만유인력의법칙——Labster理论gydF4y2Ba

아이작뉴턴은1686년출간된그의유명한저서gydF4y2Ba 에서만유인력의법칙을발표했습니다。이법칙은gydF4y2Ba질량이있는모든물체는질량이있는모든다른우주의물체를끌어당기고있다gydF4y2Ba*는것을설명합니다。뉴턴은거r리로분리된질량m과m이있는2개의물체사이에작용하는중력FgydF4y2Ba_GgydF4y2Ba를계산하는표현도제시합니다(그림1참조)。gydF4y2Ba

* *그림1:gydF4y2Ba거리r(질량중심사이의거리)에의해분리된질량m과m이있는2개의물체사이에작용하는중력의도식。gydF4y2Ba

“파생법칙:gydF4y2Ba

다음을통해물체(질량m),지구(질량m),달과관련된몇가지기본원칙을관찰함으로써,뉴턴이어떻게법칙을고안했는지이해하고자합니다。gydF4y2Ba

米(FgydF4y2Ba_GgydF4y2Ba= mg)과관계없는g:gydF4y2Ba

갈릴레오는중력가속도g m과가관계없다는점을들어지구(질량m)에서모든물체는공기저항력과무관하게동일한속도로떨어진다는것을증명했습니다。m따라서중력은반드시에비례해야하며FgydF4y2Ba_GgydF4y2Ba=毫克(gydF4y2Ba뉴턴의제2법칙gydF4y2Ba)로표시할수있습니다。는중력가속도g m과무관하지만g는m과r로표현된함수라는점에서m과r에의존할수있습니다。gydF4y2Ba

(g ~ M)에비례하는g:gydF4y2Ba

g M에대한의의존성은측정하기어렵지만(지의구米질량을변경할수있는가상실험실에있는경우제외)지구(질량M)가질량M으로물체에가하는힘은물체(질량M)가지구에가하는힘(질량M)과동일한크기여야한다는점은gydF4y2Ba뉴턴의제3법칙gydF4y2Ba에의해알려져있습니다。그러므로FgydF4y2Ba_ggydF4y2Ba그리고g는반드시에M비례해야합니다。gydF4y2Ba

1 / rgydF4y2Ba^2gydF4y2Ba(역제곱법칙)에비례하는g:gydF4y2Ba

뉴턴의위대한생각은중력을일반화해서지구상의물체뿐만아니라달을포함해,모든물체에그개념을적용하는것이었습니다。gydF4y2Ba

그림2:gydF4y2Ba뉴턴은사과뿐만아니라달도지구에끌려야한다고생각했습니다。이로부터그는달의중력가속도를추론함으로써역제곱법칙을제시하였습니다。gydF4y2Ba

그는달을궤도에머무르게하는gydF4y2Ba구심력gydF4y2Ba으로중력가속도를파악했습니다。그리고지구반지름의약60배인거리(지구-달)로중력가속도를추정해낼수있었습니다。원궤도를가정하는가속도는지구표면보다약3600배작습니다。이사실로부터그는역제곱법칙을”추리”하였습니다。gydF4y2Ba

위의고려사항들을결합하고gydF4y2Ba중력상수GgydF4y2Ba를추가하면중력FgydF4y2Ba_GgydF4y2Ba표현식인gydF4y2Ba

에이를수있습니다。gydF4y2Ba

뉴턴은이런법칙을적용하여행성들의궤도를계산할수있었고가장일반적인유한구간궤적(또는궤도)은gydF4y2Ba케플러의행성운동첫번째법칙gydF4y2Ba과일치하는타원형이라는점을발견했습니다。또한,뉴턴은gydF4y2Ba원뿔진자gydF4y2Ba로설명될수있는중력장에서물체의모든가능한궤적을수학적으로도나타내었습니다。gydF4y2Ba

벡터형태:gydF4y2Ba

지금까지는단순히중력의크기만을고려했습니다。벡터형태에서질의량米끌림으로인해질량M에작용하는중력은gydF4y2Ba

으로gydF4y2BargydF4y2Ba_{毫米gydF4y2Ba}이제시되는경우gydF4y2BargydF4y2Ba_米gydF4y2Ba-gydF4y2BargydF4y2Ba_米gydF4y2Ba和rgydF4y2Ba_{毫米gydF4y2Ba}= |gydF4y2BargydF4y2Ba_米gydF4y2Ba-gydF4y2BargydF4y2Ba_米gydF4y2Ba|으로나타납니다。gydF4y2Ba