Gravitation Theory E

8/12/2019 Gravitation Theory E

1/26

TH[UKEU

"ofgkshcuofrphyskes.kg" 6

^ QKZ ZKMG6. KGZ^MJ\EZKMG

Zhl omtkmg ma elilstkfi nmjkls sueh fs thl sug, thl ommg, thl lfrth fgj thl pifglts lte. hfs nllg f sunblet

ma afsekgftkmg skgel tkol koolomrkfi. Kgjkfg fstrmgmolrs ma thl fgeklgt tkols hfvl jmgl nrkiikfgt wmrc kg thks

aklij, thl omst gmtfnil fomg` thlo nlkg` Fryf Nhftt thl akrst plrsmg tm fsslrt thft fii pifglts kgeiujkg` thllfrth rlvmivl rmugj thl sug.

F okiilggkuo iftlr thl Jfgksh fstrmgmolr Zyemnrfhl (6>6

r

oomr A ?

>

>6

r

oo@

whlrl @ ? 1.10 6466 Go> c`> ks thl ugkvlrsfi `rfvktftkmgfi emgstfgt. Zhks ifw hmijs `mmj krrlspletkvl

ma thl gfturl ma twm mnblets (skzl, shfpl, ofss lte.) ft fii pifels fgj fii tkols. Zhft ks why kt ks cgmwg

fs ugkvlrsfi ifw ma `rfvktftkmg.

Jkolgskmgfi amrouif ma @ 2

A ?>6

>

oo

Ar?

_OS

_IS_OIZS>

>>

? SO6 I5 Z> _

Glwtmg's Ifw ma `rfvktftkmg kg vletmr amro 2

6>A

? >>6

r

o@o6>r &

6>A

? >>6

r

o@o>6r

Phlrl6>

A

ks thl amrel mg ofss o6lxlrtlj ny ofss o

>fgj vkel-vlrsf.

Gmw >66> rr , Zhus 6>>>6

>6 rr

oo@A

. Emopfrkg` fnmvl, wl `lt >66> AA

Kopmrtfgt ehfrfetlrkstkes ma `rfvktftkmgfi amrel

(k) @rfvktftkmgfi amrel nltwllg twm nmjkls amro fg fetkmg fgj rlfetkmg pfkr k.l. thl amrels frl lqufi

kg of`gktujl nut mppmsktl kg jkrletkmg.

(kk) @rfvktftkmgfi amrel ks f elgtrfi amrel k.l. kt fets fimg` thl ikgl bmkgkg` thl elgtlrs ma thl twm

kgtlrfetkg` nmjkls.

(kkk) @rfvktftkmgfi amrel nltwllg twm nmjkls ks kgjlplgjlgt ma thl gfturl ma thl oljkuo, kg whkeh

thly ikl.

(kv) @rfvktftkmgfi amrel nltwllg twm nmjkls jmls gmt jlplgj upmg thl prlslgel ma mthlr nmjkls.

(v) @rfvktftkmgfi amrel ks gl`ik`knil kg efsl ma i k ht nmjkls nut nlemols fpprlekfnil kg efsl ma

ofsskvl nmjkls ikcl stfrs fgj pifglts.

(vk) @rfvktftkmgfi amrel ks img` rfg`l-amrel k.l., `rfvktftkmgfi amrel nltwllg twm nmjkls ks laaletkvl

lvlg ka thlkr slpfrftkmg ks vlry ifr`l. Amr lxfopil, `rfvktftkmgfi amrel nltwllg thl sug fgj thl

lfrth ks ma thl mrjlr ma 64 >0 G fithmu`h jkstfgel nltwllg thlo ks 6.< 64 0 co

Lxfopil 6.

Zhl elgtrls ma twm kjlgtkefi sphlrls frl ft f jkstfgel 6.4 o fpfrt. Ka thl `rfvktftkmgfi amrel nltwllg thlo ks

6.4 G, thlg akgj thl ofss ma lfeh sphlrl. (@ ? 1.10 6466 o5 c`6 sle6)


2/26

TH[UKEU

"ofgkshcuofrphyskes.kg" >

Umiutkmg @rfvktftkmgfi amrel A ? >r

o.@o

mg sunstktutkg` A ? 6.4 G , r ? 6.4 o fgj @ ? 1.10 64 66 o5 c`6 sle6

wl `lt o ? 6.>>< 64< c`

Lxfopil >.

Zwm pfrtkeils ma ofssls o6 fgj o>, kgktkfiiy ft rlst ft kgakgktl jkstfgel armo lfeh mthlr, omvl ugjlr thl fetkmg

ma outufi `rfvktftkmgfi puii. Uhmw thft ft fgy kgstfgt thlkr rliftkvl vlimekty ma fpprmfeh ks ^/)oo(@> >6 ,

whlrl ^ ks thlkr slpfrftkmg ft thft kgstfgt.

Umiutkmg Zhl `rfvktftkmgfi amrel ma fttrfetkmg mg o6jul tm o

>ft f slpfrftkmg r ks

A6

?>

>6

r

oo@

Zhlrlamrl, thl feelilrftkmg ma o6ks f

6? >

>

6

6

r

o@

o

A

Ukokifriy, thl feelilrftkmg ma o>jul tm o

6ks f

>? >

6

ro@

thl gl`ftkvl sk`g nlkg` put fs f>

ks jkrletlj mppmsktl tm f6. Zhl rliftkvl feelilrftkmg ma fpprmfeh ks

f ? f6 f

>? >

>6

r

)oo(@ .... (6)

Ka v ks thl rliftkvl vlimekty, thlg f ?jt

jv?

jt

jr

jr

jv.

Nut jt

jr? v (gl`ftkvl sk`g shmws thft r jlerlfsls wkth kgerlfskg` t ).

f ? jr

jvv.. .... (>)

Armo (6) fgj (>), wl h fvl v jv ? >>6

r

)oo(@ jr

Kgtl`rftkg , wl `lt>

v >?

r

)oo(@ >6+ E

Ft r ?, v ? 4 (`kvlg), fgj sm E ? 4.

v> ?r

)oo(@> >6

Ilt v ? v^

whlg r ? . Zhlg v^

?

^

)oo(@> >6

Trkgekpil ma suplrpmsktkmgZhl amrel lxlrtlj ny f pfrtkeil mg mthlr pfrtkeil rlofkgs ugfaaletlj ny thl prlslgel ma mthlr glfrny

pfrtkeils kg spfel.


3/26

TH[UKEU


Zmtfi amrel fetkg` mg f pfrtkeil ks thl vletmr suo ma fii thl amrels fetlj upmg ny thl kgjkvkjufi ofssls

whlg thly frl tfclg fimgl.

.......AAAA 5>6

Lxfopil 5.

Amur pmkgt ofssls lfeh ma ofss 'o' frl pifelj mg thl emrglr ma squfrl ma skjl 'f' . Efieuiftl of`gktujl ma

`rfvktftkmgfi amrel lxplrklgelj ny lfeh pfrtkeil.

Umiutkmg 2 Ar? rlsuitfgt amrel mg lfeh pfrtkeil ? >A ems :

>

>

>

)f>(

@o

>

6

f

o.@>

? >

>

f>

o.@

)6>>(

Lxfopil :.

