erovnuli saswavlo olimpiada fizika - naec · 2016. 8. 21. · xi_xii klasebis moswavleebi. mocemul...

erovnuli saswavlo olimpiada

f i z i k a

2 0 1 0

1

S e s a v a l i

saqarTvelos ganaTlebisa da mecnierebis saministros iniciativiT 2009 wels

Catarda SemoqmedebiTi olimpiada sxvadasxva saganSi, maT Soris, fizikaSi.

olimpiadis mizani iyo gamorCeuli codniTa da niWiT dajildoebuli

moswavleebis gamovlena, maTi waxaliseba da saswavlo procesis stimulireba.

moswavleebs moeTxovebodaT rogorc sagnis faqtobrivi codna, aseve am codnis

gamoyeneba rogorc Teoriuli, aseve praqtikuli Sinaarsis amocanebis

amosaxsnelad. 2009 wels fizikaSi erovnuli saswavlo olimpiada or etapad

Catarda. masSi monawileobda oTxi asakobrivi jgufi: VIII klasis, IX klasis, X

klasis da XI_XII klasebis moswavleebi.

2010 wels fizikaSi erovnuli saswavlo olimpiada sam etapad Catarda da

monawileobda sami asakobrivi jgufi: IX, X da XI_XII klasebis moswavleebi.

2010/11 saswavlo wels fizikaSi erovnuli saswavlo olimpiada kvlav sam

etapad Catardeba da monawileobas miiRebs sami asakobrivi jgufi: IX, X da

XI_XII klasebis moswavleebi.

mocemul krebulSi warmodgenilia 2010/11 saswavlo wels fizikaSi erovnuli

saswavlo olimpiadis I turis programa, aseve 2009 da 2010 wlebis olimpiadebis

yvela turis davalebebi, swori pasuxebiTa da amoxsnebiT.

I turis programa

IX klasi

wrfivi Tanabari moZraobis siCqare, saSualo siCqare.

wrfivi TanabaraCqarebuli moZraoba. siCqare, aCqareba da gadaadgileba

TanabaraCqarebuli moZraobis dros.

mrudwiruli moZraoba da misi parametrebi.

moZraobis fardobiToba, siCqareTa Sekreba.

masa, simkvrive.

niutonis kanonebi. msoflio mizidulobis kanoni. simZimis, xaxunis, drekadobis Zalebi, hukis kanoni. sxeulTa wona.

martivi meqanizmebi. meqanikis oqros wesi.

sxeulTa wonasworoba, Zalis momenti, wonasworobis pirobebi. simZimis centri.

wneva siTxeebsa da airebSi. atmosferuli wneva. ziarWurWeli. paskalis kanoni.

hidravlikuri manqana.

amomgdebi Zala, arqimedes kanoni. curvic piroba.

impulsi, impulsis mudmivobis kanoni.

X klasi






masa, simkvrive.

2








meqanikuri muSaoba da energia. potenciuri da kinetikuri energia. simZlavre,

simZlavris erTeuli. m.q.k. energiis mudmivobis kanoni.

meqanikuri rxeva da misi parametrebi. rezonansi.

ganivi da grZivi talRebi. bgera da misi gavrceleba. eqo. dopleris efeqti.

siTburi gafarToeba, wylis anomalia. siTbos raodenoba, kuTri siTbotevadoba.

wvis siTbo. nivTierebis gadasvla erTi agregatuli mdgomareobidan meoreSi.

duRilis temperatura. siTburi balansis gantoleba..

sinaTlis sxivis gavrceleba, gardatexa, arekvla, STanTqma, dispersia. Semkrebi da

gambnevi linzebi. brtyeli sarke.

damuxtuli sxeulebis urTierTqmedeba. kulonis kanoni.

XI - XII klasebi






masa, simkvrive.








meqanikuri muSaoba da energia. potenciuri da kinetikuri energia. simZlavre,

simZlavris erTeuli. m.q.k. energiis mudmivobis kanoni.

meqanikuri rxeva da misi parametrebi. rezonansi.

ganivi da grZivi talRebi. bgera da misi gavrceleba. eqo. dopleris efeqti.

siTburi gafarToeba, wylis anomalia. siTbos raodenoba, kuTri siTbotevadoba.

wvis siTbo. nivTierebis gadasvla erTi agregatuli mdgomareobidan meoreSi.

duRilis temperatura. siTburi balansis gantoleba..

sinaTlis sxivis gavrceleba, gardatexa, arekvla, STanTqma, dispersia. Semkrebi da

gambnevi linzebi. brtyeli sarke.

denis Zala, Zabva, gamtaris winaRoba. omis kanoni wredis ubnisaTvis. gamtarTa

mimdevrobiTi da paraleluri SeerTeba.

3

2009 VIII klasi

I etapi

1. ori matarebeli moZraobs erTmaneTis Sesaxvedrad 36 km/sT da 54 km/sT siCqareebiT. pirvel matarebelSi myofma mgzavrma SeniSna, rom meore

matarebelma mas Cauara 6 wamSi. ra sigrZisaa meore matarebeli?

amoxsna

pirvel matarebelSi mjdom mgzavrs meore matarebeli Cauvlis siCqariT,

romelic matareblebis siCqareTa jamis tolia: 21 VVV +=

t droSi am siCqariT gavlili manZili tolia ( )tVVVtL 21 +== ricxviTi mniSvnelobis misaRebad saWiroa siCqareTa ganzomilebis

gadayvana SI erTeulebSi: 36V1 = km/sT 10= m/wm; 54V2 = km/sT=15m/wm; =V 25m/wm.

meore matareblis sigrZea =L 25 m/wm · 6 wm =150 m

2. 1 m radiusis mqone wrewirze erTi wertilidan erTnairi mimarTulebiT Tanabar moZraobas iwyebs ori sxeuli 3 m/wm da 1 m/wm

siCqareebiT. ra minimaluri drois Semdeg aRmoCndebian es sxeulebi

isev erT wertilSi?

amoxsna

swrafad moZravi sxeuli Sordeba nela moZravs siCqariT, romelic

tolia =−= 21 VVV 3 m/wm – 1 m/wm = 2 m/wm

erT wertilSi es sxeulebi aRmoCndebian isev mas Semdeg, rac swrafi

sxeuli gaivlis wrewiris sigrZis toli manZiliT mets, vidre nela

moZravi.

wrewiris sigrZe tolia R2L π= =6,28 m. amitom es dro tolia

=−π

==21 VV

R2V/Lt 3,14 wm

3. adamiani uZravi navidan 10 m/wm siCqariT agdebs horizontalurad 20 kg masis tvirTs. ra siCqares miiRebs amis Sedegad navi, Tu navis,

adamianis, da tvirTis masa erTad aris 220 kg?

amoxsna

navi, adamiani da qva warmoadgenen sistemas, romlis impulsis gegmili

horizontalur mimarTulebaze inaxeba. vinaidan tvirTis gadagdebamde

navi uZravia, amitom tvirTis gadagdebis Semdegac am sistemaSi Semavali

sxeulebis impulsTa horizontaluri gegmilebis jami nulis toli unda

iyos:

4

( ) 0mVVmM =− , sadac m – qvis masaa, M – navis masaa adamianTan erTad tvirTis gadagdebamde, 0V tvirTis gadagdebis siCqarea, xolo V – siCqarea, romelsac miiRebs navi. saZiebeli siCqarisaTvis miviRebT

( ) =−= mMmVV 0 1 m/wm

4. sxeuli wyalSi tivtivebs ise, rom wyalSi moTavsebuli nawilis moculoba mTeli moculobis 1/4-ia. gansazRvreT am sxeulis simkvrive,

Tu wylis simkvrivea 103 kg/m3.

amoxsna

sxulis tivtivis pirobaa masze moqmedi amomgdebi da simZimis Zalebis

toloba. amomgdebi Zala tolia gVF ρ= , sadac ρ aris wylis simkvrive,

g = 9,8 m/wm2 – Tavisufali vardnis aCqareba, xolo V _ motivtive sxeulis wyalqveSa nawilis moculoba. sxeulze moqmedi simZimis Zala tolia

gVmg 00ρ= , sadac 0ρ da 0V aris sxeulis simkvrive da moculoba. am ori Zalis tolobidan miviRebT:

4V4g

gV0

ρ=

ρ=ρ = 250 kg/m3

5. rZiani Cais momzadebisas bavSvma 120 g wyals Seuria 80 g rZe. risi toli aRmoCnda miRebuli Cais simkvrive? wylis simkvrivea 1,0 g/sm3,

xolo rZisa ki 1,40 g/sm3.

amoxsna

rZisa da wylis Serevisas maTi masebi da moculobebi ikribeba:

,mmm 21 += 21 VVV += , amitom narevis simkvrivea 21

21

VVmm

++

=ρ . wylisa da

rZis moculobebi tolia 1

11

mVρ

= da 2

22

mVρ

= . sabolood vipoviT narevis

simkvrives: ≈

ρ+

ρ

+=ρ

2

2

1

1

21

mmmm

1,13 g/sm3.

5

2009 - VIII klasi

II etapi

1. 20 m sigrZis Toki gadadebulia uwonad WoWonaqze. sawyis momentSi Toki WoWonaqze simetriulad kidia da uZravia. patara biZgis Sedegad

Toki iwyebs Camosrialebas. risi toli iqneba Tokis siCqare misi

bolos WoWonaqidan Camosrialebis momentSi? rogoria Tokis

moZraoba Camosrialebis procesSi (Tanabari, araTanabari,

TanabaraCqarebuli da a. S.)? pasuxi daasabuTeT. WoWonaqis diametri

Tokis sigrZeze gacilebiT naklebia. xaxuni Toksa da WoWonaqs Soris

didia.

amoxsna

Tokis aCqareba dakavSirebulia masaTa sxvaobasTan WoWonaqis or mxares.

vinaidan Tokis CamosrialebasTan erTad izrdeba es sxvaobac gavlili

manZilis proporciulad, amitom mamoZravebeli Zalis saSualo

mniSvneloba aris saSualo ariTmetikuli Zalis sawyisi da saboloo

mniSvnelobebis. cxadia, rom igivea samarTliani moZraobis saSualo

aCqarebasaTvisac. aCqarebis sawyisi mniSvneloba nulis tolia, saboloo

mniSvneloba ki Tavisufali vardnis aCqarebis. amrigad, saSualo aCqareba

a= g/2. Tokis mier gavlili gza srul Camosrialebamde aris L/2, sadac L Tokis sigrZea. kinematikis formulebis Tanaxmad

=== 2/2/2 gLaLV 10m/wm an:

Tokis simZimis centri sawyisi momentidan srul Camosrialebamde daiwevs

5 metriT. Tokis kinetikuri energia sruli Camosrialebis momentSi mV2/2, Tokis potencialuri energiis cvlilebis mgh-is tolia, sadac h=5m, V=√2gh=10m/wm

2. turistTa aTkaciani jgufi apirebs tivis Sekvras tbis gadasacurad. tivis Semadgeneli xis yoveli moris diametria 50 sm, sigrZe ki 4 m-

ia. morebis ra umciresi raodenobidan unda Sekran tivi turistebma,

raTa gadacuron tba fexebis dausveleblad, Tu yoveli turistis

masa bargTan erTad saSualod Seadgens 90 kg-s? wylis simkvrivea 1000

kg/m3, xisa ki 800 kg/m3.

amoxsna

morebis minimaluri raodenoba ganisazRvreba im pirobiT, rom tivze da

turistebze moqmedi simZimis Zala unda gawonaswordes tivze moqmedi

ZaliT im mdgomareobaSi, rodesac tivi TiTqmis mTlianad CaZirulia

wyalSi:

MgmgF += 10 (1) sadac F – amomgdebi Zalaa, M da m – tivisa da yoveli turistis masebia. es amomgdebi Zala tolia

gVF 0ρ= , gVM ρ= (2)

6

sadac 0ρ da ρ _ wylisa da xis simkvriveebia, xolo V – tivis

moculobaa. D diametrisa da L sigrZis N moridan Semdgari tivisaTvis

LDNV4

2π= (3)

(2) da (3) CavsvaT (1)-Si. sabolood miviRebT: ( ) 7,540

02 ≈−

=ρρπ LD

mN

amrigad, tivi unda Seikras 6 morisgan.

3. ori sxeuli vardeba sawyisi siCqaris gareSe sxvadasxva simaRlidan da dedamiwas erTdroulad ecema. pirveli sxeulis vardnis droa 3

wm, meoresi ki _ 1 wm. ra simaRleze iyo pirveli sxeuli, rodesac

meorem daiwyo vardna? Tavisufali vardnis aCqareba CaTvaleT 10 m/wm2-

is tolad.

amoxsna

pirveli sxeulis vardnis simaRlea 2

21

1gtH =

rodesac daiwyo vardna meore sxeulma, pirveli sxeuli ukve moZraobda

21 ttt −= wamis ganmavlobaSi da hqonda gavlili ( )

2

221

1ttgh −= manZili.

maSasadame, is imyofeboda 11 hHh −= simaRleze. ricxviTi mniSvnelobebis Casmis Sedegad miviRebT h=25 m.

