giorgi gedeniZe eTer lazaraSvili

inteleqti
inteleqti
giorgi gedeniZe
eTer lazaraSvili
IX
klasis
saxelmZRvanelo
giorgi gedeniZe,
eTer lazaraSvili
fizika
IX
klasis saxelmZRvanelo
grifi mieniWa 2012 wels ssip ganaTlebis xarisxis ganviTarebis
erovnuli centris (brZaneba ##375, 18.05.2012) mier
gamomcemloba inteleqti
Tbilisi 2012
giorgi gedeniZe, eTer lazaraSvili
fizika IX
Giorgi Gedenidze, Eter Lazarashvili
PHYSICS IX
INTELEKTI PUBLISHERS
Tbilisi 2012
redaqtorebi gvanca gedeniZe
naTia mamulaSvili
Editor
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Natia Mamulashvili
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© gamomcemloba `inteleqti~, 2012
Sinaarsi
I Tavi. Tanabari moZraoba. niutonis pirveli kanoni..................................................... 6
1.1.
1.2.
1.3.
1.4.
1.5.
sxeulis mdebareoba. aTvlis sxeuli. meqanikuri moZraoba............................................ 7
moZraobis traeqtoria da gavlili manZili.......................................................................... 9
nivTieri wertili.............................................................................................................................. 11
meqanikis ZiriTadi amocana. aTvlis sistema. gadaadgileba........................................... 13
moqmedebebi veqtorebze. veqtorebis Sekreba, gamokleba,
veqtoris namravli skalarze....................................................................................................... 16
1.6. veqtoris gegmili.............................................................................................................................. 19
1.7. amocanis amoxsnis nimuSi................................................................................................................ 21
1.8. daamzadeT sawveTuriani urika.................................................................................................... 23
1.9. Tanabari da araTanabari moZraoba............................................................................................ 24
1.10. siCqare. siCqaris erTeulebi. meqanikis ZiriTadi
amocanis gadawyveta Tanabari moZraobisas......................................................................... 27
1.11. amocanis amoxsnis nimuSi.............................................................................................................. 29
1.12. moZraobis gamosaxva grafikulad............................................................................................. 31
1.13. moZraobis fardobiToba............................................................................................................... 34
1.14. amocanis amoxsnis nimuSebi.......................................................................................................... 36
1.15. CaatareT, gamoikvlieT, ivaraudeT, daadgineT................................................................... 39
1.16. aristotele da galilei MmoZraobis Sesaxeb........................................................................ 40
1.17. SeamowmeT Tqveni codna............................................................................................................... 43
II Tavi. araTanabari moZraoba. niutonis II da III kanonebi............................................. 44
2.1.
2.2.
2.3.
2.4.
ra da rogor iswavleT, SeamowmeT Tqveni codna............................................................... 45
araTanabari moZraoba. saSualo siCqare. meyseuli siCqare............................................ 47
TanabaraCqarebuli moZraoba. aCqareba..................................................................................... 49
siCqaris da aCqarebis modulebisa da gegmilebis droze
damokidebulebis grafikebi.......................................................................................................... 51
2.5. meqanikis ZiriTadi amocanis gadawyveta TanabaraCqarebuli moZraobisas................ 54
2.6. amocanis amoxsnis nimuSi................................................................................................................ 56
2.7. sxeulTa Tavisufali vardna........................................................................................................ 58
2.8. amocanis amoxsnis nimuSi................................................................................................................ 60
2.9. niutonis meore kanoni.................................................................................................................... 63
2.10. niutonis mesame kanoni.................................................................................................................. 66
2.11. amocanis amoxsnis nimuSi.............................................................................................................. 69
2.12. SeamowmeT Tqveni codna............................................................................................................... 72
III Tavi. mrudwiruli moZraoba....................................................................................................... 73
3.1.
3.2.
3.3.
3.4.
