r tsst ü r r tt 3 t ss ä r r s t3 tt s r tr

Transkript

1 s rs tät ü r tät t s rs tät ü st t t ür str s P 2s s äs Pr r t r r t P r tsst ü r r tt 3 t ss ä r r s t3 tt s r tr ❼ P r ss rst r t r r r t st r s s s r t r tr r r st r P Pr r t r r t P P P st t t ür str s 2s s äs ü

2

3 str t t t s r s t s t2 st 2 t t ss t2 t t t r ss s t s t3 t r r tr s r t rst rt t s r t s r s r 2 r tr s r t s t t t r 2 t s t3 r t s s t s s s t t r tr s r ts t s r r ss rt r t r r 2 q st s t r 2 t t r 2 r 1 t s r t t t t 2 tr s rt t s st q t t r s t t s r 2 r t s t t t t t r r t r s r s r t t q t t s r s r 2 t r t s t t s r 2 r q s t ss r2 r t s t t s r t s t r t t r t rs s r s r t r t rs s t s r s r 2 t r r s t r t2 s t s t t r t t t r r r t t s t s r t t t tr 1 s t r s s s st r r s t t t s r t2 2 t ss str t t r r r s r t s t t tr2 s t s t t s t3 r r 2r s s 2r s sts 2 q r t r s s t s t t r s s r r s t r s t r t s t s t s t r t t r t t r r s s t r s t s r t2 t t t rst rt t s t t 2r s sts r 3 t r s s r r s t r 3 t 2 rs r s 2 rs r t r t s t s t r t2 t t t s t s t s 2 s rt t 2 r r2 2 r t s r t s r s t s t t r t2 s s s t s rt t s t t t t r t2 s 2r t s r s

4 s ss s r r t r r sst ü r r ss 3 t s s ä r r s t3 tt s r tr s r ss r ü rt 3 ört r P r r ÿ ss ss s r r r s t3 st tt ür s P st s ss t s r3 t r 3 r s3 r ür st s t r t r t st t r üss r t tr r s t3 ärt r 3 ss r r P üss r s rt ö t räts t P rs 3 r s ss t s rü t r t t s 3 r st ÿ r r ür s ss öt t räts t tr t 3 ss r r s r t ss ür t r t ür rät P rs s r t r r s ss t s r r öt t r t r s ü r P r ss r ü rt ür s ss rs t r t r 3 s t3 t r t r s t s r s t s s ss t r rt s r r ss r rt r r t r s r r t s ÿ r s r t tt s r t t rü t r s ss r ä r s t3 ss r r rs r ss r ss st t t r t r ss r ö r r r t rr r r st s r st r r s ss t tr 1 st r s r r s s r ü rt r t r ä r 3 s ss s r r ä r ü r s s t ss t t t r t r 3 t s st r s ss r r s t3 t rs t r rst ü r r ä r s t3 t 3 r s t 3 r s r t r r ä tr r P2r rst t s P2r st t s t r s s r r r r r t r st ss t r Pr s r st t r r r t Pr s r ss r3 ss 3 ss r ss s rt r s r r 3 r ä r r t rst r t r P2r rst t s r 3 t t r s st t s Pr s s st ss t rst s ss r r r P2r r r r t r 3 r r st ss t r st t ts r r r 3 r r r t r s st t r ö t ss s r r r 3 t st ss t t ä rt s r 3 t s r t r 3 t st ss t r r ÿ ür t t r r t r t 3 s r r s r r 3 r r 3 r

5 ts r3 s t t r t r t r t r tr 1 t r t r st r r s rt r t r t r P r r ss s ss t s r r r ss t 2 tr s r t st r r s ss ss r r r ss s rt r ss s r ss rt rst s ö ss s ss ss ss t r r r r r s Pr s s t t t r P2r s r 3 t t r s t r P2r s r r ür s ss s t r t r

6 s r3 s ss ss t r t s r r r r 3 s ss t rs ö r t r r r tr r r ss r t s r s tr 1 t r t rs t s r s st r t r t rs r r t r s t3 3 s t3 s t s ts tt r tt s r r ä s t s ss r P 2 3 r tr s ö st r st t rs rs str s t r t r tr 1 r r ss r tt r r r r ss Pr r r r r s r rs tät ss tt s ss r t 3 t rr t r ür st r ss r 3 3 s 3 t rt s 3 t r r 2 r s s t3 3 s t3 st Pr s t3 t r s rt tr 1 sst r str r ss s t3 tt tt 2 r s t3 r t r t s tr 1 r2 tr s ör r 3 r r s s t r s ts ss r r r s r t s 2st s t r s t P2r s r 3 t Pr s r r r 3 Pr s P t r t kg m 3 r r r 3 Pr s P t r t kg m 3 r r r s t P2r 3 P t ss r s t s ür t t rt r P2r t r 3 r t r r r r ss rs r r 3 ür t rt s t r t ts r t r r r ss rs r r 3 ür P2r s ür

7 t rt s t r t ts r t r r r ss rs r r 3 ür t rt s t r t ts r t r r r ss rs r r 3 ür t rt t r t r r r r 3 t r P2r ss r r 3 s s rt r r 3

8 r3 s s ät3t t N 0 3 t r t d t r s ss 3 r t s r r r t r t r t r r 3 r r 3 r t r t r r 3 r r 3 r t r t

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10 t r tr st t s3 r 2s s äs r 2s r t s r r s r 3 ss 3 3 t s r r r r r r t t t t r r r3 t r t t s r t st 3 s r t r ss r rs t r ör r t r ss m 1 m 2 3 s s t t r r t F 1,2 s r s t s r t t s s t3 s r r t r t t s st t r st 3 s m 1 m 2 e : ts t r F 1,2 = G m 1m 2 r 2 e r F 1,2 = F 2,1 r t t s r t r r P t P t st r r t F = G M r 2 e r t t r t r ss t s ss ör rs 3 r ör r r t t s r t r ä t s r t t s t t P t P 1 V(P) = G l dm = G 1 ρ dω dm =ρdω l Ω s ss t s ρ t s ss t s st 3 s ss t P t P Ω Ω s r t t s t t st st 3 s ss t P t P ä r r t t s r t r t P t 3 ss r r r r ä tr s r r r t t tst t s tr t t s P t s P t t Z(P) = 1 2 ω2 r 2 cos 2 φ ω s t P t r P t φ ± st s tr t t

11 t r r s r t t s t t tr t t r t s s r t t 1 W(P) = V(P)+Z(P) = G l dm+ 1 2 ω2 r 2 cos 2 φ t r s r t r t r P r ä t r s P t P ( ) δ g = GradW = δx,δ δy,δ δz r s t 2st t r t s tés t m s 2 r äs 2s t3t t = cm = cm s 2 s 2 r t r r ässt s s r r r s st s s r r rs rs ss s t t 3 ❼ s t t t ö r r 3s2st s ❼ r r ä ❼ r rs 2 s r Pr 3 ss r t t t s s s r t r ❼ 3 3 r t är tr s rt 3 r ❼ r ss s r ss s s rstä s s r r t r ässt s r Pr 1 s r sät3 r 3 t t 3 ss ❼ t t t P s r s r r s t r t t rs ös s s r ss t t P s r t r st t t r t t r r s 3 s st ässt s r t r r r r t t r t r üss r ä rä t r rst r st t s är str s r t rt s ÿ r r t t t r s r tt s r t r t t r s t t r t s ❼ r tr s r s r t r ü rt s t r t s t t s r t t t rr str s r ss r s t r t r t r r r t t r r t ür Ω

