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Eaeoeey 1
Eeiaiaoeea.

Aaaaeaiea.

Iaoeie iau/ii iacuaatho iiiuoee nenoaiaoece?iaaoue noiio ciaiee ia
ie-?oaeathuai ian iaoa?eaeueiii ie?a, i naiii /aeiaaea e i ?acoeueoaoao
aai aeayoaeue-iinoe. Nai oa?iei «iaoea» a ianoiyuaa a?aiy oiio?aaeytho
ai iiiaeanoaaiiii /enea, iiae/a?eeaay iiiaia?aiiinoue yoiai iiiyoey. A
yoii niunea iaoee ?ac-aeaeytho ia aoiaieoa?iua e anoanoaaiiua. Iauei
aeey anao iaoe yaeyaony iayca-oaeueiia eniieueciaaiea oi?iaeueiie eiaeee
eae oieaa?naeueiiai iaoiaea iao/iiai iuoeaiey.

Oeceea eae ioaeaeueiay iaoea eco/aao iaeaieaa iauea caeiiu oi?ie?iaaiey
e ?acaeoey ie?oaeathuae ian iaoa?ee a aa iaeaieaa i?eieoeaiuo oi?iao,
eioi?ua i?eiyoi iacuaaoue iaaeeaie i?e?iaeie. Iiyoiio iiaeii
ooaa?aeaeaoue, /oi oeceea yaeyaony ooiaeaiaioii anao anoanoaaiiuo iaoe,
a /anoiinoe aaia?aoee.

A OIX e OO aaeao oeceea ia?aaeeea ao?iue ?anoeaao, oece/aneea ciaiey e
oece/aneee iaoiae enneaaeiaaiey iieo/eee aieueooth ecaanoiinoue e iaoee
i?e-iaiaiey a ?acee/iuo aniaeoao /aeiaa/aneie aeayoaeueiinoe.

Nouiinoue yoiai iaoiaea ninoieo a oii, /oi a iniiao e?eoe/aneie ioeaiee
anao ?ac?aaioaiiuo oece/aneeo oai?ee iieiaeai yenia?eiaio.

Ia ?aiieo noaaeeyo ?acaeoey iaoee oeceee naie caeeth/aiey no?ieee ia
iniiaa ?aaeueiuo iaaethaeaiee ?acee/iuo i?e?iaeiuo yaeaiee, iai?eia?,
oaeeo eae a?ica (A. O?aieeei e A.A. ?eoiai). Iicaeiaa /aeiaae iao/eeny
eneonnoaaiii aini?iecaiaeeoue yoe yaeaiey a eaai?aoi?iuo oneiaeyo –
«noaaeoue iao/iua yenia?eiaiou». ssnii, /oi ie iaeia eaai?aoi?ey ia a
neeao iaania/eoue iieiia aini?iecaaaeaiea anao i?e?iaeiuo oneiaee
iaaethaeaiee eaeiai-eeai yaeaiey. Iiyoiio aeey i?aaeeueiie iinoaiiaee
oiai eee eiiai oece/aneiai yenia?eiaioa iaiaoiaeeii i?iaanoe i?aaeeueiue
aiaeec eco/aaiiai yaeaiey, auaeaeeoue aai iaeaieaa nouanoaaiiua nayce n
inoaeueiui ie?ii. Oaeei ia?acii eco/aiea yaeaiey eee iauaeoa anaaaea
i?iaiaeeony a iaeioi?ii i?eaeeaeaiee, eiaaea enneaaeiaaoaeue
niciaoaeueii eee iainiciaiii ioa?anuaaao iaeioi?ua aeaoaee
aini?iecaiaeeiiai yaeaiey. Iieo/ea yenia?eiaioaeueiua aeaiiua,
iaaethaeaoaeue aeey eo iauyniaiey nicaeaao ia iniiaa eiathueony o iaai
i?aaenoaaeaiee iooai neioaca ?aai/oth aeiioaco, eioi?ay iiaeao
iauynieoue ia oieueei iaeei, ii e oeaeoth a?oiio iiaeiaiuo
yenia?eiaioia. Aaaeii ioiaoeoue, /oi iniuneaiea ?acoeueoaoia
yenia?eiaioa eaeao a iaeioi?ii oi?iuaiiii eee, eae i?eiyoi aiai?eoue, a
iiaeaeueiii i?aaenoaaeaiee, o.a. naii yaeaiea caiaiyaony aai iaeioi?ui
oi?iuaiiui i?aaenoaaeaieai eee iiaeaeueth.

Anee ?ac?aaioaiiua i?aaenoaaeaiey ieacuaathony ni?aaaaeeeauie aeey
aeinoaoi/ii oe?ieiai eeanna yaeaiee, oi i?eiyoi aiai?eoue i
aicieeiiaaiee oece/aneie oai?ee. Ioaeaeueiua iieiaeaiey yoie oai?ee
iinyo iacaaiey oece/aneeo caeiiia, i?e oneiaee eo auiieiaiey aeey anaai
eeanna eco/aiiuo iauaeoia e yaeaiee.

Aaaeiie iniaaiiinoueth oece/aneie iaoee yaeyaony eniieueciaaiea
eiee/anoaaiiuo oa?aeoa?enoee ioaeaeueiuo naienoa oece/aneeo iauaeoia.
Yoe oa?aeoa?enoeee ii?aaeaeythony iooai ecia?aiee, e aeey onoaiiaeaiey
acaeiinayce iaaeaeo ?acee/iuie oece/aneeie ia?aiao?aie i?eiaiyaony
eiee/anoaaiiay eiaeea, o.a. iaoaiaoeea. Iaoaiaoeea yaeyaony iiuiui
n?aaenoaii aeey aiaeeoe/aneiai i?aaenoaaeaiey oece/aneeo caeiiia e
neaaenoaee ec ieo. Ethaay oece/aneay oai?ey aeieaeia auoue ni?aaaaeeeaie
aeey anao yaeaiee i?e?iaeu, a i?ioeaiii neo/aa oai?ey iineo eeoue
/anoiue (ia?aie/aiiue) oa?aeoa?. Anee iiyaeythony iiaua
yenia?eiaioaeueiua oaeou, eioi?ua ia iauyniythony n oi/ee c?aiey
?ac?aaioaiiie oai?ee, oi yoi eae ?ac e oeacuaaao ia ia?aie/aiiinoue
oai?ee. A yoii neo/aa noaiiaeony i/aaeaeiie iaiaoiaeeiinoue iino?iaiey
iiaie oai?ee, a eioi?ie iiaue yenia?eiaioaeueiue iaoa?eae iaoiaeeo naia
anoanoaaiiia iauyniaiea (i?eia? – iaoaieea Iuethoiia e oai?ey
ioiineoaeueiinoe Yeiooaeia). Caeanue aaaeii iiae/a?eiooue oio oaeo, /oi
e?eoa?eai ioeaiee ni?aaaaeeeainoe oiai eee eiiai eiae/aneiai iino?iaiey
aunooiaao yenia?eiaio. Eiaiii ii yaeyaony naiaia?aciui «aa?oiaiui
noaeueae», auiinyuei naie «i?eaiai?» ioiineoaeueii eaeie-eeai oai?ee.

Iaeiaei oeaii/a/iay naycue «yenia?eiaio – aeiioaca – caeii – oai?ey –
yenia?eiaio» ia icia/aao, /oi oece/aneay oai?ey ea?aao eeoue
iienaoaeueioth ?ieue, e aa i?ecaaiea ninoieo oieueei a iauyniaiee
i?iaaaeaiiuo yenia?eiaioia. Nithc oai?ee e yenia?eiaioa iineo oai?/aneee
oa?aeoa?: aoiiiay oai?ey no?iaiey aauanoaa iieo/eea anaiauaa i?eciaiea
caaeieai aei oiai, eiaaea noaei aiciiaeii iaiin?aaenoaaiiia iaaethaeaiea
ioaeaeueiuo aoiiia.

Eo?n iauae oeceee, eioi?ue aoaeao /eoaoueny aeaa naiano?a, ?ann/eoai ia
oi?ie?iaaiea oece/aneiai ie?iaicc?aiey, nicaeaiey anoanoaaiii-iao/iie
aacu aeey i?aaeeueiiai iiieiaiey anao yaeaiee ie?oaeathuaai ian ie?a.

O?aaeeoeeiiii ?anniio?aiea iauae oeceee ia/eiaaony n ?acaeaea
«Iaoa-ieea».

( 1-1. Iniiaiua iiiyoey eeiaiaoeee.