3.5.
mrudwiruli moZraoba. kuTxuri siCqare .............................................................................. 74
centriskenuli aCqareba.................................................................................................................. 76
mzis sistemis planetebis moZraoba.......................................................................................... 79
amocanis amoxsnis nimuSi................................................................................................................ 82
SeamowmeT Tqveni codna................................................................................................................. 84
IV Tavi. mudmivobis kanonebi meqanikaSi................................................................................. 85
4.1. ra da rogor iswavleT, SeamowmeT Tqveni codna............................................................... 86
4.2. impulsis mudmivobis kanoni.......................................................................................................... 89
3
4.3.
4.4.
4.5.
4.6.
4.7.
reaqtiuli moZraoba......................................................................................................................... 91
amocanis amoxsnis nimuSi................................................................................................................ 94
meqanikuri muSaoba. simZimis Zalis muSaoba.......................................................................... 97
meqanikuri energia – kinetikuri da potenciuri............................................................... 99
drekadobis Zalis muSaoba. drekadad deformirebuli sxeulis
potenciuri energia......................................................................................................................... 101
4.8. potenciuri da kinetikuri energiebis urTierTgar­
daqmna.
meqanikuri energiis mudmivobis kanoni................................................................................... 103
4.9. meqanikuri energia da xaxunis Zala.......................................................................................... 106
4.10. amocanis amoxsnis nimuSi.............................................................................................................. 109
4.11. SeamowmeT Tqveni codna............................................................................................................... 111
V Tavi. bunebis Zalebi.meqanikisa da mudmivobis kanonebis gamoyeneba............. 112
5.1. ra da rogor iswavleT, SeamowmeT Tqveni codna............................................................... 113
5.2. bunebis Zalebi.................................................................................................................................... 114
5.3. xaxunis Zala......................................................................................................................................... 116
5.4. srialis xaxuni.................................................................................................................................... 119
5.5. xaxuni bunebasa da teqnikaSi........................................................................................................ 122
5.6. eleqtromagnituri urTierTqmedeba _ drekadobis Zala............................................... 125
5.7. hukis kanoni......................................................................................................................................... 128
5.8. amocanis amoxsnis nimuSi................................................................................................................ 130
5.9. msoflio mizidulobis Zala......................................................................................................... 132
5.10. gravitaciuli mudmivas gansazRvra.Tavisuflad vardnili
sxeulis aCqareba............................................................................................................................. 135
5.11. aCqarebulad moZravi sxeuli...................................................................................................... 138
5.12. horizontisadmi kuTxiT gasrolili sxeulis moZraoba................................................. 141
5.13. nasvretebiani boTli da wyali. saSinao cdebi.................................................................. 143
5.14. saSinao cdebSi miRebuli Sedegebis Teoriuli axsna...................................................... 144
5.15. pirveli da meore kosmosuri siCqare.................................................................................... 146
5.16. sxeulis moZraoba daxril sibrtyeze..................................................................................... 148
5.17. gadabmuli sxeulebis moZraoba................................................................................................. 150
5.18. SeamowmeT Tqveni codna............................................................................................................... 153
VI Tavi. geometriuli optika.......................................................................................................... 154
6.1. CaatareT, gamoikvlieT, ivaraudeT, daadgineT..................................................................... 155
6.2. sinaTlis gavrceleba....................................................................................................................... 157
6.3. sinaTlis arekvlis kanonebi. brtyeli sarke......................................................................... 160
6.4. CaatareT, gamoikvlieT, ivaraudeT, daadgineT..................................................................... 163
6.5. sinaTlis gardatexa. sruli Sinagani arekvla...................................................................... 164
6.6. amocanis amoxsnis nimuSi................................................................................................................ 166
6.7. sinaTlis dispersia........................................................................................................................... 168
6.8. linza....................................................................................................................................................... 171
6.9. gamosaxulebis ageba linzaSi........................................................................................................ 174
6.10. optikuri xelsawyoebi. Tvali da mxedveloba.................................................................... 177
6.11. optikuri xelsawyoebi................................................................................................................... 180
6.12. amocanis amoxsnis nimuSi.............................................................................................................. 183
6.13. SeamowmeT Tqveni codna............................................................................................................... 185
ganmartebiTi leqsikoni.......................................................................................................................... 186
saganTa saZiebeli....................................................................................................................................... 189
4
rogor vixelmZRvaneloT saxelmZRvaneloTi
paragrafis nomeri
sayrdeni farTobis mqone sxeulTa wonasworoba
paragrafis saxelwodeba
individualuri samuSao
wyvilebad muSaoba
informacia suraTis
mixedviT
daukvirdiT suraTebs, isaubreT ra informacias iRebT
suraTebidan.