4. ori erTnairi mcire zomis metalis burTula damuxtulia sapirispiro niSnis muxtebiT ise, rom erT-erTis muxti sididiT

cxrajer metia meorisaze. manZili am burTulebs Soris 2 m-ia.

wertilovani muxti moTavsebulia ise, rom is wonasworobaSia.

burTulebi Seaxes erTmaneTs da isev imave wertilebSi daabrunes.

ramdeniT unda gadavaadgiloT wertilovani muxti imisaTvis, rom igi

isev wonasworobaSi aRmoCndes?

amoxsna

imisaTvis, rom wertilovani muxti wonasworobaSi iyos, is aucileblad

unda imyofebodes burTulebis SemaerTebel wrfeze. vinaidan burTulebis

muxtebi sapirispiro niSnebisaa, wertilovani muxti unda ganlagdes

burTulebs gareT. konkretulobisaTvis davuSvaT, rom cxrajer metia

dadebiTi muxti da wertilovani muxtic dadebiTia. mniSvnelovania, rom

saboloo Sedegi ar aris damokidebuli am daSvebebze. aRvniSnoT manZili

burTulebs Soris L-iT, xolo manZili dadebiTad damuxtuli burTuladan wertilovan muxtamde x-iT (ix. nax.).

x L +Q -q2 +q1

7

maSin wertilovani muxtis wonasworobis pirobaa

( ) 2

22

1

xQqk

xLQqk =+

aqedan

2/Lx = burTulebis Sexebis Sedegad maTze gaCndeba toli erTniSna muxtebi.

amitom wertilovani muxti iqneba wonasworobaSi burTulebis

SemaerTebeli monakveTis Sua wertilSi.

amrigad, wertilovani muxti unda gadavaadgiloT L/2+L/2 = 2 m-iT.

5. 60 sm sigrZis uwonad berketze gaawonaswores erTnairi masis mqone rkinisa da aluminis burTula. ramdeniT da saiT unda gadavwioT

berketis Camokidebis wertili imisaTvis, rom burTulebis wyalSi

CaSvebis Semdeg berketi isev wonasworobaSi iyos? aluminis, rkinisa

da wylis simkvriveebia Sesabamisad 2,7g/sm3, 7,8 g/sm3 da 1 g/sm3.

amoxsna

vinaidan burTulebis masebi tolia, haerSi berketis Camokidebis wertili

misi Suawertilia. aluminis burTulis simkvrive naklebia rkinisaze,

amitom misi moculoba meti iqneba da wyalSi CaSvebisas masze moqmedi

amomgdebi Zala meti iqneba, vidre rkinis burTulaze. amis Sedegad rkinis

burTulaze moqmedebs meti tolqmedi Zala, radgan simZimis Zalas

akldeba naklebi amomgdebi Zala aluminis burTulasTan SedarebiT.

amitom berketis Camokidebis wertili unda gadavwioT rkinis

burTulisaken.

aRvniSnoT saZiebeli manZili X-iT, xolo berketis sigrZe L-iT. maSin berketis wonasworobis piroba burTulebis wyalSi CaSvebis Semdeg

iqneba:

( ) ( ) ⎟⎠⎞

⎜⎝⎛ −−=⎟

⎠⎞

⎜⎝⎛ +− XLFmgXLFmg

22 21

sadac F1 da F2 _ Sesabamisad aluminis da rkinis burTulebze moqmedi amomgdebi Zalebia. amomgdebi ZalebisTvia CavwerT:

mggVFρρ

ρ 00 ==

0ρ _wylis simkvrivea. am gamosaxulebis gamoyenebiT miviRebT gantolebas

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎠⎞

⎜⎝⎛ +=⎟⎟

⎠

⎞⎜⎜⎝

⎛−⎟

⎠⎞

⎜⎝⎛ −

1

0

2

0 12

12 ρ

ρρρ XLXL

ricxviTi mniSvnelobebis gamoyenebiT da algebruli gardaqmnebis

Sedegad miviRebT

≈X 4,8 sm

8

2009 IX klasi

I etapi

1. 15 km/sT siCqariT moZravi avtomobilis samuxruWo manZili 5 m-ia. gamoTvaleT am avtomobilis samuxruWo manZili im SemTxvevaSi,

rodesac misi siCqarea 90 km/sT.

amoxsna

manZili S, gavlili avtomobilis mier gaCerebamde, dakavSirebulia damuxruWebis a aCqarebasTan da sawyis V siCqaresTan formuliT:

a2VS

2

= . vinaidan damuxruWebis aCqareba orive SemTxvevaSi erTia, miviRebT:

2

22

1

21

S2V

S2V

= aqedan: == 121

22

2 SVVS 180 m

2. 200 l tevadobis abazanaSi Seuries 10°C temperaturis civi da 60°C temperaturis cxeli wyali. ra moculobis cxeli da civi wyali unda

SevurioT, rom wylis temperatura gavsebul abazanaSi iyos 40°C?

amoxsna

aRvniSnoT cxeli da civi wylis moculobebi da maTi temperaturebi

Sesabamisad 1V , 2V , 1t da 2t , abazanis moculoba V da wylis saboloo temperatura t . maSin siTburi balansis gantolebas eqneba aseTi saxe:

( ) ( )2211 ttcVttcV −ρ=−ρ sadac ρ da c _ wylis simkvrive da kuTri siTbotevadobaa. imis gaTvaliswinebiT, rom VVV 21 =+ , sabolood saZiebeli masebisTvis

miviRebT: VttttV

21

21 −

−= , da V

tttt

V21

12 −

−= . ricxvTi mniSvnelobebis CasmiT

miviRebT:

1V = 120 l da =2V 80 l

3. ra manZilze unda movaTavsoT sagani 12 sm fokusuri manZilis mqone Semkrebi linzidan, rom misi namdvili gamosaxuleba 3-jer meti iyos

saganze?

amoxsna

Txeli linzis formulaa fdF111

+= . d -ze gamravlebiT da

gamadideblobis formulis df

=Γ gamoyemebiT miviRebT Γ

+=11

Fd

. aqedan

⎟⎠⎞

⎜⎝⎛

Γ+=

11Fd sabolood d =16 sm.

9

4. ra siRrmeze iqneba tbaSi wneva 3-jer meti, vidre wylis zedapirze? atmosferuli wneva normaluria (760 mm. vercxliswylis sveti). (wylis

simkvrivea 103 kg/m3, vercxliswylis simkvrivea 13,6*103 kg/m3)

amoxsna

p tbaSi h siRrmeze tolia atmosferuli 0p wnevisa da hidrostatikuri wnevis ghρ jamis. ghpp 0 ρ+= , sadac ρ aris wylis simkvrive. amocanis pirobidan 00 p2ghp3p =ρ⇒= . Tu gavixsenebT, rom normaluri atmosferuli wneva tolia 105 pa, xolo wylis simkvrivea 103kg/m3,

ukanaskneli tolobidan SegviZlia saZiebeli h siRrmis povna. ricxviTi gamoTvlebis gasamartiveblad SegviZlia gamoviyenoT Tanafardoba:

000 ghp ρ= , sadac =0ρ 13,6 g/sm3 da 0h =76 sm aris vercxliswylis svetis simaRle normaluri atmosferuli wnevis dros. amrigad, miviRebT

766,132 ⋅⋅=h sm≈20,7 m

5. moswavles aqvs sami erTnairi liTonis damuxtuli burTula,

romelTa muxtebia q1 =+4*10-9k, q2 =+1*10-8k da q3 =-3*10-9k. man jer Seaxo pirveli burTula meores da daaSora. Semdeg Seaxo meore mesames,

daaSora da bolos Seaxo pirveli mesames. rogori muxtebi eqnebaT am

burTulebs amis Semdeg?

amoxsna

gaviTvaliswinoT, rom burTulebi erTnairia. es imas niSnavs, rom ori

burTulis yoveli Sexebis Sedegad maTze arsebuli muxtebi Tanabrad

gadanawildeba am or burTulas Soris. amitom pirveli Sexebis Semdeg

pirvel da meore burTulaze aRmoCndeba toli Q`=+7·10-9k muxti. meore Sexebis Semdeg meore da mesame burTulebze iqneba toli Q``=+2·10-9k. da bolos mesame Sexebis Semdeg pirveli da mesame burTulebs eqnebaT

muxtebi Q```=+4,5·10-9k. sabolood burTulebis muxtebi iqneba Q1=+4,5·10-9k, Q2=+2·10-9k da Q3=+4,5·10-9k.

10

2009 IX klasi

II etapi

1. axlomxedvel adamians SeuZlia teqstis wakiTxva, Tu misi daSoreba Tvalebidan ar aRemateba 18 sm-s. ra optikuri Zalis mqone saTvale

unda ataros am adamianma?

amoxsna

amocanis pirobis Tanaxmad, axlomxedveli Tvalis akomodaciis wertili

Tvalidan 18 sm manZilzea, maSin rodesac janmrTeli Tvalis akomodaciis

wertili usasrulobaSi mdebareobs. amrigad, gamosayenebelia iseTi

saTvale, romelic usasrulobidan dacemul sxivebs (anu sxivTa

paralelur konas) gadaaqcevs iseT konad, romelic Tvals moeCvenaba 18

sm manZilidan gamosulad. amrigad, es unda iyos gambnevi linza, romlis

fokusuri manZilia – 18 sm. misi optikuri Zala – 5,6 dioptris tolia.

2. wyalqveSa navi eSveba vertikalurad mudmivi siCqariT. misi

hidrolokatori gamoscems 0τ xangrZlivobis mokle bgeriT impulsebs Sveulad horizontaluri fskerisaken. navze miRebuli areklili

impulsebis xangrZlivoba aRmoCnda τ -s toli. risi tolia navis siCqare? bgeris siCqare wyalSi c-s tolia.

amoxsna

naxatze moyvanilia bgeriTi impulsis

amplitudis droze damokidebulebis

gamartivebuli grafiki. drois aTvlis

sawyisi momenti davamTxvieT impulses wina

frontis gamosxivebas.

SemoviRoT Semdegi aRniSvnebi:

H - navis daSorebaa fskeridan drois sawyis momentSi T – gamosxivebuli impulses wina frontis dabrunebis dro, maSin 0ττ −+T aris impulsis ukana frontis dabrunebis momenti bgeris mier gavlili manZilebia:

wina frontisaTvis signalisTvis:

VTHVTHHcT −=−+= 2 sadac V aris navis moZraobis siCqare

impulsis ukana frontisaTvis:

( ) { } )(2)()( 000 ττττττ +−−=+−+−=−+ VVTHTVHVHTc am ori gantolebis gamoyenebiT advilad miviRebT:

ττττ

+−

=0

0cV

misarebia amoxsna meore gziTac. kerZod gadasvla navTan dakavSirebul

aTvlis sistemaSi.

0 τ0

crA

t

11

3. pirvelad navma gadacura mdinare umoklesi gziT, xolo meored uswrafesiT (anu im gziT, romlis gavlas umciresi dro sWirdeba).

SeadareT am or SemTxvevaSi navis mier gavlili manZilebi da

daxarjuli droebi (ipoveT maTi Sefardebebi). dinebis siCqarea, xolo

navis siCqare wylis mimarT V0 -is tolia (V0 > U).

amoxsna

gasagebia, rom umoklesi iqneba traeqtoria, romelic mdinaric napirebis

marTobulia. amitom siCqareTa veqtorebi

Semdegnairad iqneba ganlagebuli (nax. 1).

am SemTxvevaSi: nax. 1

2201 UVV −= , ,1 LS = 22

0

1UV

Lt−

=

sadac V1, S1 da t1 aris am SemTxvevaSi kateris moZraobis siCqare, gavlili manZili da daxarjuli dro.

meore SemTxvevaSi siCqareTa ganlageba Semdegnairia (nax. 2)

2202 UVV += ,

02 V

Lt = , 2200

2 UVVLS +=

sabolood miviRebT:

22

0

0

2

1

UV

VSS

+= da

220

0

2

1

UV

Vtt

−=

4. 20 sm diametrisa da 1 m simaRlis cilindruli WurWeli avsebulia wyliT da Camokidebuli uWimvad da uwonad Tokze. WurWlis simaRlis

naxevarze mis kedelSi gaCnda 2 sm2 farTobis naxvreti, romlidanac

gamoedineba wyali. vertikalidan ra kuTxiT gadaixreba Toki, sanam

wylis masa mniSvnelovnad Semcirdebodes, Tu gadaxris Semdeg wyali naxvretidan gamoedineba 4,4 m/wm siCqariT? (ipoveT am kuTxis romelime

trigonometruli funqciis mniSvneloba). WurWlis masa wylis

masasTan SedarebiT mcirea.

amoxsna

WurWlis wonasworobis piroba warmoadgens Camokidebis O wertilis mimarT simZimisa da reaqciis Zalebis

momentebis tolobas.