Iaoaie/aneei aeaeaeaieai iacuaaaony eciaiaiea iieiaeaiey i?aaeiaoa
ioiineoaeueii caaeaiiie nenoaiu ion/aoa. Iiiyoea nenoaiu ion/aoa
aeeth/aao a naay oaei ion/aoa e nenoaio eii?aeeiao. Aeey aieueoeinoaa
caaea/ iaoaai eo?na aeinoaoi/ii ia?aie/eoueny i?yiioaieueiie nenoaiie
eii?aeeiao e aua?aoue a ea/anoaa oaea ion/aoa Caieth. I?inoaeoei
iauaeoii aeey eco/aiey iaoaie/aneiai aeaeaeaiey iiaeao neoaeeoue
iaoa?eaeueiay oi/ea. Aeey iienaiey iieiaeaiey iaoa?eaeueiie oi/ee

ioiineoaeueii aua?aiiie nenoaiu ion/aoa i?eiyoi eniieueciaaoue
aaeoi?iia i?aae-

Y

( A ( l

o rA rB ( A

o O

iacuaaaony eeiaiaoe/aneei caeiiii aeaeaeaiey: r = r ( t ). Eii?aeeiaou
oi/ee a yoii neo/aa oaeaea yaeythony ooieoeeyie a?aiaie: o = o(t), o =
o(t) (ni.?en.1) e z = = z(t), eioi?ua iiaeii ?anniao?eaaoue eae
ia?aiao?e/aneea o?aaiaiey aeaeaeaiey. Anee ca a?aiy ( t oi/ea
ia?aianoeeanue ec iieiaeaiey A a iieiaeaiea A (ni.?en.1),

oi ?aaeeon – aaeoi? (l, i?iaaaeaiiue ec A a A, iacuaaaony ia?aiauaieai
oi/ee ca a?aiy ( t. Ec ?en. 1 aeaeii, /oi (l = rB – rA = ( r. Aeey
iaeaieaa oi/iiai iienaiey aeaeaeaiey iaiaoiaeeii auae?aoue a?aiy ( t eae
iiaeii iaiueoa. A yoii neo/aa e?e-

(s

(l

)

Ae?oaie ecaanoiie oa?aeoa?enoeeie iaoaie/aneiai aeaeaeaiey
oi/ee neoaeeo nei?inoue. N?aaeiyy nei?inoue ( v ( ca i?iiaaeooie a?aiaie
( t ii?aaeaeyaony eae:

. ( 1- 1 )

ssnii, /oi i?e oaeii ii?aaeaeaiee nei?inoe aa cia/aiea caaeneo io auai?a
aaee/eiu a?aiaiiiai eioa?aaea ( t e , eae neaaenoaea, io aaee/eiu ( l
. Iaeiaei i?e oiaiueoaiee aaee/eiu ( t ioiioaiea (1-1) no?aieony e
iaeioi?iio i?aaeaeo, eioi-?ue i?eiyoi iacuaaoue nei?inoueth
iaoa?eaeueiie oi/ee a aeaiiue iiiaio a?aiaie:

, ( 1- 2 )

.

Eae e ethaie aaeoi?, aaeoi? nei?inoe iiaeii i?aaenoaaeoue a aeaea noiiu
ninoaaeythueo ii eii?aeeiaoiui inyi:

, ( 1-3 )

aaea i , j , k yaeythony aaeeie/iuie aaeoi?aie, iai?aaeaiiuie
niioaaonoaaiii aaeieue inae X,Y e Z. N ae?oaie noi?iiu ?aaeeon aaeoi? r
oaeaea iiaeii i?aaenoaaeoue a

aeaea noiiu:

r = x i + y j +
z k, ( 1-4 )

aaea x,y e z i?aaenoaaeytho niaie i?iaeoeee ?aaeeona-aaeoi?a ia
iai?aaeaiea niioaaonoaothueo eii?aeeiaoiuo inae . Aeeooa?aioee?oy
oi?ioeo ( 1-4 ) e n?aaieaay ?acoeueoao aeeooa?aioee?iaaiey n
au?aaeaieai (1- 3 ), iieo/ei:

= z , (1- 5 )

eioi?ua icia/atho, /oi nei?inoe aeaeaeaiey i?iaeoeee oi/ee aaeieue
eii?aeeiaoiuo inae ?aaiu i?iaeoeeyi aaeoi?a nei?inoe ia niioaaonoaothuea
ine. Ec au?aaeaiey (1-5) neaaeoao, /oi ii ecaanoiie caaeneiinoe
eii?aeeiao oi/ee io a?aiaie ( ecaanoiiio caeiio aeaeaeaiey ) x(t), y(t)
e z (t) i?inoui aeeooa?aioee?iaaieai iiaeii iaeoe i?iaeoeee vx , vy , vz
aaeoi?a nei?inoe ia eii?aeeiaoiua ine, a neaaeiaaoaeueii e nai aaeoi?
nei?inoe a ethaie iiiaio a?aiaie. Aaee/eia aaeoi?a nei?inoe (aai
iiaeoeue) eae e aaee/eia ethaiai aaeoi?a iaoiaeeony eae ei?aiue
eaaae?aoiue ec noiiu eaaae?aoia niioaaonoaothueo i?iaeoeee:

. ( 1- 6 )

+ o0 , aaea o0 – eii?aeeiaoa oi/ee a ia/aeueiue iiiaio a?aiaie ( i?e t
= 0 ). Caaeneiinoue io a?aiaie ae?oaeo eii?aeeiao iaoiaeeony aiaeiae/iui
niiniaii.

E?iia oiai, ec oi?ioeu (1-3) auoaeaao, /oi nei?inoue ethaiai aeaeaeaiey
iiaeii i?aaenoaaeoue eae ?acoeueoao neiaeaiey o?ao i?yiieeiaeiuo
aeaeaeaiee aaeieue eii?aeeiaoiuo inae X,Y e Z ,o.a. ethaia neiaeiia
aeaeaeaiea iiaeii i?aaenoaaeoue eae noiio i?yiieeiaeiuo aeaeaeaiee (
i?eioeei noia?iiceoeee aeaeaeaiee ). I?eia?ii i?eiaiaiey yoiai i?eioeeia
iiaeao neoaeeoue au/eneaiea oae iacuaaaiie ia?aie einie/aneie nei?inoe,
o.a. oaeie nei?inoe, eioi?ith iaaei niiaueoue ethaiio oaeo ia?aeeaeueii
caiiie iiaa?oiinoe, /oiau iii ieeiaaea ia oiaei ia Caieth. A i?aia-

A vI ( t N

RC B

RC

O

?en.3. E auaiaeo ia?aie einie/aneie nei?inoe. a?aaeaiee nii?ioeaeaieai
aicaeooa caaea/a iiaeao auoue ?aoaia neaaeothuei ia?acii. Aeaeaeaiea
oaea, a?ioaiiiai aaeieue caiiie iiaa?oiinoe iiaeii i?aaenoaaeoue eae
noiio aeaoo aeaeaeaiee: ?aaiiia?iiai ai?eciioaeueiiai aeaeaeaiey ni
nei?inoueth a?inaiey vI e naiaiaeiiai iaaeaiey

+ (1/2)g((t)2

( i?aaeiieaaaaony, /oi a?aiy (t aeinoaoi/ii iaei e i?iaeoeeae nei?inoe
vI ia iai?aaeaiea AI iiaeii i?aiaa?a/ue). Caiaiyy noi?iiu ( AIN ia
iniiaaiee i?eaaaeaiiuo ?aaainoa, eiaai:

. (1- 7 )

. I?e (t 0 au?aaeaiea aeey ia?aie einie/aneie nei?inoe
i?eia?aoaao oaeie aeae:

. (1- 8 )

Eae aeaeii ec auaiaea au?aaeaiey aeey ia?aie einie/aneie nei?inoe,
ethaia oaei, aeaeaaynue aie?oa Caiee, iaoiaeeony a naiaiaeiii iaaeaiee,
ii oiaiueoaiea auniou iieaoa i?e naiaiaeiii iaaeaiee ia Caieth a
oi/iinoe eiiiaine?oaony ca n/ao i?e?auaiey ?annoiyiey aei Caiee i?e
aeaeaeaiee ii eanaoaeueiie.

Iaeiaei neo/ae, eiaaea oaei nio?aiyao naith nei?inoue iaeciaiiie,
e?aeia ?aaeee. Iaiai?io, a iauai neo/aa nei?inoue eciaiyaony eae ii
aaee/eia, oae e ii iai?aaeaieth. Aeey oa?aeoa?enoeee auno?iou eciaiaiey
nei?inoe aaiaeeony iiiyoea onei?aiey. Onei?aieai a aeaiiue iiiaio
a?aiaie iacuaaaony i?aaeae ioiioaiey i?e?auaiey nei?inoe e eioa?aaeo
a?aiaie, ca eioi?ue i?iecioei yoi i?e?auaiea:

. (1- 9 )

Aaeoi? onei?aiey iiaeii oaeaea ?aceiaeeoue ii eii?aeeiaoiui inyi:

a = a x i + a y j +
a z k . ( 1-10 )

Iiaeoeue aaeoi?a onei?aiey ?aaai:

. ( 1- 11 )

I?yiui aeeooa?aioee?iaaieai aiaeiae/ii eiiiiiaioai aaeoi?a nei?inoe

iiaeii iaeoe, /oi eiiiiiaiou aaeoi?a onei?aiey ?aaiu:

a x = v x = x ; a y = v y = y ;
a z = v z = z . ( 1-12 )

Anee ecaanoiu caaeneiinoue io a?aiaie aaeoi?a onei?aiey e ia/aeueiia
cia/aiea aaeoi?a nei?inoe, oi aaeoi? nei?inoe a ethaie iineaaeothuee
iiiaio a?aiaie iooai eioaa?e?iaaiey. Iai?eia?, aeey i?iaeoeee v x :

, ( 1-13 )

aaea v x0 – i?iaeoeey nei?inoe ia inue O a ia/aeueiue iiiaio a?aiaie.
?aiaa oeacuaaeinue, /oi ii ecaanoiie caaeneiinoe v (t) iiaeii iaeoe
caeii aeaeaeaiey. Neaaeiaaoaeueii, ii ecaanoiiio onei?aieth, ciay
ia/aeueiua cia/aiey iieiaeaiey oi/ee e aa nei?inoe, iiaeii iaeoe aa
caeii aeaeaeaiey. N oi/ee c?aiey i?aeoeee aaeoi? on-

D

vA B vB

(v

A (vn

E ( vt C

?en.4. Ii?iaeueiay e oaiaaioeeaeueiay

ninoaaeythuea eciaiaiey nei?inoe. ei?aiey oaeiaiaa i?aaenoaaeyoue a
aeaea aeaoo ninoaaeythueo, iaeia ec eioi?uo iai?aaeaia ii eanaoaeueiie e
o?aaeoi?ee, a ae?oaay ii ii?iaee, i?iaaaeaiiie a oi/eo eanaiey. Ionoue
ca a?aiy (t oi/ea ia?aianoeeanue ec A a A, e ca yoi a?aiy aa nei?inoue
eciaieeanue io vA aei vB .