jgufebad muSaoba
amocanebi logikuri
msjelobis gamoyenebiT
 moZraobas, romlis drosac sxeuli drois nebismier
tol SualedSi erTnair gadaadgilebebs asrulebs, Tanabari
moZraoba ewodeba.
axali terminebi
umniSvnelovanesi
daskvnebi
amgvarad, rom ganvsazRvroT sxeulis mdebareoba drois nebismier
momentSi saWiroa, vicodeT sxeulis sawyisi mdebareoba da drois am
momentisaTvis Sesrulebuli gadaadgilebis gegmilebi: x=x0+Sx, y=y0+Sy.
saSinao davaleba
saSinao cdebis Catarebamde aucileblad dakvirvebiT waikiTxeT paragrafi.
masSi mocemul SekiTxvebze damo­ukideblad pasuxis gacemiT ganimtkiceT codna.
SeasruleT samuSao rveulSi miTiTebuli davalebebi.
5
am TavSi gaixsenebT
da gaecnobiT
I Tavi
Tanabari
moZraoba.
niutonis
pirveli kanoni
 meqanikur moZraobas;
 aTvlis sxeuls;
 traeqtorias;
 gavlil manZils;
 sawveTuriani urikis gamoyenebas;
 Tanabar da araTanabar moZraobas;
 siCqares, saSualo da meyseul
siCqares;
 siCqarisa da koordinatis droze
damokidebulebis grafikebs;
 siCqaris, koordinatis, gavlili
manZilis da moZraobis drois
gansazRvras.
 moZraobis fardobiTobas.
qaSueTis taZari
sionis taZari
mamadavi­
Tis
taZari
sur. 6.1
6
6.1 suraTze saqarTvelos dedaqalaqis erT-erTi raionis (mTawminda-krwanisis)
sqematuri rukis nawilia gamosaxuli. daaTvaliereT igi. gaarkvieT, erTi da
igivea Tu ara Tavisuflebis moednis mdebareoba mamadaviTis, sionis da qaSueTis
taZrebis mimarT? suraTze am taZrebidan Tavisuflebis moednisken mimarTulebebi
miuTiTeT isriT.
rukaze miuTiTeT umoklesi samanqano gza, romelic Tavisuflebis moednidan
sionis taZramde migiyvanT. vTqvaT, Tqven mier miTiTebuli gziT megobarTan erTad avtomobiliT sionis taZarTan mixvediT. Seicvala Tu ara Tqveni mdebareoba
Tavisuflebis moednis mimarT? avtomobilis mimarT? megobris mimarT?
sxeulis mdebareoba. aTvlis
sxeuli. meqanikuri moZraoba
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1.1
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
sad mdebareobs Tqveni skola?
Tavisuflebis moedans mamadaviTis, sionisa da qaSueTis taZrebis mi­
marT sxvadasxva mdebareoba aqvs.
maSasadame, Tavisuflebis moednis mdebareoba gansxvavebulia sxvadasxva
sxe­
ulis mimarT. asevea nebismieri sxva sxeulisTvisac.
1. moiyvaneT am faqtis damadasturebeli sxva magaliTi.
daadgineT saklaso oTaxSi Tqveni magidis mdebareoba sxva­
dasxva sxeulis (karebis, fanjris, dafis da sxva) mimarT.
SeiZleba davaskvnaT: sxeulis mdebareoba gansxvavebulia sxva­
dasxva
sxe­ulis mimarT. es niS­navs, rom mdebareoba fardobiTia (SedarebiTia).
sxeuls, romlis mimarTac gani­xileba sxva sxe­ulis mdeba­reoba, aTvlis
sxeuli ewodeba.