( ) ( )

MgF

hLFhLMg2sin

2/sin2/

=

⇒+=+

α

α

sadac M –wylis masaa, F – reaqciis Zalaa, L – Tokis sigrZea da h- WurWlis simaRle. wylis masisaTvis CavwerT

4

2 ρπ hdM = (ρ-wylis simkvrivea).

V1 V0

U

V2 V0

U

F

mg

Oα

12

reaqciis ZalisaTvis: t

mVF = , sadac m aris t droSi V siCqariT

gadmoRvrili wylis masa, romlisTvisac SVtm ρ= (S – naxvretis farTobia). sabolood

2SVm ρ= . saZiebeli kuTxisaTvis vipoviT:

≈=hgd

SV2

24sinπ

α 1,22·10-2 rad

5. pirvel WurWelSi wyalSi dacuravs yinulis naWeri, romelzec devs

tyviis safanti, meore WurWelSi ki dacuravs yinulis naWeri,

romelzec devs xis naWeri. yinuli gadna. SeadareT wylis doneebi

WurWlebSi amis Semdeg sawyis doneebs. pasuxi daasabuTeT.

amoxsna

arqimedes kanonis Tanaxmad, orive WurWelSi Tavdapirvelad gamodevnili

wylis moculobaa erTnairia: 0

0 ρmMV += , sadac M da m – yinulis da

safantis (an xis naWris) masebia, xolo 0ρ wylis simkvrivea. vinaidan yinulis dnobis Sedegad miRebuli wylis masa yinulis masis tolia,

amitom orive WurWelSi momatebuli wylis moculoba iqneba 0/ ρM . pirvel WurWelSi safanti CaiZireba da gamodevnili wylis moculobaa

ρ/m , sadac ρ tyviis simkvrivea ( 0ρρ > ). vinaidan meore WurWelSi xis naWeri itivtivebs, am WurWelSi gamodevnili wylis moculoba, arqimedes

kanonis Tanaxmad, iqneba 0/ ρm . sabolood pirvel WurWelSi gamodevnili wylis moculobaTa sxvaoba

iqneba

000

0 >−=−− ρρρρmmmMV

amrigad, pirvel WurWelSi wylis done daiwevs pirvandelTan SedarebiT.

meore WurWelSi gamodevnili wylis moculobaTa sxvaoba iqneba

000

0 =−− ρρmMV . maSasadame, meore WurWelSi wylis done ar Seicvleba.

13

2009 X klasi

I etapi

1. hidravlikuri wnexiT 2 tona masis tvirTis awevisas Sesrulda 40 j muSaoba. am dros mcire dguSi gadaadgilda 1 metriT. miiReT didi

dguSis farTobis mcire dguSis farTobTan Sefardebis saangariSo

formula da ipoveT am fardobis ricxviTi mniSvneloba.

amoxsna

hFA = , sadac F patara dguSze moqmedi Zalaa, 40=A j, 1=h m. aqedan

hAF =

hidravlikur wnexSi did dguSze moqmedi Zalis Sefardeba patara dguSze

moqmed ZalasTan dguSebis farTobebis Sefardebis tolia.

did dguSze moqmedi Zalaa gm , sadac kgt 20002 ==m .

maSasadame, 5001

2 ===A

hgmF

gmSS

2. dedamiwis zedapiridan garkveuli simaRlidan wmm10 siCqariT vertikalurad zeviT asrolili sxeuli dedamiwaze dacemamde m40 manZils gadis. gansazRvreT, ra simaRlidan aisroles sxeuli.

amoxsna

wmm10v0 = , 2

wmm10=g , m40=l .

vertikalurad zeviT gavlili manZilia g

h2v20= .

vertikalurad qveviT gavlili manZilia Hh + , sadac H saZebni simaRlea.

sul gavlili manZilia Hh += 2l , saidanac m30v220 =−=−=

ghH ll

3. horizontalur iatakze m8,1 simaRlidan vardeba burTi. iatakTan yoveli dajaxebisas igi kargavs Tavisi kinetikuri energiis 40%-s.

gansazRvreT, ra vertikaluri sawyisi siCqare unda mivaniWoT burTs,

rom iatakTan ori dajaxebis Semdeg man sawyis simaRles miaRwios.

amoxsna

pirveli dajaxebis Semdeg darCenili energiaa 60%, ori dajaxebis Semdeg

darCenili energia ki_ sawyisis 36%. sawyisi energiaa 2v2mhgm + ,

ori dajaxebis Semdeg energia pirobis Tanaxmad unda gaxdes hgm .

hgmmhgm =+ )2v(36,0

2

, saidanac

wm

m818

64v == hg

14

4. Coj500 siTbotevadobis kalorimetrSi asxia 200 g masis wyali. maTi temperaturaa Co12 . wyalSi CauSves Co100 temperaturis 250 g masis foladis naWeri. ipoveT wylis saboloo temperatura. wylis kuTri

siTbotevadobaa Co⋅kgj4200 , foladis _ Co⋅kgj460 . garemosTan siTbocvla SegiZliaT ugulebelyoT.

amoxsna

C500C oj= , C12t1o= , g200m1 = , C4200c1

o⋅= kgj C100t2o= , g250m2 = ,

Cc o⋅= kgj4602 . saboloo temperatura t . siTburi balansis gantolebaa

( ) ( ) ( ) 0ttmcttmcttC 2221111 =−+−+− C19mcmcCtmctmctCt

2211

2221111 o≈++++

=

5. linzis meSveobiT viRebT sagnis namdvil, orjer Semcirebul gamosaxulebas. Tu sagans 21 sm-iT mivuaxloebT linzas, miviRebT

sagnis namdvil, orjer gadidebul gamosaxulebas. ipoveT linzis

fokusuri manZili.

amoxsna

11 f1

d1

F1

+= , 21

df

1

1 = F3d1 =⇒

22 f1

d1

F1

+= , 2df

2

2 = 2

32

Fd =⇒

pirobis Tanaxmad sm21dd 21 =− . am tolobaSi wina Sedegebis SetaniT miiReba, rom sm14=F .

15

2009 X klasi

II etapi

1. uwono berketis sxvadasxva wertilSi dakidebulia foladis da aluminis sxeulebi. berketi wonasworobaSia haerSic da sxeulebis

wyalSi mTlianad CaSvebis Semdegac. gansazRvreT, romel sxeulSia

siRrue da sxeulis moculobis ra nawilia is. aluminis simkvrivea

3m

kg3107,2 × , xolo foladis _ 3

m

kg3108,7 × .

amoxsna

davweroT berketis wonasworobis piroba haerSi

21 dgVdgV alalff ρρ = , radganac berketi wonasworobaSia sxeulebis wyalSi CaSvebis Semdegac,

amitom erTmaneTis toli unda iyos amomgdebi Zalebis momentebi

2211 dgVdgV wywy ρρ =

am tolobebidan miiReba 21 VV

VV alalff ρρ = (1). pirobis Tanaxmad, an

alf VVVV >= 21 , , an alf VVVV => 21 , . advili misaxvedria, rom SesaZlebelia

mxolod meore varianti, (radganac alf ρρ > ) anu siRrue foladis

sxeulSia. (1) formulidan miiReba f

alf

ρρ

=1V

V anu 65,01

1

≈−=f

alsiR

ρρ

VV

.

2. WurWelSi, romelSic 20 Co temperaturis 2,8 l wyali asxia, Cades 460 Co temperaturis 3 kg masis foladis naWeri. wyali gaTba 60 Co -mde, nawili ki aorTqlda. ipoveT aorTqlebuli wylis masa. wylis

kuTri siTbotevadobaa Co⋅kg

j4200 , foladis kuTri siTbotevadobaa

Co⋅kgj

460 , wylis orTqladqcevis kuTri siTboa kg

j6103,2 × . WurWlis

siTbotevadoba da siTburi danakargebi atmosferoSi ugulebelyaviT.

amoxsna

xm masis wyali, romelic aorTqlda, gaTba duRilis temperaturamde anu Co100 –mde, xolo danarCeni wyali gaTba Ct o60= -mde.

davweroT siTburi balansis piroba

( ) ( )( ) ( ) 02211 =−+−−++− ttmcttmmcmLttmc xxx fwyduRwy aq Ct o201 = , Ct

o4602 = , kg32 =m foladis naWris masaa, kg8,21 =m 2,8 l wylis masaa.

dawerili gantolebidan

16

( ) ( )

( ) gkgduRwywyf 33033,01122 =≈+−

−−−=

Lttcttmcttmc

mx

3. gvaqvs ampermetri danayofis fasiT mka100 =i . mis skalas aqvs 100=n danayofi. xelsawyos winaRobaa omi50=r . ra damatebiTi winaRoba mivuerToT ampermetrs da rogor, rom miviRoT voltmetri, romlis

gazomvis zRvaria v2000 =U ? rom miviRoT ampermetri, romlis gazomvis zRvaria ma8000 =I ?

amoxsna

voltmetris misaRebad damatebiTi winaRoba mimdevrobiT unda mivuerToT

da ise SevarCioT, rom xelsawyoSi 0in maqsimaluri denis Zala gadiodes

0U maqsimaluri gasazomi Zabvis dros. gvaqvs omi5

0

0

0

01 102×=≈−= in

Urin

UR .

didi denis gamzomi ampermetris misaRebad damatebiTi winaRoba (Sunti)

unda mivuerToT paralelurad ise, rom xelsawyoSi 0in deni gadiodes 0I

maqsimaluri gasazomi denis dros. gvaqvs omi06,00

0

00

02 =≈−=

Irin

inIrinR .

4. gluv horizontalur zedapirze devs M masis firfita. masze kideSi devs m masis Zelaki. ipoveT firfitaze horizontalurad moqmedi maqsimaluri 0F Zala, romlis drosac Zelaki jer kidev ar

misrialebs masze. ra droSi Camosrialdeba

Zelaki firfitidan, Tu masze moqmedi

horizontalurad mimarTuli F Zala metia 0F -ze. firfitis sigrZea L . Zelaksa da

firfitas Soris xaxunis koeficientia µ .

amoxsna

Tu Zelaki ar srialebs, maSin mas da firfitas toli aCqareba aqvT. mas

sistemisaTvis niutonis meore kanonis gamoyenebiT vipoviT. mM

Fa+

= .

Zelaks am aCqarebas uZraobis xaxunis Zala aniWebs. gamoviyenoT

ZelakisaTvis niutonis II kanoni

mM

FmamF+

==xax

(1)

uZraobis xaxunis Zala akmayofilebs utolobas

NF µ≤xax

, (2)

sadac Zelakze moqmedi reaqciis Zalaa. is tolia

gmN = (3) (1-3) formulebidan

17

gmmM

Fm µ≤+

anu ( )gmMF +≤ µ maSasadame,

( )gmMF += µ0 Tu 0FF > , maSin Zelaki srialebs firfitaze da isini urTierTqmedeben srialis xaxunis ZaliT, romelic tolia gmF µ=

xax.

niutonis II kanonis Tanaxmad:

M

gmFa µ−=1 (firfitisaTvis)

gm

gma µµ ==2 (ZelakisaTvis)

Zelakis aCqareba firfitis mimarT aris (moduliT)

( )M

gmMFgM

gmFaaa +−=−−=−= µµµ21

Zelakis sawyisi siCqare firfitis mimarT nulis tolia, xolo

Camovardnamde firfitis mimarT Zelakis gavlili manZilia L e.i.

2

2taL = , saidanac ( )gmMFML

aLt

+−==

µ22

5. linzis saSualebiT ekranze miiRes sagnis 2-jer gadidebuli mkafio gamosaxuleba. manZili sagansa da ekrans Soris 1,6-jer gazardes da

igive linziT ekranze isev miiRes sagnis mkafio gamosaxuleba.

gansazRvreT, risi tolia gadideba am SemTxvevaSi.

amoxsna

ekranze miiReba mxolod namdvili gamosaxuleba, e.i. linza Semkrebia.

gvaqvs formulebi Lfd =+ , df

=Γ , fdF111

+= , sadac L sagansa ekrans

Soris manZilia. am formulebidan gamomdinareobs ( )

LF Γ+Γ

=211.

gamoviyenoT es formula or SemTxvevaSi: ( )

11

21 11

LF Γ+Γ

= , ( )

22

22 11

LF Γ+Γ

= .

pirobis Tanaxmad: 12 6,1 LL = da 21 =Γ . amis gaTvaliswinebiT miiReba kvadratuli gantoleba 2Γ -saTvis: 012,5 2

22 =+Γ−Γ .

gantolebas aqvs amonaxsnebi: 52 =Γ da 51

2 =Γ . orive SemTxvevaa

SesaZlebeli. swored asec unda iyos sxivTa svlis Seqcevadobis gamo.