12 r tr r t t 3 rt t rr str s r t r ä t 3 s P tt r t r P tt r s r t ür ss s s sss2st r r r 3 t t ❼ rr str s ss r ö P t rt r r ä s r s s r r t t r tr s t r tr 3 s t3 r t r tr r r r t rs 3 r r r P t ss r r ss r t 3 t ä r r3 t r s t r tr r t s r tr st t P t 3 st 1 rä ös r s t t t st r r s ä s r q t sss s t r r t ö s rä t r r t t t rr str s t s r t t t tr t t r ü r 3 ö ös r t rä ä r rt r r tr s r ss s s t r r st t ❼ 2s s ö äs ❼ ü s üss t str t r r r 2s ❼ t 2s 1 r t tör ör rs ❼ t ss r t r ä s r P t ä t r t r t 3 ss str t r st t t t r r t r s P t s r 3 r ss r3 ss st t r ä s r s P t s 3 r ss s t t3t ü s üss t str t r r r r r 3 3 r r r t st s r r r rs t r t s t ss ä r s t r r s r 3 ss r3 r r ss s t3 r tr s 3 r tt 3 r 3 r ts s r r s 3 r t r P t r s ts tt r st s rr r s ü t r P t t r rt s 3 s r r s t s ss

13 ss ss t r t s r r r ss Pr 3 ss r r ss t r r 3 r r rt s ss t 3 t r ss 3 r t r ss t s r s P t st st s t r 3 r t s r 3 s μ ü r μ s t r r ss r ärt r r rs t r 3 ss r r ÿ rs ä t r ärt r

14 r 3 s ss t rs ö r t r r r s t st r t r t s ü r r ts t s s 3 r rs t s s t ö r 3 t ÿ r 3 t s rs s r r ts st r ä r s 3 ss r rs ä r s r ss s t3 s r r ÿ ss r s s t3t st r t 3 ss s r ö s r t r r rü r r t r t t s r r t r 3 t t tät s2st r s P t r s t3 t rs t 3 r r ss t t r t r tr 1 s t r t r P t s r ü rt r s rt r ss t r t r s r s s r t s r t r t t s r s t t r t s 3 ört ss r3 r s r ss r t r r r

15 tr r r ss r t r r r t s 3 t ss s 2 r s s r s tt s t r tr 3 t st s s s r r t r tst ü r ö t r Pr 1 s ür t 2 r s r s t3 t tt s t r t r r ü rt r t s s t s ss r s ss s r t s 3 ss rs r s r ss s t t r r P r ss st rt P rs ss str t 3 t st 3 s ät3 ä t s ss t s t ss r t s r s t r r r s rü r ä r t 3 ss r ä r s t3 ö t3t r ö r t Pr 3 r

16 t r t r s ss str t ür t s r r t r t r r t ür s ss ö t ü r t r 3 r r ü st tr 1 3 st r r t r st t t ür str s P 2s s äs r ü st r r s ür r ss 3 r 2 r s r ss s t st r ö t 3 r r ü st t t r t r tr 1 tr 1 r t r t s s r s t ss2st t r rt r s r r3 s r ä t st ss r r t r t s ss r r tr st t s ü st r t t t rt P s t r st ss s t r r r r ä rt r 3 t r t st t tt r t r r 3 ss st s r t ä t r st s r r r s r t ss r r r r s r ÿ s r r 3 r t r t t s ü ss t r t r s P tt s t r r3 t r tr st t s r t st ss sü t s P s t 3 rü 3 r ü ss r rt 3 r t r rt ss t st r t s t s 3 t 3 s rt r t s r s tr 1 t r t rs t µgal 3 ä r st ss r r P s t sä r r st ss t r t m r ss r r 3 t t tät 10 5 r t rs r s tr 1 r ÿ ss ss s s r r r s r r st t r t r t s t t ss s tr 1 t r r t tr rs r t r ss s r t r 3 t r st ss t t 3 r st r r s s r

17 t r r t r t rä r ö t r t r rr t r 3 t s rt r r s tr 1 r ü t ü r s ä ss2st s t3 r ä r st t r r 3 s s r rt s rt t r t str t s s rt r t t s t rr t r 3 t r st r r r st t r t s r rr rt r t r r t r st t r t r s 3 r 3 rü s t t r r t s r t rs t s s r str t t r t r r r r st r t r 3 r t r s r 3 ss s r t ä ÿ r r t r üss r rs ütt r s sss2st s r3 t r 3 P t P t ür r mg P = k(l P l 0 ) mg Q = k(l Q l 0 ) Δ g P g Q = k m (l P l Q ) Δ r t rs l 0 l P,l Q k m g P,g Q : ä r st t r ä r st t r P t P 3 r st t st ss r s P 3 t r t r st r st r r t r s r t t t ss2st r st ss st t st r r r s rä r ä t s

18 t s r s st r t r t rs r st r r t r t s s s t st s rt r t s r st t r st t r t r t r ür stä s t ss s s t r t s 2st s s s s r t r r r t rü tr r t s rs2st s 3 s s t3t s s r t r s r ss s r röÿ rt 3 r rt r s s 2st r r t t st röÿ r rä r s r ts r s r st ÿ ür r r r r ss r r r s str t s r r r t t r st ts s t s rr t r rst r sss r r t r r t t r sss st ÿ ür r s r t r t r r t r str t s sst r s t t rt st t rt r s r t rs st ss r röÿt r s st r r t rs st ä r s 3 t s r s ss s s r r st t t st r r t r r ü ü r ss r s t ts r r t t mgasin(α δ) = kbd l l 0 l sinα st ss r st t

19 l,l 0 : ä r st t 3 r st t r ürα+δ = 90 l 0 = 0 mga = kbd sinα g = kbd ma sinα kbd cosα α ma s t tät rr t s sα ür α st s r r s rt r t r t r r r t r t r t r ss r ÿ r rt r r s rt t r 2st r s t ö st r 3 t r 2st r s t r s s t s r t rs tst t r s s str t s s rt s rs2st t s rt ts r r st t r r 3 rü ä r r t rt s t r t r ö st r s ütt r s r tr s rt rt r s t r s st r r r t ö st r r s r ü r ö r st r s s r r s t s str t s s tr s rt rt r ss s rs ütt r r t s ä r r t rt s ä r s s 3 ÿ r r st r r t r r r rr t rt st t r r r s rt ss r rr t r s 3 s r t r rst st r 3 t r r s str t t rr t rt r

20 P P r r ss r P r ss t 3 P rs ss str t 3 t st 3 s ät3 ä t s ss t s t ss r t s r s t r r r s rü r ä r t 3 ss r ä r s t3 ö t3t r ö r t Pr 3 r s r ä r r ä t rt s ss t s ss s r r s r s t3 3 s r t t r ss ü r s t3 tt s r r s rt t r s 3 t r t r st t r ü rt r s ss rt s st 3 ä st t s rst ü r 3 rs s s rr r t s ss rt s rü t r r ÿ rt t str t r P rs r st t s st s r t t P s rr r rt s rr t s t3 tt s r r s t r s t r r rt 3 s s ü r t t r ss s t3 tt ö t s 3 s t3 tt r r s rt P rs str t 3 rs t st t r r ö r t r s t3 3 s t3

21 P 3 s r s t ü r rü t r s s sr t t ö är r t s r s st t ss r r t st t t r s st sst tt t s t r t s 3 ÿ st t r str r r t r r r t ÿ r s rü t r r ür st s t r t rs ür r ü r r ss t t st r t st t 3 röÿt s tt r tt ür s r ts s t ür st s r t rs st st r str t s s t s s r ü r st r ü s t s ts tt r tt r r r ss t ür r ss r st ss t öt t t r t r s r st ä m 2 st t r t t rs ütt r r t sst r r ÿ r ö t s r t r rö 3 tt r r s ä r s s t s t r t s 3 r st r r 3 t r s r t rs ü rt s ss t r t s s s ür r s ss 3 s ät r t t r r s 3 3 r s t3 ür r t st r3 s s rü s s s t s ür r ss 3 s r r s t t s r s rt st r st s 3

22 P st t ür st s r t rs t s ss st s r r r s st t t r t r s r s rs röÿ r ö rt rrs t st r s r r s s r ö t P r r st s t st s tt ss t s s ö t s s t t r ÿ s t3 tt ö st r t r t r t s s st t s r st 3 ä r t st 3 ö t r stä 3 s t s s t st t s r r 3 s s st s s r r ss stä r 3 ü r ss st t st r ü r 3 ö ö s r r r ät3 r stä r s s s r 3 ä s r t r t är s ä r tt 3 r tt s ss r ü rt r s t ssst r r ö st t t s r st ür st t3t r r äÿ st 3 s ss 3 t stä ö t rs 3 s ss t ö 3 s ät r t t tt s 2 tr r st t r t r t s ä s r r t s st ö t rs r t st r t r ö t rs s r r ä r r r ss t 3 r r s ÿ 3 s 3 tt r tr 1 t3t r r r r r r ä r r ss t ö t t P t rt r s ä r r ss r r r 2 tr s r 3 3 är 3 s ä s r r 2 tr s r t st ür st r r t r r ss t t 2 tr s ss r ü r 3 r ss t r ü rt r s