Tavisuflebis moednidan sionis taZarTan avtomobiliT misvlisas Tqveni
mdebareoba Tavisuflebis moednis mimarT Seicvala. avtomobilisa da
megobris mimarT ki _ ara.
drois ganmavlobaSi siv­
rceSi sxeulis mde­
bareobis cvlilebas sxva
sxe­
ulis mimarT meqa­
nikuri moZ­
raoba ewodeba.
2. gamoiyeneT meqanikuri moZraobis ganmarteba da upasuxeT, moZraobdiT Tu ara Tavisuflebis moednis (dedamiwis
zedapiris) mimarT? avtomobilis mimarT? megobris mimarT?
pasuxi daasabuTeT.
imisaTvis, rom vupasuxoT SekiTxvas, moZraobs Tu ara sxeuli, saWiroa
gavarkvioT, icvleba Tu ara misi mdebareoba aTvlis sxeulis mimarT. Tu
sxeulis mdebareoba aTvlis sxeulis mimarT icvleba, igi moZraobs; Tu
ar icvleba _ uZravia. maSasadame, erTi da igive sxeuli SeiZleba erTi
aTvlis sxeulis mimarT icvlides mdebareobas, meoris mimarT ki ara. es
imas niSnavs, rom es sxe­uli erTi aTvlis sxeulis mimarT moZraobs, meoris mimarT _ uZravia. maSasadame, moZraoba da uZraoba fardobiTia.
SevTanxmdeT, rodesac aTvlis sxeuli miTiTebuli ar aris, maSin moZraoba gani­xileba dedamiwis zedapiris mimarT.
3. matarebeli icvlis
mdeba­r eobas dedamiwis
zedapiris mimarT. va­
gonSi, magidaze Zevs wigni (sur. 7.1).
moZraobs Tu ara wigni ma­gidis, vagonis
ked­lebis, relsebis, xeebisa da vagonrestoranSi mimavali mgzavris mimarT?
TiToeul SemTxvevaSi miuTiTeT aTvlis sxeuli. pasu­xebi CawereT samuSao
rveulSi miTiTebul cxrilSi.
sur. 7.1
7
4. daasaxeleT sxeuli, romlis
mimarTac tvirTi ar icvlis
mdebareobas (sur. 8.1).
5. daasaxeleT sxeuli, romlis
mimarTac tvirTi icvlis mdebareobas.
6. daasaxeleT sxeulebi, romelTa mimarT
moZraobs tvirTi da is sxeulebi, romelTa
mimarT uZravia.
7. SesaZleblad migaCniaT Tu ara, rom
tvirTi romelime sxeulis mimarT icvlides
mdebareobas da mis mimarT uZravi iyos? axseniT, ratom fiqrobT ase?
sur. 8.1 tvirTi dedamiwis zedapiridan aswies.
rodesac avtomobili icvlis mdebareobas romelime sxeulis, vTqvaT, xis
mimarT, maSin aTvlis sxeulia xe. TviT xe avtomanqanis mimarT moZraobs,
dedamiwis zedapiris mimarT ki ar icvlis mdebareobas, amitom igi dedamiwis
zedapiris mimarT uZravia.
pirvel SemTxvevaSi avtomanqanaa aT­
vlis sxeuli, meore SemTxvevaSi _
dedamiwa. igive xe mzis mimarT dedamiwasTan erTad moZraobs, aTvlis sxeulia mze.
aTvlis sxeuli, meqanikuri moZraoba, fardobiTi
SekiTxvas _ sad mdebareobs sxeuli? _ pasuxi rom gavceT aucile­
belia, avirCioT sxva sxeuli, romlis mimarTac ganvixilavT mocemu­
li sxeu­lis mdebareobas. am sxva sxeuls aTvlis sxeuls uwodeben.
aTvlis sxeulad nebismieri sxeuli SeiZleba avirCioT. sxeulis
mdebareoba gansxvavebulia sxvadasxva aTvlis sxeulis mimarT. es niS­
navs, rom sxeulis mdebareoba fardobiTia.