18

2009 XI-XII klasebi

I etapi

1. tyviis sawyisi siCqarea wmm600 , masa ki _ g10 . traeqtoriis umaRles wertilSi tyviis kinetikuri energiaa j450 . gansazRvreT, horizontisadmi ra kuTxiTaa gasrolili tyvia. ugulebelyaviT haeris

winaaRmdegobis Zala.

amoxsna

wm

m600v0 = , kgg 01,010 ==m , j450=E . wm

m300mE2v

2vmE

2

==⇒= , sadac v

tyviis siCqarea traeqtoriis umaRles wertilSi. moZraobis dros ar

icvleba tyviis siCqaris gegmili horizontalur RerZze, amitom

αcosvv 0= . maSasadame, o605,0

vvcos

0

=α⇒==α .

2. gansazRvreT, ra siCqariT moZraobda tyviis burTula, Tu zustad iseTive uZrav burTulasTan dajaxebis Semdeg, orive burTula

moZraobs erTad da maTi temperatura Co5,2 -iT gaizarda. tyviis kuTri siTbotevadobaa C130 o⋅kgj .

amoxsna

impulsis mudmivobis kanonis Tanaxmad: 2

vvvm2vm 00 =⇒= . energiis

mudmivobisa da gardaqmnis kanonis Tanaxmad: 4vm

2vm2

2vm

Q20

220 =−= .

gamoyofili siTbos raodenoba dakavSirebulia temperaturis

cvlilebasTan formuliT tmc2Q ∆= bolo ori formulidan

gamomdinareobs wm

m51tc8v0 =∆= .

3. maTematikuri qanqara erT SemTxvevaSi mcire kuTxiT gadaxares wonasworobis mdebareobidan, meore SemTxvevaSi ki qanqaras burTula

vertikalurad aitanes maRla sakidamde. romel SemTxvevaSi iqneba

burTulas sawyis mdebareobaSi dabrunebis dro meti da ramdenjer?

amoxsna

g24Tt1

lπ==

g2t

2tg

2

22 ll =⇒= . vxedavT, rom 21 tt > anu burTula sawyis

mdebareobas ufro swrafad meore SemTxvevaSi ubrundeba. 1,122t

t

2

1 ≈π

=

4. ekransa da sagans Soris manZilia 100 sm. maT Soris moTavsebuli linza ekranze gvaZlevs sagnis mkveTr gamosaxulebas. rodesac

19

linzas 20 sm-iT gadavaadgilebT ekranisaken, masze isev miiReba am

sagnis mkveTri gamosaxuleba. gansazRvreT am linzis fokusuri

manZili.

amoxsna

ekranze miiReba namdvili gamosaxuleba, e.i. linza Semkrebi yofila.

vTqvaT, pirvel SemTxvevaSi linzidan sagnamde manZilia d , xolo linzidan gamosaxulebamde _ f . sxivTa Seqcevadobis gamo aSkaraa, rom

meore SemTxvevaSi linzidan sagnamde manZilia f , xolo linzidan gamosaxulebamde _ d . gvaqvs, rom adf,fd =−=+ l , sadac

smsm 20a,100 ==l , saidanac 2

af,2

ad +=−= ll . linzis formulis

Tanaxmad: f1

d1

F1

+= . sabolood miiReba sm244

aF22

=−

=l

l.

5. ipoveT denis Zala AB gamtarSi, Tu oms4004 =R , oms100321 === RRR da Zabva

momWerebs Soris 26 v-ia. AB gamtaris winaRoba nulis tolia.

amoxsna

vinaidan AB gamtaris winaRoba nulis tolia, amitom mis sigrZes mniSvneloba ar aqvs da nulamde SemcirebiT sxva gamtarebSi denis

Zalebi ar Seicvleba. miiReba

suraTze moyvanili sqema.

omi50RR

RRR

31

31 =+

=′ ,

omi80RR

RRR42

42 =+

=′′ ,

omi130RRR =′′+′= , a2,0RUI == ,

v10RIU1 =′= ,

v16UUU 12 =−= , a1,0RU

I1

11 == , a16,0R

UI2

22 == davubrundeT sawyis sqemas.

aSkaraa, rom AB gamtarSi deni miedineba B–dan A–sken da a06,0III 12BA =−= .

I

1U

1I 2I

2R

4R3R

1R

2U

B

A

3R

2R

4R

1R

20

2009 XI-XII klasebi

II etapi

1. omi100 winaRobis rezistorSi denis Zala drois mixedviT icvleba

kanoniT tkI = , sadac wm

a1=k . gansazRvreT, sawyisi momentidan ra

drois ganmavlobaSi gamoiyo rezistorSi 1,8 kj siTbos raodenoba.

amoxsna

rezistorSi mudmivi denis gavlisas τ droSi gamoyofili siTbos raodenoba gamoiTvleba formuliT τRIQ 2= . Cvens SemTxvevaSi denis Zala cvladia, amitom gamoyofili siTbos raodenoba ganisazRvreba τ droSi denis Zalis kvadratis saSualo mniSvnelobiT ( ) τRIQ saS2= . pirobis Tanaxmad, denis Zalis kvadrati drois wrfivi funqciaa tkI 22 = , amitom denis Zalis kvadratis saSualo mniSvneloba misi sawyisi da

saboloo mniSvnelobebis saSualo ariTmetikulis tolia:

( ) ( ) ( )22

2222 τkIII =

+=

sabsawysaS

amis gamoyenebiT gvaqvs

2

22 RkQ τ= ,

saidanac

RQ

k21

=τ .

pirobis Tanaxmad: wm

a1=k , omi100=R , jkj 18008,1 ==Q . ricxviTi

mniSvnelobebis Casma iZleva wm6=τ .

2. erTmaneTisgan 110 m-iT daSorebuli ori wertilovani wyaro asxivebs 7,5 mghc sixSiris radiotalRebs toli sawyisi faziT. gansazRvreT,

wyaroebis SemaerTebeli monakveTis romel wertilebSia

interferenciuli maqsimumebi. miuTiTeT manZilebi erT-erTi wyarodan.

radiotalRis siCqarea wm

m8103× .

amoxsna

vinaidan sawyisi fazebi tolia, amitom interferenciuli maqsimumebi

gvaqvs wertilebSi sadamdec talRebis mier gavlil manZilTa sxvaoba

talRis sigrZis jeradia. davweroT es piroba wyaroebs Soris aRebuli

wertilisaTvis

antena antena x−l x

21

( ) Ll ,2,1,0, ±±==−− mmxx λ . Cven gvainteresebs x -is mxolod is mniSvnelobebi, romlebic wyaroebis SemaerTebeli monakveTis wertilebs

eTanadeba anu l

22

gantolebebidan miiReba 32cos =α da

32 lg

=υ . iatakTan dajaxebis

momentSi qveda burTulas siCqare horizontaluradaa mimarTuli. misi

moduli iyos 1υ . zeda burTulas 2→

υ siCqare mimarTulia horizontisadmi kuTxiT. Reros sigrZis ucvlelobis gamo am siCqaris horizontaluri

gegmili 1υ –s tolia. iatakis sigluvis gamo mudmivia hantelis impulsis

horizontaluri gegmili: 12cos υαυ mm = , saidanac 272

2cos

1lg

==αυυ .

energiis mudmivobis kanonis Tanaxmad: 22

22

21 υυ mmgm +=l . sabolood

gveqneba 3

1332

27522 212

lll

ggg ==−= υυ .

4. Tboizolirebul cilindrul WurWelSi fskeridan 0h simaRleze gaCerebulia dguSi. WurWelSi imyofeba 0T absoluturi temperaturis erTatomiani idealuri airi. dguSs zemodan adevs sawoni, romlis

masa dguSis masis tolia. cilindri vakuum-kameraSia. dguSsa da

cilindris kedels Soris xaxuni Zalze mcirea. risi toli gaxdeba

airis temperatura da fskeridan dguSis daSoreba, Tu sawons aiReben

dguSidan da dguSis gaCerebis Semdeg isev dadeben masze? axseniT

miRebuli Sedegi energetikuli TvalsazrisiT. erTatomiani airis

Sinagani energia ganisazRvreba formuliT VpTRU23

23

== ν , sadac ν

molebiT gazomili nivTierebis raodenobaa, R _ airis mudmiva, p _ airis wneva, V _ airis moculoba. ugulebelyaviT WurWlisa da dguSis siTbotevadobebi.

amoxsna

sawonis aRebis Semdeg dguSi amoZravdeba zeviT da garkveuli

raodenobis rxevebis Semdeg gaCerdeba axal wonasworul mdebareobaSi.

dguSs aCerebs mcire xaxunis Zala bevri rxevis Semdeg, maram wnevis

gansazRvrisas mcire xaxunis Zalas ar gaviTvaliswinebT. dguSis axali

daSoreba fskeridan aRvniSnoT 1h -iT. dguSis (agreTve sawonis) masa iyos m , WurWlis fskeris farTobi (agreTve dguSis farTobi) _ S . dguSis moZraobis procesSi sistema WurWeli+ dguSi+ airi Caketilia. energiis mudmivobisa da gardaqmnis kanonis Tanaxmad,

111000 23

23 hgmVphgmVp +=+

CavsvaT am gantolebaSi S

gmp 20 = , romelic airis Tavdapirveli wnevaa,

00 hSV = _ airis Tavdapirveli moculoba, Sgmp =1 _ airis wneva axal

23

wonasworul mdebareobaSi, 11 hSV = _ airis moculoba axal wonasworul

mdebareobaSi. miiReba 01 58 hh = .

sawonis dadebis Semdeg dguSi amoZravdeba qveviT da garkveuli

raodenobis rxevebis Semdeg gaCerdeba axal wonasworul mdebareobaSi

fskeridan h simaRleze. energiis mudmivobisa da gardaqmnis kanonis Tanaxmad:

hgmhSS

hgmhgmhSSgm 22

232

23

11 +=+ ,

saidanac 01 2528

107 hhh == .

klapeironis kanonis Tanaxmad TVp

TVp

=0

00 , sadac T airis saboloo

temperaturaa. 0pp = , hSV = , 00 hSV = , amitom gvaqvs 02528 TT = .

dguSze sawoni ido Tavdapirveladac da saboloodac, dguSi maRlaa

aweuli, airis temperatura gazrdilia, e.i. sistemis energia momatebulia.

saidan movida Warbi energia? energia sistemas gadaeca gare wyarodan

sawonis awevisas. is xom dguSidan aiRes nakleb simaRleze da Semodes

masze met simaRleze.

5. adamiani uyurebs Tevzs, romelic imyofeba wyliT savse 20sm radiusis sferul akvariumSi diametralurad mopirdapire wertilSi. wylis

gardatexis maCvenebelia 34 . akvariumi Txelkedliania. gansazRvreT manZili Tevzis gamosaxulebasa da Tevzs Soris.

amoxsna

adamianis TvalSi xvdeba Tevzidan akvariumis diametris gaswvriv

gavrcelebuli da misi maxlobeli sxivebi. gamosaxulebis asagebad

gamoviyenoT diametris gaswvriv gavrcelebuli da erT-erTi misi

maxlobeli sxivi. TvalsaCinoebisaTvis naxazze es sxivebi Zalian

maxlobeli ar unda iyos. radganac akvariumi Txelkedliania, amitom

masSi gavlisas sinaTlis sxivi ise vrceldeba TiTqos wylidan haerSi

gadavida. S Tevzia, xolo S ′_ misi gamosaxuleba. saZebni sididea xSS =′ .

αβ − θ

B

A

S ′ S

ϕ α

β

α

24

α , β , θ da ϕ mcire kuTxeebia. BS didi sizustiT R2 –is tolia.

gardatexis kanonis Tanaxmad, n1

sinsin

=βα

. kuTxeebis simciris gamo αα =sin ,

ββ =sin , e.i. αβ n= . vinaidan 34

=n , amitom βααϕ =>= n2 . maSasadame,

Tevzis gamosaxuleba warmosaxviTia (gamosaxuleba namdvili iqneboda, Tu

n orze meti iqneboda). erTi mxriv αα RtgRAB 22 == , meore mxriv ( ) ( )θθ xRtgxRAB +=+= 22 .

( ) ( )αβααβαθ n−=−=−−= 22 . am tolobebidan miiReba ( )( )αα nxRR −+= 222 ,

34

=n -is Setanis Semdeg miiReba sm20== Rx .