Akgj `rfvktftkmgfi amrel lxlrtlj ny pmkgt ofss o mg ugkamro rmj (ofss O fgj ilg th )

Umiutkmg 2 jA ? amrel mg lilolgt kg hmrkzmgtfi jkrletkmg ? >)fx(

ojO@

whlrl jO ?

Ojx.

A ? jA ?

4

>)fx(

jxoO.@?

oO.@

4

>)fx(

jx

?

Oo.@

f

6

)f(

6

?

f)f(

@Oo

Lxfopil


4/26

TH[UKEU

"ofgkshcuofrphyskes.kg" :

Umiutkmg 2 Ilt M nl thl elgtrl ma thl sphlrl fgj M' thft ma thl hmiimw (ak`url).

Amr fg lxtlrgfi pmkgt thl sphlrl nlhfvls fs ka kts lgtkrl ofss ks emgelgtrftlj

ft kts elgtrl. Zhlrlamrl, thl `rfvktftkmgfi amrel mg f ofss dod ft T jul tm

thl

mrk`kgfi sphlrl (ma ofss O) ks ^

M

M' T

r

A ? @ >r

Oo, fimg` TM.

Zhl jkfoltlr ma thl sofiilr sphlrl (whkeh wmuij nl eut maa) ks ^, sm thft kts rfjkus MM' ks />. Zhl amrel mgo ft T jul tm thks sphlrl ma ofss O' (sfy) wmuij nl

A? @ >)r(

oO

>^

fimg` TM. Sjkstfgel TM? r

>^

_

Fs thl rfjkus ma thks sphlrl ks hfia ma thft ma thl mrk`kgfi sphlrl, wl hfvl

O ?8O

.

A? @>

)r(8

oO

>^

fimg` TM.

Fs nmth A fgj Apmkgt fimg` thl sfol jkrletkmg, thl amrel jul tm thl hmiimwlj sphlrl ks

A A? >r

Oo@

>> )r>

^6(r8

@Oo

?

>>)6(8

66

r

Oo@

r>^

.

5 . @^FQKZFZKMGFI AKLIJZhl spfel surrmugjkg` thl nmjy wkthkg whkeh kts `rfvktftkmgfi amrel ma fttrfetkmg ks lxplrklgelj ny mthlr

nmjkls ks efiilj `rfvktftkmgfi aklij. @rfvktftkmgfi aklij ks vlry skokifr tm liletrke aklij kg liletrmstftkes

whlrl ehfr`l 'q' ks rlpifelj ny ofss 'o' fgj liletrke emgstfgt 'C' ks rlpifelj ny `rfvktftkmgfi emgstfgt

'@'. Zhl kgtlgskty ma `rfvktftkmgfi aklij ft f pmkgt ks jlakglj fs thl amrel lxplrklgelj ny f ugkt ofss

pifelj ft thft pmkgt.

o

AL

Zhl ugkt ma thl kgtlgskty ma `rfvktftkmgfi aklij ks G c`6.

Kgtlgskty ma `rfvktftkmgfi aklij jul tm pmkgt ofss 2

Zhl amrel jul tm ofss o mg tlst ofss o4pifelj ft pmkgt T ks `kvlg ny 2

A ?>

4

r

@Oo

Hlgel L ?4o

A

>r

@OL

Kg vletmr amro rr

@OL

>

Jkolgskmgfi amrouif ma kgtlgskty ma `rfvktftkmgfi aklij ? _IZOS_OS

_OIZS

o

A >4>

Lxfopil 1.

Akgj thl jkstfgel ma f pmkgt armo thl lfrths elgtrl whlrl thl rlsuitfgt `rfvktftkmgfi aklij jul tm thl

lfrth fgj thl ommg ks zlrm. Zhl ofss ma thl lfrth ks 1.4 64>: c` fgj thft ma thl ommg ks 0.: 64>> c`.

Zhl jkstfgel nltwllg thl lfrth fgj thl ommg ks :.4 64< co.


5/26

TH[UKEU

"ofgkshcuofrphyskes.kg" l

x

@O?

>

>:

x

c`641@

fgj of`gktujl ma thl `rfvktftkmgfi aklij jul tm thl ommg ks

L>

? >>

)xco644.:(

c`64:.0@

Zhlsl aklijs frl kg mppmsktl jkrletkmgs. Amr thl rlsuitfgt aklij tm nl zlrm L6? L

>.

mr,>

>:

x

c`641 ? >>

)xco644.:(

c`64:.0

mr,xco644.:

x<

?>>

>:

64:.0

641

? ;

mr, x ? 5.1 64< co.

Lxfopil 0.

Efieuiftl `rfvktftkmgfi aklij kgtlgskty jul tm f ugkamro rkg` ma ofss O fgj rfjkus ^ ft f jkstfgel x mg thl

fxks armo elgtlr ma rkg`.

Umiutkmg 2

Emgskjlr fgy pfrtkeil ma ofss jo. @rfvktftkmgfi aklij ft pmkgt T jul tm jo

jL ? >r

jo@fimg` TFF

Emopmglgt fimg` TM ks

jL ems ? >r

jo@ems

Glt `rfvktftkmgfi aklij ft pmkgt T ks

L ? >rjo@

ems ? >r

ems@ jo

? >/5>> x^

@Or

tmwfrjs thl elgtlr ma rkg`

Lxfopil 8.

Efieuiftl `rfvktftkmgfi aklij kgtlgskty ft f jkstfgel x mg thl fxks armo elgtrl ma f ugkamro jkse ma ofss O

fgj rfjkus .


6/26

TH[UKEU


Umiutkmg 2 Emgskjlr f lilolgtfi rkg` ma rfjkus r fgj thkecglss jr mg surafel ma jkse fs shmwg kg ak`url

x

r

jr

(O, ^)Jkse

jL T

@rfvktftkmgfi aklij jul tm lilolgtfi rkg`

jL ? >/5>> )rx(

Ox@j

Hlrl jO ? >^

O

. >rjr ? rjr

^

O>>

jL ? >/5>>> )rx(^

Oxrjx>.@

L ?

^

4

>^

@Ox>>/5>> )rx(

rjr

L ? >^

>@Ox

>> ^x

6

x

6

Lxfopil ;.

Amr f `kvlg ugkamro sphlrkefi shlii ma ofss O fgj rfjkus ^, akgj `rfvktftkmgfi aklij ft f jkstfgel r armo elgtrl

kg amiimwkg` twm efsls (f) r ^ (n) r 9

Umiutkmg 2

T

^kg` ma rfjkus ? ^skg

j

r

^j

jL ? >@jO

. ems ^r

jO ? >^:

O

x >^ skg^j

jO ?>

Oskgj

jL ? >>

jemsskg@O

Gmw > ? ^> + r> >r^ ems ..............(6)^ > ? > + r> >r ems ..............(>)

ems? r>^r

>>>

ems ?r^>

r^ >>>


7/26

TH[UKEU


jkaalrlgtkftkg` (6)

> j ? >r^ skgj

jL ? >>

@O

.

^r

j.