sxeuli, romelic drois ganmavlobaSi aTvlis sxeulis mimarT mde­
bareobas icvlis, moZraobs.
erTi da igive sxeuli SeiZleba erTi aTvlis sxeulis mimarT icv­
lides mdebareobas, meoris mimarT ki ara. es imas niSnavs, rom es
sxeul­i erTi aTvlis sistemis mimarT moZraobs, meoris mimarT – uZ­
ravia. maSasadame, moZraoba da uZraoba fardobiTia.
saSinao cda
ramdenjerme dawereT furcelze Tqveni sayvareli fraza da daakvirdiT
fanqris wveris moZraobas. rogor moZraobs fanqris wveri furclis mimarT?
SemCneuli CawereT samuSao rveulSi.
8
1.2
moZraobis traeqtoria da
gavlili manZili
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
wirs, romelsac sxeulis rome­
lime wer­
tili
moZraobisas Semowers, am wertilis moZra­
obis
traeq­toria ewodeba.
traeqtoria SeiZleba xiluli iyos.
magaliTad, carciT werisas dafaze rCeba carcis wve­
ris kvali _ misi moZra­
obis traeqtoria.
traeqtoriis formis mixed­
v iT moZraoba
SeiZleba iyos wrfivi an mrudwiruli.
traeqtoria xilulia?
sur. 9.1. Ramis caze meteor­
is mnaTi kvali misi moZrao­
bis tra­eq­toriaa.
1. CaatareT cda: Zafze dakidebuli burTula
gada­xareT (sur. 9.2). jer ivaraudeT, Semdeg cdiT
SeamowmeT, ra moxdeba Tu burTulas gaaTavisuflebT? gaaTavisufleT da daakvirdiT. daxazeT
burTulis moZraobis traeqtoria. rogor icvleba burTulis
mdebareoba A wertilis mi­marT? gamarTlda Tqveni hipoTeza?
zambaraze
CamokideT
B
sxeuli
CamoswieT sxe­
uli da gaaTa­
visufleT
sur. 9.3
C
A
sur. 9.2
2. daxazeT zambaraze da­
kidebuli sxeulis moZraobis traeqtoria.
rogor icvleba sxeulis
mdebareoba B wertilis
mimarT (sur. 9.3)?
D
moZraobas, romlis drosac sxeuli raRac mdebareobis mimarT xan
erT, xan meore mxares gadaixreba, rxeviTi moZraoba ewodeba.
3. moiyvaneT yofa-cxovrebidan rxeviTi moZraobis magaliTebi.
4. rogoria 9.4 suraTze gamosa­
xuli borblis wertilebis moZ­
raobis tra­eqtoria?
wylis wveTebis?
gavaanalizoT saSinao cdaSi SemCneuli movle­
nebi: fanqris wveri furclis mimarT asrulebs
rTul rxeviT moZraobas. misi gadaadgileba
furclis mimarT drois nebismier momentSi ar
aRemateba ≈6 mm-s.
sur. 9.4
9
fanqris wveris traeqtoria furclis mimarT emTxveva wirs, romelsac is
tovebs furcelze, im momentebis gamotovebiT, rodesac fanqari ar exeba
furcels.
drois raime SualedSi aRwerili traeqtoriis nawilis sigrZes dro­
is am SualedSi gavlili manZili ewodeba. amasTan, traeqtoriis erTi
da igive ubani sxeulma ramdenjerac aRwera, imdenjer unda aviRoT
Sesakrebad misi sigrZe.
gavlili manZili fizikuri sidi­dea. igi izomeba sigrZis erTeu­lebiT.
5. A da B qalaqebi er­
TmaneTTan dakavSi­rebu­
lia rogorc samanqano
gziTY , ise rkinigziT (sur.
10.1). TiTo­eul gzaze gan­saz­RvreT
A-dan B qalaqSi Cas­
vlisas gav­
lili manZili.
masStabi
1
10 km
sur. 10.1
6. gaivlis Tu ara erTnair manZils
avtomobilis marjvena da marcxena
borblebi mosaxvevSi (sur. 10.2)?