Aeey oiai, /oiau iaeoe eciaiaiea (v ia-

?aianai aaeoi? vB a oi/eo ia/aea aaeoi?a vA. Oiaaea ?aciinoue aeaoo
aaeoi?ia vB – vA

iiaeao auoue i?aaenoaaeaia a aeaea aaeoi?a (v = DC. A naith i/a?aaeue,
aaeoi? (v ii-

aeii i?aaenoaaeoue oiaea eae noiio aeaoo ninoaaeythueo (v = (vn + (vt ,
aaea aaeoi? (vt iaoiaeeony eae ?aciinoue AN-AA ( AA=AD, AN= vB ), o.a.
eae ?aciinoue iiaeoeae aaeoi?ia vB e vA. Aaeoi? (vn oa?aeoa?ecoao
eciaiaiea iai?aaeaiey aaeoi?a vA , o.e. vA = AA = AD. O?aoaieueiee DAE
?aaiiaaae?aiiue, iiyoiio i?e oiaiueoaiee eioa?aaea a?aiaie (t aei ioey
((t 0) oaie DAE oaeaea no?aieony e 0, a (ADA 900,

e (vn ieacuaaaony ia?iaiaeeeoey?iui iai?aaeaieth nei?inoe. A oi aea
a?aiy ynii,

/oi iai?aaeaiea aaeoi?a (vt i?e (t 0 i?eaeeaeaaony e iai?aaeaieth
eanaoaeueiie a oi/ea A. Iiyoiio

. (1- 14 )

Ia?aia ec neaaaaiuo a (1- 14 ) iacuaatho ii?iaeueiie ninoaaeythuae
onei?aiey eee i?inoi ii?iaeueiui onei?aieai, a aoi?ia –
oaiaaioeeaeueiui. Oaeei ia?acii

, (1- 15 )

. (1- 16 )

Iiaeoeue iieiiai onei?aiey ii?aaeaeyaony neaaeothuei au?aaeaieai:

. ( 1-17 )

( 1 – 2. Eeiaiaoeea a?auaoaeueiiai
aeaeaeaiey.

vA

vA ( (v

( l vB

r

(

?en.5. E auaiaeo oeaio?ino?a-

ieoaeueiiai onei?aiey *anoiui i?eia?ii ii?iaeueiiai onei?aiey
neoaeeo

oeaio?ino?aieoaeueiia onei?aiea, aicieeathuaa i?e

?aaiiia?iii aeaeaeaiee oi/ee ii ie?oaeiinoe. Anee

ca iaeue i?iiaaeooie a?aiaie (t oi/ea oniaaaao ii-aa?iooueny ia oaie (,
oi eae aeaeii ec ?en.5, iaaeaeo

ia?aiauaieai (l , ?aaeeonii r , i?e?auaieai (v e

naiie nei?inoueth v iiaeii caienaoue neaaeothuaa niioiioaiea:

. ( 1-18 )

Ec yoiai niioiioaiey i?e?auaiea nei?inoe (v ?aaii:

( 1-19 )

Aeaey au?aaeaiea ( 1-19 ) aeey i?e?auaiey nei?inoe ia i?iiaaeooie
a?aiaie (t, eiaai:

. (1- 20 )

Aeey neo/ay a?auaoaeueiiai aeaeaeaiey iieaciuie ieacuaathony oaeea
aeiiieieoaeueiua eeiaiaoe/aneea oa?aeoa?enoeee eae oaeiaay nei?inoue e
oaeiaia onei?aiea. Aaee/eia oaeiaie nei?inoe ( ii?aaeaeyaony eae
ioiioaiea oaea ((, eioi?ue iienuaaao ?aaeeon-aaeoi? oi/ee ca a?aiy (t,
o.a.

. ( 1-21 )

(

v

r

(( (s

?en.6.E ii?aaeaeaieth iai?aa-

eaiey oaeiaie nei?inoe. I?e yoii oaeiaie nei?inoe
i?eienuaaaony ii?aaeaeaiiia iai?aaeaiea, eioi?ia ii?aaeaeyaony
neaaeothuei ia?acii: iai?aaeaiea ion/aoa oaea ii?aaeaeyaony iai?aaeaieai
a?auaiey, a iai?aaeaiea ( ii?aaeaeyaony i?aaeeii i?aaiai ao?aa/eea – iii
niaiaaeaao n aeaeaeaieai ine ao?aa/eea, eiaaea ii a?auaaony a
iai?aaeaiee a?auaiey iaoa?eaeueiie oi/ee ( ni. ?en.6 ). Aaeoi? oaeiaiai
onei?aiey ( ii?aaeaeyaony /a?ac eciaiaiea oa-

eiaie nei?inoe a?auaiey ca a?aiy (t. I?e yoii iai?aaeaiea ( niaiaaeaao n
iai?aaeaieai (, anee ca a?aiy (t i?ienoiaeeo oaaee/aiea nei?inoe ( e
iai?aaeaiea ( i?ioeaiiieiaeii aaeoi?o (, anee ca a?aiy (t oaeiaay
nei?inoue oiaiueoaaony. Oaeei ia?acii

. ( 1- 22 )

I?e a?auaoaeueiii aeaeaeaiee iaaeaeo eeiaeiie nei?inoueth oi/ee,
iai?aaeai-

iie ii eanaoaeueiie e ie?oaeiinoe a?auaiey nouanoaoao ii?aaeaeaiiay
acaeiinaycue. Aeaenoaeoaeueii

[( r ] , ( 1-23 )

aaea eaaae?aoiua neiaee iaicia/atho aaeoi?iia i?iecaaaeaiea aeaoo
aaeoi?ia – ( e r.

Eae ecaanoii, aeaa aaeoi?a iiaoo auoue ia?aiiiaeaiu aeaoiy niiniaaie –
neaey?ii e aaeoi?ii. Iineieueeo i?e neaey?iii i?iecaaaeaiee aaeoi?ia
iieo/aaony /enei (neaey?), a nei?inoue ii ii?aaeaeaieth – aaeoi?, oi
inoaaony oieueei aaeoi?iue

niinia ia?aiiiaeaiey aaeoi?ia ( e r. Iai?aaeaiea aaeoi?iiai
i?iecaaaeaiey oae-aea ii?aaeaeyaony ii i?aaeeo i?aaiai ao?aa/eea: ia?aue
aaeoi? ( a iaoai neo/aa -yoi aaeoi? ( ) a?auaaony ii e?ao/aeoaio
iai?aaeaieth e aoi?iio aaeoi?o ( a iaoai neo/aa – yoi ?aaeeon – aaeoi? r
); aeaeaeaiea ine ao?aa/eea i?e oaeii a?auaiee iieaaeao iai?aaeaiea
aaeoi?iiai i?iecaaaeaiey ( ni. ?en.6 ).

Eaeoeey 2
Aeeiaieea iaoa?eaeueiie oi/ee.

( 2-1. Ia?aue caeii
Iuethoiia.

Eeiaiaoeea onoaiaaeeaaao caeiiu aeaeaeaiey iaoa?eaeueiie oi/ee, ii ia
oeacuaaao i?e/eiu aucaaaoea yoi aeaeaeaiea, a oaeaea oaeoi?u, aeeythuea
ia aa?eaoeee eeiaiaoe/aneeo ia?aiao?ia aeaeaeaiey. Caeiiu Iuethoiia,
noi?ioee?iaaiiua aieaa 300 eao iacaae , yaeeenue ?acoeueoaoii iaiauaiey
aieueoiai eiee/anoaa iaaethaeaiee e yenia?eiaioia. Yoe caeiiu eiatho
ooiaeaiaioaeueiia cia/aiea e a iaoa a?aiy. Ia?aue caeii ooaa?aeaeaao,
/oi nouanoaotho oaeea nenoaiu ion/aoa, a eioi?uo anyeia oaei nio?aiyao
ninoiyiea iieiy eee ?aaiiia?iiai i?yiieeiaeiiai aeaeaeaiey aei oao ii?,
iiea aicaeaenoaey ni noi?iiu ae?oaeo oae ia canoaayo aai eciaieoue yoi
ninoiyiea. Naienoai oaea nio?aiyoue naia ninoiyiea iaeciaiiui iacuaatho
eia?oeeae, a nenoaiu ion/aoa, a eioi?uo auiieiyaony yoio caeii, –
eia?oeeaeueiuie. Oece/aneee niune caeiia ninoieo a oii, /oi aeey
iaoaieee iao ?acee/ey iaaeaeo ninoiyieai iieiy e ?aaiiia?iiai
i?yiieeiaeiiai aeaeaeaiey. Ii iiae/a?eeaaao ioiineoaeueiinoue
aeaeaeaiey. No?iai aiai?y,