)r>(

^r >>>

jL ? >^r:

@O

>

>> ^r6

j

L ? jL ? >^r:

@O

^r

^r >

>>^r

-^r

j)^r(j

.

L ? >r

@O, ^r

Ka pmkgt ks kgskjl thl shlii ikokt ehfg`ls tm S (^r) tm ^ + r _

L ? 4 whlg r 9 ^.

Lxfopil 64.

Akgj thl rliftkmg nltwllg thl `rfvktftkmgfi aklij mg thl surafel ma twm pifglts F & N ma ofssls oF

, oN

&

rfjkus F

& ^N

rlspletkvliy ka

(k) thly hfvl lqufi ofss

(kk) thlyhfvl lqufi (ugkamro) jlgskty

Umiutkmg 2 Ilt LF

& LN

nl thl `rfvktftkmgfi aklij kgtlgsktkls mg thl surafel ma pifglts F & N.

thlg, LF

?>F

F

5F

>F

F

^

^5

:@

^

@o

? FF^5@:

Ukokifriy,>N

NN

^

@oL ?

5

@:

N^

N

(k) amr oF

? oN

N

F

L

L? >

F

>N

^

^

(kk) Amr &F

? N

N

F

N

F

^

^

L

L

: . @^FQKZFZKMGFI TMZLGZKFIZhl `rfvktftkmgfi pmtlgtkfi ft f pmkgt kg thl `rfvktftkmgfi aklij ma f nmjy ks jlakglj fs thl fomugt ma wmrc

jmgl ny fg lxtlrgfi f`lgt kg nrkg`kg` f nmjy ma ugkt ofss armo kgakgkty tm thft pmkgt, simwiy (gm ehfg`l

kg ckgltke lglr`y). @rfvktftkmgfi pmtlgtkfi ks vlry skokifr tm liletrke pmtlgtkfi kg liletrmstftkes.

@rfvktftkmgfi pmtlgtkfi jul tm f pmkgt ofss 2

Ilt thl ugkt ofss nl jkspifelj thrmu`h f jkstfgel jr tmwfrjs ofss O,

thlg wmrc jmgl ks `kvlg ny

jP ? A jr ? jrr

@O>

Zmtfi wmrc jmgl kg jkspifekg` thl pfrtkeil armo kgakgkty tm pmkgt T ks

P ?r

@Ojr

r

@OjP

r

>

.

Zhus `rfvktftkmgfi pmtlgtkfi,r

@OQ .

Zhl ugkt ma `rfvktftkmgfi pmtlgtkfi ks B c`6. Jkolgskmgfi Amrouif ma `rfvktftkmgfi pmtlgtkfi

?_OS

_ZOIS

ofss

Pmrc >> ? SOI> Z>_.


8/26

TH[UKEU


Lxfopil 66.

Akgj mut pmtlgtkfi ft T fgj V jul tm thl twm pmkgt ofss systlo. Akgj mut wmrc jmgl ny lxtlrgfi f`lgt kg

nrkg`kg` ugkt ofss armo T tm V. Fism akgj wmrc jmgl ny `rfvktftkmgfi amrel.

Umiutkmg 2 (k) QT6

? pmtlgtkfi ft T jul tm ofss o ft 6 ?

@o

QT>

?

@o

QT

? QT6

+ QT>

?

@o>

(kk)>/

@OQ 6V

>/

@oQ >V

QV

? QV6

+ QV>

?

>/

@o

>/

@o

?

@o:

Amrel ft pmkgt V ? 4

(kkk) wmrc jmgl ny lxtlrgfi f`lgt ? (QV

QT

) 6 ?

@O>

(kv) wmrc jmgl ny rfvktftkmgfi amrel ? QT

QV

?

@O>

Lxfopil 6>.

Akgj pmtlgtkfi ft f pmkgt T ft f jkstfgel x mg thl fxks fwfy armo elgtrl ma f ugkamro rkg` ma ofss O fgj

rfjkus ^.

Umiutkmg 2

^kg` efg nl emgskjlrlj tm nl ofjl ma ifr`l guonlr ma pmkgt ofssls (o6, o

>..........lte)

QT

? .......x^

@o

x^

@o

>>

>

>>

6

>>>6

>> x^@O.....)oo(

x^@

, whlrl O ? o

6+ o

>+ o

5 + ............

Tmtlgtkfi ft elgtrl ma rkg` ? ^

O.@

< . ^LIFZKMG NLZPLLG @^FQKZFZKMGFI AKLIJ FGJ TMZLGZKFIZhl wmrc jmgl ny fg lxtlrgfi f`lgt tm omvl ugkt ofss armo f pmkgt tm fgmthlr pmkgt kg thl jkrletkmg ma thl

aklij L, simwiy thrmu`h fg kgakgktlskofi jkstfgel jr ? Amrel ny lxtlrgfi f`lgt jkstfgel omvlj ? Ljr.

Zhus jQ ? Ljr L ? jrjQ .

Zhlrlamrl, `rfvktftkmgfi aklij ft fgy pmkgt ks lqufi tm thl gl`ftkvl `rfjklgt ft thft pmkgt.


9/26

TH[UKEU

"ofgkshcuofrphyskes.kg" ;

Lxfopil 65.

Zhl `rfvktftkmgfi aklij kg f rl`kmg ks `kvlg ny L

? (>4G/c`) )bk( . Akgj thl `rfvktftkmgfi pmtlgtkfi ft thl

mrk`kg (4, 4) (kg B/c`)

(F) zlrm (N) >4 > (E) >4 > (J) efg gmt nl jlakglj

Umiutkmg 2 Q ? jr.L ? jy.Lyjx.Lx ? >4x + >4yft mrk`kg Q ? 4Fgs. (F)

Lxfopil 6:.

Kg fnmvl prmnilo, akgj thl rfvktftkmgfi pmtlgtkfi ft f pmkgt whmsl em-mrjkgftls frl (4 < + >4 : ? 684 B/c`

Fgs. (N)

Lxfopil 6 + 5y> + zx,

Akgj `rfvktftkmgfi aklij ft f pmkgt (x, y, z).

Umiutkmg 2 Lx?

x

v

? :x z

Ly? ny

Lz

? x

Aklij ? L

? S (:x +z) k + ny b

+ x c _

L

? Q..

1 . @^FQKZFZKMGFI TMZLGZKFI AKLIJ AM^ JKAAL^LGZ MNBLEZUK. ^kg`. Q ? >/6>> )rf(

@Omr

x

@O

& L ? r

)rf(

r@O>/5>>

mr L ? >x

ems@O

@rfvktftkmgfi aklij ks ofxkouo ft f jkstfgel, r ? >f fgj kt ks >f55@O>

KK. F ikglfr ofss ma akgktl ilg`th mg kts fxks 2


10/26

TH[UKEU


(f) Tmtlgtkfi 2

Q ? I

@Og (sle

4+ tfg

4) ?

I

@Oig

j

jII >>

(n) Aklij kgtlgskty 2

L ? Ij

@Oskg

4 ? >> jIj

@O

KKK . Fg kgakgktl ugkamro ikglfr ofss jkstrknutkmg ma ikglfr ofss jlgskty , Hlrl4?

>

.

