作者Honor1984 (希望愿望成真)
看板Physics
标题Re: [题目]类似球壳定理的证明
时间Sun Nov 8 22:09:54 2009
※ 引述《r02fsm39hecf (哈哈卫)》之铭言：
: [领域] 大ㄧ普物
: [来源] 习题
: [题目] Prove a uniformly dense spherical shell exerts no gravitation
: force on a particle located anywhere inside it .
: [瓶颈] 试着切两半来解 , 但要怎样推广到 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.
--
※ 发信站: 批踢踢实业坊(ptt.cc)
◆ From: 122.124.102.140
1F：→ Honor1984:only one equation can proof it. 11/09 13:59
2F：→ Honor1984: prove 11/09 13:59