7. moZraobs Tu ara avtomobilis borblis
sur. 10.2
centri
a. avtomobilis Zaris mimarT?
b. gzatkecilis mimarT? rogori SeiZleba iyos
borblis centris traeqtoria TiToeul SemTxvevaSi?
8. fardobiTia Tu ara sxeulis moZraobis traeA
qtoria da gavlili manZili? pasuxi daasabuTeT.
B
9. rogoriAaA 10.3 suraTze gamosaxuli wamzomis
wamebis maCvenebeli isris A da B wertilebis moZraobis traeqtoria wamzomis korpusis mimarT?
10. SeadareT, drois erT da imave SualedSi rosur. 10.3
melia meti A Tu B wertilis mier korpusis mimarT
gavlili manZili da axseniT, ratom fiqrobT ase?
11. rogor moZraobs A wertili B wertilis mimarT? pasuxi daasabuTeT.
traeqtoria, gavlili manZili, rxeviTi moZraoba
amgvarad, yvela sxeuli moZraobisas `tovebs kvals~, romelsac
traeqtorias uwodeben. traeqtoria SeiZleba iyos xiluli, SeiZleba
uxilavi.
traeqtoriis mixedviT moZraoba an wrfivia, an mrudwiruli.
Tu gvsurs gavigoT, ra manZili gaiara sxeulma drois SualedSi,
unda gavzomoT drois am SualedSi aRwerili traeqtoriis sigrZe.
traeqtoria da gavlili manZili fardobiTi sidideebia.
10
nivTieri wertili
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1.3
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1. Tqveni azriT, sxeulis moZraobis Sesas­wavlad aucilebelia Tu ara misi mdebareobis gansazRvra? axseniT, ratom
fiqrobT ase?
sur. 11.1. yvela sxeuls gaaCnia forma da moculoba. sxeulis sxvadasxva nawili
sivrcis sxvadasxva adgilas imyofeba.
2. Tqveni azriT, sxeulis mdebareobis gansazRvrisas
unda mivaqcioT Tu ara yuradReba im faqts, rom sxeulis
sxvadasxva nawili sivrcis sxvadasxva adgilas imyofeba?
aRmoCnda, rom, zogjer, realurad arsebuli sxe­
ulis moZraobasTan
dakavSirebuli zogierTi sakiT­
xis Seswavlisas sxeulis zoma SeiZleba ar
gaviTvaliswinoT.
sxeuls, romlis zoma amocanis mocemul pirobebSi SeiZleba ugule­
belvyoT, nivTieri wertili ewodeba.
gansazRvrebaSi sityvebiT `mocemul pirobebSi~ vazustebT, rom am
konkretul pirobebSi sxeuli SeiZleba CaiTvalos nivTier wertilad. sxva
pirobebSi ki _ ara.
magaliTad, Tbilisidan soxumSi avtobusis Casasvlelad saWiro drois
gansazRvrisas avtobusi SeiZleba miviCnioT nivTier wertilad. avtobusis
moZraobis procesSi, daZvrisas, xidze gadasvlisas, gaCerebisas sifrTxilis mizniT saWiroa avtobusis zomebis gaTvaliswineba, amitom igi nivTier
wertilad ver CaiTvleba.
rodesac gvainteresebs mzis irgvliv planetebis moZraobis (sur. 12.1)
maxasiaTebeli sidideebis _ koordinatis, garSemovlis periodis, gavli­
li manZilis, siCqaris, aCqarebis da sxvaTa gansazRvra, maSin dedamiwa da
sxva planetebi SeiZleba CaiTvalos nivTier wertilebad, magram mzeze
an planetebze mimdinare sxvadasxva procesebis Seswavlisas, rogoricaa
dRe-Ramis monacvleoba, weliwadis droebis cvlileba, mzisa da mTvaris
dabneleba, qaris warmoqmna, vulkanebis amofrqveva da mravali sxva, maSin
ciuri sxeulebis nivTier wertilebad ganxilva ar SeiZleba.
