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Вычисление координат центра тяжести плоской фигуры

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Ieienoa?noai iauaai e i?ioanneiiaeueiiai ia?aciaaiey ?inneeneie
oaaea?aoeee.

O?aeueneee Ainoaea?noaaiiue Oaoie/aneee Oieaa?neoao – OIE.

?aoa?ao ii oaia:

AU*ENEAIEA EII?AeEIAO OeAIO?A OssAEANOE IEINEIE OEAO?U.

Auiieiee:

Nooaeaio a?oiiu O-149

Iie?ianeee I.A.

I?iaa?ee:

I?aiiaeaaaoaeue eaoaae?u AI e OIO

Ie?iaianeay EI.

Aeaoa?eiao?a.

1999.

Eii?aeeiaou oeaio?a oyaeanoe.

Ionoue ia ieineinoe Oxy aeaia nenoaia iaoa?eaeueiuo oi/ae

P1(x1,y1); P2(x2,y2); … , Pn(xn,yn)

c iannaie m1,m2,m3, . . . , mn.

I?iecaaaeaiey ximi e yimi iacuaathony noaoe/aneeie iiiaioaie iannu mi
ioiineoaeueii inae Oy e Ox.

Iaicia/ei /a?ac xc e yc eii?aeeiaou oeaio?a oyaeanoe aeaiiie nenoaiu.
Oiaaea eii?aeeiaou oeaio?a oyaeanoe iienaiiie iaoa?eaeueiie nenoaiu
ii?aaeaeythony oi?ioeaie:

Yoe oi?ioeu eniieuecothony i?e iouneaiee oeaio?ia oyaeanoe ?acee/iuo
oeao? e oae.

Oeaio? oyaeanoe ieineie oeao?u.

Ionoue aeaiiay oeao?a, ia?aie/aiiay eeieyie y=f1(x), y=f2(x), x=a, x=b,
i?aaenoaaeyao niaie iaoa?eaeueioth ieineoth oeao?o. Iiaa?oiinoiith
ieioiinoue, oi anoue ianno aaeeieoeu ieiuaaee iiaa?oiinoe, aoaeai
n/eoaoue iinoiyiiie e ?aaiie ( aeey anao /anoae oeao?u.

, oi ianna iieinee aoaeao i?eaeeaeaiii ?aaia

(i = 1, 2, … ,n).

I?eaeeaeaiii oeaio? oyaeanoe yoie iieinee aoaeao iaoiaeeoueny a oeaio?a
niioaaonoaothuaai i?yiioaieueieea:

Caiaiyy oaia?ue eaaeaeoth iieineo iaoa?eaeueiie oi/eie, ianna eioi?ie
?aaia ianna niioaaonoaothuae iieinee e nin?aaeioi/aia a oeaio?a oyaeanoe
yoie iieinee, iaeaeai i?eaeeaeaiiia cia/aiea oeaio?a oyaeanoe anae
oeao?u:

, iieo/ei oi/iua eii?aeeiaou oeaio?a oyaeanoe aeaiiie oeao?u:

Yoe oi?ioeu ni?aaaaeeeau aeey ethaie iaeii?iaeiie (o.a. eiathuae
iinoiyiioth ieioiinoue ai anao oi/eao) ieineie oeao?u. Eae aeaeii,
eii?aeeiaou oeaio?a oyaeanoe ia caaenyo io ieioiinoe ( oeao?u (a
i?ioeanna au/eneaiey ( nie?aoeeinue).