23 P ss r r s s t s ss P 2 3 s r s rt r s r ü r 3 ö r 3 P t t t r t rt r ÿ t öt t r tr är t r t P t s r st rt r rt P t t t r t 3 r ö s tt s ä r r ä r rt s t rt r ür P 2 3 öt t 3 Pr s 2 t r s r r P t A,1,2...n st t r β i t tr s i t t s r t r P t P t s r ss t ö t r P t t t ss r s t s ss r P 2 3 s r t r P t P ss s t r ss tr s A1 β A r t P t s r t P t s s 1,2 β 1 s r ÿ r ❼ r r t s t A1 = t P1 ±200g +β A t 12 = t A1 ±200g +β 1. t ne = t n 1,n ±200g +β n

24 P ❼ r r r t y 1 = y A +s A1.sint A1 x 1 = x A +s A1.cost A1 y 2 = y 1 +s 12.sint 12 x 2 = x 1 +s 12.cost 12. y E = y n +s ne.sint ne x E = x n +s ne.cost ne ö r P t P i 3 st s 3 sät3 3 t β i tr S i r t Z 2 t r ö Pr s ö t rä str d 3 ä st P t t t r r t t s ss r tr s ö st r st t rs rs ss s t r t ö s P t s P 3 A r tr tr s ö st ö r 3 P t st

25 P Δ = d.cosz A,E Δ =Δ +i t r r t st s s r t 2 t r s Pr s 3 s 3 tt s Pr s s 3 3 s t s s s 2 t r 3 r s r tt s r r t s st s Pr s ss ä r t ss 3 t ä s Pr s 3 3 ö t s s s Pr s 3 t t 2 t r t t r t r s 3 s r Pr s st s sst öt t ür s ss sst tt ❼ t r t r ür ss r r t rs t3 tt r rät ❼ r t r ür t r ss ❼ 2 t r t rs t3 tt r rät 3 Pr s ❼ ä r s t P t r ä s s r s ❼ t rstä ür ss r rät ö s r t r s 2 t r ❼ r s r 2 t r r 1 ür s r r r P t ür s ät r ss r är t r t rt ❼ sr t r t r 3 s t rs t3 tt r ❼ st r r ts r r s s ❼ ü r r t s r r s ss ür r ss r s t3 t rs r t r 3 s t3 s s rs rst r s t rs r t t rs r s r s 3 ü rt ss s str t r r r 3 3 s s P t ss ö t s 3 rü st r r s r str t r r r st s t r r str t 3ü s st r s s tr 1 3 st s r t r st s r ä t s r ss r r 3 3 rs t r t r 3 s s P t

26 P C i = Δ i Δ CG ❼ Δ i r t r t r ss r r 3 ❼ Δ CG3 tr 1 ss r r 3 r r t r r r ts s t r t r tr 1 s P ss t s t r t r r rt s s r ä r 3 t r t s t r ss r ssstr s t3 s r t r r r s sr st t r rt t r t 1, s t s st 3 3 ss t s t r t r st s t ä st t r 3 t t tät r r t r s rät s 3 r r t r s r ü r r ü r ö t str s t r t r tr 1 t t r t r s s t s t r t r t r st ö r t r t r ä s t r tr t r t r t s r rät rs rst r t rs t s ö t r rt s r r r r r ss s t3 t r str s s t3 r s ss r ür st r r t r r t t t r ü r ss r s s r s t 3 sät3 t r ÿ r t str s ss t ss r 3 ss ÿ r t ür rr r t r

27 P r r röÿ r r r 3 s ö r t r r r s t s r rstr r s t3 s r t r ÿ r r 3 3 s t r ÿ r 3 t s s r r str r t r t s r s rt ö t 3 s r s ss r r r ss t r t r r s rt st t s t s t ö st st s t t rt 3 ss t ststr rt r str t s r P t st t ür st r rät t ❼ s str t ss t st s ss str t ö s ss t ss r ❼ r t s r t r r 3 t r ❼ str t ö r t t rst s 3 r r t s rät s ss tr ❼ t rr t r s t r t s rät s r s t r ü rt s r t t r t t s t r r3 t r ss tr ❼ r ss r r t r s r ö ss tr ❼ s str t ss t t rr t rt st s rs2st s ä t s str t r r r ö t ür tr 1 rät t ❼ t t ss t st s ÿ str t ö s ss t ss r ❼ rs t s t3 s s str t r 3 t rt r 3 t s r rät t r 3 t rr s t t s 1 2 ts ❼ s t s rät s P t r ss st rt t r ❼ r ss r r t r s r ö str t ö s 3 r r t ss tr ä r r ss s öt rs ütt r r t r t r r r s t s rät r 3 st öt 3 r r t r rs ütt r s s r rs ä r r ss s r t t r ss s r ss t r t r 2st r s t 3 t ür r t st s ss t str t 3 3 t rs t3t s t3t r

28 P rt r r t ür r r t s r ö s ss r r rt r r t r r t r t st ÿ ür r r r s t r ö s r r t s t r t r r r ö s r s r t t mgal m t r r s ö r äÿ r t st r ör rs trä t r t r t s rt r r t mgal m s r rt r rt r ä r rt r ss ss s rt r r t r t ss r r tt r P t ss ü r s P t t r t ö r ü rt ö t r r r ss st s s t r tt r r s 3 t r r r ss r tt r r s s ür ststr ür r r ss t r r ss t s r s s t r tt r r r t r t ss t rt t ür rät t ss ü r P t t s 3 rü P t s t r 3 P t s ÿ r ö t str t r r ss t rt r r t ss r s ststr tür r r 3 r ss t 3 r ö t r t st s s s t t r ❼ str t s s P t 3 ststr ss ❼ ü ss t 3 s t3 s r t rs 3 r ss ❼ r r rt r r t ss ür ö r t r ü r r r r3 t stä 3 s 3 ss s P t t s r r rt r t s r t rs t stä 3 r s s sät3 3 r t ü r s t ss r r r s t3 P t r ss r sst r

29 P r s ss r r s ür s ss r t r st s s t Pr r r s st t s r ss rst s 3 t3t P t t3t ü r s ss t 3 rü s 3 rst ss t s s r r r t ü r ä r t stä r sst r r r ss Pr r r s r r ss s Ü ür t r s s t t str t rtr t s s t ö r tr rt r s ät r r ss s r s r r ü rt r s rt r ss r r ss r ss rt r s rt r ss t r ä t rt ür üss r r s st r r t r r t r s s s r t r t 3 rst s t r t r s ss r r ü rt r s s t t r t ür st r r t r 3 t s ür r r r s r rs tät ür s rt s st r r t r 3 t rt s r t r t r st st s t r s r 3 t ts r t r r t r t 3 rt r s r s r t t r 3 s

30 P rt rt r t r rt t ür st r rät r rst r t rt tr 1 ss s s rät s s rät s rt s t3 3 st r ss rt t r rt r ss r t r ss rt t r t r s str t s 1 2 t r t r s s r 3 t rr t r 3 t P 3 t r t r st t t r ss rt r t rs s r t t s rr rt r r t st 3 t r t rr t r t t s ss rt r t r s r Pr t t s t1t t r t r 3 ss ös r P t r ä r s s ä r r ss s r ss tt s ss r t r r ss rt r st r r t r s t r r ss rt s tr 1 üss s t r 3 rt r 3 t r t 3 t r tst t tsä r s r r ss t r ss s s st s r ör rs tr rä t r r r t t r3 t r 3 t 1 t t ±100μ s r ä r s st ts t r t üss r s s s s ss s t rr rt r ür öt t 3 t s 3 t r t 3 st ür st r r t r r 3 rst s ts r t t r s 3 t rt r r ss st tt t r