25

fizikis olimpiadis I turi 2010

IX klasi

1. sastarto xazidan orma morbenalma erTdroulad aiRo starti 400

metrian wriul sarben bilikze. maTi siCqareebia 7m/wm da 5m/wm. ipoveT

umciresi rkalis sigrZe sastarto xazsa da im wertils Soris,

romelSic:

a) swrafi morbenali pirvelad daeweva Tavis metoqes.

b) swrafi morbenali meored daeweva Tavis metoqes.

amoxsna:

a) swrafi morbenali pirvelad daeweva Tavis metoqes startis

aRebidan im t droSi, romelSic is erTi sruli wris toli manZiliT _ 400m _iT mets gairbens, vidre misi metoqe.

S1-S2 = V1t- V2t = (V1- V2)·t = 400m

t = 200wm S1 = V1t = 1400m = 3·400m + 200m an S2 = V2t = 1000m = 2·400m + 200m

pasuxi: 200m

b) swrafi morbenali meored daeweva Tavis metoqes startis aRebidan

im t droSi, romelSic is ori sruli wris toli manZiliT _ 800m _iT mets gairbens, vidre misi metoqe.

S1-S2 = V1t- V2t = (V1- V2)·t = 800m

t = 400wm S1 = V1t = 2800m = 7·400m + 0m an S2 = V2t = 2000m = 5·400m + 0m

pasuxi: 0m, meored daeweva zustad sastarto xazze

2. or sxvadasxva masis urikas maTze modebuli erTi da igive sididis

Zala aniWebs a1 da a2 aCqarebebs. a) ra a aCqarebas mianiWebs igive Zala am urikebs, Tu maT gadavabavT? b) risi tolia urikebis masaTa Sefardeba m2 / m1, Tu cnobilia mxolod is, rom Sefardeba a1 /a=k

26

amoxsna:

a) a =F/(m1 +m2) , sadac m1 = F/a1 m2 = F/a2 . masebis gamosaxulebebis Setanis Semdeg pirvel formulaSi miviRebT:

a = a1· a2 /(a1 +a2)

b) Sefardeba m2 / m1 = a1 /a2 . amocanis a) punqtis pasuxidan gamomdinare

a1 /a = (a1 +a2) /a2 = a1 /a2 +1 = m2 /m1+1= k, saidanac

m2 /m1 = k _1

3. 10m siRrmis Widan 8 kg masis wyliT savse vedro amoaqvT bagiriT,

romlis TiToeuli metris masa 400 g-ia.

a) ra muSaoba unda Sesruldes vedros amosatanad?

b) ra muSaoba unda Sesruldes vedros amosatanad Wis siRrmis

naxevramde?

amoxsna:

a) mxolod vedros amotanaze Sesrulebuli muSaoba A1 = mv·h. bagiris simZimis centri imyofeboda h/2 siRrmeze. amitom mis amotanaze Sesrulebuli muSaoba A2 = mb·h/2 A = A1+A2 = mv·g·h + mb·g ·h/2 = 800j +200j = 1000j

b) Wis siRrmis naxevramde mxolod vedros awevaze Sesrulebuli

muSaoba A1 = mv·g ·h/2. bagiris sigrZis qveda naxevari, romelic ar amovida Widan,

gadaadgilda vertikalurad zeviT h/2 manZiliT. A2 = (mb/2) ·g ·h/2 bagiris sigrZis zeda naxevris simZimis centri imyofeboda h/4 siRrmeze. amitom mis amotanaze Sesrulebuli muSaoba A3= (mb/2) ·g ·h/4 A = A1+A2 +A3 = mv·g ·h/2 + (mb/2) ·g ·h/2 +(mb/2) ·g ·h/4 = 400j +100j +50j = 550j

27

4. zRvaSi motivtive yinulis wyalzeda nawilis moculobaa 180 m3,

mTeli moculoba ki 2000 m3. gamoTvaleT yinulis wyalzeda nawilis

masa, Tu wylis simkvrivea 1000kg/m3.

amoxsna:

moc. v = 180 m3, v + V = 2000 m3, ρwy = 1000kg/m3.

yinulis wyalzeda nawilis masa m = ρy· v ρy = My /(v + V) yinulze moqmedi amomgdebi Zala tolia yinulis wonis.

ρwy·g · V = My·g My = ρwy· V ρy = ρwy· V/(v + V) m = ρwy· v ·V/(v + V) = 163 800 kg

5. erTi katarRidan gagzavnili eqolotis bgeriTi signali meore

katarRaze orjer miiRes. pirvelad pirdapiri, xolo meored –

fskeridan arekvlili – 2 wamis dagvianebiT. ipoveT zRvis siRme, Tu

katarRebs Soros manZili 12km-ia, xolo bgeris gavrcelebis siCqare

wyalSi 1500 m/wm-ia. CaTvaleT, rom orive signali wyalSi vrceldeboda.

amoxsna:

moc. 2S = 12km, 2tL _ 2tS = 2wm, v =1500 m/wm. h = ?

aq tL da tS signalis L da S manZilebze gavrcelebis droebia.

tS = S/v = 4wm tL = tS + 1wm =5 wm

h = √L S = v · t t = 1500 ·3 m = 4500m

v

V

L

SS

L h

28


X klasi

1. sadguridan gasuli matarebeli iwyebs TanabaraCqarebul moZraobas. pirveli vagonis dasawyisTan mdgomma damkvirvebelma CainiSna, rom am

vagonma mis Tvalwin gaiara moZraobis dawyebidan 5 wamis ganmavlobaSi.

ra dros moandomebs mis Tvalwin gavlas mecxre vagoni? vagonebs Soris

manZili SegiZliaT ugulebelyoT. √2 ≈ 1,42

amoxsna: t1 = 5wm = 2L/a sadac L vagonis sigrZea, a ki matareblis aCqareba.

mecxre vagoni damkvirvebels Cauvlis t = t9 _ t8 droSi, sadac t9 = 2 · 9L/a _ 9 vagonis Cavlis droa, t8 = 2 · 8L/a _ ki 8 vagonis Cavlis dro.

t = ( √9 _√8) · 2L/a = ( √9 _√8) ·5wm = 0,8wm

2. 2 t masis TviTmfrinavi moZraobs horizontaluri mimarTulebiT 50 m/wm siCqariT. 420 m simaRlidan igi iwyebs daSvebas gamorTuli

ZravebiT. aerodromis bilikze dajdomisas misi siCqarea 30 m/wm.

gansazRvreT haeris winaaRmdegobis Zalis mier Sesrulebuli muSaoba

TviTmfrinavis planirebisas.

amoxsna: TviTmfrinavis sruli meqanikuri energia Zravebis gamorTvis

momentSi iyo E1 = mV12/2 +mgh1, bilikze dajdomisas ki E2 = mV22/2. TviTmfrinavis mier dakarguli energia daixarja haeris

winaaRmdegobis Zalis daZlevaze.

A = E1_ E2 = mV12/2 + mgh1 _ mV22/2 = 10 000 000j

3. Tboizolirebul WurWelSi siTbur wonasworobaSia 500 g wyali da amdenive yinuli. am WurWelSi CauSves 10g 100-gradusiani orTqli.

ramdeni wyali iqneba WurWelSi siTburi wonasworobis damyarebis

Semdeg? wylis orTqladqcevis kuTri siTboa 2,3·106j/kg, kuTri siTbotevadobaa 4200j/kg·grad, yinulis dnobis kuTri siTboa 3,3·105j/kg

amoxsna: siTbos raodenoba, romelic gamoiyofa m=10g orTqlis kondensirebisas da Semdeg 00 C-mde gacivebisas, mTlianad moxmardeba WurWelSi yinulis dnobas.

mL+ mc(t1 – t0) = M λ , sadac M damdnari yinulis masaa, t0 =00 C, t1 =1000 C

M = (mL+ mc(t1 – t0))/ λ = (0,01·2,3·106 +0,01·4200·100)/ (3,3·105)kg ≈ 0,082 kg=82g

sabolood WurWelSi aRmoCndeba 500g+10g+82g = 592g wyali

4. mnaTi wertili mcire zomis brtyeli sarkidan daSorebulia 60sm-iT, rogorc naxazzea naCvenebi.

a) ra manzilze unda imisaTvis, rom manZili am wertilidan mis

gamosaxulebamde orjer Semcirdes?

b) ra kuTxiT unda gadavxaroT sarke, imisaTvis, rom manZili am

wertilidan mis gamosaxulebamde orjer Semcirdes?

amoxsna: wertilis gamosaxuleba brtyel sarkeSi mdebareobs am

wertilidan sarkis sibrtyeze (da ara aucileblad sarkis zedapirze)

daSvebul perpendikularze da imyofeba igive manzilze am sibrtyidan,

rac TviTon wertili.

a) manZili wertilidan mis gamosaxulebamde tolia 120 sm. am manZilis

orjer Sesamcireblad wertili unda mivuaxlovoT sarkes 30 sm-ze.

b) wertilidan sarkis sibrtyemde

manZilis orjer Sesamcireblad sarke

unda gadavxaroT 600 –iT vertikalidan.

29

60 sm

30 30 wertilis gamosaxuleba

600

30060 sm

30 sm

30 sm

wertilis

gamosaxuleba

30

5. liTonis ori erTnairi burTula damuxtulia +5·10-16k da -3·10-16k muxtebiT. manZili maT Soris 60sm-ia. es burTulebi Seaxes erTmaneTs da

Semdeg daaSores 40sm-ze. rogor da ramdenJer Seicvala amis Sedegad

burTulebis urTierTqmedebis kulonuri Zala?

amoxsna: sxvadasxva niSniani muxtebiT damuxtuli burTulebi miizidebian.

kulonis kanonis Tanaxmad F1 = k , = k · , 10 ZaliT.

Sexebis Semdeg maTi jamuri muxti +5·10-16k +(-3·10-16k) = +2·10-16k tolad ganawildeba orive burTulaze da TiToeulis muxti gaxdeba +1·10-16k.

40sm-ze maTi daSorebis Semdeg maTi ganzidvis Zala toli gaxdeba

F2 = k , = k · , 10 F1F2

= _20/3 = 6,6(6)

burTulebis urTierTqmedebis kulonurma Zalam Seicvala niSani da

Semcirda 6,6(6) -jer.

31


XI-XII klasebi

1. vagonis Werze dakidebul 1 m sigrZis Zafze mimagrebulia patara

burTula. vagonis moZraobis ra siCqarisas iqneba burTulas rxevebis

amplituda maqsimaluri? relsis sigrZe 12,5 m-ia.

amoxsna: qanqaras sakuTari rxevebis periodia g

T lπ20 = , sadac l Zafis

sigrZea. qanqara ganicdis biZgebs borblebis mier relsebs Soris

Sualedebis gadavlis momentebSi. biZgebis ganmeorebis periodia vagonis

mier relsis sigrZis gavlis dro: VLT = , sadac L relsis sigrZea, V ki _

vagonis siCqare. qanqara asrulebs iZulebiT rxevebs. rxevebis amplituda

maqsimaluri rezonansis dros iqneba: 0TT = . aqedan miiReba:

wm

m62

2 ==⇒=l

l gLVgV

Lπ

π

2. 0,5 m sigrZis Zafze dakidebuli burTula horizontalur sibrtyeSi 0,3 m radiusian wrewirze brunavs. gansazRvreT misi kuTxuri siCqare.

32

amoxsna: niutonis II kanonis Tanaxmad: →→→

=+ amgmT . gadavideT gegmilebze (ix. nax.):

( X geg.) amT =αsin (Y geg.) 0cos =− gmT α aqedan miiReba:

gatg =α

gamoviyenoT centriskenuli aCqarebis formula: ra 2ω= .

naxatidan vxedavT, rom 22 r

rtg−

=l

α

bolo sami formulis gamoyenebiT miiReba:

wm

rad5

22=

−=

rg

lω

3. horizontalurad moZravi 20 g masis tyvia moxvda uwonad Zafze

dakidebul qviSiT savse 980 g masis yuTs da Sig CarCa. amis Sedegad yuTis

simZimis centrma 20 sm-iT aiwia. gansazRvreT tyviis siCqare moxvedris

momentSi.

amoxsna: tyviis masa aRvniSnoT m –iT, misi siCqare 0V –iT, qviSiani yuTis masa _ M -iT, tyviis dajaxebis Sedegad mis mier SeZenili siCqare _ V -iT,

yuTis awevis simaRle _ h –iT. dajaxebis procesSi mudmivia impulsi:

( )VmMVm +=0 yuTis gadaxris procesSi mudmivia meqanikuri energia (haeris

winaaRmdegobas ugulebelvyofT):

( ) ( ) hgmMVmM +=+

2

2

aqedan miiReba:

( )

wm

m1002

0 =+

=m

hgmMV

33

4. gansazRvreT gambnevi linzis optikuri Zala, Tu cnobilia, rom

linzidan 40 sm-iT daSorebuli sagnis gamosaxuleba oTxjeraa

Semcirebuli.

amoxsna: gambnev linzaSi gamosaxuleba warmosaxviTia. linzis formulas

Semdegi saxe aqvs:

fdF

D 111 −==

aq D linzis optikuri Zalaa, F _ fokusuri manZili, d _ linzidan sagnamde manZili, f _ linzidan gamosaxulebamde manZili (moduliT).

gadidebisaTvis gvaqvs formula df

=Γ , saidanac df Γ= . amis SetaniT

linzis formulaSi miiReba:

d

DΓ−Γ

=1

40=d sm = 0,4m ; 41

=Γ , amitom 5,7−=D dptr.