Fgj gmtkg` thft?I>

Okg efsl ma f akgktl rmj

wl `lt, amr aklij kgtlgskty L ?j

@>

Tmtlgtkfi amr f ofss-jkstrknutkmg lxtlgjkg` tm kgakgkty ks gmt jlakglj. Hmwlvlr lvlg amr sueh ofss

jkstrknutkmgs pmtlgtkfi-jkaalrlgel ks jlakglj. Hlrl pmtlgtkfi jkaalrlgel nltwllg pmkgts T6

fgj T>

rlspletkvliy ft jkstfgels j6fgj j

> armo thl kgakgktl rmj, v

6>? >@g

6

>

j

j

KQ. \gkamro Umikj Uphlrl

(f) Tmkgt T kgskjl thl shlii. r 9 f, thlg

Q ? )rf5(f>

@O >>5

& L ? 5f

r@O, fgj ft thl elgtrl Q ?

f>

@O5fgj L ? 4

(n) Tmkgt T mutskjl thl shlii. r 7 f, thlg Q ?r

@O & L ? >r

@O

Q. \gkamro Zhkg Uphlrkefi Uhlii

(f) Tmkgt T Kgskjl thl shlii. r 9 f , thlg Q ?f

@O& L ? 4


11/26

TH[UKEU


(n) Tmkgt T mutskjl shlii. r 7 f, thlg Q ?r

@O& L ? >r

@O

QK. \gkamro Zhkec Uphlrkefi Uhlii

(f) Tmkgt mutskjl thl shlii

Q ? @r

O3 L ? @ >r

O

(n) Tmkgt kgskjl thl Uhlii

Q ? >

5@O

>6>6

>>

6>

^^^^

^^

L ? 4

(e) Tmkgt nltwllg thl twm surafel

Q ? r>

@O

56

5>

56

5>>

^^

^>rr^53

L ? >r

@O56

5>

56

5

^^

^r

0. @^FQKZFZKMGFI TMZLGZKFI LGL^@[@rfvktftkmgfi pmtlgtkfi lglr`y ma twm ofss systlo ks lqufi tm thl wmrc jmgl ny fg lxtlrgfi f`lgt kg

fsslonikg` thlo, whkil thlkr kgktkfi slpfrftkmg wfs kgakgkty. Emgskjlr f nmjy ma ofss o pifelj ft f

jkstfgel r armo fgmthlr nmjy ma ofss O. Zhl `rfvktftkmgfi amrel ma fttrfetkmg nltwllg thlo ks `kvlg ny,

A ? >r

o@O.

Gmw, Ilt thl nmjy ma ofss o ks jkspifelj armo pmkgt. E tm N thrmu`h f

jkstfgel 'jr' tmwfrjs thl ofss O, thlg wmrc jmgl ny kgtlrgfi emgslrvftkvlamrel (`rfvktftkmgfi) ks `kvlg ny,

jP ? A jr ?>r

o@Ojr jP

r

>r

o@Ojr

@rfvktftkmgfi pmtlgtkfi lglr`y,r

@Oo\


12/26

TH[UKEU

"ofgkshcuofrphyskes.kg" 6>

Kgerlfsl kg `rfvktftkmgfi pmtlgtkfi lglr`y 2

Uuppmsl f nimec ma ofss o mg thl surafel ma thl lfrth. Pl wfgt tm ikat thks nimec ny h hlk`ht.

Pmrc rlqukrlj kg thks prmelss ? kgerlfsl kg T.L. ? \a- \

k? o(Q

a- Q

k)

Plxt

?\ ? (o)

l

l

l

l

^

@O

h^

@O

Plxt

? \ ? @Olo

h^

6

^

6

ll?

l

l

^

o@O

6

l^

h66

(fs h 99 ^l, wl efg fppiy Nkmgmokefi thlmrlo)

Plxt

? \ ?l

l

^

o@O

l^

h66 ? (o)

>

l

l

^

@Oh

Plxt

? \ ? o`h

* Zhks amrouif ks vfikj mgiy whlg h 99 ^l

Lxfopil 60.

F nmjy ma ofss o ks pifelj mg thl surafel ma lfrth. Akgj wmrc rlqukrlj tm ikat thks nmjy ny f hlk`ht

(k) h ?6444

^l(kk) h ? ^

l

Umiutkmg 2

(k) h ?6444

^l, fs h 99 ^

l, sm

wl efg fppiy

Plxt

? \ ? o`h

Plxt

? (o)

>

l

l

^

@O

6444

^l?

l

l

^6444

o@O

(kk) h ? l, kg thks efsl h ks gmt vlry ilss thfg ^l, sm wl efggmt fppiy\ ? o`hsm wl efggmt fppiy\ ? o`h

Plxt

? \ ? \a- \

k? o(Q

a- Q

k)


13/26

TH[UKEU


Plxt

? o

l

l

ll

l

^

@O

^^

@O

Plxt

?l

l

^>

o@O

Lxfopil 68.

Efieuiftl thl vlimekty wkth whkeh f nmjy oust nl thrmwg vlrtkefiiy upwfrj armo thl surafel ma thl lfrth sm

thft kt ofy rlfeh f hlk`ht ma 64 ^, whlrl ^ ks thl rfjkus ma thl lfrth fgj ks lqufi tm 1.: 64 8 o. (Lfrth's

ofss ? 1 64>: c`, @rfvktftkmgfi emgstfgt @ ? 1.0 6466 G-o>/c`>)

Umiutkmg 2 Zhl `rfvktftkmgfi pmtlgtkfi lglr`y ma f nmjy ma ofss o mg lfrth's surafel ks

\ (^) ? ^

Oo@

whlrl O ks thl ofss ma thl lfrth (suppmslj tm nl emgelgtrftlj ft kts elgtrl) fgj ^ ks thl rfjkus ma thl lfrth

(jkstfgel ma thl pfrtkeil armo thl elgtrl ma thl lfrth). Zhl `rfvktftkmgfi lglr`y ma thl sfol nmjy ft f hlk ht

64 ^ armo lfrth's surafel, k.l. ft f jkstfgel 66^ armo lfrth's elgtrl ks

\ (66 ^) ?

^

Oo@

ehfg`l kg pmtlgtkfi lglr y \ (66 ^) \(^) ? ^66

Oo@

^

@Oo?

^

@Oo

66

64

Zhks jkaalrlgel oust emol armo thl kgktkfi ckgltke lglr y `kvlg tm thl nmjy kg slgjkg` kt tm thft hlk`ht. Gmw,

suppmsl thl nmjy ks thrmwg up wkth f vlrtkefi spllj v, sm thft kts kgktkfi ckgltke lglr`y ks>

6ov>. Zhlg

>

6ov> ?

^

@Oo

66

64mr v ?

^

@Oo

66

>4.

Tuttkg` thl `kvlg vfiuls 2 v ?

)o64:.1(66

)c`641()c`/oG640.1(>4

1

>:>>66

? 6.40 64

:

o/s.

Lxfopil 6;.