31 P 3 t rt 3 t t r ss r s s t r s st t ss 3 s s r ss 3 t rr rt 3 t rt r 3 t rr t r δ t s rü r Pr t ä r s t s r tr s str s Pr t r r rs t r s s s tt s r t r t r st r st r st r tr 1 ss rt str rst t t r 3 s 3 t rt r ss st r r t r rr rt r sst r s t ÿ rt r 3 t rr t r s 3 s s r t r st t s 3 t r 3 t rr t r ür st r ss r 3 t r t ür st r ss r s 3 t t3t s P 3 r r ü st t r tr 1 ss r t t r rt 3 t t t s r 3 rt s 3 t rr t r ss t r 3 r üss 3 rst 3 t rr t r s ss tr 1 r s r t r s ÿ ö 3 t rt s 3 t st t s r t r t s ss rt s tr 1 s st r r t r t s 3 t r 3 rt 3 t t rs 3 s rr t r tr 1 3 t 3 t ür s Pr t r tr s str s t r ss 3 t rr t r s tr 1 t röÿ r s s s 3 t

32 P r 3 3 s 3 t rt s 3 t t r st t ö r r t rs s r ö r t rs s rät üss r ss s r ö s r t rs t r 3 rt r r3 r ä r r r ss str t ö s 3 r r t s rät s ss ür t r str t ö s r 3 r ❼ t r st t ö ür st r t r t r δg H = g mess +(δg h (h 0,15)) ❼ t r st t ö ür tr 1 t r t r δg H = g mess +(δg h (h 0,22)) reduziert str t ö r 3 rt r mess ss r st 3 s r t s rät s s r

33 P δg h ss str t ö s 3 r r t rt r r r t mgal m δg h = g 2 g 1 h 2 h 1 = dg dh [mgal m ] st g 1 r ss r rt r t ö h 1 g 2 r ss r rt r t ö h 2 r rt r r t t P t ss r t r t r rt t r r t r t r r ss sst üss ss s rr rt r t t t röÿ r r t rr rt r r rs t r r r r r ä ür s rr t r r r ür rät r t ❼ r t r P n s r ö s r ö Ü P n = 1013,25(1 0,0065H 288,15 )5,2559 ❼ t r rr t r δg p = 3(P Pn)10 3 [mgal] P ss r r ss t P n r t r P s r ö t ss s r t t 3 rr t r s ÿ r r 3 s ss g reduced = g mess δg t δg H δg p [mgal]

34 P s r ss rt t r rt t r t 3 t t r 3 s ät3 r r tt s t3t t s t s 3 t t Z i = g i N 0 d (t i t 0 ) SZ i t t s r r s Pr t r tr s str s s s ❼ g i : r t r rt t P t ❼ N 0 t r ü r rät 3 s t t ü rät ❼ d r t s räts t ür ü rät ❼ S s t r r ür st r rät s tr 1 r st r rst t t r s t s s t t t s t r ä t s tr 1 r röÿ 1 ö t t s t s r t s 3 t t s r r r s tr 1 s s t3t r tr 1 r s t3t st s t s r ss r s t t Z i + ˆV i = ĝ i ˆN 0 ˆd(t i t 0 ) ŜZ r tr 1 st t s t r t s t t röÿ 3 r t 1 3 r t A = 1 1 (t 1 t 0 ) Z (t i t 0) Z i

35 P st s st st s s r 3 tr 1 Q bb st tr 1 st t s r t r t r t t r ss s rr t t r r s t röÿ 1 3 r t Q bb = diag(σ 2 Scintrex,σ 2 LCR195,σ 2 LCR087,σ 2 LCR587) tr 1 str t r t r r t s 3 ör ss r t s s s t3 3 r ss t st r r t r s t t t ür s r r s t3 r P t ür n i=1 ss t r r t s = ( x x i) 2 n 1 ät3 r t ˆx r r ss r ˆv r s t ˆb t t ˆx = (A P bb A) 1 A P bb Z mit P bb = Q 1 bb ˆx = x+ ˆx ˆV = A ˆx Z ˆb = Z + ˆV ät3 s r 3 t rs r r 3 tr 1 r t r t r t ˆσ 2 0 = ˆV P bbˆv r mit r = n u r 3 3 r t 3 r t ˆK bb = ˆσ 2 0Q bb ät3 r r 3 tr 3 r t Qˆxˆx r r ss r Qˆvˆv r s t Qˆbˆb ss s t r Qˆxˆx = (A P bb A) 1 ˆKˆxˆx = ˆσ 2 0Qˆxˆx

36 P QˆV ˆV = Q bb AQˆxˆx A ˆKˆV ˆV = ˆσ 2 0QˆV ˆV Qˆbˆb = Q bb QˆV ˆV ˆKˆbˆb = ˆσ 2 0Qˆbˆb t r s t s t r ˆx r s s ät3t r rt r t t r st t t 3 s rt r ä t t r r3 r r tr 1 Qˆxˆx 3 t r ss s Pr ts r tr s str s 3 t r st r rät rät N 0 d 1+ S tt rt tr 1 s ät3t t N 0 3 t r t d t r s s t ss t r ür st r rät 3 r rt tt rt r st r t rt ä st 3 st r st r t tt rt stär st t

37 P rst s ö ss s ür r r s s r str r r 3 ssstr 3 t ü t r t r 3 r r ü st är s s ssstr ü str 3 t r r str r ss r st s s 3 P rs t r t r t t r ss ür r r r t st tt ü 3 s r t t r ÿ s t3 tt r r 3 s r r s r ss r t t s ts r str 3 ss r 3 ü r r t3 t 3 rä r r tr r r r r 2 r s s t3 3 s t3 st ä r ÿ s 3 r r r s s trä t 3 t r r r s 3 t r ä t t t r r r s t r 3 3 s t3 tt ä t t s s r t s s t ä t s tr s r ü t t r s t r str t r ä t

38 P Pr s t3 t r ss r tsä r 3 t sst r t 2st r s t r r r str t r t s r t r s t s s t ss r t s s r s ss ü r st t t r r P t r ü rt r ö ss rt P t t r r r r s r r t r t r ss r ss t 3 r3 s rät r s rt t st r r s r r 2st r s t rr rt r s rt rt s rs2st s r t rs t s rt 3 r 3 rü s r t st str t s 3 s rr rt t r r s rt s 2st r s rt ö s 3 µgal/h r s rt tr r r ä r r st 3 t st s 3 r r ss r 2st r s ü rt 3 r rr r r 3 r st r sst s r r ä rt st 3 t r3 t t ür ss r st s3 t r s är t 2st r s t r r µgal 3 r s r rü s t st ss r r t stä r ü rt ö st r s rt s t t t r rs t 2st r s t 3 t 3 t 2st r s s 3 sst r s rt t r t r st t s ss r r r 3 3 s 3 ss t 3 r str t r t r ss r trä t t r st r r 2st r s t t r 2st r s r äÿ 2st r s t s st r r r 3 r r rt

39 P t r s rt tr 1 sst r r ss tt s t r t r ör r ss r ❼ t s r t 3 P t ❼ r s ss st t P t ür st s r t r s ❼ ss r rt r r t s ss s r r ä s s ü r s s t r sst 3 s s öt t r r 3 ä t ür r ss r s t ssstr sät3 3 r ss3 t üss r r r t s t t 3 r t r r s r t rt s r rs ss r rä t r s r t r r s s ss ä r 3 r r s 3 s tt r ö r ss r ä t rt ä r r rtä ss r s r r s s t r t ür ss r ss t r ss r r r s rst s ü r ss rät ss ss t t r t t ü r ss P t rr s r r s s s r P t s ür r ss s t rt t s P t s r r ss r rst t3t r ss r ü r s

