88.2 Conservative force and the potential energy W=F·△r -mgj·(xr-x)+(y-y)j+(乙 -2) mgf mgy or y b W=「Fd -(mgi)(dxi +dyi+dzk) dy -mgdy Hmgy -mgy 88.2 Conservative force and the potential energy @the gravitational force F GMm W=Fd=r-Gmm pdFi l、 arcos 0 mM G dr l-c mM -G mM M mM mM W G .drr
11 y a L2 L1 f y i y r′ r r r r r d mg r b o z x dy ( ) d )ˆ d ˆ d ˆ ) (d ˆ ( d f i b a b a f i y y mgy mgy mg y mgj xi yj zk W F r = − − = − = − ⋅ + + = ⋅ ∫ ∫ ∫ r r ( ) ]ˆ ( ) ˆ ( ) ˆ [( ) ˆ f i f i f i f i mgy mgy mgj x x i y y j z z k W F r = − − = − ⋅ − + − + − = ⋅∆ r r or §8.2 Conservative force and the potential energy 2the gravitational force r r GMm F ˆ 2 = − r [( ) ( )] d d cos d ˆ d 2 2 2 f i r r b a b a b a r mM G r mM G r r mM G r r mM G r r r mM W F r G f i = − − − − = − = − = ⋅ = − ⋅ ∫ ∫ ∫ ∫ θ r r r r r r dr r′ r a b M L ir r f r r F r r r d or ∫ ∫ = − ⋅ = − f i r r b a r r mM r rr G r mM W G ˆ d ˆ d 2 2 §8.2 Conservative force and the potential energy
88.2 Conservative force and the potential energy ③ spring force OooOOOGOC F=-ki 0 xi x dr= dri =∫"(6分,d=kx=k2-k2 3. Potential energy To exploit the position dependence of the work done by conservative force on a system, we introduce the concept of the potential energy. 88.2 Conservative force and the potential energy define Wonsr =-(PE-PE =-4PE Gravitational potential energy near the earth Local=-(mgy -mi=PE-PEi PEocal=mgy y=0, PElocaI0 @gravitational potential energy mM mM W=-(-G—)-(-G— mM PE 5 一GF PE
12 3spring force i x f 0 x r xi F kxi ˆ d d ˆ = = − r r ) 2 2 d ( ˆ ) d ˆ ( 2 2 i f x x x x x k x W kxi xi k x x k f i f i = − ⋅ = − = − − ∫ ∫ 3. Potential energy To exploit the position dependence of the work done by conservative force on a system, we introduce the concept of the potential energy. §8.2 Conservative force and the potential energy define Wconsrv = −(PEf − PEi) = −∆PE 1gravitational potential energy near the earth 2gravitational potential energy §8.2 Conservative force and the potential energy ( ) ( ) Wlocal = − mgyf −mgyi = − PEf − PEi PE = mgy local y=0, PElocal=0. [( ) ( )] f ir mM G r mM W = − −G − − r mM PEgrav = −G r=∞, PEgrav=0