yoio caeii yaeyaony /enoie aano?aeoeeae, ii iiuo anaai /aeiaa/anoaa ca
i?ioaaeoea o?e n eeoiei aaea iiaeoaa?aeaeaao aai ni?aaaaeeeainoue.
I?e/eia eciaiaiey ninoiyiey oaea, o.a. iiyaeaiea onei?aiey naycaia n
iiiyoeai neeu. Neea – eiee/anoaaiiay ia?a aicaeaenoaey ia aua?aiiia
iaie oaei ni noi?iiu ae?oaeo oae. Aiiaua aiai?y, yoi aicaeaenoaea iiaeao
auoue aeinoaoi/ii neiaeiui, ii a yoii neo/aa aai iiaeii ?aceiaeeoue ia
oae iacuaaaiua i?inoua aicaeaenoaey. Iiyoiio neeie iacuaatho
eiee/anoaaiioth ia?o i?inoiai aicaeaenoaey ia oaei ni noi?iiu ae?oaeo
oae, a ai a?aiy aeaenoaey eioi?iai oaei eee aai /anoe iieo/atho
onei?aiey. Eae iieacuaaao iiuo, aaee/eia iieo/aiiiai onei?aiey caaeneo
io naienoa acaeiiaeaenoaothueo oae, io ?annoiyiey iaaeaeo ieie e io eo
ioiineoaeueiuo nei?inoae. Neeo i?eiyoi ecia?youe ( a iaaeaeoia?iaeiie
nenoaia aaeeieoe NE ) a Iuethoiiao ( I ). Ia oa??eoi?ee iaoae no?aiu yoa
nenoaia aaeeieoe yaeyaony Ainoaea?noaaiiui Noaiaea?oii n 1977 aiaea.
Iaeiaei aei neo ii? nouanoaotho iao?e/aneea aianenoaiiua aaeeieoeu:
a?aii, eeeia?aii e oiiia. Yoe aaeeieoeu eniieuecothony i?e ii?aaeaeaiee
aana oaea. Ia i?aeoeea aeey ecia?aiey aaee/eiu neeu eniieuecotho
aeeiaiiiao? – oa?e?iaaiioth ( a?aaeoe?iaaiioth) i?oaeeio, niaaaeaiioth
oeaeie.

( 2-2. Aoi?ie caeii
Iuethoiia.

Iiuo iieacuaaao, /oi iaeia e oa aea neea niiauaao ?acee/iui oaeai
?aciua onei?aiey. Aieaa ianneaiua oaea i?eia?aoatho iaiueoea onei?aiey.
Aeey oa?aeoa?enoeee niiniaiinoe oae i?ioeainoiyoue aeaenoaeth neeu
eniieuecoaony iiiyoea iannu . *ai iaiueoa onei?aiea, eioi?ia iieo/aao
oaei, oai aieueoa aai ianna, o.a.

onei?aiey oae ia?aoii i?iii?oeeiiaeueiu eo iannai:

. ( 2-1 )

I?eiya eaeoth-eeai ianno ca yoaeii, n iiiiuueth yoiai niioiioaiey
iiaeii ecia?youe ethaoth ianno.

Aaee/eia onei?aiey, eioi?ia iieo/aao oaei ii?aaeaeaiiie iannu, caaeneo
io aaee/eiu neeu, – /ai aieueoa neea F, oai aieueoa onei?aiea ( a ( F )
, ii ae?oaiio a = k F, aaea k – eiyooeoeeaio i?iii?oeeiiaeueiinoe. N
o/aoii (2-1) eiaai:

. ( 2-2 a )

Auai? eiyooeoeeaioa i?iii?oeeiiaeueiinoe caaeneo io auai?a nenoaiu
aaeeieoe. A ianoiyuaa a?aiy ai anao nouanoaothueo nenoaiao aaeeieoe
i?eiyoi n/eoaoue

k = 1, o.a.

. ( 2-2 a)

Onei?aiea – aaeoi?, ianna – aaee/eia neaey?iay ( /enei ), iiyoiio neea
oiaea aaeoi?, iai?aaeaiea eioi?iai niaiaaeaao n iai?aaeaieai onei?aiey.
Anee ia oaei aeaenoaoao ianeieueei nee, oi onei?aiea oaea
i?iii?oeeiiaeueii eo aaiiao?e/aneie noiia:

. ( 2-3 )

O?aaiaiea ( 2-3 ) i?aaenoaaeyao iaeio ec oi?i caiene aoi?iai caeiia
Iuethoiia. A iaoaieea yoi o?aaiaiea i?eiyoi iacuaaoue o?aaiaieai
aeaeaeaiey . Yoi o?aaiaiea – aaeoi?iia, e aai iiaeii caiaieoue o?aiy
neaey?iuie, i?iaeoe?oy iii/a?aaeii( 2-3 ) ia ine eii?aeeiao X, Y e Z.
Aoi?ie caeii Iuethoiia iiaeao auoue noi?ioee?iaai ianeieueei ae?oaei
niiniaii n iiiiuueth iiiyoey eiioeuena oaea. Eiioeuenii i?eiyoi
iacuaaoue aaee/eio p = mv, aaea v – nei?inoue oaea. A iuethoiiianeie
iaoaieea i?aaeiieaaaaony, /oi ianna oaea iinoiyiia e ia caaeneo io
nei?inoe, iiyoiio:

. ( 2-4 )

N o/aoii ( 2-4 ) o?aaiaiea ( 2-3 ) i?eieiaao oaeie aeae :

. ( 2-5 )

( 2-3. O?aoee caeii
Iuethoiia.

Iiiyoea neeu ii?aaeaeaii eae ia?a acaeiiaeaenoaey oae, o.a. i?e
?anniio?aiee aeaeaeaiey eaeiai-ieaoaeue oaea o/eouaaaony oieueei iaeia
noi?iia yoiai acaeiiaeaenoaey. ssnii, iaeiaei, /oi ana oaea iaaei
?anniao?eaaoue eae ?aaiii?aaiua, o.a. anee aoi?ia oaei aicaeaenoaoao ia
ia?aia, oi e ia?aia oaei aicaeaenoaoao ia aoi?ia. O?aoee caeii Iuethoiia
onoaiaaeeaaao niioiioaiea iaaeaeo yoeie aicaeaenoaeyie.

Neeu, n eioi?uie aeaa oaea aeaenoaotho ae?oa ia ae?oaa, ?aaiu ii
aaee/eia e iai?aaeaiu i? iaeiie i?yiie a ?aciua noi?iiu.

I?eia?: eieaa eaaeeo ia noiea; iia i?eoyaeaaaony e Caiea e aneaaenoaea
yoiai aeaaeo ia noie. Iaeiaei eieaa ia i?iaaeeaaaony e oeaio?o Caiee,
o.e. noie ni naiae noi?iiu aeaenoaoao ia eieao n neeie ?aaiie ii
aaee/eia neea aeaaeaiey eieaa ia noie. Yoa neea ni noi?iiu noiea iineo
iacaaiea ?aaeoeee iii?u. E naiie eieaa i?eeiaeaii aeaa neeu: neea
i?eoyaeaiey e neea ?aaeoeee iii?u. Iie ?aaiu ii aaee/eia e
i?ioeaiiieiaeii iai?aaeaiu, o.a. eo noiia ?aaia ioeth, iiyoiio eieaa
ieeoaea ia aeaeaaaony.

( 2-4. I?e?iaea iaoaie/aneeo
nee.

Ec eeiaiaoeee ecaanoii, /oi ciaiea aaee/eiu e iai?aaeaiey onei?aiey
iicaieyao au/eneeoue cia/aiey ?aaeeona – aaeoi?a iaoa?eaeueiie oi/ee a
ethaie iineaaeothuee iiiaio a?aiaie, o.a. i?aaeneacaoue iieiaeaiea
oi/ee. Caeiiu aeeiaieee iicaieytho naeaeaoue yoi, anee ecaanoia i?aaay
/anoue o?aaiaiee (2-3) eee (2-5). Ae?oaeie neiaaie, ioaeii oiaoue
ii?aaeaeyoue neeu, aeaenoaothuea ia oaei, iieiaeaiea eioi?iai o?aaoaony
iienaoue. Acaeiiaeaenoaea iaaeaeo iae?ineiie/aneeie oaeaie oeceea
naiaeeo e acaeiiaeaenoaeth iaaeaeo yeaiaioa?iuie /anoeoeaie. Oaeeo
yeaiaioa?iuo /anoeoe a ianoiyuaa a?aiy ecaanoii aieaa nioie. N?aaee ieo
iaeaieaa iiioey?iu yeaeo?ii, i?ioii e iaeo?ii. Aeey oa?aeoa?enoeee anao
/anoeoe aaiaeyony oaeea iiiyoey eae ianna iieiy, yeaeo?e/aneee ca?yae,
nianoaaiiue iaoaie/aneee iiiaio ( niei ), a oaeaea /aoiinoue,
no?aiiinoue, e?aneainoue, aa?eiiiue ca?yae, oeaaoiaie ca?yae, neaaue
ca?yae e o.ae. Onoaiiaeaii, /oi iaaeaeo yeaiaioa?iuie /anoeoeaie
nouanoaoao /aou?a ooiaeaiaioaeueiuo acaeiiaeaenoaey: neeueiia, neaaia,
yeaeo?iiaaieoiia e a?aaeoaoeeiiiia. N?aaieoaeueiua oa?aeoa?enoeee yoeo
acaeiiaeaenoaee i?eaaaeaiu a oaaeeoea 1.

Oaaeeoea 1.