40 P t 3 t r rs r t r 3 st r r ss rr ö s r 3 r r 3 s ssstr s r t r tr s t rr r r t3t r t s r r r ss s s t3 tt t r ä s r str t t r s ö r r s s tt t r ssstr s t ss t ❼ r ä rt t t r ÿ st t st r s s s r t ss P t r ä s t3t ss s s ss P t t ä s r ss r ü rt r ü s r s ss P t ür r t st r ü rt r s ss r t r 3 r r s 3 rü r 3 rü s r r ä st st s r s r P rs st s s r t r 3 tr ❼ r ä rt t t r ss st t st r s s r ü rt ss P t t r ä ü r P t s 3 P t t r ä 3 rü s P t ür r t st r s t3t r r 3 r r s 3 rü ❼ r ü r t P t t r ä s s r r r s s 3 P t 3 ÿ s r r r s t s r t t r t r t ü s P t ür r t st 3 t s ss r ❼ r st rt t ss s t3 tt t ä r t t s s 3 P t t r ä r ü s P t t3t ür r t st 3 t s ss r ❼ r ä t ss r r s 3 r P t r r t ü r P t s r r t t r r r r s r ss s t3 tt tr t ü s P t rr 3 t s ss r r ss 3 s s tt s r 3 r r s 3 rü r rt t3t ss P t rr r ü r ss 3 t s r r s tt s r rst ä st tr 3 rü 3 r r s sst s s r

41 P t r r r ssstr t s s t ür r r s r P t rr s t rt r ss r t r ❼ r ü r t s tt r r s s 3 s 3 t s r t t ä r s st ss r ü r s ss r ü rt r ❼ r ü rt ss P t s t3 tt s ü P t r s ss ❼ r t s 3 st r r ü r t P t t r ä s ü P t 3 t ss r ü r ❼ r ä rt t t r ss st t r s s ü rt ss P t t r ä s ü ö r ür r t st P t ss r t r ❼ r ä rt t st t st r s rt s ü r t s s tt s ü P t ss r ss t s r ss üss ss s rt r r t ür ö r t r ü rt r s ss t s t r t r P t s r s st s ss r tt r3 t 3 s r s ss s r r t t ss t r ü rt r s r r tt ss t t ätt r ü P t r str s r r t ss P t 3 s ö tt s t r t st t r 3 t s r s ss

42 t s r ss r s tt ss t ä r s ss r ÿ s t3 tt s t3 tt s t3 tt 3 r r s r r s r r s t s r ss r s tt ss t ä r s ss r r s t ss t 3 r ss t ss r rt r r t r s tt ssp t ä 3 r ss s t3 tt s t3 t 3 s t r r s ss 3 r t s r r r t P

43 P str r ss s t3 tt tt 2 r s t3

44 P r r ss st r t3t rt r t s r 3 t t ür t ür st s rät s ss ür r ür r ss t t öt 3 P s r3ö r ür 3 sät3 t r ss ür r t t s r ür ü s 3 s s t tr t r t ss r 3 ü r 3 r ss r r tr r st rt t ss s ss r P t rr r r s r r ü rt r s ss t r r r r ss r ü rt r t 3 r r 3 ü r s 3 s t3 t 3 t ür r t st s r s ss r r P t s t3t s r t t s r t s r r t 3 r rät ü r ss3 t t t r r t r rät r s 3 t r s r t ü r t r st r rs ütt r s r t rs r s rt ü 3 r t r ss t st r 3 r äÿ r t r t r t r r rü r r t r2 r t s r ss st r r r ü rt r r P r s s ss t r t r r t 3 st s r r r r t t ü r sst ür rät st st t r 3 s μ μ tr 1 r r rät s t r r rr t r μ r t r tät r r t 3 rü 3 t r tät r r t 3 r r 3 sät3 P t r s t3t s r s s r r r r t r t r ä rr t s 3 μ 3 sät3 r rs r r t r t s tr 1 3 s r s rt t t ü 3 st t3 3 rt r s rt r s st t3 s st st t r tr 1 st r s r t ä r r ä r t rt ü st t3 r t s ür3 r t rt ÿ ärs r s st t3 s s

45 P r t r t s tr 1 r2 t s 3 r st r str t r ü r r r ss r s tt ss r r r r ä t rt öt ss 3 r t r tr r ü rt r ö s st s rt t r r s rt r 3 rst t 3 r r t r r t r ss 3 r r t r t t r r 3 r t t r r t ❼ 3 t r t ❼ t str t ö ❼ t r r t r s r t ö r 3 rt ss rt tt s r s s s r ss rt s r s ss s 3 r ü rt r rst r r t r t r r s 3 ss rs r ss st tt 3 t ss r t r ü rt r ss r st s r ss r r s r t r st s s 3 r ss r st ö t s r t r r t r s t3 3 t t t r ts tt r st tt rt tä t r t r rs ö

46 t r r r ä r rst ü r r ä r ss t 3 r s t t r r tt s t r ü rt s t s s ss rü r s r t s t r ss r r r t ür s ss t 3 s r r r ä tr öt t s r r r r t r 3 t s P2r s r r t r s st t s Pr s st s r ss s r ss ä r r t ä r s rt r s 3 r r r r r r Pr s t s t ü rt r s Pr s t r s t r tör t s ä ö r ss t 3 st r r r s r r ss t tr ä r ü t s ä r 3 r s s s r t 3 t ä st s s tr 1 ö rt r t tt r r st t s st r rü 3 rü s t ür ss r s t r t r s r s r t 3 r ür Pr s r t s rt s s r t s2st 3 s ts r r s s tr s r 3 r t ü r r t r rt r P t t r r 3 r tr s ör r 3 r r s s t r s ts ss r r r r s t r s t s r s r t s t s r tr s r P2r t r ässt r ss rst s ÿ r st s t s r s rt r s s tr t t r

47 r r s Pr s s t ür s P t t s t r s s t st t r t r t s2st t P(x p, y p, z p ) r ˆ ˆ ˆ [ V(P) = Gρ (xp x) 2 +(y p y) 2 +(z p z) 2] 1 2 dxdydz x y z r r t t 3 t r ä t r t t s s V z ˆ ˆ ˆ V z = Gρ x y z z p z dxdydz [(x p x) 2 +(y p y) 2 +(z p z) 2 ] 3 2 t r ös s s t r s r t s r r V z s t r s s 3 r r V z = Gρ xln(y +r)+yln(x+r) zarctan xy zr x y z x 1 y 1 z 1 r = x 2 +y 2 +z 2 r rt r r t V zz ässt s r t V zz = Gρ zarctan xy zr x y z x 1 y 1 z 1 ρ r r t t s st t 6, m3 kg.s 2 t kg m 3 r t r t s Pr s s st 3 s rs r s r t s2st s r s t t s Pr s s t r r t r r 3 r r t s Pr s s rs r s r t s2st s s s Pr s t r r P t P(x p, y p, z p ) 3 r ss s Pr s 3 s

48 t P t P s rs r ss r P t P r t rs r t s üss r t r t s Pr s s r t s P t s P 3 r x t y t z t = x y z x p y p z p r s r r s r t ö tr s r rt r t r t s Pr s s (x t, y t, z t ) ür r r = x 2 +y 2 +z 2 r t r s ÿ r s t3t r r r 3 r r t 3 r V z 3 V zz s Pr s s t P r t m s 2 r s r r t s rt ss t r t r 3 r t r t r t t t t s s r r s r t rt s t ss s r r s r t t r r P t P r t t s t r t rs r t r st 3 s P 3 t s Pr s s t s r r s r t ÿ r t s tr 3 s s Pr s s t rä rt s t V z s t rt P r s Pr s s t t r 3 s r t t r ä t ür r P t P t r s Pr s s ätt V z t rt t r 3 s r t s ä t

49 r s r t s 2st s t r s t t t t r P2r s r 3 t t r s ür s t r s rst s P2r s r 3 t Pr s t t s r 3 t st ss t tr t t r ö 3 r r t r tt t (x M,y M,z M ) r Pr s st t Pr s r r tt t r t r s 1 2 r t t rs s r r 3 r t s ÿ r t ä d s rst t r Pr s s st t r r ö h r 3 Pr s t s t3 3 r t ä st t s r t x M y M rt s 3 r t st t r ö r t ä d s t rst Pr s s trä t t r ö t Pr s ö h r Pr s trä t t Pr s st t t s röÿ d, h r t r tt t ss s r t r t (x E,y E,z E ) r 3 Pr s r x E y E z E = x M y M z M ± d 2 d 2 h 2 r ä t t r t r r r r Pr s 3 r r Pr s ässt s ä r 3 r tt t ä rt t ts r t ä