5. kalis dnobis kuTri siTbos gasagebad kalorimetrSi, romelSic 7 Co temperaturis 330g wyali esxa, Caasxes 350g gamyarebis temperaturamde

miyvanili kala. amis Semdeg kalorimetrSi, romlis siTbotevadoba iyo 100

j/ Co , damyarda 32 Co temperatura. cdis monacemebze dayrdnobiT gansazRvreT kalis dnobis kuTri siTbo.

wylis kuTri siTbotevadobaa Co⋅kgj4200 , kalis _ Co⋅kgj230 . kalis dnobis temperaturaa Co232 .

amoxsna: SemoviRoT aRniSvnebi: kgg 33,03301 ==m , Cto71 = ,

Cc

o⋅=

kg

j42001 , C

100Co

j= , C32t o=

t2 = ,C232 o kgg 35,03502 ==m , Cc

o⋅=

kg

j2302 ,.

kala Caasxes, maSasadame, is Txevadi iyo. gamyarebis temperatura dnobis

temperaturis tolia. procesis dros wyali da kalorimetri gaTba, xolo

kala jer gamyarda da Semdeg gacivda.

CavweroT siTburi balansis piroba:

( ) ( ) ( )ttmcmttCttmc −+=−+− 22221111 λ

aqedan miiReba: ( )( ) ( )

kg

j60143

mttmcttCmc

2

222111 =−−−+

=λ

34

fizikis olimpiadis II turi 2010 IX klasi

1. kalorimetrSi asxia 100C temperaturis 0,4 kg wyali. wyalSi moaTavses 0,6 kg yinuli, romlis temperaturaa -400C. gamoTvaleT wylisa da yinulis masaTa Sefardeba siTburi wonasworobis damyarebis Semdeg. kalorimetris siTbotevadobas da siTbur

danakargebs nu gaiTvaliswinebT. wylisa da yinulis kuTri siTbotevadobebia 4200

j/kg·grad da 2100j/kg·grad, yinulis dnobis kuTri siTboa 0,33 mgj/kg.

amoxsna: imis dasadgenad, Tu ramdeni yinuli iqneba siTbur wonasworobaSi, unda

Sefasdes Sesabamisi siTbos raodenobebi:

wylis 00-mde gacivebisas gamoyofili siTbo:

01680tmcQ 1111 == j; yinulis 00-mde gasaTbobaT siTboa:

05040tmcQ 2222 == j.

12 QQ > niSnavs, rom wylis nawili mainc gaiyineba. mTeli wylis gacivebisas gamoiyofa 200000λmQ 13 == j. vinaidan 3121 QQQQ +

35

2. zeTisa da wylis sazRvris doneze II muxlSi da igive doneze I muxlSi

wnevebis toloba ase Caiwereba: ρ0ghwy= ρghz. (simaRleebi aiTvleba sazRvris donidan).

sabolood hz_hwy= hz(1_ρ/ ρ0)= (V/2 S )(1_ρ/ ρ0)

3. saerTo mTavari optikuri RerZis mqone ori

Semkrebi linzis fokusuri manZilebia F1 da F2. L aris manZili maT Soris da L

36

( ) =−−

=23

22

23

XX2VVa 2,5m/wm2 da X0 = 1,2m, saidanac miiReba gantoleba

V2 = 5X _ 6 2. am formulis gamoyenebiT:

X4 = (V42+6)/5 = 4,4m

3. X1=1 wertilSi V1siCqarisTvis miviRebT:

V12 = _1, rac imas niSnavs, rom am wertilSi sxeuli ar yofila, radganac sxeulis

siCqare nulis toli iyo X=1,2m wertilSi.

5. vertikalur L sigrZis Zafze kedlze kidia tyviis burTula. burTula gadaxares kedlidan ise, rom Zafma kedelTan Seadgina α kuTxe da gauSves xeli. kedlidan burTulis arekvlis Semdeg Zafi kedlidan β kuTxemde gadaixara. ramdeni gradusiT gaTba burTula?

kedelsa da haerSi gadacemuli siTbo ugulebelyaviT.

tyviis kuTri siTbotevadobaa c.

amoxsna.

kedelTan Sejaxebis momentSi burTulas gaaCnia kinetikuri energia, romelic

potenciuli energiis cvlilebis tolia.

Ekin=∆Epot=mg∆h=mgL(1-cosα).

kedelTan Sejaxebis Sedegad meqanikuri energiis nawili gadava Sinegan energiaSi da

Zafi naklebi kuTxiT gadaixreba (β

37

fizikis olimpiadis II turi 2010 X klasi

1. L sigrZis Zafze dakidebuli m masis burTula brunavs horizontalur sibrtyeSi wrewirze ise, rom Zafi SveulTan α kuTxes adgens. 1. ra siCqariT moZraobs burTula?

2. risi tolia Zafis daWimulobis Zala?

3. SeiZleba Tu ara am moZraobis periodi iyos 3 wamis toli, Tu Zafis

sigrZe 1 metria? pasuxi daasabuTeT.

amoxsna:

ZalTa samkuTxedi naxazzea moyvanili.

1. naxazidan miviRebT: αgtga = . centriskenuli aCqarebis formulidan advili misaRebia:

ααα tggLgRtgV sin==

2. amave naxazidan: αcos

mgF =

3. moZraobis periodisaTvis:

gL

gL

tggLL

VRT παπ

αααππ 2cos2

sinsin22

38

3. naxazze moyvanil wredSi R1=3 omi, R2=9 omi R3= R4= R6=6 omi, R5=4 omi. ipoveT am wredis saerTo winaRoba.

amoxsna:

amocanis pirobaSi moyvanili sqema

SegviZlia gadavxazoT Semdegnairad:

exla winaRoba advili dasaTvlelia

da sabolood R=3 omi.

4. zambariani qanqara asrulebs harmoniul rxevas maqsimaluri gadaxris A da B wertilebs Soris. risi tolia tvirTis saSualo siCqare AB monakveTze, Tu cnobilia, rom misi moZraobis maqsimaluri siCqarea 10 m/wm?

amoxsna:

saSualo siCqaris formulidan ( ) TA

TABV max4

2/== , sadac T da Amax aris am rxevis

periodi da amplituda.

energiis mudmivobis kanonis 22

2max

2max kAmV = da periodis

kmT π2= formulis

gamoyenebiT saSualo siCqarisaTvis miviRebT ≈=π

max2VV 64 m/wm

5. linzis gadaadgilebisas ekransa da sagans Soris, aRmoCnda, rom sagnis gamosaxuleba ekranze mkveTria linzis ori mdgomareobisaTvis. miRebuli

gamosaxulebebis simaRleebia h1 da h2. risi tolia sagnis simaRle?

amoxsna: advili misaxvedria, rom Tu pirvel mdgomareobaSi linzis

gamadidebloba iyo Γ , maSin meore mdgomareobaSi gamadidebloba iqneba Γ1-s

toli. gamadideblobis ganmartebidan hH

=Γ , sadac H da h aris

gamosaxulebisa da sagnis simaRleebi, miviRebT: hh

hh 12 :1= . sabolood

21 hhh ⋅=

R6R5 R4

R3 R2 R1

R6

R5

R4

R3 R2 R1

39

fizikis olimpiadis II turi 2010 XI-XII klasebi

1. gluv horizontalur magidaze devs patara burTula. mas drekadad daejaxa wveroTi win moZravi gluvi soli (ix. nax.). solis

horizontisadmi daxris kuTxea α . gansazRvreT, risi tolia solis masis

fardoba burTulas masasTan, Tu dajaxebis

Semdeg burTula magidaze solis wverosTan

davarda. rogori α kuTxeebisaTvis SeiZleba momxdariyo es?

amoxsna:

dajaxebisas soli burTulaze moqmedebs solis sibrtyis marTobuli

mimarTulebis ZaliT, amitom burTula am mimarTulebis siCqares SeiZens

(ix. nax.). naxatze miTiTebulia solisa da burTulas siCqareebi

dajaxebis Semdeg momentSi.

dajaxebisas mudmivia meqanikuri energia da horizontalur RerZze

impulsebis gegmilTa jami:

222

2220 vv umMM +=

αsinvv0 umMM += am gantolebaTa sistemis amoxsna gvaZlevs

αα

2

2

0 sinsin

vvmMmM

+

−= da

αα

20

sinsin2 v

mMM

u+

= (1)

dajaxebis Semdeg soli moZraobs Tanabrad, xolo burTula moZraobs

rogorc kuTxiT gasrolili sxeuli. burTula solis wverTan davardeba,

Tu burTulas siCqaris horizontaluri gegmili da solis siCqare

erTmaneTis tolia:

αsinv u= (2) (1) da (2)-dan miiReba, rom

α

α2

2

sin21sin

−=

mM

gamosaxulebas fizikuri azri aqvs mxolod maSin, rodesac o45

40

2. m masisa da µ moluri masis idealuri airis gaTbobisas misi absoluturi temperatura da moculoba icvlebian

2VT α= kanoniT, sadac α mocemuli koeficientia. gansazRvreT airis mier Sesrulebuli muSaoba moculobis 1V -dan 2V -mde gazrdisas. idealuri airis qceva

emorCileba mendeleev-klapeironis gantolebas: TRmVpµ

= , sadac p

airis wnevaa, xolo R idealuri airis cnobili mudmiva.

amoxsna:

airis muSaoba geometriulad gamoisaxeba wnevis moculobaze

damokidebulebis grafikis qvemoT moculobaTa RerZamde moTavsebuli

farTobiT. wnevis moculobaze damokidebulebis kanons vipoviT, Tu 2VT α= -s CavsvamT mendeleev-klapeironis gantolebaSi. miiReba, rom

VRm

pµ

α=

am damokidebulebis grafikia koordinatTa saTaveSi gamavali wrfe. airis

muSaoba naxazze gamosaxuli trapeciis farTobis tolia. trapeciis

fuZeebia 11 VRm

pµ

α= da 22 V

Rmp

µα

= , xolo simaRle ( )12 VV − .

( ) ( )µ

α 212

212

21

2VVRm

VVpp

A−

=−+

=

3. denis wyaros miuerTes 1R winaRobis rezistori da gazomes masze gamoyofili simZlavre, gamorTes da miuerTes 2R winaRobis rezistori,

2R ≠ 1R . 2R _ze gamoyofili simZlavre 1R _ze gamoyofili simZlavris toli aRmoCnda.

1. gansazRvreT denis wyaros Siga winaRoba.

2. gansazRvreT, ra winaRobis rezistori unda mivuerToT am denis

wyaros, rom rezistorSi gamoyofili simZlavre maqsimaluri iyos.

p p2 p1

V1 V2 V

41

amoxsna:

denis wyaros em Zala iyos E , Siga winaRoba _ r , denis Zala wredSi _ I , Zabva wyaros momWerebze _ U . pirobis Tanaxmad gvaqvs

2221

21 RIRI =

anu

( ) ( )222

2

21

12

rRR

rRR

+=

+

EE

saidanac advilad miiReba, rom

21 RRr =

ganvixiloT denis wyarosTan SeerTebuli R winaRobis rezistori. rezistorSi gamoyofili simZlavrea

UIP = magram

rIU −= E amitom

( ) rIIrIIP 2−=−= EE rogorc vxedavT simZlavre denis Zalis kvadratuli funqciaa. misi

grafikia parabola StoebiT qveviT. cxadia, rom simZlavre maqsimaluria

parabolas wveroSi anu rodesac rI 2E

= . aseTi denis Zala maSin gveqneba,

rodesac rR = .