Iacaaiea aca-eiiaeaenoaey Ioiineoaeue-iay eioainea-iinoue
*anoeoea,«ia?a-iinyuay» acae-iiaeaenoaea Oa?aeoa?enoe-ea /anoeoeu

Neeueiia 1 (-iaciiu (aethiiu)(8 oeiia ) m ( 250
myeaeo

?aciiia?aciua

Yeaeo?iiaa-

ieoiia 10-2 oioii E= h(

Neaaia 10-13 W – /anoeoeu

Z – /anoeoeu A (102 n2 m i?ioii

aeiioaoe/iu

A?aaeoaoeeii-

iia 10-40 a?aaeoii

aeiioaoe/ai

A eeanne/aneie oeceea n/eoaaony, /oi yeaeo?iiaaieoiia e
a?aaeoaoeeiiiia acaeiiaeaenoaey inouanoaeythony iin?aaenoaii iiey. Iiea
– yoi iniaue aeae iaoa?ee, oa?aeoa?iue oai, /oi eaaeaeie oi/ea
i?ino?ainoaa iiaeii i?eienaoue ii?aaeaeaiiia cia/aiea iiey. Oece/aneia
iiea – iai?a?uaii. Iaeiaei, nia?aiaiiay oeceea, aace?othuayny ia
eaaioiauo i?aaenoaaeaieyo, n/eoaao aeene?aoiie ethaoth oece/aneoth
aaee/eio, eioi?ay iiaeao eciaiyoueny oieueei ii?aaeaeaiiuie ii?oeeyie –
eaaioaie. Iia i?eienuaaao iieyi aeene?aoiue oa?aeoa?, eiaaea eciaiaiea
iiey ?anniao?eaaaony eae eceo/aiea eee iiaeiuaiea iaeie /anoeoeu,
?ani?ino?aiythuaeny n eiia/iie nei?inoueth ( ia aieueoa nei?inoe naaoa
n). Ae?oaeie neiaaie, a eaaioiaie oeceea acaeiiaeaenoaey naiaeyony e
iaiaio oaie eee eiuie /anoeoeaie, ia?aiinyueie eaaio aeaenoaey. Anee
eaaio aeaenoaey yeaeo?iiaaieoiiai iiey

oi?ioi ecaanoai iiae iacaaieai oioii, oi eaaio a?aaeoaoeeiiiiai
acaeiiaeaenoaey

inoaaony ii neo ii? iaioe?uoui, oioy ii oaea iieo/ee iacaaiea a?aaeoiia.

No?iai aiai?y, neeu a iaoaieea iiaoo auoue naaaeaiu e yoei aeaoi
acaeiiaeaenoaeyi, oai aieaa, /oi aeaa ae?oaeo oeia iienuaatho
acaeiiaeaenoaey, nouanoaothuea oieueei a iee?iie?a. A /anoiinoe,
neeueiia acaeiiaeaenoaea iiaeao iauynieoue iaee/ea yaea?iuo nee,
ioaaonoaaiiuo ca onoie/eainoue aoiiiiai yae?a. Neaaua acaeiiaeaenoaey
aicieeatho iaaeaeo iee?i/anoeoeaie, iaeaaeathueie oae iacuaaaiui neaaui
ca?yaeii. Aei 1983 aiaea yoio oei acaeiiaeaenoaey ?anniao?eaaeny oieueei
oai?aoeeaie, ii a yoii aiaeo yenia?eiaioaeueii auea ioe?uoa W+ –
/anoeoea n yia?aeae 81 AyA ( Aeaa – 109, yeaeo?ii – Aieueo – aaeeieoea
ecia?aiey yia?aee, ?aaiay 1,6(10 -19 Aeaeioey), oae /oi neaaia
acaeiiaeaenoaea iieo/eei iiuoiia iiaeoaa?aeaeaiea.

Ec oaaeeoeu 1 aeaeii, /oi a?aaeoaoeeiiiua neeu yaeythony neaaaeoeie ec
anao ooiaeaiaioaeueiuo acaeiiaeaenoaee, iaeiaei iie iaeaaeatho
naienoaaie aaeaeeoeaiinoe e aeinoeaatho cia/eoaeueiuo aaee/ei a
einie/aneii ianooaaa ( i?eoyaeaiea Eoiu, no?iaiea Nieia/iie nenoaiu e
o.i.). Aaee/eia a?aaeoaoeeiiiie neeu i?eoyaeaiey aeaoo oi/a/iuo iann m1
e m2 ii?aaeaeaia Iuethoiiii e ecaanoia eae caeii anaie?iiai oyaioaiey:

, ( 2-6 )

aaea r – ?annoiyiea iaaeaeo iannaie, a G = 6,67 10 -11 I( i2/ea2 –
a?aaeoaoeeiiiay iinoiyiiay. *oiau iiae/a?eiooue, /oi neea – aaeoi?,
caeii caienuaatho ianeieueei eia/a, ?anniao?eaay neeo, aeaenoaothuoth ia
m2 ni noi?iiu m1:

r12 , ( 2-7 )

ioeoaea aeaeii iai?aaeaiea neeu ( iia iai?aaeaia aaeieue i?yiie,
niaaeeiythuae acaeiiaeaenoaothuea iannu). Iiaeoeue neeu i?eoyaeaiey P
oaea iannu m e Caiea , eioi?oth iacuaatho neeie oyaeanoe iiaeii
caienaoue oae:

( 2-8 )

– onei?aiea naiaiaeiiai iaaeaiey, IC- ianna Caiee, a RC – ?aaeeon
Caiee. Ec au?aaeaiey g aeaeii, /oi iii ia caaeneo io iannu aua?aiiiai
oaea e iiyoiio iaeeiaeiai aeey anao oae a ii?aaeaeaiiie oi/ea caiiie
iiaa?oiinoe.

N

a

?

?en.7. E ii?aaeaeaieth

aana oaea.

Aaaeii iiae/a?eiooue ?acee/ea aeaoo iiiyoee – neeu oyaeanoe e aana
oaea: ia?aay neea nouanoaoao anaaaea, eiaaea anoue i?eoyaeaathuay ianna
IC, oiaaea eae aoi?ay, i?aaenoaaeythuay ia?o aicaeaenoaey oaea ia
iiaenoaaeo eee ieoue iiaeaana, aiiaua aiai?y iiaeao eciaiyoueny. Aeey
iiyniaiey neacaiiiai iieacii ?anniio?aoue iieacaiey aania, ia eioi?uo
noieo ae?y. A iaiiaeaeaeiii ninoiyiee ia

ae?th aeaenoaoao aeaa neeu – neea oyaeanoe ? e neea ?aaeoeee iii?u (
aania )N, i?e/ai ? – N = 0. Anee aanu aeaeaeoony

aiec n onei?aieai a (ni ?en.7), oi o?aaiaiea aoi?iai caeiia Iuethoiia,
caienaiiia a iaiiaeaeaeiie nenoaia eii?aeeiao, eiaao aeae:

ma = P – N ,
( 2-9 )

ioeoaea N = P – ma = mg – ma = m( g – a
). ( 2-10 )

). Iiyoiio aan

= 0, o.a. ana naiaiaeii iaaeathuea oaea ie/aai ia aanyo. Neea oyaeanoe
ia iiaa?oiinoe Caiee ia yaeyaony iinoiyiiie ii aeaoi i?e/eiai:
ai-ia?auo, Caiey, eae ecaanoii ia yaeyaony eaeaaeueiui oa?ii ( iia
niethniooa ia

r

(

R

?en.8. Eciaiaiea ?a

nis(, iai?aaeaiiia a oo aea noi?iio, /oi e g. Iiyoiio (n?. n ?en.7) aan
oae aoaeao iaiueoa oai, aaea ?aaeeon a?auaiey aieueoa, o.a. ia yeaaoi?a
oaea eiatho iaeiaiueoee aan.

, ( 2-12 ) ae

aaea ( iaicia/aao aaee/eio oi?oaie aeaoi?iaoeee, a e – eiyooeoeeaio
i?iii?oeeiiaeueiinoe, caaeneiue io naienoa aeaoi?ie?oaiiai oaea e aeaea
aeaoi?iaoeee. *anoiui i?eia?ii i?iyaeaiey oi?oaeo nee neoaeao neeu
?aaeoeee iii?, iai?aaeaiea eioi?uo n/eoaaony anaaaea ii?iaeueiui (
ia?iaiaeeeoey?iui ) e aeaoi?ie?oaiie iiaa?oiinoe. Ae?oaei i?eia?ii
aeaenoaey oi?oaeo nee iiaoo neoaeeoue oae iacuaaaiua neeu nayce ( neeu
iaoyaeaiey ).

?anniio?aiea nee o?aiey iiaeii ia?aie/eoue aeaoiy i?eia?aie : neeaie
nooiai e neeaie ayceiai o?aiey. Neea nooiai o?aiey neieueaeaiey ecaanoia
ec oeieueiiai eo?na oeceee: Fo? = -( N, aaea ( – eiyooeoeeaio o?aiey,
oa?aeoa?ecothuee naienoaa acaeiiaeaenoaothueo iiaa?oiinoae, a N – oae
iacuaaaiay neea ii?iaeueiiai aeaaeaiey . A ioee/ea io nee ayceiai o?aiey
yoa neea ia caaeneo io nei?inoe aeaeaeaiey oaea. Neea ayceiai o?aiey,
iai?ioea, caaeneo io aaee/eiu nei?inoe, i?e/ai noaiaiue caaeneiinoe
iaiyaony ii ia?a aic?anoaiey nei?inoe. Aeey n?aaieoaeueii iaaieueoeo
nei?inoae iia iiaeao auoue i?aaenoaaeaia a oaeii aeaea:

. ( 2-13 )

Aaee/eia eiyooeoeeaioa b caaeneo eae io naienoa naiiai oaea, eioi?ia
aeaeaeaony a ayceie n?aaea, oae e io naienoa n?aaeu. Eiiaaea yoo neeo
o?aiey oaeiaiaa i?aaenoaaeyoue a oaeii aeaea:

, ( 2-14 )

aaea S – ieiuaaeue nii?eeiniiaaiey oaea ni n?aaeie, ( – eiyooeoeeaio
aioo?aiiaai o?aiey n?aaeu, a aaee/eia i?iecaiaeiie, aoiaeyuae a
au?aaeaiea aeey neeu, iineo iacaaiea a?aaeeaioa nei?inoe, iienuaathuaai
auno?ioo eciaiaiey nei?inoe neiaa n?aaeu, oaeaeaaiuo oaeii, a
iai?aaeaiee, ia?iaiaeeeoey?iii iai?aaeaieth nei?inoe oaea.