50 r Pr s ä t P2r t s t s 1 1 s P2r s r 3 t Pr s röÿ r P2r 3 r t r s r s 3 t rü ür t s ä t ä r 3 t 3 r t r röÿ r r P2r r r ö r 3 r t r s r 3 t r ä rt t st ss t s t r s r P2r r ä tr t s ÿ r r 3 r st ss t st t P t r t r 3 ss st s t ρ r t t s st t 3 3 Pr s r r t3t 3 3 st3 st t t ür t r s kg m 3 ss r t r Pr s t t t P r tr s r r s r r s t3 s r t s s r s ür t t r t P

51 r r r 3 Pr s P t r t kg m 3 s t r r r 3 Pr s t P ässt r ss r r r s rst t rst Pr s s t rt t rt t s Pr s t r s t s r s Pr s t r s P t s t r s t rt r r s r Pr s t rt s r s P t s P ÿ r st r r r t r t r 3 s Pr s t 3 r Pr s r r ö r r r ss s 3 r s ä r r ü rt t3t t P r ö st tt ür rst t r r s Pr s t r s s t s t rt s t r stät t 3 t

52 r r r 3 Pr s P t r t kg m 3 s t ss r r s Pr s t r r t ö s t rt t r ö röÿ r r t s ä r ss t st röÿ r r r r s P t s P r t r t s r r t r r t 3 st ür r r r s t P2r t r s r 3 r r t r t r r t r r r r 3 Pr s t r 3 ö r s ür t 3 t r r r s t P2r rs t s tt r r t 3 r Pr s r st s st t t kg m r

53 r r r s t P2r 3 P t s s t ss r r r s t P2r r r r s 3 Pr s rst ü P t t rt st r rst P t s s r st r r r s Pr s t r s s P t s s t r s t s t s r r s P t t t rt r r r s Pr s r s s P t s t tr röÿ r s s s t r r s Pr s t r s s P t s s t ür rst ü t rst s t r s t s Pr s s t s r r r röÿ r s r tr r r t r r s Pr s r s P t s st t s t rt ür r r ÿ r s t r t t r r r s t r ÿ st 3 t r röÿ r Pr s r ärt r 3 r t 3 s r Pr s r r r sü 3 t s t ür t s r st t s t s t s ❼ r t 2r s t r t s ts(x M,y M,z M ) t ä d Pr s ö h r t r t r 3 ü r r t r s r ❼ r t 3 r r s r t r Pr s s ss t r t rs r r s s t ür t Pr s P2r 3 tr s r rt r

54 ❼ f z s t r t s tr s r rt r t r t t s st t r t ρ r r V z r r t V zz s 3 Pr s t t t r s s ÿ r

55 X= xm ± d Y= ym ± d Z= zm ± h Koordinaten der Mittelpunkte xm, ym, zm Kantenlänge d Höhe h Generate- Pyramide.m XT = XT X x p x p YT = YT Y Y y p y p ZT = ZT Z z p z p Koordinaten Koordinaten der der Eckpunkte Eckpunkte X, X, Y, Y, Z Koordinaten der Aufpunkte x p, y p, z p Trafo.m Transformierte Koordinaten in Cells gespeichert Dichte Dichte ρ Grav.Konst. Grav.Konst. G fz.m Schwerewirkung gz Schweregradinet gzz ss r s t s ür t

56 s r t s t t r t ür r ss s r r t ä r r rä r s ss r ts st t r 3 Pr s ässt s t r t ä r r r rs P t r P2r r r rs t ss r r s t3 r t t 3 s r rr r t tt r r tät t t ür r s s t3 ss st t ü r 3 s t t r r t kg m 3 s r t t r r t kg cm st s rt P r t t r s r t s s t t s st s 3 ρ D = (2900 0,746)+(2710 0,254) = 2851,7 kg m 3 r r r s t3 t r ö rrs t P r r st P r st s t ss r ü t t s t s r ö r t r t r s ρ Wasser = 1000 kg m 3 0,0442 = 44,2 kg m 3 ρ W =ρ D +ρ Wasser = 2895,9 kg m 3 s r t t ρ D ür tr s st ρ W ür t ss r sätt t s st r t3t s t3t r s r 3 r ä r ä t rs t ❼ ür t r t r t t s rt P r r st r st t r t t kg r t r t r t m t kg 3 r s rt s ä r 3 r t m s t3 r t r t r r r t r s tρ W r t r tρ D rs s s r s r r P t r t ❼ ür r t r s 3 r t t s rt r r t r st ü rt 3 r 3 r t r r 3 3 P r r st r s ÿt s 3 ss r r t r st ss t r rs t s t ss r r s r t r t r r Pr s tρ W rs s r ts s rt Pr s s s tρ D s ü t tρ W rs 3 s t r P r r st r s ÿ s ss rs t r st ss t r 3 s r s ÿ r r r s P t s r s 3 r t r s r 3 t P t r r r t r st t r t r rt r s r P2r

57 P t r r t r t t rt r P2r t r 3 r t 3 t t rt t r r t Pr s t r r ö r t ρ W r s t Pr s r t ρ D r r r r ä r r 3 r s r ss

58 r r r r ss rs r r 3 ür s t ss r r ä 3 rst t r ö 3 t r ö t ❼ P t s s r r st t r r s P t 3 P t r ss t r s r P t s t 3 s r t t stär r r t ❼ P t ü s t s P t r r r 3 r r r t r st 3 t r röÿ r Pr s t röÿ r ss r 3 s r t ä r 3 s rä t s t r röÿ r Pr s ÿ r s t r r 3 ss s t rt r rs r ö r t Pr s t s t rt r ä t röÿ r s Pr s r t r t s r ss röÿ r 3 s r t r t s t r t r P t

59 r r t r rs r ss t r röÿ rt s r st 3 r r P t s 3 r r r P t s ü rt P t t r ö t s röÿt s t r 3 t s t3t tt r t r s P t t r rs r t Pr s t r t t 3 t rt r r r 3 s P t r V z2 V z1 r r 3 ❼ t r t rrs t t t ❼ r 3 r t s t s Pr s s r ö t s t r r t t s r r 3 r t rst r 3 r 3 P r r st t rt s t r t ts r t

60 r r r ss rs r r 3 ür s t ÿ r r 3 rst r P t s t3 3 s t rt s t r ÿ t3t r 3 Pr s t r s r t r ss r t t s t st t stär r r r t ü t t r r 3 t rt tt r t r s s t s ss rs t r r röÿ r s st st tt t s t r r r t3t r t r rs t t 3 t rt r r r 3 ür s P t r V z2 V z1 r r 3 t s r t t r 3 r r ss r r r P2r s r r s r s t r r r 3 r rs r t ρ W r r Pr s 3 t r röÿ r Pr s r t ss 3 s r P2r s t st s t s 2st r t ü r ss ä r r P2r r r

61 ä st r tt ss r t s 2st s rt t r rs r t s t ss r P2r t t s ss 2st r ü r 3 ö r r 3 t t r ÿ Pr s t t s s t ρ W Pr s 3 ä st r s t ss r P2r rs r t r P2r s r r ür ür s t r rs t r t P t r t (x M,y M,z M ) s s ür tt t r 3 ür rt r r 3 t r st t s st ss t rst 3 r ür t r r ö r r s ss P2r tst t s 3 ür röÿ 1 1 röÿ r P2r trä t 1 1 r r r ÿ 3 r ür ür s st r P2r öt t s ür r s r ÿ t t 3 3 t t 3 ä r st 3 ür t 3 s r r r ür s t t r r r s t P2r t st r r r r r 3 ür t r t r t ür s t 3 t P t r r t r t P2r s ür