3. Eii?aeeiaou oeaio?a oyaeanoe ieineie oeao?u

A i?aaeuaeouae aeaaa oeacuaaeinue, /oi eii?aeeiaou oeaio?a oyaeanoe
nenoaiu iaoa?eaeueiuo oi/ae P1, P2, . . ., Pn c iannaie m1, m2, . . .,
mn ii?aaeaeythony ii oi?ioeai

.

eioaa?aeueiua noiiu, noiyuea a /eneeoaeyo e ciaiaiaoaeyo ae?iaae,
ia?aeaeoo a aeaieiua eioaa?aeu, oaeei ia?acii iieo/athony oi/iua oi?ioeu
aeey au/eneaiey eii?aeeiao oeaio?a oyaeanoe ieineie oeao?u:

(*)

Yoe oi?ioeu, auaaaeaiiua aeey ieineie oeao?u n iiaa?oiinoiie
ieioiinoueth 1, inoathony a neea e aeey oeao?u, eiathuae ethaoth
ae?oaoth, iinoiyiioth ai anao oi/eao ieioiinoue (.

Anee aea iiaa?oiinoiay ieioiinoue ia?aiaiia:

oi niioaaonoaothuea oi?ioeu aoaeoo eiaoue aeae

Au?aaeaiey

e

iacuaathony noaoe/aneeie iiiaioaie ieineie oeao?u D ioiineoaeueii inae
Oy e Ox.

au?aaeaao aaee/eio iannu ?anniao?eaaaiie oeao?u.

Oai?aiu Aoeueaeaia.

Oai?aia 1.

Ieiuaaeue iiaa?oiinoe, iieo/aiiie i?e a?auaiee aeoae ieineie e?eaie
aie?oa ine, eaaeauae a ieineinoe yoie e?eaie e ia ia?anaeathuae aa,
?aaia aeeeia aeoae e?eaie, oiiiaeaiiie ia aeeeio ie?oaeiinoe, iienaiiie
oeaio?ii oyaeanoe aeoae.

Oai?aia 2.

Iauai oaea, iieo/aiiiai i?e a?auaiee ieineie oeao?u aie?oa ine, ia
ia?anaeathuae aa e ?aniieiaeaiiie a ieineinoe oeao?u, ?aaai
i?iecaaaeaieth ieiuaaee yoie oeao?u ia aeeeio ie?oaeiinoe, iienaiiie
oeaio?ii oyaeanoe oeao?u.

II.I?eia?u.

1)

Oneiaea: Iaeoe eii?aeeiaou oeaio?a oyaeanoe iieoie?oaeiinoe X2+Y2=a2,
?aniieiaeaiiie iaae inueth Ox.

,

Iaeaeai oaia?ue i?aeeiaoo oeaio?a oyaeanoe:

2)

Oneiaea: Ii?aaeaeeoue eii?aeeiaou oeaio?a oyaeanoe naaiaioa ia?aaieu
y2=ax, ionaeaaiiai i?yiie, o=a (?en. 2)

iiyoiio

(oae eae naaiaio neiiao?e/ai ioiineoaeueii ine Ox)

3)

Oneiaea: Ii?aaeaeeoue eii?aeeiaou oeaio?a oyaeanoe /aoaa?oe yeeeina
(?en. 3)

iieaaay, /oi iiaa?oiinoiay ieioiinoue ai anao oi/eao ?aaia 1.

?aoaiea: Ii oi?ioeai (*) iieo/aai:

4)

Oneiaea:

.

?aoaiea:

aeeeia aeoae

Neaaeiaaoaeueii,

5)

Oneiaea:

Iieuecoynue oai?aiie Aoeueaeaia iaeoe eii?aeeiaou oeaio?a oyaeanoe
/aoaa?oe e?oaa

.

?aoaiea:

NIENIE ENIIEUeCIAAIIIE EEOA?AOO?U

Aeaiei I.A., Iiiia A.A., Eiaeaaieeiaa O.ss. «Aunoay iaoaiaoeea a
oi?aaeiaieyo e caaea/ao», /anoue 2, «Aunoay oeiea», Iineaa, 1999.

Ieneoiia I.N. «Aeeooa?aioeeaeueiia e eioaa?aeueiia en/eneaiey aeey
aoocia», oii 2, «Iaoea», Iineaa, 1965

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