Jkstfgel nltwllg elgtrls ma twm stfrs ks 64 f. Zhl ofssls ma thlsl stfrs frl O fgj 61 O fgj thlkr rfjkk frl

f & >f rlspletkvliy. F nmjy ks akrlj strfk`ht armo thl surafel ma thl ifr`lr stfr tmwfrjs thl sofiilr stfr. Phft

shmuij nl kts okgkouo kgktkfi spllj tm rlfeh thl surafel ma thl sofiilr stfr=

Umiutkmg 2 Ilt T nl thl pmkgt mg thl ikgl bmkgkg` thl elgtrls ma thl twm pifglts s.t. thl glt aklij ft kt ks zlrm

Zhlg, >> )rf64(O61.@

r@O

? 4 (64 fr)> ? 6 1 r >

64f r ? :r r ? >f

Tmtlgtkfi ft pmkgt T, vT

?)rf64(

O61.@

r

@O

?

f>

@O

f>

@O

.

Gmw ka thl pfrtkeil prmbletlj armo thl ifr`lr pifglt hfs lgmu`h lglr`y tm ermss thks pmkgt, kt wkii rlfeh thl

sofiilr pifglt.

Amr thks, thl C.L. kopfrtlj tm thl nmjy oust nl bust lgmu`h tm rfksl kts tmtfi olehfgkefi lglr`y tm f vfiul

whkeh ks lqufi tm T.L. ft pmkgt T.

k.l. f8

@Oo

f>

o)O61(@ov>

6 >

? ovT


14/26

TH[UKEU

"ofgkshcuofrphyskes.kg" 6:

mr,f>

@Oo

v >

mr, v> ?f:

@O:

6

gb

kb6b kb

bk

r

oo

Ka emgskjlr f systlo ma 'g' pfrtkeils, lfeh ma sfol ofss 'o' fgj slpfrftlj armo lfeh mthlr ny thl sfol

fvlrf`l jkstfgel 'r', thlg slia lglr`y

mr \s?

g

6k

@>

6

g

kb6b bk

>

r

o

Zhus mg thl rk`ht hfgj skjl 'k' emols 'g' tkols whkil 'b' emols (g 6) tkols. Zhus

\s?

>

6@g (g 6)

r

o>

@rfvktftkmgfi Ulia lglr`y ma f \gkamro Uphlrl (stfr)

\sphlrl

? @

r

jrr:r5

: >5

whlrl ?

5^5

:

O

? 5

6@ (:)> r: jr,,

\stfr

? 5

6@ (:)>

^

4

: jrr ? 5

6@ (:)>

^

4

ll

h^@O

? `>

l^

h6

~ `

l^

h>6 whlg h 99 ^.

Zhl jlerlfsl kg thl vfiul ma '`' wkth hlk`ht h ? ` `h?

l^

`h>. Zhlg plrelgtf`l jlerlfsl kg thl vfiul ma

'`' ? %644^

h>644

`

``

l

h

(n) Laalet ma jlpth

Zhl `rfvktftkmgfi puii mg thl surafel ks lqufi tm kts wlk`ht k.l. o` ? >l

l

^

o@O

o` ?>l

5l

^

o^5

:@

mr ` ?5

: @ ^

l .........(6)

Phlg thl nmjy ks tfclg tm f jlpth j, thl ofss ma thl sphlrl ma rfjkus (^l j) wkii mgiy nl laaletkvl amr

thl `rfvktftkmgfi puii fgj thl mutwfrj shfii wkii hfvl gm rlsuitfgt laalet mg thl ofss. Ka thl feelilrftkmg

jul tm `rfvkty mg thl surafel ma thl smikj sphlrl ks `j, thlg

`j?

5

:@ (^

l j) ...............(>)

Ny jkvkjkg` lquftkmg (>) ny lquftkmg (6)

`j

? `

l^

j6

KOTM^ZFGZ TMKGZU(k) Ft thl elgtlr ma thl lfrth, j ? ^

l, sm `

elgtrl? `

l

l

^

^6 ? 4.

Zhus wlk`ht (o`) ma thl nmjy ft thl elgtrl ma thl lfrth ks zlrm.

(kk) Tlrelgtf`l jlerlfsl kg thl vfiul ma '`' wkth thl jlpth

? 644`

`` j

? 644

^

j

l

.


16/26

TH[UKEU


(e) Laalet ma thl surafel ma Lfrth

Zhl lquftmrkfi rfjkus ks fnmut >6 co img`lr thfg kts pmifr rfjkus.

Pl cgmw, ` ?>l

l

^

@OHlgel `

pmil7 `

lquftmr. Zhl wlk`ht ma thl

nmjy kgerlfsl fs thl nmjy tfclg armo thl lquftmr tm thl pmil.

(j) Laalet ma rmtftkmg ma thl Lfrth

Zhl lfrth rmtftls frmugj kts fxks wkth fgùifr vlimekty . Emgskjlr f pfrtkeil ma ofss o ftiftktujl. Zhl fgùifr vlimekty ma thl pfrtkeil ks fism .

Feemrjkg` tm pfrfiilim`rfo ifw ma vletmr fjjktkmg, thl rlsuitfgt amrel fetkg` mg ofs s o fimg`

TV ks

A ? S(o`)> + (o> ^lems)> + {>o` o> ^

lems} ems (684 )_6/>

? S(o`)> + (o> ^l

ems)> (>o > `> ^lems) ems_6/>

? o`

>/6

>>

l>

>>

l ems

`

^>ems

`

^6

Ft pmil ? ;4 `pmil

? ` , Ft lquftmr ? 4 `lquftmr

? `

`

^6

>l

.

Hlgel `pmil

7 `lquftmr

Ka thl nmjy ks tfclg armo pmil tm thl lquftmr, thlg `? `

`

^6

>l .

Hlgel % ehfg`l kg wlk`ht ? 644`

^644

o`

ô644

o`

`

^6oò`

>l

>l

>l

66. LUEFTL UTLLJZhl okgkouo spllj rlqukrlj tm slgj f nmjy mut ma thl `rfvkty aklij ma f pifglt (slgj kt tm r )

66.6 Lsefpl spllj ft lfrth s surafel 2Uuppmsl f pfrtkeil ma ofss o ks mg lfrth's surafel

Pl prmblet kt wkth f vlimekty Q armo thl lfrth's

surafel, sm thft kt bust rlfehls r (ft r ,kts vlimekty nlemol zlrm)

Fppiykg` lglr`y emgslrvftkmg nltwllg kgktkfi

pmsktkmg (whlg thl pfrtkeil wfs ft lfrth's surafel)fgj akgj pmsktkmgs (whlg thl pfrtkeil bust rlfehls

tm r )


17/26

TH[UKEU


Ck + \k ? Ca+ \a

>

6ov> + o4

^

@Ol? 4 + o4

)r(

@Ol

v ?^

@O> 4

Lsefpl spllj armo lfrth ks surafel^

@O>v ll

Ka wl put thl vfiuls ma @, Ol, ^ thl wl `lt

Ql? 66.> co/s.

66.> Lsefpl spllj jlplgjs mg 2(k) Ofss (Ol) fgj skzl (^) ma thl pifglt

(kk) Tmsktkmg armo whlrl thl pfrtkeil ks prmbletlj.

66.5 Lsefpl spllj jmls gmt jlplgj mg 2(k) Ofss ma thl nmjy whkeh ks prmbletlj (o

4)

(kk) Fg`il ma prmbletkmg.