62 r ä t t rs ss rs s s r t 2st t rs t ❼ s r 3 r r 3 3 P r r st r s rt 3 t r s s ür s 3 r t r ür ÿ t rt s t r t ts r t s s r t rt r r r r r 3 r t 3 t r ss 3

63 r r r ss rs r r 3 ür s t s r r r 3 s ss rs r s r st 3 t rt r r r 3 3 s P t r V z2 V z1 r r 3 s r 3 t ss r ss t r üss ss r r r 3 3 r t ❼ r t s 3t tt r r ür r t rst P r r st r 3 ÿt r t r r ür s

64 t rt s t r t ts r t 3 t r r r r 3 ür s t t

65 r r r ss rs r r 3 ür s t ss r r 3 t rt ÿ r r tt P t r s t rt st rst 3 P t s ss r ss rs s P t ä rt s 3 r tär r r r ü rt röÿ r t rt t s ä r r r 3 s ss t r r r P2r st t r r tt P t t r 3 s t rt s r 3 t r s s P t s ss s t 3 t rt r r r 3 3 s P t r V z2 V z1 r r 3 r r t r t3t t r P s t r t s rt t t3t r P2r

66 t r s t3t tr r P2r 3 t t t s r3 r r r t r r t r t 3 t P t r r t r t t rt t r t s t r ss r r rt 3 r s t

67 r r r r 3 t r P2r s t ss t3t r rt t r P2r röÿ r s s t ss r ärt r t3t t r tt r ü r P t s 3 r r 3 ü rt s t3 3 s t t r r r 3 r röÿ r t r 3 s t t ss s rs s r r t s 3 t ss r r t r r s s rt r P2r ä st s r r P s t r t P t r r P2r r t r stär r s P t r P2r

68 s ss s P r ss t 3 t ÿ ss r r r t r s t3 r s r st r r s rt ö t r r s t3 r ö s s r r s rt s str t r s P rs s r s ss ts ss r r ü t s ss t ü r ü t st s ö t r t r t r t r t ä r r ss t t s r r s s r r t r r r tt s Pr s s rt r t 3 r 3 rst s 2st r t ss ä r r P2r r r r t rt 3 s r Pr s st t r 3 t st ss t r r s r s t r r r r r t r t r s t s s r t r s st t ss rs r t 3 t r röÿ r Pr s s r s t s s r P2r r t r t r r t ü r ss r r s t3 r r s ss s 2st rst t r s s rs r s t ss s rt r t s r t r r r 3 t Pr s r ÿ ür t s s t r ss r P2r ä r t r t t 3 t ss t ä t r ä r r ss r s t r st t r ö rt s 3 1 tät r r ö r st ss t 3 r ür r P2r s r r t 3 Pr s rst r P2r s Pr s t st t ss ö r r r r 3 r r ss ÿ r r r ss rt s rt t r üÿ r r r s r r t t rü s t t ür r st s r t öt t r t s ü r s s t3 t 3 ört r s ür t 3 r t ss r öt t r r t ü r rt r st s rt t ss ü r s s rt t 3 ör s r st s rt s t r s t ätt s r ö ss r r äs s t r är s r t t rst ür r s ÿ r s t ss ü r s r t s r ss rs s t3 t ür s t s rt 3 ö t r rst ür 2 r ss ts t r r s t r s t ö rü s t t r s r t r ss rs ä s t3 t rt r r t ss s t r t 3 r r t st st s ä r r r3 t r r Pr r rs r s t s r ä t ü r t r s ss 3 t ö st r3 t r

69 s s r r t r ä t r r r r t r r r ss s t3 t r ü rt ss r r r 3 s 3 s r 3 s ss s r t rt s 3 ür r r 3 r ss t t r P2r s r r ür s rt r r 3 3 r t r r r r 3 s rs rst r P t ss r r 3 s s rt r r 3 s t ä r r ss s rt r r 3 r rst P t trä r r r 3 s t t s r s rt r r 3 r s P t ss r P t t rt r r r 3 r t r ss rs s s t ss r P2r s rt r ss r P t s t r r 3

70

71 t r t r r3 s r r r tt 3 t ss ä r s t3 r r s s s tt r 2 r s t3 t r r r r ö s2st st r 2 r ss rt t 2 r s r ss s t r tt P t t t t tr 1 r t r r t r t t2 r s ts tr s t t s 2 r 2 t r s r r2 r t r r s t ss t t rst s r s s st t3 s st r r ss rt t ss P r t öt r t t r r t2 t r t s t 3 ts t t t s r t t r s r t s t tät 3 t rä r r t r s 2st r s s t3 r t 2 r t2 s st 2 t P r r Pr t r 3 t t s t t st r rs 2 Pr st r str P s s rt rs2t2 P rt str tt s t r tr tr s r 2 2s s s s t2 st 2 r t r r s r t r2 st r s s s r t s r rt r r t r r tr2 r r t r r 2t r r

72 r r t s r r ür t t3t r s r st t s t r t r t t r rt s t r 3 rst tt t r t r Pr s rt ür r 3 t Pr s st tt t rt s r P2r r s r rr t r t ür 1 2 t r ür 3 r t s t3t r r ö r tt t t t ö r tt t r t P t ö r ür P2r s r ür r r t s r 3 ür r 3 t t st s r r ü P t tt r 3 rst r tt t r t r ür rst t r P t tt r r t r ö r r t r tt t r r r s r t r t r Pr s t r t r t P2r r t s t r t r P2r t t t 3 t t r t ö t3t rt r 3 s t r t r t t r t r tr s r rt r ss t s t r st t r ür Pr s t r 3 ä rt r r r 3 r r t 3 r t 3 r tt r t st t ä r s r

73 %Koordinaten der Mittelpunkte der horizontalen Prismen definieren x=250; y=250; z= 12.5:25:487.5; Kantenlaenge = 500:-25:25; % Kantenlänge der einzelnen Prismen h=12.5; % halbe Höhe der Prismen (siehe Beschreibung von generatepyramide ) %%%% Koordinaten der Eckpunkte des Prismen Koord=generatePyramide(x,y,z,h,Kantenlaenge);%Aufrufen der Funktion generatepyramide für die Berechnung der Eckpunktkoordinaten der Prismen [n,m,k]=size(koord); edges=cell(1,k); for i=1:k end edges{i}=koord(:,:,i); % edges : Koordinaten der Eckpunkte der Prismen in cells speichern (ein cell für ein Prisma) % Koordinaten der Aufpunkte definieren z=25:25:500; x_p=10.5:12.5:248; y_p=ones(length(z),1)*250; z_p=z'; %%%% Zeichnen der Pyramide figure (1) faces = [ ; ; ; ; ; ]; for k = 1:length(Koord) patch('vertices',koord(:,:,k),'faces',faces,'facealpha',0.2%intesitätderoberflächenfarbe,...'faceco lor','flat','facevertexcdata',hsv(6)) % faces definiert Oberflächen der Prismen durch 4 Punkte % Patch zum färben der Oberflächen der Prismen grid on xlabel('x [m]') ylabel('y [m]') zlabel('z [m]') end hold on

74 hold off view(125,22) hold on for k=18:20 patch('vertices',koord(:,:,k),'faces',faces,'facealpha',1,... 'FaceColor','flat','FaceVertexCData',hsv(6)) grid on end hold on for k=16 patch('vertices',koord(:,:,k),'faces',faces,'facealpha',1,... 'FaceColor','flat','FaceVertexCData',hsv(6)) grid on end view(125,22) %%%% Koord Trafo bezüglich Aufpunkte edges_t=cell(length(z_p),1); for i=1:length(z) edges_t{i} = Trafo(edges,x_p(i),y_p(i),z_p(i));% Aufrufen der Funktion Trafo für die Trafo der Prismen bezüglich der Aufpunkten end % Berechnung der Dichte % Dolomit=2900kg/m³ 74,6% Gesteinsanteil %Calz.Carbonat= 2710 kg/m³ 25,6% Gesteinanteil % Porenanteil =4,4442 % Dichte_g=(2900*0.746)+(2710*0.254); %Dichte des Gesteins Masse_Wasser= 1000*0.0442; % Masse des von Poren aufgenommene Wassers Dichte_W=Dichte_g+ Masse_Wasser; %%%% Berechnung der Schwerewirkung G= *10^-11; density_1=ones(1,length(edges)).*dichte_g; %Kg/m³ Vektor: ein Element=Dichte eines Prismas. Hier kann man einzelnen Prismen Diche_g bzw. Dichte_w zuweisen density_1(17:20)=dichte_w; [Vz_1,grad_1]= fz(edges_t,density_1,g,koord); % Aufrufen der Funktion fz zur Berechnung der Schwerewirkung

