Re: [題目]類似球殼定理的証明

[Honor1984 (希望願望成真)]

※ 引述《r02fsm39hecf (哈哈衛)》之銘言：
: [領域] 大ㄧ普物
: [來源] 習題
: [題目] Prove a uniformly dense spherical shell exerts no gravitation
:        force on a particle located anywhere  inside it .
: [瓶頸] 試著切兩半來解 , 但要怎樣推廣到 anywhere



F = GMm/r^2

Put the test mass on the z-axis with the coordinate (0,0,d),

and set the radius of spherical to be R. (d < R)

Because of the symmetry, we can only consider the z component of the force

exerted on the test mass by the suface element of the shell.

Denote r as

      - Gσm R^2 sinθdθ                Rcosθ - d
2π∫------------------------ *  -------------------------
     [R^2 + d^2 - 2Rdcosθ]       √[R^2 + d^2 - 2Rdcosθ]


u = √[R^2 + d^2 - 2Rdcosθ]


  -2πGσmR   R^2 - d^2 - u^2
= ----------∫----------------du
     2d^2          u^2



  -2πGσmR    -(R^2 - d^2)  R+d           R+d
= ---------- [ ------------ [     +   (-u)[     ]
     2d^2          u         R-d           R-d


= 0     QED


There are another one or two methods to prove it.

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