I?aeoe/anee aaaeiia cia/aiea eiaao neea o?aiey iieiy , aicieeathuay
iaaeaeo nii?eeanathueieny oaeaie. Iaeneiaeueioth aaee/eio yoie neeu
iau/ii ioeaieaatho ii oi?ioea aeey neeu o?aiey neieueaeaiey, oioy a
aeaenoaeoaeueiinoe iie ianeieueei ioee/athony ae?oa io ae?oaa.

( 2- 5. Aeeiaieea a?auaoaeueiiai aeaeaeaiey
iaoa?eaeueiie oi/ee.

N

v

mg

r

?en.9. Neeu i?e

a?auaoaeueiii

aeaeaeaiee. Niaoeeoeea oaeiai aeaeaeaiey ninoieo a oii, /oi aeey
aai iienaiey i?eoiaeeony i?eaaaaoue e iaeioi?ui ooeu?aieyi aeey auai?a
nenoaiu ion/aoa, a eioi?uo iiaeii caienaoue o?aaiaiea aeaeaeaiey. Anee
auae?aoue iau/ioth iaiiaeaeaeioth nenoaio eii?aeeiao, oi iai?aaeaiey
nei?inoae e onei?aiey oi/ee aoaeoo aaeanaeoiaeii eciaiyoueny
ioiineoaeueii eii?aeeiaoiuo inae, /oi ia nianai oaeiaii. Iiyoiio
iia?e?otho n oae iacuaaaiie neaaeyuae nenoaiie eii?aeeiao, o.a. n oaeie
nenoaiie,

ia/aei eioi?ie iaiiaeaeaeii e niaiaaeaao a aua?aiiue iiiaio a?aiaie n
aeaeaeouaeny iaoa?eaeueiie oi/eie, a iai?aa-

eaiey aa inae niaiaaeaao n iai?aaeaieai nei?inoe oaea a yoio iiiaio
a?aiaie e n

iai?aaeaieai ?aaeeona a?auaiey, i?iaaaeaiiiai a oi/eo, aaea ?aniieiaeaii
oaei a yoio aea iiiaio a?aiaie. Aaaeii ioiaoeoue, /oi aua?aiiay oaeei
ia?acii nenoaia

ion/aoa yaeyaony iaiiaeaeaeiie ioiineoaeueii eia?oeeaeueiie nenoaiu
ion/aoa (ia-i?eia?, Caiee), e a iae ni?aaaaeeeau caeiiu Iuethoiia.

?anniio?ei a ea/anoaa i?eia?a aeaeaeaiea aaoiiaoeiu ii auioeeiio iinoo,
?aaeeon eioi?iai r (ni. ?en.9) .Iai?aaei iaeio ec inae neaaeyuae nenoaiu
eii?aeeiao e oeaio?o iinoa, a ae?oaoth – aaeieue iai?aaeaiey nei?inoe v.
O?aaiaiea aeaeaeaiey a yoii neo/aa eiaao aeae ( a i?iaeoeee ia
aa?oeeaeueioth inue):

maoe = mg – N,
( 2-15 )

aaea /a?ac N iaicia/aia neea ?aaeoeee iinoa, a mg – neea oyaeanoe.
?aoay yoi o?aaiaiea ioiineoaeueii N, iieo/aai :

), ( 2-16 )

= g neea ?aaeoeee iinoa aoaeao ?aaia 0 . Ii yoi icia/aao, /oi
aaoiiaoeia a yoio iiiaio a?aiaie ia ieacuaaao ieeaeiai aeaaeaiey ia
iino, o.a. iia iaoiaeeony a ninoiyiee iaaaniiinoe.

Eaeoeey 3 Aeeiaieea
nenoaiu iaoa?eaeueiuo oi/ae.

( 3 – 1. Oeaio? iann nenoaiu iaoa?eaeueiuo
oi/ae.

Y

m1

A (

r1= l1 (

R l2 ( A

r2 m2

X

?en.10. E ii?aaea-

eaieth oeaio?a

iann. Oeaio?ii iann aeaoo iaoa?eaeueiuo oi/ae A e A n iannaie m1
e m2 niioaaonoaaiii iacuaaaony oi/ea N, eaaeauay ia io?acea,
niaaeeiythuai A e A, ia ?annoiyieyo l1 e l2 io A e A, ia?aoii
i?iii?oeeiiaeueiuo iannai oi/ae (ni. ?en.10.), o.a.

. ( 3-1 )

Anee iieiaeaiey oi/ae A e A caaeathony ?aaeeon-aaeoi?aie r1 e r2 , oi
iieiaeaiea oeaio?a iann ii?aaeaeyaony ?aaeeonii – aaeoi?ii R. Ec ?en.10
neaaeoao, /oi

Oiiiaeay ia?aia ec yoeo o?aaiaiee ia m1, a aoi?ia – ia m2 e neeaaeuaay
eo, iieo/ei:

. ( 3-3 )

Ec ?en.10 e ?aaainoaa ( 3-1 ) neaaeoao, /oi m2l2 = – m1l1. N o/aoii
yoiai niioiioaiey ec au?aaeaiey ( 3-3 ) iiaeii ii?aaeaeeoue cia/aiea
?aaeeona – aaeoi?a R:

. ( 3-4 )

Iaiauay yoi au?aaeaiea aeey i?iecaieueiiai /enea iaoa?eaeueiuo oi/ae,
iieo/ei:

, ( 3-5 )

= I – iieiay ianna nenoaiu oi/ae.

Nei?inoue oeaio?a iann oaeie nenoaiu ii?aaeaeyaony aeeooa?aioee?iaaieai
( 3-5 ):

. ( 3-6 )

Aaee/eiu mivi i?aaenoaaeytho niaie eiioeuenu ioaeaeueiuo oi/ae,
iiyoiio o?aa-iaiea ( 3-6 ) iiaeii ia?aienaoue a neaaeothuai aeaea:

= ?, ( 3-7 )

aaea /a?ac ? iaicia/ai noiia?iue eiioeuen nenoaiu. Aeeooa?aioee?oy ( 3-7
), iaoiaeei au?aaeaiea aeey onei?aiey oeaio?a iann nenoaiu A:

. ( 3-8 )

( 3 -2 Caeii eciaiaiey eiioeuena nenoaiu iaoa?eaeueiuo
oi/ae.

Aeey i?inoiou ?anniio?ei aeaeaeaiea nenoaiu, ninoiyuae ec o?ao oi/ae,
ia

eaaeaeoth ec eioi?uo aeaenoaotho aioo?aiiea neeu fik e aiaoiea – Fi ,
aaea eiaeaen i i?aaenoaaeyao iiia? oi/ee. O?aaiaiey aeaeaeaiey aeey
eaaeaeie oi/ee eiatho aeae:

( 3-9 )

Neeaaeuaay yoe o?aaiaiey, iieo/ei:

( 3-10 )

Ii o?aoueaio caeiio Iuethoiia aioo?aiiea neeu iiia?ii ?aaiu ii aaee/eia
e i?ioeaiiieiaeiu ii iai?aaeaieth ( iai?eia?, f12 = -f21). Iioiio noiia
anao aioo?aiieo nee ?aaia ioeth, e

, ( 3-11 )

aaea /a?ac ? iaicia/ai noiia?iue eiioeuen nenoaiu. Iaiauay ( 3-11 )
aeey ethaiai /enea iaoa?eaeueiuo oi/ae, iiaeii caienaoue neaaeothuaa
au?aaeaiea:

, ( 3-12 )

eioi?ia i?eiyoi iacuaaoue caeiiii eciaiaiey eiioeuena nenoaiu
iaoa?eaeueiuo oi/ae. Eae aeaeii ec yoiai au?aaeaiey, eciaiaiea
noiia?iiai eiioeuena ii?aaeaeyaony ?aaiiaeaenoaothuae anao aiaoieo nee,
aeaenoaothueo ia nenoaio. Anee aea yoa ?aaiiaeaenoaothuay ?aaia ioeth (
eee ia nenoaio ia aeaenoaotho ieeaeea aiaoiea neeu), oi noiia?iue
eiioeuen nenoaiu inoaaony iinoiyiiui. Yoi neaaenoaea o?aaiaiey ( 3-12 )
iacuaaaony caeiiii nio?aiaiey eiioeuena. Ae?oaei neaaenoaeai
?anniio?aiiiai caeiia eciaiaiey eiioeuena neoaeeo oai?aia i aeaeaeaiee
oeaio?a iann, eioi?ay ooaa?aeaeaao, /oi oeaio? iann nenoaiu
iaoa?eaeueiuo oi/ae iiae aeaenoaeai aiaoieo nee aeaeaeaony eae
iaoa?eaeueiay oi/ea noiia?iie iannu, e eioi?ie i?eeiaeaiu ana aiaoiea
neeu, e caienuaaaony a oaeii aeaea:

. ( 3-13 )

Aeieacaoaeuenoai yoiai ooaa?aeaeaiey neaaeoao ec n?aaiaiey ii?aaeaeaiey
onei?aiey oeaio?a iann( 3-8 ) e au?aaeaiey ( 3-13 ).