75 density_2= ones(1,length(edges)).*dichte_g; %Kg/m³ Vektor: ein Element=Dichte eines Prismas. Hier kann man einzelnen Prismen Diche_g bzw. Dichte_w zuweisen density_2(18:20)=dichte_w; density_2(12)=dichte_w; [Vz_2,grad_2]= fz(edges_t,density_2,g,koord); % Aufrufen der Funktion fz zur Berechnung der Schwerewirkung % für die Pyramide aus mehreren Würfeln wrden die mittelpunktkoordinaten der Würfel so erstellt x0=1:2:19; y0=1:2:19; z0=ones(length(x0),1)'*1; [X0,Y0,Z0]=meshgrid(x0,y0,z0); %%scatter3(x0(:),y0(:),z0(:)) X0=X0(:,:,1); Y0=Y0(:,:,1); Z0=Z0(:,:,1); X0=reshape(X0,100,1); Y0=reshape(Y0,100,1); Z0=reshape(Z0,100,1); % % % % 1 % % % x1=2:2:18; y1=2:2:18; z1=ones(length(x1),1)'*3; [X1,Y1,Z1]=meshgrid(x1,y1,z1); %scatter3(x1(:),y1(:),z1(:)) X1=X1(:,:,1); Y1=Y1(:,:,1); Z1=Z1(:,:,1); X1=reshape(X1,81,1); Y1=reshape(Y1,81,1); Z1=reshape(Z1,81,1); %usw bis X9=X9(:,:,1); Y9=Y9(:,:,1); Z9=Z9(:,:,1); X9=reshape(X9,1,1); Y9=reshape(Y9,1,1); Z9=reshape(Z9,1,1); X=vertcat(X0,X1,X2,X3,X4,X5,X6,X7,X8,X9);%Vektor mit allen x-koordinaten Y=vertcat(Y0,Y1,Y2,Y3,Y4,Y5,Y6,Y7,Y8,Y9);%Vektor mit allen y-koordinaten Z=vertcat(Z0,Z1,Z2,Z3,Z4,Z5,Z6,Z7,Z8,Z9);%Vektor mit allen z-koordinaten % dannach wird genauso vorgegangen wie bei der Pyramide mit horizontalen Prismen

76 function Koord = generatepyramide(x,y,z,h,kantenlaenge) %Imput:x, y, z = Koordinaten der Mittelpunkte der einzelnen Prismen d= Abstand von Mittelpunkt zu den vertikalen Oberflächen der Prismen h= Abstand von Mittelpunkt zu den horizontalen Oberflächen der Prismen Output: Koord= Eckpunktkoordinaten der einzelnen Prismen d= Kantenlaenge/2; % Berechnung der Koordinaten der Eckpunkte Koord(:,1,:)=[x-d;x+d;x+d;x-d;x-d;x+d;x+d;x-d]; Koord(:,2,:)=[y-d;y-d;y+d;y+d;y-d;y-d;y+d;y+d]; Koord(:,3,:)=[z-h;z-h;z-h;z-h;z+h;z+h;z+h;z+h]; faces = [ ; ; ; ; ; ]; function edges_t = Trafo(edges,x_p,y_p,z_p) %input : % edges = Koordinaten der Eckpunkte % x_p, y_p, z_p = koordinaten der Aufpunke % Output: % edges_t = Eckpunktkoordinaten transformiert bezüglich Aufpunkte for i = 1:length(edges) edges_t{i}(:,1)= edges{i}(:,1)-x_p; edges_t{i}(:,2)= edges{i}(:,2)-y_p; edges_t{i}(:,3)= edges{i}(:,3)-z_p; end function [Vz,grad]= fz(edges_t,density,g,koord) % imput: bezüglich Aufpunkten Tansformierte koordinaten der Eckpunkte "edges_t" der einzelner prismen, Dichte einzelner Prismen "density" und Konstante G % Output: Schwerteinwirkung "Vz" und Schweregradient "grad" im Aufpunkt % ra: Abstände des Aufpunktes zu den Ecken der Prismen % at: atan zu dem Aufpunkt t=length(koord); for i=1:length(edges_t) for k=1:t for j=1:8

77 ra{i}{k}(j) = sqrt(((edges_t{i}{k}(j,1)).^2+(edges_t{i}{k}(j,2)).^2+(edges_t{i}{k}(j,3)).^2)); if (edges_t{i}{k}(j,3)*ra{i}{k}(j)) ~= 0 at{i}{k}(j) = atan2((edges_t{i}{k}(j,1)*edges_t{i}{k}(j,2))/(edges_t{i}{k}(j,3)*ra{i}{k}(j)),1); else at{i}{k}(j) = pi/2; end check1{i}{k}(j) =edges_t{i}{k}(j,2)+ra{i}{k}(j); check2{i}{k}(j) =edges_t{i}{k}(j,1)+ra{i}{k}(j); if check1{i}{k}(j) == 0, check1{i}{k}(j) = eps; end if check2{i}{k}(j) == 0, check2{i}{k}(j) = eps; end end end end for i=1:length(edges_t) for k=1:t for j=1:8 Vz1(i,k) =density(k)*g* (((edges_t{i}{k}(7,1)*log(check1{i}{k}(7)))+(edges_t{i}{k} (7,2)*log(check2{i}{k}(7)))-(edges_T{i}{k}(7,3)*at{i}{k}(7)))-... ((edges_t{i}{k}(3,1)*log(check1{i}{k}(3)))+(edges_t{i}{k}(3,2)*log(check2{i} {k}(3)))-(edges_t{i}{k}(3,3)*at{i}{k}(3)))+... ((edges_t{i}{k}(4,1)*log(check1{i}{k}(4)))+(edges_t{i}{k}(4,2)*log(check2{i} {k}(4)))-(edges_t{i}{k}(4,3)*at{i}{k}(4)))-... ((edges_t{i}{k}(8,1)*log(check1{i}{k}(8)))+(edges_t{i}{k}(8,2)*log(check2{i} {k}(8)))-(edges_t{i}{k}(8,3)*at{i}{k}(8)))+... ((edges_t{i}{k}(2,1)*log(check1{i}{k}(2)))+(edges_t{i}{k}(2,2)*log(check2{i} {k}(2)))-(edges_t{i}{k}(2,3)*at{i}{k}(2)))-... ((edges_t{i}{k}(6,1)*log(check1{i}{k}(6)))+(edges_t{i}{k}(6,2)*log(check2{i} {k}(6)))-(edges_t{i}{k}(6,3)*at{i}{k}(6)))+... ((edges_t{i}{k}(5,1)*log(check1{i}{k}(5)))+(edges_t{i}{k}(5,2)*log(check2{i} {k}(5)))-(edges_t{i}{k}(5,3)*at{i}{k}(5)))-... ((edges_t{i}{k}(1,1)*log(check1{i}{k}(1)))+(edges_t{i}{k}(1,2)*log(check2{i} {k}(1)))-(edges_t{i}{k}(1,3)*at{i}{k}(1)))); grad1(i,k) = density(k)*g*((-at{i}{k}(7))-... (-at{i}{k}(3))+... (-at{i}{k}(4))-... (-at{i}{k}(8))+... (-at{i}{k}(2))-...

78 (-at{i}{k}(6))+... (-at{i}{k}(5))-... (-at{i}{k}(1))); end end end for i=1:length(edges_t) Vz(i,1)=sum(Vz1(i,:)); % Summe der Schwerewirkungen aus allen Prismen auf die Aufpunkte Vz(i)=Vz(i)*10^5; % Umrechnung in mgal end grad(i,1)=sum(grad1(i,:)); grad(i)=grad(i)*10^5;

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