4. linzidan 2 m manZiliT daSorebuli sagnis gamosaxuleba linzaSi 0,02 m simaRlis miiReba, igive sagnis gamosaxuleba 3 m manZilidan ki _

0,012 m simaRlis. gansazRvreT linzis optikuri Zala.

amoxsna:

gvaqvs m21 =d , m02,01 =H , m32 =d , m012,02 =H . sagnis zoma iyos h , gamosaxulebamde manZilebi Sesabamisad 1f da 2f . gamoviyenoT formulebi:

11

11fd

D += , h

Hdf 1

1

1 = , 22

11fd

D += , h

Hdf 2

2

2 = .

aqedan miiReba

11

11 11 H

hfd

dD +=+= , 22

22 11 H

hfd

dD +=+=

saidanac

1

2

2

1

11

HH

dDdD

=−−

am gantolebis amoxsna gvaZlevs

dpt22211

21 =−−

=HdHd

HHD

42

5. liTonze daasxives jer m7100,3 −× , xolo Semdeg m7100,6 −× talRis sigrZis sinaTle. aRmoCnda, rom amotyorcnili eleqtronebis maqsimaluri

siCqareebi erTmaneTisagan 2-jer gansxvavdebodnen. gansazRvreT

liTonidan eleqtronis gamosvlis muSaoba. plankis mudmivaa

wmj ⋅× −34106,6 , vakuumSi sinaTlis siCqarea wm

m8100,3 × .

amoxsna:

orive SemTxvevisaTvis vwerT fotoefeqtis gantolebebs. sinaTlis

sixSires gamovsaxavT talRis sigrZiTa da siCqariT.

2

21

1

vmA

ch+=λ , 2

22

2

vmA

ch+=

λ

eleqtronebs ufro maRali siCqariT amoagdebs ufro maRali sixSiris,

anu naklebi talRis sigrZis mqone sinaTle, e.i 21 vv 2= . moyvanili sami gantolobidan miiReba

( )j

19

21

21 102,234 −×=

−= λλ

λλchA

ricxviTi pasuxi moyvanilia ori niSnadi cifris sizustiT, vinaidan

sawyisi monacemebic am sizustiT iyo mocemuli

43

fizikis olimpiadis III turi 2010 IX klasi

1. m1=2kg da m2=1kg masebisa da S=10sm2 erTi da igive fskeris farTobis mqone ori cilindruli Wiqidan

amotumbulia haeri. TiToeuli maTgani hermetulad

daxurulia uxaxunod moZravi uwonadi dguSiT,

romlebsac ikavebs ori k=50n/m sixistis erTnairi zambara. es Wiqebi ori WoWonaqis saSualebiT

CaSvebulia wyalSi ise, rom Zafi daWimulia (ix.nax.).

1. risi tolia WurWlebis fskerTa doneTa sxvaoba

wonasworobis mdgomareobaSi?

2. iqneba Tu ara es wonasworoba mdgradi? pasuxi daasabuTeT.

3. risi toli unda iyos zambarebis sixiste imisaTvis, rom wonasworobaSi

Wiqebis fskerebi aRmoCndnen erT doneze? g=10m/wm2

amoxsna: wonasworobaSi Zafis daWimulobis Zala T tolia: T= m1g_ρgl1S = m2g_ρgl2S xolo Wiqebze moqmedi ZalebisaTvis miviRebT:

(ρgh1+ p0 ) S = m1g+k(l0-l1) da (ρgh2+ p0 ) S = m1g+k(l0-l2) am gantolebebSi l0-aradeformirebuli zambarebis sigrZea, l1 da l2 – deformirebuli zambarebis sigrZeebi, h1 da h2 Wiqebis fskerebis doneebia wylis zedapiridan, ρ wylis simkvrivea, p0 ki atmosferuli wneva. pirveli gantolebidan advilad miviRebT: m1-m2=ρS(l1-l2) (1) gantolebaTa meore wyvilidan: ρgS(h1-h2)= (m1-m2)g+k(l2-l1) (2) (1) da(2) gantolebebidan (l2-l1) sididis gamoricxviT vpoulobT saZiebel doneTa sxvaobas:

gS

mmg

Skhh

ρρ21

12−

⎟⎟⎠

⎞⎜⎜⎝

⎛−=− (∗)

vinaidan m1>m2, miviRebT, rom amocanis pirobebSi h2>h1 da wonasworobaSi SedarebiT mZime Wiqa aRmoCndeba msubuqze maRla (!).

mocemuli ricxviTi mniSvnelobebis CasmiT miviRebT:

h2-h1= 4m miRebuli wonasworobis mdgomareoba aramdgradia, radgan misi darRvevis

Sedegad (magaliTad, zeda Wiqa cotaTi zeviT gadainacvlebs) zeda Wiqaze

moqmedi amomgdebi Zala gaizrdeba, qvedaze ki Semcirdeba (dguSebze moqmedi

wnevaTa cvlilebis gamo) da Wiqebi kidev ufro daSordebian wonasworobis

mdgomareobas.

(∗) gantolebidan miviRebT pasuxs amocanis mesame kiTxvazec. kerZod, imisaTvis, rom Wiqebi erT doneze aRmoCdnen, zambarebis sixiste unda

iyos

k=ρgS=10n/m

cxadia, rom es wonasworobac aramdgradia

44

2. sakidze, L=1.4m sigrZis ZafiT Camokidebul sxeuls mianiWes V0=7m/wm siCqare horizontaluri mimarTulebiT.

1. sakidis donidan sxeulis asvlis ra simaRleze iqneba Zafis daWimuloba nulis

toli?

2. gansazRvreT sxeulis asvlis maqsimaluri simaRle.

amoxsna:

1. Zafis daWimulobis Zala

T = mV2/L _ mg*sin α xolo energiis mudmivobis kanonidan

)sin1(mgL2

mV2

mV 220 α++= da T =0 pirobidan

advilad miviRebT rom sin α = 1/2

pasuxi: h1= L*sin α = 0,7 m

2. h1 simaRlidan Zafis daWimulobis Zala 0-is toli xdeba da amocana daiyvaneba horizontisadmi 600 kuTxiT, V siCqariT gasrolili sxeulis asvlis simaRlis

povnaze. h2=V

. sadac Vvert =V·sin600 = · √ , saidanac h2 = = 0,2625

sabolood h= h1+ h2= 0,9625

3. 20 sm fokusuri manZilis mqone linzidan 40sm manZilze mTavar optikur RerZze moTavsebulia

sinaTlis wertilovani wyaro, romelzec midgmulia

20 sm sigrZis SuSis marTkuTxa naWeri, romlis

gardatexis maCvenebelia n=3/2. linzidan ra manZilze unda movaTavsoT ekrani

sinaTlis wyaros mkafio gamosaxulebis misaRebad?

wyarodan linzaSi moxvedrili sxivebi mTavar optikur RerZTan mcire kuTxes

qmnis.

amoxsna:

sinaTlis wyarodan gamosuli sxivebi gardatydebian SuSis naWris marjvena

waxnagze ise, TiTqos wertilovani wyaro miuaxlovda linzas da ganlagebulia

SuSis naWris SigniT. mTavar optikur RerZTan kuTxis simciris gaTvaliswinebiT

miviRebT, rom es axali wyaro daSorebulia marjvena waxnagidan L/n manZilze, sadac L – SuSis naWris sigrZea, xolo n – misi gardatexis maCvenebelia. amocanis monacemebis gamoyenebiT advili misaRebia, rom manZili linzidan am

wertilamde tolia (100/3)sm. Txeli linzis formulis gamoyenebiT miviRebT, rom

manZili linzidan ekranamde tolia 50sm.

mg*sin α

mg

V0

α

45

4. dedamiwis zedapiridan ra simaRleze unda avitanoT maTematikuri qanqara, rom misi rxevis periodi 1%-iT gaizardos? ramdeni procentiT unda SevamciroT

qanqaris sigrZe, rom am simaRleze misi periodi ucvleli darCes?

(dedamiwis radiusia 6400km.)

amoxsna: qanqaras rxevis periodi dedamiwis zedapirTan aris

0

00 2 g

LT π= , sadac L0 aris qanqaras sidrZe, xolo 20 RGMg = (M da R – dedamiwis

masa da radiusia). dedamiwis zedapiridan H simaRleze: gLT 02π= , sadac

2)( HRGMg+

= . aqedan R

HRTT +

=0

da amocanis pirobis T=1,01T 0 gamoyenebiT miviRebT

H=0,01R=640km

Tu periodi ucvlelia, maSin 0

0

gL

gL= , L damoklebuli qanqaras sigrZea. aqedan

2

22

0

0

0

0

)()(HR

RHRg

ggL

LL+

−+=

−=

−. da radganac R+H=1.01R

%20

0 ≈−

LLL

5. avtomobili, romlis masaa 1000 kg, horizontalur gzaze aviTarebs 72km/sT siCqares da 1km gzaze xarjavs 100g benzins. ra siCqariT imoZravebs avtomobili,

drois erTeulSi benzinis igive xarjviT, aRmarTze, romlis daxriloba (daxris

kuTxis sinusi) 0,03 tolia? motoris m.q.k 30%-ia?

CaTvaleT, rom winaaRmdegobis Zala orive SemTxvevaSi erTnairia. benzinis wvis

kuTri siTbo q=46mgj/kg

amoxsna: m masis benzinis dawvis dros gamoiyofa qm siTbos raodenoba da

motoris margi simZlavre 00 FV

SqmV

tqmN === ηη . es simZlavre pirvel

SemTxvevaSi mTlianad xmardeba F winaaRmdegobis Zalebis daZlevas. meore SemTxvevaSi am simZlavris nawili xmardeba avtomobilis potenciuri

energiis gazrdas. drois erTeulSi avtomobilis potenciuri energiis

namati iqneba αsinMgV , sadac α gzis daxrilobis kuTxea, xolo M avtomobilis masaa. amrigad, meore SemTxvevaSi

αη sinMgVS

qmVN += aqedan

≈+

=αη

ηsin

0

MgSqmqmVV 59km/sT

46

fizikis olimpiadis III turi 2010 X klasi

1. eleqtroCaidani CarTes 110 v Zabvis qselSi. wyali CaidanSi mivida duRilis temperaturamde, magram ver aduRda. amis Semdeg Zabva gazardes 220 v–mde da 8

wuTSi aorTqlda wylis raRac mcire raodenoba. ra droSi aorTqldeboda wylis

igive raodenoba Zabva rom 250 v-mde gazrdiliyo?

amoxsna:

eleqtroCaidnis simZlavrea: P=U2/R (*)

im momentidan rodesac wylis temperatura 100oC damyardeba mTeli energia midis danakargebze

P0=U02/R .

Semdeg etapze

P1t1= = P0t1+ mL (2) sadac m aris drois t1 intervalSi aorTqlebuli wylis masa. L- wylis orTqladqcevis kuTri siTboa.

bolo etapze P2t2= = P0t2 +mL (3)

(*) formulidan P1=4P0 da P2= 121625 P0

(2) da (3) formulebidan advili misaRebia

≈=−−

= 1113

122 504

363 ttPPPPt 5,8wT

2. horizontalur sibrtyeSi ω = √5 radiani /wm kuTxuri siCqariT mbrunav umaso Reroze, brunvis RerZidan 1,2m manZilze mimagrebul Zafze Camokidebuli 0,1 kg

masis burTula brunavs 1,5 m radiusis wrewirze.

xaxunisa da haeris winaaRmdegobis Zalebis moqmedebiT RerZi da burTula

gaCerda. ra muSaoba Seasrules am Zalebma?

amoxsna: moc: R=1,5m r = 1,2m, ω = √5 radiani /wm m = 0,1 kg burTulas kinetikuri energia Ek = (mV2)/2 = m ω 2R2/2 = 0,5625 j

burTulas potenciuri energia Ep = mgh sadac h = L (1-cosα) = (R-r) (1-cosα) /sinα. α kuTxis dasadgenad ganvixiloT Zalebi, romliTac sxeuli moqmedebs Zafze:

tgα= mω2R /mg , ricxviTi mniSvnelobebis Casmis Semded miviRebT rom tgα=3/4, saidanac: sinα=3/5 da cosα=4/5 . Ep = mgh =0,1j

mg

mω2R α

1,5m

1,2m

47

xaxunisa da haeris winaaRmdegobis Zalebis moqmedebiT Sesrulebuli muSaoba

tolia sxeulis sawyisi sruli meqanikuri energiis:

A = Ek + Ep = 0, 6625 j

3. horizontalur zedapirze devs 1kg masis mqone sxeuli. sawyisi momentSi masze moqmedeba daiwyo orma Zalam _ horizontalurad marjvniv mimarTulma mudmivma,

5 n-is tolma Zalam da horizontalurad marcxniv mimarTulma F=αt kanoniT droSi zrdadma Zalam, sadac α=10-1n/wm. xaxunis koeficientia µ=0,2. 1. aageT horizontalurad marjvniv mimarTul RerZze am sxeulis aCqarebis

gegmilis droze damokidebulebis grafiki.