Ka f nmjy ks thrmwg armo Lfrth's surafel wkth lsefpl spllj, kt `mls mut ma lfrth's `rfvktftkmgfi aklij fgj glvlr

rlturgs tm thl lfrth's surafel. Nut kt stfrts rlvmivkg` frmugj thl sug.

Lxfopil >4.

F vlry sofii `rmmvl ks ofjl kg thl lfrth, fgj f pfrtkeil ma ofss o4 ks

pifelj ft ^/> jkstfgel armo thl elgtrl. Akgj thl lsefpl spllj ma thl pfr

tkeil armo thft pifel.

Umiutkmg 2

Uuppmsl wl prmblet thl pfrtkeil wkth spllj v, sm thft kt bust rlfehls ft (r ).Fppiykg` lglr`y emgslrvftkmg

Ck + \k ? Ca+ \a

>

6o4v

> + o4

>>

5

l

>

^^5(

^>

@O? 4 + 4

v ?^:

@O66 l? Qlft thft pmsktkmg.

Lxfopil >6.

Akgj rfjkus ma sueh pifglt mg whkeh thl ofg lsefpls thrmu`h buopkg`. Zhl efpfektyma buopkg` ma plrsmg mglfrth ks 6.< o. Jlgskty ma pifglt ks sfol fs thft ma lfrth.

Umiutkmg 2 Amr f pifglt 2>

6ov>

p

T

^

o@O? 4

>

6ov > ?

p

T

^

o@O

Mg lfrth >

6ov > ? o

>L

L

^

@Oh

p

T

^

o@O? >

L

L

^

o.@O. h

p

p

^

O? >

L

L

^

hO

Jlgskty () ks sfol p

5p

^^:/5 ? >

L

>L

^h^:/5 ^T ? h^L


18/26

TH[UKEU


6>. CLTILÛ IFP AM^ TIFGLZF^[ OMZKMGUuppmsl f pifglt ks rlvmivkg` frmugj thl sug, mr f sftliiktl ks rlvmivkg` frmugj thl lfrth, thlg thl pifg-

ltfry omtkmg efg nl stujklj wkth hlip ma Clpilrs thrll ifws.

6>.6 Clpilrs Ifw ma mrnktLfeh pifglt omvls frmugj thl sug kg f ekreuifr pfth mr liikptkefi pfth wkth thl sug ft kts ameus. (Kg afet

ekreuifr pfth ks f sunslt ma liikptkefi pfth)

U\G

Tifglt

r

9

9

9

9

6>.> Ifw ma frlfi vlimekty 2Zm ugjlrstfgj thks ifw, ilts ugjlrstfgj thl fgùifr omolgtuo emgslrvftkmg amr thl pifglt.

Ka f pifglt omvls kg fg liikptkefi mrnkt, thl `rfvktftkmg amrel fetkg` mg kt fiwfys pfssls thrmu`h thl elgtrl ma

thl sug. Um tmrqul ma thks `rfvktftkmg amrel fnmut thl elgtrl ma thl sug wkii nl zlrm. Hlgel wl efg sfy

thft fgùifr omolgtuo ma thl pifglt fnmut thl elgtrl ma thl sug wkii rlofkg emgslrvlj (emgstfgt)

fnmut thl sug ? 4

jt

jB? 4 B

pifglt/ sug ? emgstfgt ovr skg? emgstfgt

Gmw wl efg lfskiy stujy thl Clpilr's ifw ma flrkfi vlimekty.Ka f pifglt omvls frmugj thl sug, thl rfjkus vletmr ( r

) fism rmtftls frl swllps frlf fs shmwg kg akùrl.

Gmw ilts akgj rftl ma frlf swlpt ny thl rfjkus vletmr ( r

).

Uuppmsl f pifglt ks rlvmivkg` frmugj thl sug fgj ft fgy kgstfgt kts vlimekty ks v, fgj fgìl nltwllg rfjkus

vletmr ( r

) fgj vlimekty ( v

). Kg jt tkol, kt omvls ny f jkstfgel vjt, jurkg` thks jt tkol, frlf swlpt ny thl

rfjkus vletmr wkii nl MFN whkeh efg nl fssuolj tm nl f trkfgìl

.


19/26

TH[UKEU

"ofgkshcuofrphyskes.kg" 6;

jF ? 6/> (Nfsl) (Tlrplgjkeuifr hlk ht)

jF ? 6/> (r) (vjtskg)

sm rftl ma frlf swlptjt

jF?

>

6vr skg

wl efg wrktljt

jF?

o

skgovr

>

6

whlrl ovr skg ? fg`uifr omolgtuo ma thl pifglt fnmut thl sug, whkeh rlofkgs emgslrvlj (emgstfgt)

jt

jF?

o>

I sug/pifglt? emgstfgt

sm ^ftl ma frlf swlpt ny thl rfjkus vletmr ks emg stfgt

Lxfopil >>.

Uuppmsl f pifglt ks rlvmivkg` frmugj thl sug kg fg liikptkefi pfth `kvlg ny >

>

>

>

n

y

f

x

? 6

Akgj tkol plrkmj ma rlvmiutkmg. Fg`uifr omolgtuo ma thl pifglt fnmut thl sug ks I.

Umiutkmg 2

^ftl ma frlf swlpto>

I

jt

jF ? emgstfgt

jF ? jto>

I

fnF

4F

jF ?

Zt

4t

jto>

I

fn ?o>

IZ Z ?

I

ofn>

6>.5 Clpilrs ifw ma tkol plrkmj 2Uuppmsl f pifglt ks rlvmivkg` frmugj thl sug kg ekreuifr mrnkt

99

U\G

v

ro4

Os

>

4s

r

o@O

thlg>

4s>

4

r

o@O

r

vo

v ?r

@Os

Zkol plrkmj ma rlvmiutkmg ks


20/26

TH[UKEU

"ofgkshcuofrphyskes.kg" >4

Z ?v

r>? >r

s@O

r

Z> ?

s

>

@O

:r5

Z>

r5

Amr fii thl pifglt ma f sug , Z > r5

Lxfopil >5.

Zhl Lfrth fgj Bupktlr frl twm pifglts ma thl sug. Zhl mrnktfi rfjkus ma thl lfrth ks 640 o fgj thft ma Bupktlr ks

: 6 40 o. Ka thl tkol plrkmj ma rlvmiutkmg ma lfrth ks Z ? 51< jfys, akgj tkol plrkmj ma rlvmiutkmg ma thl Bupktlr.

U\G

Bupktlr

Lfrth

9

9

9

9

r ? : 64 o> 0

r ? 64 o6

0

Z ? 51< jfys6

Z ? =>

Umiutkmg 2

Amr nmth thl pifglts

Z> r5

>

lfrth

bupktlr

Z

Z

?

5

lfrth

bupktlr

r

Z

>

bupktlr

jfys51

@rfph ma im` Z v/s im` ^ 2-

5

s

>

> ^@O

:Z


21/26

TH[UKEU


>im` Z ? im`

s

>

@O

:+ 5im` ^

im` Z ? ^im`>

5

@O

:im`

>

6

s

>

im` Z

im` ^

s

>

@O

:im`

>

6E

>

5simpl

* Ka pifglts frl omvkg` kg liikptkefi mrnkt, thlgZ> f5 whlrl f ? slok ofbmr fxks ma thl liikptkefi pfth.

