physics

Chapter 6 Work and kinetic energyIn chapters 4 and 5 we studied Newtons laws of motion and applied them to various situations.In chapter 6 we shall introduce a new approach that makes the solution of mechanics problem easierThe following concepts will be introduced inetic energ! "s!mbol#$ %ork"s!mbol#% $ %e will also introduce and use the work-energy theorem "6&'$"6&($The of two vectors "also known as ) ) product$ * cosNote '#* +Note (#* , when-, The dot prscalar producoduct in termt d tso ofB ABB AB = = = = ur urur ur ur urur ur$ $$ $ vector components*+*x y zx y zx x y y z zA i A j A kB i B j B kB A B A B A B= + += + + = + +ur$ur$ur urWork-energy theorem(6-3).onsider the motion of an ob/ect of mass m along the 0&a0is from point 1 "coordinate 0'$ to point 2 "coordinate 0($.* constant net force 1net acts on the ob/ect.The velocit! at points 1 and 2 is v' and v( 2 respectivel!. . .x1x2m FnetO 1 2x-axis

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2( (( ' ( '( (( (( '' ( ' ' ( (( " $ "third e3uation of kinematics$ 4ubstitute a in the e3uation above (" $ " $( (nnetetneta x x v vFamF mv mvF x x v v x xm = = = = a. . .x1x2m FnetO 1 2x-axis

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2x"6&4$( (( '( ' ( '( (( 'netnet" $%ork&energ! theorem ( (1 %e define as work "s!mbol %$ ( (performed b! 1during the motion from point ' to point (# 5nit net netnetmv mvF x x x x xmFv mvW xx = = = = ((%e define as the kinetic energ! of a moving ob/ect

The work&energ! s# "6$5 theorem can be written as# nits#Nm Joulekg mNm Js= =netW K = (

(mvx-axism Fnet. . .O 1 2xx-axis. . .m FnetO 1 2xmFnet. . .O 1 2x-axisx. . .m FnetO 1 2x-axisx*lgebraic sign for %net 7 1net0 "all possible scenarios$"6&5$!!!!!!!"x # $Fnet # $Wnet # $x % $Fnet # $Wnet % $x # $Fnet % $Wnet %$x % $Fnet % $Wnet #$Ca&tionIf there are more than one forces "1' 2 1(2 18 2 91N$ acting on the moving ob/ect we calculate the net work %net as follows#%e first determine the work each force performs#%' 7 1' 0 2 %( 7 1( 0 2 %8 7 18 0 2 92 %N 7 1N 0 %net is simpl! the sum of all the terms abovei.e.%net 7 %' : %( :%8 : 9: %N'ote(The work&energ! theorem %net 7 applies for the network.)** forces acting on the ob/ect for which we are appl!ing the theorem m&st +e included"6&6$,xamp*e (6-1)page 1-6 * car of mass m 7 '(,, kg falls a vertical distance h 7 (4 m starting from rest.1ind the velocit! v( of the car before it hits the water...12wateryymgh

2"6&;$5( (( ''(( (( "& $"& $ '(,, -.< (4 (.< ',6"work&energ! theorem$&,( (((( -.< (4('.; m=s netnet netnetnetnetF mgW F y mg h mghWW Kmv mvK vmvW v ghv== = == = = = = = == =%ork&energ! theorem when we havemotion of an ob/ect of mass m in a plane from point ' to point (under the action of a constant force >isplacement 7 %ork&energ! theorem# netFrW K= urr"6&ivide the path into segmentsr .alculate the work % 7 Frfor each element 4um all the contributions and take the limit asr , "6&'8$ The limit of the sum gives# This t!pe of integral is known as )line intergal)BArrW F dr = uuruurur r%ork performed during uniform circular motion.The net force F points towards the center . of the orbit "centripetal force$.1or the path segment ds the work d% is#. CConc*&sion(No work is done on an ob/ect that undergoes uniform circular motion "6&'4$cos"-, $ , , dW F ds Fds W dW = = = = =ur r)/123OxyC*assi0ication o0 0orces %ork % performed b! a force F as it moves an ob/ect along one of the three paths from point * to point +#* force is called @conseratieA if % does not depend on the path but onl! on the coordinates of the start and finish points. In this case# %' 7 %( 7 %8* force is called @non-conseratieA if % depends not onl! on the coordinates of the start and finish points but on the path as well.In this case %' %(%8 "6&'5$BArrW F dr = uuruurur r)/12O xyC1If the forceF is conservative than % along an! closed path is Bero.This statement can be used as an alternative definition of a conservative force.onsider a closed path *.+>*.This can be divided into two different paths that take us from point * to point +.Cath ' "*.+$2 and path ( "*>+$. %*.+ 7 %*>+"6&'6$("along path$ ,AACBDA ACB BDA BDABA BBDA ADB ACBDA ACB ADBB AW W W W F drW F dr F dr W W W W = + = = = = = = ur rur r ur rD0ample of a conservative force# The gravitational force%e shall prove that the work gone b! 1g along path ' and path ( is the same. 2ath 1( %' 3Fgr' %' 7mg?cos"-,&$cos "-, & $ 7 sin %' 7 mg?sin 2ath 2(%( 7 %*. : %.+%*. 7 mghcos, 7 mgh h 7 ?sin %*. 7 mg?sin %.+ 7 mg?cos"-,$ 7 ,%( 7 mg?sin7 %'"6&';$r'2ath 12ath 2D0ample of a non-conseratie force# friction f%e shall calculate the work the work done b! friction as it moves the cup along a closed path that starts at point * and ends at point *.>uring the trip we appl! a force F 7 &0 so that the net force on the cup2 and thus its acceleration a is Bero"6&'