2. gamoTvaleT am grafikis maxasiaTebeli wertilebis parametrebi

3. risi tolia am sxeulis siCqare t=100wm drois momentSi? amoxsna:

sawyis momentSi sxeuli iwyebs

aCqarebul moZraobas marjvniv klebadi

aCqarebiT da sawyisi aCqareba a0=(F0/m)- µg=3m/wm2. t1 moomentamde, romelic ganisazRvreba pirobidan F(t1)=F0-µmg ⇒ t1=30wm sxeuli moZraobs aCqarebulad. t1-dan t2-mde moZraoba Senelebulia da t2 ganisazRvreba pirobidan V(t2)=0 ⇒ F(t2)=2(F0-µmg) ⇒ t2=2t1=60wm. vinaidan F(t2)=6n< F0+µmg=7n, amitom V=0 t3 momentamde, romlisTvisac F(t3)= F0+µmg ⇒ t3=70wm. amis Semdeg iwyeba aCqarebuli moZraoba marcxvniv. me-100

wamisTvis sxeulis siCqare

ricxvobrivaT tolia grafikze

daStrixuli figures farTobis. gamoTvla gvaZlevs V=45m/wm

4. F fokusuri manZilis mqone Semkrebi linzis mTavar fokusSi moTavsebulia gambnevi linza.

sagani imyofeba Semkrebi linzis meore mxares,

misgan 3F manZilze. ras udris gambnevi linzis fokusuri manZili, Tu es sistema gvaZlevs

sagnis namdvil da orjer gadidebul

gamosaxulebas.

3F F F

t3 t1

3

ax(m/wm2)

t(wm)

100 t2 0

48

amoxsna:

A1B1 warmoadgens AB sagnis gamosaxulebas Semkreb linzaSi da amave dros A2B2 sagnis warmosaxviT gamosaxulebas gambnev

linzaSi. Semkrebi linzis formulaSi

d=3F Casmis Sedegad miviRebT f=(3/2)F da (f/d)=1/2. e.i. A1B1 gamosaxuleba orjer Semcirebulia. gambnevi linzis

formulaSi 111

111dfF

−=− da

211FFfd =−= . amocanis pirobis tanaxmad Fdf 24 11 == sabolood gambnevi linzis

formulidan miviRebT:

3

21

FF =

5. R radiusis mavTulis wriul rgolze Tanabrad ganawilebulia q muxti. ipoveT rgolis simetriis RerZze wertili, sadac velis daZabuloba maqsimaluria da

gansazRvreT misi mniSvneloba.

(miniSneba: y = x3 _ x funqciis minimumi da maqsimumi, rodesac x icvleba (_1, 1) intervalSi, miiRweva x = 1

√3 da x = _ 1

√3 wertilebSi)

amoxsna.

davyoT rgoli 2n nawilad, sadac n imdenad didi ricxvia, rom rgolis TiToeuli nawili SeiZleba

CaiTvalos q/2n wertilovan muxtad. yoveli aseTi muxtisaTvis moiZebneba misi diametrulad

mopirdapire wyvili. am ori muxtis mier Seqmnili

velis daZabuloba iqneba (ix.nax.):

αααα 2221 sincos)2/(cos2cos2 ⋅===

nRkq

ACnqkEE E

aseTi n wyvilis mier Seqmnili daZabulobebis

x

y

1√3

1√3

R

E

E2 E1 C

O B A

α

O

B2

A2

B1

A1

B

A F

49

SekrebiT miviRebT:

)cos(coscossin 322

20 αααα −=⋅= Rkq

RkqE

es daZabuloba maqsimaluria, rodesac 31cos2 =α . maSin

2RctgROC =⋅= α . am wertilSi velis daZabulobaa 2max 33

2R

kqE =

50

fizikis olimpiadis III turi 2010 XI-XII klasebi

1. E em Zalis da r Siga winaRobis denis wyaros miuerTes arawrfivi elementi, romlis volt-amperul maxasiaTebels Semdegi saxe aqvs:

2IAU = . ra aris A koeficientis erTeuli SI -Si? gamosaxeT es erTeuli ZiriTadi erTeulebiT. gansazRvreT denis Zala wredSi.

amoxsna:

[ ]33

2

22222wma

mkg

awmawm

mmkg

awma

mn

ak

j

a

v

⋅

⋅=

⋅⋅⋅

⋅⋅=

⋅⋅

⋅=

⋅==A

2IArIUrI +=+=E

02 =−+ EIrIA

AArr

I 242 E++−

=

2. siTburi Zravis muSa sxeuli idealuri airia. is asrulebs ciklur process, romelic ori izoqoruli da ori

izoTermuli ubnisagan Sedgeba. izoTermuli

procesebis dros absolutur temperaturaTa

Sefardeba 3-is tolia (ix. nax.) izoqoruli

gaTbobis dros airi iRebs imdenive siTbos

raodenobas, ramdensac izoTermuli

gafarToebisas. gansazRvreT Zravis margi

qmedebis koeficienti. izoTermuli procesis

dros mudmivia temperatura, izoqorulis

dros _ moculoba. idealuri airis qceva

emorCileba mendeleev-klapeironis gantolebas: TRm

Vpµ

= , sadac p airis

wnevaa, V _ moculoba, T _ absoluturi temperatura, m _ masa, µ _ moluri masa, xolo R _ idealuri airis mudmiva.

amoxsna: miRQA

=η , sadac A cikluri procesis ganmavlobaSi airis

Sesrulebuli muSaobaa, xolo miRQ cikluri procesis ganmavlobaSi airis miRebuli siTbos raodenobaa. QQ 2=miR

Termodinamikis I kanonis Tanaxmad, izoTermuli procesis dros

idealuri airis mier Sesrulebuli muSaoba tolia miRebuli siTbos

raodenobis. aqedan gamomdinareobs, rom izoTermuli gafarToebisas

airis Sesrulebuli muSaoba Q -s tolia. izoTermuli SekumSvisas, mendeleev-klapeironis kanonis Tanaxmad, yoveli moculobis dros airis

wneva samjer naklebia, vidre gafarToebisas, amitom airis muSaoba

moduliT samjer naklebi iqneba, vidre gafarToebisas. amasTan,

SekumSvisas muSaoba uaryofiTi niSnisaa. amrigad, SekumSvisas airis

muSaobaa ⎟⎟⎠

⎞⎜⎜⎝

⎛−

3Q

. izoqoruli procesis dros airis muSaoba nulis tolia.

51

maSasadame, cikluri procesis ganmavlobaSi airis muSaobaa

32

3QQ

QA =⎟⎟⎠

⎞⎜⎜⎝

⎛−+= . sabolood miiReba, rom %33

31

232

≈==Q

Qη .

3. gamWvirvale biprizma moTavsebulia sinaTlis erTgvarovan vertikalur nakadSi

ise, rom zeda waxnagi horizontaluria (ix.

nax.).

a) miiReT gantoleba, romelic akavSirebs biprizmaSi gavlisas sinaTlis θ gadaxris kuTxes biprizmis gardamtex α kuTxesTan da misi nivTierebis n gardatexis maCvenebelTan;

b) gansazRvreT sinaTlis intensivoba (sxivebis marTob erTeulovan farTobSi erTeulovan

droSi gasuli sinaTlis energia), Tu misi moqmedebiT biprizma

gaCerebulia haerSi. biprizmis nivTierebis simkvrivea ρ . is mTlianad sinaTlis erTgvarovan nakadSia moTavsebuli. sinaTlis arekvla da

STanTqma ugulebelyaviT. sinaTlis θ gadaxris kuTxe cnobilad CaTvaleT. biprizmis erT-erTi zoma miTiTebulia naxatze. vakuumSi

sinaTlis siCqarea c . Tavisufali vardnis aCqarebaa g . fotonis

impulsi ganisazRvreba formuliT cE

p fotfot = , sadac fotE fotonis

energiaa.

amoxsna: a) pirvel zedapirze gardatexisTvis gvaqvs

n=βα

sinsin

am formulidan gamomdinareobs, rom

nαβ sinarcsin=

naxatidan Cans, rom βαγ −= meore zedapirze gardatexisTvis

gvaqvs

γθ sinsin n= am formulebidan gamomdinareobs,

rom

⎟⎠⎞

⎜⎝⎛ −=

nn ααθ sinarcsinsinsin

b) Fmg = (prizmis wonasworobis piroba) (1)

sadac m prizmis masaa, xolo F sinaTlis nakadidan prizmaze moqmedi Zalaa.

tppF∆

−=

θcos (2)

sadac p biprizmaSi t∆ droSi gasuli sinaTlis impulsia, xolo θ

α α

a

θ

α α

β γ

52

sinaTlis gadaxris kuTxea. cEp = (3), sadac E biprizmaSi t∆ droSi

gasuli sinaTlis energiaa, xolo c vakuumSi sinaTlis siCqarea. tabIE ∆= 0 (4), sadac a da b biprizmis zeda waxnagis gverdebis sigrZea.

(2), (3) da (4) formulebis gamoyenebiT miiReba, rom

( )c

abIF θcos10 −= (5)

biprizmis masa ganisazRvreba formuliT

4221 2 αραρ tgbabtgaam =⋅⋅⋅⋅⋅= (6)

(1), (5) da (6) formulebidan miiReba, rom

( )θαρ

cos140 −=

tggacI (7)

4. sinaTlis wertilovani wyaro mudmivi →

0v siCqariT moZraobs wrfeze, romelic mcire α kuTxes adgens Semkrebi linzis mTavar optikur RerZTan. wyaros traeqtoria mTavar optikur RerZs ormag fokusSi kveTs.

gansazRvreT wyaros mimarT gamosaxulebis minimaluri fardobiTi

siCqare, agreTve gamosaxulebis siCqare im momentSi, rodesac wyaros

mimarT gamosaxulebis fardobiTi siCqare minimaluria.

amoxsna:

linzaSi gardatexis Semdeg, wyaros traeqtoriis gaswvrivi sxivi

gadis linzis meore ormag fokusSi. cxadia, rom am sxivis gaswvriv

moZraobs gamosaxuleba. es sxivic α kuTxes adgens Semkrebi linzis mTavar optikur RerZTan (nax. 1). maSasadame, gamosaxulebisa da wyaros

siCqareebs Soris kuTxe aris α2 . fardobiTi siCqaris povna gamosaxulia nax. 2-ze. gamosaxulebis siCqare icvleba nulidan usasrulobamde.

naxatidan advili misaxvedria, rom fardobiTi siCqare maSinaa

minimaluri, rodesac is gamosaxulebis siCqaris marTobulia (ix. nax. 3).

nax. 3-ze dayrdnobiT miiReba, rom

α2sin0min. vv =fard α2cos0vv =

53

5. m da M masebis Zelakebi SeerTebulia L sigrZis rezinis zonriT da moTavsebulia gluv horizontalur zedapirze. Zelakebis zomebi bevrad

naklebia zonris sigrZesTan SedarebiT. zonris masa bevrad naklebia

Zelakebis masebTan SedarebiT. zonris sixistea k . Tavdapirvelad Zelakebi uZravia, zonari moSvebuli ar

aris, magram arc gaWimulia. erT-erT

Zelaks swrafi biZgiT mianiWes 0υ siCqare zonris gaswvriv, naxatze naCvenebi

mimarTulebiT. gansazRvreT, biZgidan ra

drois Semdeg daejaxebian Zelakebi erTmaneTs.

amoxsna: Zelakebis masaTa centri Tanabrad moZraobs marcxniv siCqariT

Mm

mc +

= 0υ

υ

am siCqariT gadataniTad moZravi aTvlis sistema inerciulia da

masSi Tamamad SegviZlia meqanikis CvenTvis cnobili kanonebis gamoyeneba.

am aTvlis sistemaSi masaTa centri uZravi darCeba, marcxena Zelakis

sawyisi siCqare mimarTulia marcxniv da moduliT tolia

MmM

c +=−= 001

υυυυ , xolo marjvena Zelakis sawyisi siCqare mimarTulia

marjvniv da moduliT tolia Mm

mc +

== 02υ

υυ . Zelakebis Tavdapirveli

daSoreba masaTa centridan aris MmLM

+=1l da Mm

Lm+

=2l (ix. nax.).

zonris pirveli nawilis sixistea ( )

MMmkLk

k+

==1

1 l, xolo meoresi

( )m

MmkLkk

+==

22 l

. advili misaxvedria (advili Sesamowmebelicaa), rom

Zelakebi iwyebs rxeviT moZraobas toli periodiT

( )MmkMm

kM

km

T+

=== πππ 22221

naxevari periodis Semdeg Zelakebs eqnebaT Tavdapirveli mdebareoba

da Tavdapirvelis toli, magram sawinaaRmdegod mimarTuli siCqareebi.

amis Semdeg zonari moeSveba da Zelakebi erTmaneTisaken Tanabrad

imoZraveben dajaxebamde. amis gamo miiReba

( ) 0212 υπ

υυL

MmkMmLT

t ++

=+

+=

erovnuli saswavlo olimpiada fizika - naec · 2016. 8. 21. · xi_xii klasebis moswavleebi. mocemul...

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