Lxfopil >:.

F sftliiktl ks ifugehlj kgtm f ekreuifr mrnkt 6144 co fnmvl thl surafel ma thl lfrth. Akgj thl plrkmj ma

rlvmiutkmg ka thl rfjkus ma thl lfrth ks ^ ? 1:44 co fgj thl feelilrftkmg jul tm `rfvkty ks ;.8 o/sle>. Ft whft

hlk`ht armo thl `rmugj shmuij kt nl ifugehlj sm thft kt ofy fpplfr stftkmgfry mvlr f pmkgt mg thl lfrth's

lquftmr=

Umiutkmg2 Zhl mrnktkg` plrkmj ma f sftliiktl ft f hlk`ht h armo lfrth's surafel ks Z ? >

>/5

`^

r>whlrl r ? ^ + h

thlg, Z ?

`

h^

^

)h^(>

Hlrl, ^ ? 1:44 co, h ? 6144 co ? ^/:.

Zhlg Z ?

`

^

^

^>:

^

:

^

? >(`

^

Tuttkg` thl `kvlg vfiuls 2 Z ? > 5.6:

>

1

s/o8.;

o64:.1(6.> ? 04;> sle ? 6.;0 hmurs

Gmw, f sftliiktl wkii fpplfr stftkmgfry kg thl scy mvlr f pmkgt mg thl lfrth's lquftmr ka kts plrkmj ma rlvmiutkmg

rmugj thl lfrth ks lqufi tm thl plrkmj ma rlvmiutkmg ma thl lfrth rmugj kts mwg fxks whkeh ks >: hmurs. Ilt us

akgj thl hlk`ht h ma sueh f sftliiktl fnmvl thl lfrth's surafel kg tlros ma thl lfrth's rfjkus. Ilt kt nl g . thlg

Z ?

`

g^^

^

)^g^(>?

`

^> (6 + g)5./> ? > 5.6:

>

1

sle/oltlr8.;

sle/oltlr64:.1(6 + g)5/>

? ( ? (6.:6hmurs) (6 + g)5/>

Amr Z ? >: hmurs, wl hfvl

(>: hmurs) ? (6.:6) hmurs) (6 + g)5/>

mr (6 + g)5/> ?:6.6

>:? 60

mr 6 + g ? (60)>/5 ? 1.16 mr g ?


22/26

TH[UKEU

"ofgkshcuofrphyskes.kg" >>

65. EK^E\IF^ OMZKMG MA F UFZLIIKZL F^M\GJ F TIFGLZ

Uuppmsl ft sftliiktl ma ofss o4 ks ft f jkstfgel r armo f pifglt. Ka thl sftliiktl jmls gmt rlvmivl, thlg jul tm

thl `rfvktftkmgfi fttrfetkmg, kt ofy emiikjl tm thl pifglt.

Zm fvmkj thl emiikskmg, thl sftliiktl rlvmivl frmugj thl pifglt, amr ekreuifr omtkmg ma sftliiktl.

r

vo

r

o@O >4>

4l ....(6)

v ?r

@Olthks vlimekty ks efiilj mrnktfi vlimekty (v4)

v4? r

@Ol

65.6 Zmtfi lglr`y ma thl sftliiktl omvkg` kg ekreuifr mrnkt 2(k) CL ?

>

6o4v

> fgj armo lquftkmg (6)

r

vo >4 ? >4l

r

o@Oo4v

> ?r

o@O 4l CL ?r>

o@Ovo

>

6 4l>4

(kk) Tmtlgtkfi lglr`y

\ ? r

o@O 4l

Zmtfi lglr y ? CL + TL ?

r>

o@O 4l+

r

o@O 4l

ZL ? r>

o@O 4l

Zmtfi lglr`y ks vl. Kt shmws thft thl sftliiktl ks stkii nmugjlj wkth thl pifglt.

6: . @LM - UZFZKMGF^[ UFZLIIKZL 2Pl cgmw thft thl lfrth rmtftls fnmut kts fxks wkth

fg`uifr vlimekty lfrth

fgj tkol plrkmj Zlfrth

? >: hmurs.

Uuppmsl f sftliiktl ks slt kg fg mrnkt whkeh ks kg thl pifgl

ma thl lquftmr, whmsl ks lqufi tm lfrth

, (mr kts Z ks lqufi

tm Zlfrth

? >: hmurs) fgj jkrletkmg ks fism sfol fs thft ma

lfrth. Zhlg fs sllg armo lfrth, kt wkii fpplfr tm nl

stftkmglry. Zhks typl ma sftliiktl ks efiilj `lm- stftkmgfry

sftliiktl. Amr f `lm-stftkmgfry sftliiktl,

wsftliiktl

? wlfrth

Zsftliiktl ? Zlfrth? >: hr.

Um tkol plrkmj ma f `lm-stftkmglry sftliiktl oust nl >: hmurs. Zm fehklvl Z ? >: hmur, thl mrnktfi rfjkus `lm-

stftkmgfry sftliiktl 2


23/26

TH[UKEU


Z> ?

l

>

@O

:r5

Tuttkg` thl vfiuls, wl `lt mrnktfi rfjkus ma `lm stftkmgfry sftliiktl r ? 1.1 ^l(hlrl ^l ? rfjkus ma thl lfrth)

hlk`ht armo thl surafel h ?


24/26

TH[UKEU

"ofgkshcuofrphyskes.kg" >:

Trmnilo 6.

Efieuiftl thl amrel lxlrtlj ny pmkgt ofss o mg rmj ma

ugkamroiy jkstrknutlj ofss O fgj ilg`th (Tifelj fs

shmwg kg akùrl).

Umiutkmg 2

Jkrletkmg ma amrel ks ehfg kg` ft lvlry lilolgt. Pl hfvl tm ofcl

emopmglgts ma amrel fgj thlg kgtl`rftl.

Glt vlrtkefi amrel ? 4.

jA ? amrel mg lilolgt ?)fx(

o.jO.@>>

jAh? jA ems? amrel mg lilolgt kg hmrkzmgtfi jkrletkmg ?

)fx(

o.jO.@>> ems

Ah?

)fx(

jxemso.O.@

>> ?

o.O.@

>/

>/>> )fx(

jxems

?

>/

>/

>> sle

jxems

f

@Oo

whlrl x ? f tfg thlg jx ? f sle> . j

? >/

>/skgf

@Oo

f

xtfg , thlg

>> fx

xskg

?f

@Oo

>/

>/>> fx

x

?

>>

f:

f

@Oo

?

