Chapter3Work and Energy
Chapter 3 Work and Energy
$ 1.WorkmF=F(t)F(t)dt → transport momentumF=F(r))What does F transport?dvdiF(r)·drF(r)m:dr =(dv)·mmdtdt transport energyd2dvdvdv=2m7dtdtdtdtd2=2.dcovariant under Galileao'sTransformation
F F(r) = m dt dv F r ( ) = m dr dv vm dt dv F r dr ( ) = = ( ) transport momentum . §1. Work F = F(t) F(t)dt What does F transport? transport energy. dt dv v v dt dv dt dv v dt dv = + = 2 2 dv v dv = 2 2 covariant under Galileao’s Transformation
Fm Theorem of Kinetic energyF(r).dr = d(_mv2)一drF(1)·dr - Work d(mv2) Kinetic Energy*2112["2 F(r).drmimi=Ek2-Ekl2(n)12）2Definition of workm But not r .dF Why?dw = F(r)·drF.drrelated toenergy,momentumdwdr= F(r)·dvF(r)Power:Pdtdt
) 2 1 ( ) ( 2 F r dr d mv = Theorem of Kinetic energy: ) 2 1 ( 2 d mv F r dr ( ) F dr Work Kinetic Energy r r EK EK r r F r dr mv mv 2 1 2 ( ) 2 ( ) 2 1 2 1 2 1 2 1 ( ) = − = − dw F r dr = ( ) But not Why? r dF F dr related to energy,momentum Definition of work: Power: F r dv dt dr F r dt dw P = = ( ) = ( )
$ 2.Law of conservation of mechanical energy2-1,Conservative force无法显示Conservative force Work done isindependent of path Non conservative forcefF.dr=0Uniform force,Gravity, Elastic force:Example(1).Work done by a uniform forcefF.dr=-'aF.dr +"F.drYh(1)(II)F=Fxi+F,j+Fzk dr=idx+jdy+kdz
§2.Law of conservation of mechanical energy 2-1. Conservative force Conservative force Non conservative force Example: (1).Work done by a uniform force. F dr • = 0 Work done is independent of path. Uniform force,Gravity, Elastic force: F dr = + b a a b r r r r F dr F dr (I) (II) F F i F j F k X y Z = + + dr i dx jdy kdz = + +
path-one一fF.dr-'(Fx+F,d+Fd)X()'(Fdx+F,dy+F.d)X(II)=0path-two(2).Work done by a central forcedrPolar coordinate (for simplicity)F=f(r)rIdr =drr+rdee
dx dy dz) F dr dx dy dz) (F F F (F F F y z r r x y z r r x b a a b + + + = + + (I) (II) = 0 path-two path-one (2).Work done by a central force. F f r r = ( ) Polar coordinate (for simplicity): ^ ^ dr = drr+ rd F dr