I?eia?aie caeiia nio?aiaiey eiioeuena iiaoo neoaeeoue ioaea/a i?e
no?aeueaa ec iaiano?aeueiiai i?oaeey, ?aaeoeaiia aeaeaeaiea, ia?aiauaiea
inueieiiaia e o.i.

Eaeoeey 4.
Aeeiaieea oaa?aeiai oaea.

( 4-1. Eeiaiaoe/aneea niioiioaiey.

Oaa?aeia oaei iiaeii ?anniao?eaaoue eae nenoaio iaoa?eaeueiuo oi/ae,
aeanoei ne?aieaiiuo ae?oa n ae?oaii. Ionoonoaea oaeiai cae?aieaiey
nouanoaaiii cao?oaeieei au iienaiea aeaeaeaiey anaai eiiaeiia?aoa oi/ae.
Aeey iieiiai iienaiey aeaeaeaiey iaeiie oi/ee iaiaoiaeeii ciaoue aa o?e
eii?aeeiaou, iiyoiio aeey N oi/ae /enei iaiaoiaeeiuo eii?aeeiao , a
neaaeiaaoaeueii, e /enei o?aaiaiee aeey eo ii?aaeaeaiey ninoaaeei au 3N.
Oae eae /enei N iiaeao auoue eae oaiaeii aieueoei, oi aiciiaeiinoe
no?iaiai ?aoaiey nenoaiu ec 3N o?aaiaiee aanueia ia?aie/aiu.

E?iia oiai oa?aeoa? aeaeaeaiey oaea eae oeaeiai iiaeao auoue ?acee/iui.
Iau/ii ?acee/atho iinooiaoaeueiia, a?auaoaeueiia e ieineia aeaeaeaiey.
I?e iinooiaoaeueiii aeaeaeaiee ana oi/ee oaea aeaeaeoony ii
ia?aeeaeueiui o?aaeoi?eyi, oae /oi aeey iienaiey aeaeaeaiey oaea a
oeaeii aeinoaoi/ii ciaoue caeii aeaeaeaiey iaeiie oi/ee. A /anoiinoe,
oaeie oi/eie iiaeao neoaeeoue oeaio? iann oaa?aeiai oaea. A yoii neo/aa
caaea/a iienaiey aeaeaeaiey oaea ?aoaaony n iiiiuueth oai?aiu i
aeaeaeaiee oeaio?a iann. I?e a?auaoaeueiii aeaeaeaiee ana oi/ee oaea
iienuaatho eiioeaio?e/aneea ie?oaeiinoe, oeaio?u eioi?uo eaaeao ia
iaeiie ine. Nei?inoe oi/ae ia ethaie ec ie?oaeiinoae naycaiu n
?aaeeonaie yoeo ie?oaeiinoae e oaeiaie nei?inoueth

a?auaiey: vi = [( ri ]. Oae eae oaa?aeia oaei i?e a?auaiee nio?aiyao
naith oi?io, ?aaeeonu a?auaiey inoathony iinoiyiiuie e

= [ (ri] . ( 4-1 )

( 4-2. Ii?aaeaeaiea iiiaioa neeu.

Aeey iienaiey aeeiaieee a?auaoaeueiiai aeaeaeaiey oaa?aeiai oaea
iaiaoiaeeii aaanoe iiiyoea iiiaioa neeu. I?e yoii iaaei ?acee/aoue
iiiyoey iiiaioa neeu

M

O f

r (

A

?en.11. Iiiaio neeu io-

iineoaeueii oi/ee. ioiineoaeueii oi/ee e ioiineoaeueii ine.
Anee neea f i?eeiaeaia e iaoa?eaeueiie oi/ea A(ni. ?en.11),oi iiiaioii
neeu I ioiineoaeueii i?iecaieueiie oi/ee I iacuaaaony aaeoi?iia
i?iecaaaeaiea ?aaeeona-aaeoi?a r, i?iaaaeaiiiai ec oi/ee I e oi/ea A, e
aaeoi?a neeu:

I = [ r f ] .
( 4-2 )

= r f sin (, a ia-

i?aaeaiea aaeoi?a I ii?aaeaeyaony i?aaeeii i?aaiai

ao?aa/eea: iai?aaeaiea ia?aiai aaeoi?a r ii e?ao/ae-

oaio iooe a?auaaony e iai?aaeaieth aoi?iai aaeoi?a f, a aeaeaeaiea ine
ao?aa/eea

z Mz

f

f

O f

r (

A

?en.12. Iiiaio neeu io-

iineoaeueii ine. i?e yoii a?auaiee iieacuaaao iai?aaeaiea
aaeoi?a I.

Iiiaioii neeu ioiineoaeueii i?iecaieueiie ine z

iacuaaaony aaeoi?iia i?iecaaaeaiea ?aaeeona-aaeoi?a r

e ninoaaeythuae f neeu f , i?eeiaeaiiie a oi/ea A:

I = [ r f ] ,
( 4-3 )

aaea ninoaaeythuay f i?aaenoaaeyao niaie i?iaeoeeth ne-

eu f ia ieineinoue, ia?iaiaeeeoey?ioth ine z e i?ioiaeyuoth /a?ac oi/eo
A , a r – ?aaeeon- aaeoi? oi/ee A, ea-

aeauee a yoie ieineinoe .

( 4-3. Iniiaiia o?aaiaiea aeeiaieee a?auaoaeueiiai
aeaeaeaiey.

I1

ri

mi

I2

?en.13 A?auaiea

oaa?aeiai oaea. Ionoue eiaaony oaa?aeia oaei
i?iecaieueiie oi?iu (ni. ?en 13), eioi?ia iiaeao a?auaoueny aie?oa ine
I1I2 . ?acaeaay oaei ia iaeua yeaiaiou, iiaeii caiaoeoue, /oi ana iie
a?auathony aie?oa ine I1I2 a ieineinoyo, ia?iaiaeeeoey?iuo ine a?auaiey
n iaeeiaeiaie oaeiaie nei?inoueth (. Aeaeaeaiea eaaeaeiai ec ioaeaeueiuo
yeaiaioia iaeie iannu m iienuaaaony aoi?ui caeiiii Iuethoiia. Aeey i
-ai yeaiaioa eiaai:

aaea fik ( k = 1,2, …N) i?aaenoaaeytho niaie aioo?aiiea neeu
acaeiiaeaenoaey anao yeaiaioia n aua?aiiui, a Fi – ?aaiiaeaenoaothuay
anao aiaoieo nee, aeaenoaothueo ia i – yeaiaio. Nei?inoue vi eaaeaeiai
yeaiaioa aiiaua aiai?y iiaeao iaiyoueny eae oaiaeii, ii iineieueeo oaei
yaeyaony oaa?aeui, oi niauaiey oi/ae a iai?aaeaiee ?aaeeonia a?auaiey
iiaeii ia ?anniao?eaaoue. Iiyoiio ni?iaeoe?oai o?aaiaiea ( 4-4 ) ia
iai?aaeaiea eanaoaeueiie e oiiiaeei iaa /anoe o?aaiaiey ia ri :

ri(fi1)t + ri(fi2)t + ….. +ri(fiN)t + ri(Fi)t . ( 4-4a )

A i?aaie /anoe iieo/eaoaainy o?aaiaiey i?iecaaaeaiey oeia ri(fi1)t
i?aaenoaaeytho niaie (niaeanii ( 4-3)) iiiaiou aioo?aiieo nee
ioiineoaeueii ine a?auaiey, o.e. ri e (f i)t acaeiii ia?iaiaeeeoey?iu.
Aiaeiae/ii i?iecaaaeaiey ri(Fi)t yaeythony iiiaioaie aiaoieo nee,
aeaenoaothueo ia i-yeaiaio. I?inoiie?oai o?aaiaiey aeae-

1 O1

(f12)

f12 r1

(

l12

f21

l21

(f21) ( .
2 r2

O2 ?en.14. Eiiiainaoeey

iiiaioia aioo-

?aiieo nee . aeaiey ii anai yeaiaioai, ia eioi?ua auei ?acaeoi
oaei.

Noiio iiiaioia aioo?aiieo nee iiaeii ?acaeoue ii ia?ai

neaaaaiuo, iaycaiiuo naiei aicieeiiaaieai acaeiiaeaenoaeth aeaoo
yeaiaioia oaea iaaeaeo niaie. Ia ?en.14 i?aae-

noaaeaia ia?a, ninoiyuay ec 1-ai e 2-ai yeaiaioia. I?iaiaey ieineinoue
/a?ac eeieth, niaaeeiythuoth yoe yeaiaiou, ia?aeeaeueii ine a?auaiey
I1I2, iao?oaeii caiaoeoue, /oi iiiaiou nee acaeiiaeaenoaey yoeo
yeaiaioia ?aaiu ii aaee/eia e i?ioeaiiieiaeii iai?aaeaiu, o.a. iie
eiiiaine?otho ae?oa ae?oaa. Aeaenoaeoaeueii, neeu f12 e f21 ?aaiu
iaaeaeo niaie; ?aaiu e eo ninoaaeythuea (f12) = (f21) . E?iia oiai
?aaiu e eo iea/e ( l12= l21 ), o. e. eaaeaeia ec ieo

ou nee I1 = ( f12) r1sin(900 – () = (f12) l12 e M2 = (f21) r2 sin(900
– () = (f21) l21 ?aaiu e i?ioeaiiieiaeii iai?aaeaiu. Ia iniiaaiee yoiai
iiaeii naeaeaoue auaiae, /oi i?e neiaeaiee anao iiiaioia aioo?aiieo nee
iie iiia?ii oie/oiaeaony. Noiia?iue iiiaio anao aiaoieo nee iaicia/ei (
Ii , aaea Mi = [ ri Fi].

