansm must exert a orce to push the bowlng balls up a 1.0-m long ramp. The ramp leads the balls to a chute 0.5 m above the base o the ramp. Approxmately how much orce must be exerted on a 5.0-kg bowlng ball? N. 50 N 3. 5 N N 5. mpossble to determne Note: The orce exerted by the mechansm tmes the dstance o 1.0 m over whch the orce s exerted must equal the change n the potental energy o the ball.

13 13 More about Gravtatonal Potental Energy The general expresson or gravtatonal potental energy s: U ( r) GM 1M where r U r ( ) 0

14 14 Example: What s the gravtatonal potental energy o a body o mass m on the surace o the Earth? U ( r R ) GM M r 1 e GM em R e

15 Conservatve Forces A orce s conservatve the work t does on an object movng between two ponts s ndependent o the path the objects take between the ponts The work depends only upon the ntal and nal postons o the object Any conservatve orce can have a potental energy uncton assocated wth t Note: a orce s conservatve the work t does on an object movng through any closed path s zero.

16 Examples o Conservatve Forces: Examples o conservatve orces nclude: Gravty Sprng orce Electromagnetc orces Snce work s ndependent o the path: W : only ntal and nal ponts c PE PE

17 Nonconservatve Forces A orce s nonconservatve the work t does on an object depends on the path taken by the object between ts nal and startng ponts. Examples o nonconservatve orces knetc rcton, ar drag, propulsve orces

18 Example: Frcton as a Nonconservatve Force The rcton orce transorms knetc energy o the object nto a type o energy assocated wth temperature the objects are warmer than they were beore the movement Internal Energy s the term used or the energy assocated wth an object s temperature

19 Frcton Depends on the Path The blue path s shorter than the red path The work requred s less on the blue path than on the red path Frcton depends on the path and so s a nonconservatve orce

20 Conservaton o Mechancal Energy Conservaton n general To say a physcal quantty s conserved s to say that the numercal value o the quantty remans constant In Conservaton o Energy, the total mechancal energy remans constant In any solated system o objects that nteract only through conservatve orces, the total mechancal energy o the system remans constant.

21 Conservaton o Energy Total mechancal energy s the sum o the knetc and potental energes n the system E K +U KE + PE E E KE + PE KE + PE Whenever nonconservatve orces do no work, the mechancal energy o a system s conserved. That s E E or ΔK -ΔU. Other types o energy can be added to mody ths equaton

22 What do you do when there are nonconservatve orces? For example, rcton s present ΔE E E W rc The work done by rcton.

23 Problem Solvng wth Conservaton o Energy Dene the system Select the locaton o zero gravtatonal potental energy Do not change ths locaton whle solvng the problem Determne whether or not nonconservatve orces are present I only conservatve orces are present, apply conservaton o energy and solve or the unknown

24 Example: A roller coaster car s about to roll down a track. Ignore rcton and ar resstance. At what speed does the car reach the top o the loop? 4 m 988 kg 40 m 0 m y0 (a) Idea: use conservaton o energy: mechancal energy s the same! U mgy + E K + 0 v U E mgy + g K + 1 ( y y ) 19.8 m/s mv

25 ConcepTest A block ntally at rest s allowed to slde down a rctonless ramp and attans a speed v at the bottom.to acheve a speed v at the bottom, how many tmes as hgh must a new ramp be?

26 ConcepTest A block ntally at rest s allowed to slde down a rctonless ramp and attans a speed v at the bottom.to acheve a speed v at the bottom, how many tmes as hgh must a new ramp be? Note: The gan n knetc energy, proportonal to the square o the block s speed at the bottom o the ramp, s equal to the loss n potental energy. Ths, n turn, s proportonal to the heght o the ramp.

27 Work Done by Varyng Forces The work done by a varable orce actng on an object that undergoes a dsplacement s equal to the area under the graph o F versus x

28 8 Example: What s the work done by the varable orce shown below? F x (N) F 3 F F 1 x 1 x x 3 x (m) The work done by F 1 s W F ( x 0) The work done by F s W F ( x ) x1 3 F3 x3 x The work done by F 3 s W ( ) The net work s then W 1 +W +W 3.

29 Potental Energy Stored n a Sprng Involves the sprng constant (or orce constant), k Hooke s Law gves the orce F - k x F s the restorng orce F s n the opposte drecton o x k depends on how the sprng was ormed, the materal t s made rom, thckness o the wre, etc.

30 Example: (a) I orces o 5.0 N appled to each end o a sprng cause the sprng to stretch 3.5 cm rom ts relaxed length, how ar does a orce o 7.0 N cause the same sprng to stretch? (b) What s the sprng constant o ths sprng? 30 F F 1 1 (a) For sprngs F x. Ths allows us to wrte. F 7.0 N 5.0 N Solvng or x : x x ( 3.5 cm) 4.9 cm. 1 x x 1 F1 (b) What s the sprng constant o ths sprng? Use Hooke s law: k x F 5.0 N 3.5 cm N/cm. Or k x F 7.0 N 4.9 cm 1.43 N/cm.

31 Example: An deal sprng has k 0.0 N/m. What s the amount o work done (by an external agent) to stretch the sprng 0.40 m rom ts relaxed length? 31 F x (N) kx 1 x m x (m) W Area under curve ( kx )( x ) kx ( 0.0 N/m)( 0.4 m) 1.6 J 1 1 1