2-(3)势能-机械能守恒定律

1rdfdWvv⋅=mABFvABumgf-=dsmg⋅-=μ∫⋅-=LdsfW0mgLμ-=2-32rrm'mGFvv3-=∫∫⋅-=⋅=BArrrmmGrFWvvvvd'd3rvm'm)(trv)d(ttr+vrvdmO'mABm'mmABFv3⎥⎦⎤⎢⎣⎡-=ABrrmGmW11'∫-=BArrrrmmGWd'2)(trv)d(ttr+vrdvφ∫∫⋅-=⋅=BArdrrmmGrdFWvvvv3'φcosrdrrdrvvv=⋅0=WBArr=)(trv)(dttr+vmO'mABβsinrdrv=drr⋅=ββrdv40d=-=∫zmgWkdzjdyidxrdvvvv++=)(ABmgzmgz--=kmgGvv-=dzmgrdGWBAzzBA)(∫∫-=⋅=vvABAzBzmgoxyzBAzz=50d=-=∫xkxWikxFvv-=∫∫-==BABAxxxxxkxxFWdd)2121(22ABkxkxW--=AxBxFvxoBAxx=6:∫∫⋅=⋅ADBACBrdFrdFvvvvABCD⎥⎦⎤⎢⎣⎡----=)1()1('ABrrmGmW)(ABmgzmgzW--=)2121(22ABkxkxW--=7--0d=⋅∫lrFvv∫∫∫⋅+⋅=⋅BDAACBlrFrFrFvvvvvvddd∫∫⋅=⋅ADBACBrFrFvvvvddABCDABCD8)3()11()2()(21)1()(22LLLababababababrrGMmAxxkAhhmgA--=--=-=9P1p2p)(EEEWΔ-=--=2p21kxE=rmmGE'p-=mgzEG=)2121(22ABkxkxW--=⎥⎦⎤⎢⎣⎡----=)'()'(ABrmmGrmmGW)(ABmgzmgzW--=10),,(ppzyxEE=∫=⋅=0)(0)(PEx,y,zPrdFx,y,zEvv00P=EPPPEEEWΔ)(0-=--=111PdErdF-=⋅vv)(kzEjyEixErddEFppppvvvvv∂∂+∂∂+∂∂-=-=aa∫⋅==→aaardFAEvv12pEzOmgzE=p20,0p==Ex0,0p==Ez0,p=∞→ErxOpE2p21kxE=xOpErmmGE'p-=13--XpEOX0XpEOX1XpEOX2xddEFpvv-=14)1()1(112kkEEA-)2()2(212kkEEA-n)()(12nknknEEA-nFv2m2Fv1Fvnm1m12Fv2nFvnF2v1nFv21FvnF1v15nAAAA+++=L21nFv2m2Fv1Fvnm1m12Fv2nFvnF2v1nFv21FvnF1v12111222][][)1()1()1()()2()1(kkkkknkkkEEEEEEEE-=+++-+++LL16AAAAEk2Ek1-(Ep2Ep1)AA+[-(Ep2Ep1)]=Ek2Ek117--E1,E2AA=(Ek2Ek1+[(Ep2Ep1)]AA=(Ek2+Ep2-Ek1+Ep1E=Ek+EPA+A=E2-E1181:--A+A=E2-E1A+A+A=Ek2-Ek1219A+A=0E2=E1Ek2+Ep2=Ek1+Ep1A+A=0E2=E1202:A+A=0F=F=01211mLmμa1:aL-a0-mv21AA2fG=+Lm=λydygdAGλ=∫=LaGgydyAλ)(2122aLg-=λ)(222aLLmgAG-=y22∫--=aLffdxA0∫--=aLxgdx0μλ2)(2aLg--=μλ2)(2aLLmg--=μaL-aLm=λy2)(2aLLmgAf--=μxgfμλ=23)(222aLLmgAG-=2)(2aLLmgAf--=μ0212-=+mvAAfG])()[(222aLaLLgv---=μaL-aLm=λy)(222aLLmg-2)(2aLLmg--μ221mv=242)(2aLLmgAf--=μ12EEAf-=221aagEλ-=22212mvLmgE+-=aL-a2512EEAf-=221aagEλ-=22212mvLmgE+-=aL-a2)(2aLLmgAf--=μ2)(2aLLmg--μ)212(2mvLmg+-=)2(agaλ--])()[(222aLaLLgv---=μ263dtdydydvm=∫∫=--vlaLvdvdygyLyg0])([λμλλ2222221)](21)([)(21LvaLaLLgaLgλμλλ=-----])()[(222aLaLLgv---=μdtdvmgyLygF=--)(μλλ27--∫∫⋅=⋅ADBACBrFrFvvvvddABCDABCD28aa∫⋅==→aaardFAEvv--E1,E2A+A=E2-E129A+A=0E2=E1Ek2+Ep2=Ek1+Ep1A+A=0E2=E1