Eaaay /anoue o?aaiaiey ( 4-4a ) n o/aoii (3 -7) i?aaenoaaeony a oaeii
aeaea:

, ( 4-5 )

i?eiyoi iacuaaoue iiiaioii eia?oeee oaa?aeiai oaea ioiineoaeueii
caaeaiiie ine. Yoa aaee/eia oa?aeoa?ecoao ?ani?aaeaeaiea iannu oaea
ioiineoaeueii ii?aaeaeaiiie ine. Eae neaaeoao ec ii?aaeaeaiey iiiaioa
eia?oeee – yoi aaee/eia aaeaeeoeaiay. Iiiaio eia?oeee oaea neeaaeuaaaony
ec iiiaioia eia?oeee aai ioaeaeueiuo yeaiaioia, eioi?ua iiaeii
?anniao?eaaoue eae iaoa?eaeueiua oi/ee, o.a.

– iiiaio eia?oeee iaoa?eaeueiie oi/ee.

I?e i?aeoe/aneii au/eneaiee iiiaioia eia?oeee aianoi noiie?iaaiey
eniieuecoaony eioaa?e?iaaiea ( noiie?iaaiea aaneiia/ii iaeuo aaee/ei).
Anee inue, ioiineoaeueii eioi?ie au/eneyaony iiiaio eia?oeee, i?ioiaeeo
/a?ac oeaio? neiiao?ee oaea, oi au/eneaiea oaeiai eioaa?aea
i?aaenoaaeyao n?aaieoaeueii ianeiaeioth caaea/o, ii a iauai neo/aa
caaea/o ?aoeoue o?oaeii. Aeey oi?iuaiey au/eneaiee iieaciie ieacuaaaony
oai?aia i ia?aeeaeueiii ia?aiina inae eia?oeee (oai?aia Atheaaina –
Ooaeia?a), oi?ioee?iaea eioi?ie aeaneo, /oi iiiaio eia?oeee
ioiineoaeueii ethaie ine ?aaai noiia iiiaioa eia?oeee ioiineoaeueii
ia?aeeaeueiie ine, i?ioiaeyuae /a?ac oeaio? iann, e i?iecaaaeaiey iannu
oaea ia eaaae?ao ?annoiyiey iaaeaeo inyie, o.a.

Ii?iec = Ioei
+ m d 2 . ( 4-6)

Aeey iaeioi?uo oae i?aaeeueiie oi?iu cia/aiea iiiaioia eia?oeee
ioiineoaeueii inae, i?ioiaeyueo /a?ac oeaio? eo neiiao?ee i?eaaaeaiu a
oaaeeoea 2.

Oaaeeoea 2.

Oi?ia oaea ?aniieiaeaiea Aaee/eia

ine iiiaioa

eia?oeee

Ia?o/ m R2

I?eia/aiea: m- ianna oaea, R – aai ?aaeeon Ia iniiaaiee
eceiaeaiiiai o?aaia-iea (4-4a) n o/aoii (4-5) i?eaiaeeony e aeaeo:

, ( 4-7 )

eioi?ia iacuaaaony o?aaiaieai aeeiaieee a?auaoaeueiiai aeaeaeaiey
oaa?aeiai oaea eee o?aaiaieai iiiaioia. Aeaei a oii, /oi eaaoth /anoue
yoiai o?aaiaiey iiaeii i?aaenoaaeoue ii ae?oaiio, o.e. ii

aiaeiaee n i?aaie /anoueth aaee/eio

iacuaatho eciaiaieai iiiaioa eiioeuena (?aaeeon ri aianai iiae ciae
aeeooa?aioee?iaaiey, o.e. ana oi/ee a?auathony ii ie?oaeiinoyi
iinoiyiiiai ?aaeeona ) . Anee

= L , oi o?aaiaiea (4-7) iiaeii ca-

. ( 4-8 )

L

O mv

r (

A

?en.15.Iiiaio eiioeue-

L = [ r p
] , ( 4-9 )

aaea cia/aiey r e ? niioaaonoaotho iaicia/aieyi ?en.12 ( n caiaiie f ia
? ). Aeey a?auaoaeueiiai aeaeaeaiey oi/ee L = [r mv] = [r m(r] = ( mr 2
= ( Ii . Aeey oaa?aeiai oaea
L = (I . (
4-10 )

( 4-4. Caeii nio?aiaiey iiiaioa
eiioeuena.

e L = const. Yoi neo/aaony, anee nenoaia caieiooa, o.a. aiaoiea neeu
aiiaua ia aeaenoaotho, eee anee iiiaiou aiaoieo nee eiiiaine?otho ae?oa
ae?oaa. Iaeiiaoe, anee aiaoiea neeu ieacuaathony oeaio?aeueiuie – eeiee
aeaenoaey anao nee ia?anaeathony a iaeiie oi/ea. Aanueia eioa?aniui
i?aaenoaaeyaony neo/ae, eiaaea iaoaie/aneee iiiaio eiioeuena i?e
a?auaiee oaea eiaao aeinoaoi/ii aieueooth aaee/eio ( ii n?aaiaieth n
iiiaioii aiaoieo nee ). Iaeaieaa y?eei i?eia?ii yoiai neoaeeo ae?ineii (
ni. ?en 16 ).

L1

d(

M L2 dL

mg

?en.16 I?aoeanney ae?i-

neiia. Ae?ineiiii i?eiyoi iacuaaoue aeinoaoi/ii ianneaiia

oaei, auno?i a?auathuaany aie?oa ine neiiao?ee. Ae?ineii cae?aieytho a
iaeiie oi/ea n iiiiuueth niaoeeaeueiiai ono?ienoaa – ea?aeaiiaa iiaeaana
. Anee ia ae?ineii aeaenoaotho aiaoiea neeu ( a?oc mg ia ?en.),

oi inue ae?ineiia ia/eiaao niauaoueny iiae aicaeaenoaeai iiiaioa neeu (
ni. ( 4-8 )), o.a. eciaiaiea iiiaioa eiioeuena niaiaaeaao n iai?aaeaiea
I. Ca iaeue i?iiaaeooie a?aiaie dt inue ae?ineiia iiaa?iao-

ny ia oaie d( oae, /oi eciaiaiea iiiaioa eiioeuena dL = L1 – L2 = Ld(. A
oi aea a?aiy ec o?aaiaiey ( 4-8 ) neaaeoao dL = M dt , eee Ld( = M
dt , ioeoaea iiaeii i?eaeoe e auaiaeo, /oi ae?ineii ia/eiaao a?auaoueny
a ieineinoe, ia?iaiaeeeoey?iie ieineinoe ?enoiea n /anoioie, eioi?ay
iacuaaaony /anoioie i?aoeannee.

. ( 4-11 )

Anee iiiaiou aiaoieo nee iaeu ii n?aaiaieth n iiiaioii eiioeuena
a?auathuaainy oaea, oi /anoioa i?aoeannee iaea, e oaei nio?aiyao
i?eaioaoeeth ine a?auaiey a i?ino?ainoaa ( i?eia? – aeiiaee?iaaiea
i?aaeiaoaie a oee?ea).

A ioee/ea io th?eaee/aneeo caeiiia, i?aaeienuaathueo oa eee eiua
i?aaeea iiaaaeaiey, oece/aneea caeiiu iinyo

iienaoaeueiue oa?aeoa? e io?aaeatho ?aaeueiua niioiioaiey iaaeaeo
?acee/iuie yaeaieyie i?e?iaeu.

Iaoa?eaeueiie oi/eie iiaeii n/eoaoue ethaie iauaeo, anee aai
aaiiao?e/aneea ?acia?u iaeu ii n?aaiaieth n oa?aeoa?enoe/aneeie
?annoiyieyie eiie?aoiie caaea/e.

O?aeoao E. Iuethoiia «Iaoaiaoe/aneea ia/aea iaoo?aeueiie oeeinioee» aue
iioaeeeiaai a 1687 a.

Aan oaea – yoi neea, n eioi?ie oaei aeaaeo ia iiaenoaaeo eee
?anoyaeaaao ieoue iiaeaana. A auoo neeo a Iuethoiiao ecia?youe ia
i?eiyoi.

Yoi ia eiaao ie/aai iauaai n oae iacuaaaiuie «i?aaeneacaieyie»
ieeoeueoiuo «iaoe».

Iieiaeeoaeueiia iai?aaeaiea ine eii?aeeiao oaeiaii iai?aaeoue aiec.

Aeey oi?iuaiey eceiaeaiey iaoa?eaea neeu o?aiey ea/aiey ia
?anniao?eaathony .

Iea/ii neeu iacuaatho aaee/eio r sin( (ci. au?aaeaiea (4-2) e
iaicia/aiey ?en.11.). Iii yaeyaony ia?iaiaeeeoey?ii, iiouaiiui ia eeieth
aeaenoaey neeu.

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