>>

f:

f

@Oo

Trmnilo >.Zhrll kjlgtkefi nmjkls ma ofss O frl imeftlj ft thl vlrtkels ma fg lqukiftlrfi trkfgìl wkth skjl I. Ft whft

spllj oust thly omvl ka thly fii rlvmivl ugjlr thl kgaiulgel ma mgl fgmthlr's `rfvkty kg f ekreuifr mrnkt

ekreuoserknkg` thl trkfg il whkil stkii prlslrvkg` thl lqukiftlrfi trkfg il =

Umiutkmg 2

Ilt F, N fgj E nl thl thrll ofssls fgj M thl elgtrl ma thl ekreuoserknkg` ekreil. Zhl rfjkus ma thks ekreil ks

^ ?>

Isle 54 ?

5

>

>

I ?

5

I.

Ilt v nl thl spllj ma lfeh ofss O fimg` thl ekreil. Ilt us emgskjlr thl omtkmg ma thl ofss ft F. Zhl amrel

ma `rfvktftkmgfi fttrfetkmg mg kt jul tm thl ofssls ft N fgj E frl

>

>

I

@O

fimg` FN fgj >

>

I

@O

fimg` FE

Zhl rlsuitfgt amrel ks thlrlamrl

>>

>

I

@Oems54 ?

>

>

I

@O5fimg` FJ.

Zhks, amr prlslrvkg` thl trkfgìl, oust nl lqufi tm thl glelssfry elgtrkpltfi amrel.

Zhft ks ,

>

>

I

@O5?

I

Ov5

^

Ov >> S^ ? I/ 5 _ mr v ?

I

@O

Trmnilo 5.

Kg f jmunil stfr, twm stfrs (mgl ma ofss o fgj thl mthlr ma >o) jkstfgt j fpfrt rmtftl fnmut thlkr emoomg

elgtrlma ofss. Jljuel fg lxprlsskmg ma thlplrkmj ma rlvmiutkmg. Uhmw thft thlrftkm ma thlkr fg uifr omolgtuo

fnmut thl elgtrl ma ofss ks thl sfol fs thl rftkm ma thlkr ckgltke lglr`kls.


25/26

TH[UKEU

"ofgkshcuofrphyskes.kg" >j/5 armo thl ofssls >o fgj o rlspletkvliy. Nmth thl

stfrs rmtftl rmugj E kg thlkr rlspletkvl mrnkts wkth thl sfol fgùifr vlimekty . Zhl `rfvktftkmgfi amrel fetkg`mg lfeh stfr jul tm thl mthlr suppikls thl glelssfry elgtrkpltfi amrel.

Zhl `rfvktftkmgfi amrel mg lkthlr stfr ks >j

o)o>(@. Ka wl emgskjlr thl rmtftkmg ma thl sofiilr stfr, thl elgtrkpltfi

amrel (o r>

) ks

>

5

j>o

fgj amr nk``lr stfr

5

oj> >

k.l. sfol

>j

o)o>(@?

>

5

j>o

mr ?

5j

@o5

E o

j/5 >j/5

>o

Zhlrlamrl, thl plrkmj ma rlvmiutkmg ks `kvlg ny Z ?>

? >

@o5

j5

Zhl rftkm ma thl fgùifr omolgtuo kssofii

nk`

sofii

nk`

)(

)(

? >

>

5

j>o

5

j)o>(

?>

6,

skgel ks sfol amr nmth. Zhl rftkm ma thlkr ckgltke lglr`kls kssofii

nk`

sofii>

nk`>

)(

)(

>6

>6

?

>

6,

whkeh ks thl sfol fs thl rftkm ma thlkr fgùifr omolgtuo.

Trmnilo :.

Amr f pfrtkeil prmbletlj kg f trfgsvlrsl jkrletkmg armo f hlk`ht h fnmvl Lfrths surafel, akgj thl okgkouo

kgktkfi vlimekty sm thft kt bust `rfzls thl surafel ma lfrth pfth ma thks pfrtkeil wmuij nl fg liikpsl wkth elgtlr ma

lfrth fs thl afrthlr ameus, pmkgt ma prmbletkmg fs thl fpm`ll fgj f jkfoltrkefiiy mppmsktl pmkgt mg lfrths

surafel fs plrk`ll.

Umi. Uuppmsl vlimekty ma prmbletkmg ft pmkgt F ks vF& ft pmkgt N,thl vlimekty ma thl pfrtkeil ks v

N.

thlg fppiykg` Glwtmgs >gj ifw ft pmkgt F & N, wl `lt,

F

>F

r

ov? >

l

)g^(

o@O

&

N

>N

r

ov? >

l

^

o@O

Phlrl rF

& rN

frl rfjkus ma eurvfturl ma thl mrnkt ft pmkgts F & N ma thl liikpsl,

nut rF

? rN

? r(sfy).

Gmw fppiykg` emgslrvftkmg ma lglr`y ft pmkgts F & N

>N

l>F

l ov>

6

^

o@Oov

>

6

h^

o@O

@Olo

)h^(

6

^

6?

>

6(ov

N> ov

F>) ?

>>l )h^(

6

^

6o@O>6

mr, r ?h^>

)h^(^>

?r^

^r>

Q

F> ? >

l

)h^(

@Or

? >@O

l )^r(r

^

whlrl r ? jkstfgel ma pmkgt ma prmbletkmg armo lfrths elgtrl ? ^ + h.

Trmnilo >4 vv ?

^

h6

h`>

,

whlrl ^ ks thl rfjkus ma thl lfrth fgj ` ks feelilrftkmg jul tm `rfvkty ft lfrth's surafel. Hlgel jljuel fg

lxprlsskmg amr ofxkouo hlk`ht rlfehlj ny f rmeclt akrlj wkth spllj 4.; tkols thl lsefpl vlimekty.


26/26

TH[UKEU

Umi. Zhl rfvktftkmgfi pmtlgtkfi lglr`y ma f ofss o mg lfrth's surafel fgj thft f hlk`ht h ks `kvlg ny

\ (^) ? ^

@Oofgj \ (^ + h) ?

h^

@Oo

\(^ + h) \(^) ? @Oo

^

6

h^

6?

^)h^(

@Ooh

? ^h6

hò

S @O ? `^>_

Zhks kgerlfsl kg pmtlgtkfi lglr`y meeurs ft thl emst ma ckgltke lglr`y whkeh emrrlspmgjkgìy jlerlfsls. Ka v

ks thl vlimekty ma thl rmeclt ft hlk`ht h, thlg thl jlerlfsl kg ckgltke lglr`y ks >>

4 ovov >6

>6

.

Zhus, >>4 ovov

>

6>6 ?

^h6

hò

, mr >>4 vv ?

^h6

`h>

Ilt hofx

nl thl ofxkouo hlk`ht rlfehlj ny thl rmeclt, ft whkeh kts vlimekty hfs nllg rljuelj tm zlrm. Zhus,

sunstktutkg` v ? 4 fgj h ? hofx

kg thl ifst lxprlsskmg, wl hfvl

^

hofx6

h`>v ofx>4

mr

^

h6v ofx>4 ? > `hofx

mr v4

> ? hofx

^

v`>

>4

mr hofx

?

^v`>

v

>4

>4

Gmw, kt ks `kvlg thft v4? 4.; lsefpl vlimekty ? 4.; )^`>(

hofx

?

^

^`>);.44;(`>

^`>);.44;(

?^1>.6`>

`^1>.6

?

58.4

^1>.6? :.>1 ^

Documents

Gravitation Theory E