11
sur. 12.1
unda gvaxsovdes, rom bunebaSi ara­
viTari
nivTieri wertili ar arse­
bobs. igi realurad
arsebuli sxeulis mo­delia. nivTieri wertilis
cneba meqa­
nikaSi bevri amocanis advilad da
sur. 12.2
moxer­
xebulad gadasawyvetad Semoitanes. aseT
SemTxvevaSi sxeulis uamravi wertilis mdebareobis gansazRvris nacvlad
sakma­
risia misi erTi wertilis mdebareobis gansazRvra.
sxeulis TiToeuli wertilis moZraobis aR­
wera arc maSinaa saWiro,
rodesac sxe­
ulis yvela wertili erTnairad moZraobs. aseT moZraobas
gadataniTi moZraoba ewodeba.
gadataniTi moZraobisas sxeulSi war­
modgeniT
gavle­
buli nebismieri wrfe sawyisi mdebareobis
paraleluri rCeba (sur. 12.2).
3. Tqveni az­
riT, sxeulis gadataniTi moZrao­bisas (sur. 12.3)
SeiZ­leba misi niv­Tier wertilad CaTvla? axseniT,
ratom fiqrobT ase?
4. ivaraudeT Tqveni skolis koordinatebi
da manZili Tqveni saxlidan skolamde. aTvlis
saTaved airCieT Tqveni saxli. X RerZi mimarTeT
aRmosavleTiT, Y _ CrdiloeTiT. varaudi
CawereT samuSao rveulSi.

sur. 12.3
nivTieri wertili, gadataniTi moZraoba
amgvarad, sxeulis sxvadasxva nawili siv­rcis sxva­dasxva adgilas mde­
bareobs. zog­jer, amocanis mocemul pirobebSi sxe­ulis zoma SeiZleba
ugulebelvyoT da igi ganvixiloT rogorc nivTieri wer­tili.
XOY koordinatTa sistemis saTave daukavSireT
Tqvens saxls. X RerZi mimarTeT aRmosavleTiT, Y RerZi
_ CrdiloeTiT. SearCieT masStabi. agebiT aCveneT skolis mdebareoba da
saxlidan skolamde Tqveni moZraobis traeqtoria. gazomeT Tqveni nabijis
sigrZe. Tu SesaZlebelia, daTvaleT nabijebis raodenoba saxlidan skolamde da gansazRvreT manZili saxlsa da skolas Soris. gamarTlda Tqveni
navaraudevi?
saSinao cda
12
1.4
meqanikis ZiriTadi amocana.
aTvlis sistema. gadaadgileba
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meqanika (berZnuli) _
mecniereba manqanebis Sesaxeb,
manqanebis agebis xelovneba.
meqanika mecnierebaa, romelic safuZ­
vlad udevs bunebismetyvelebas. meqanika
Seiswavlis meqanikur moZraobas, amitom
meqanikis ZiriTadi amocanaa sxeulis
mdebareobis (koordinatebis) gansazRvra
sivrceSi drois nebismier momentSi.
jer-jerobiT gamonaTqvamSi
`sxeuli~ nivTier wertils
vigulisxmebT.
1. ramdeni ricxviT (koordinatiT) ganisazRvreba wrfis,
sibrtyis, sivrcis nebismieri wertilis mdebareoba?
2. gansazRvreT 13.1 suraTze gamosaxuli A, B da C
wertilebis koordinatebi.
3
Y
A
2
O
X
-1
1 2 3
B
X
3. rogor miuTiTeben figuris
mdebareobas Wadrakis dafaze?
4. rogor miuTiTeben adresatis
mdeba­reobas safosto gzavnilebze?
1
2
1
O
Y
C
1
2
X
sur. 13.1
Z
8
7
6
5
4
3
2
1
A
B
C
D
E
F
G
H
sur. 13.2
sivrceSi sxeulis mdebareobis gansazRvrisaTvis sakmarisia sami ricxvis
_ koordinatis codna.
daakvirdiT 13.3 suraTs.
5. karuselis brunvisas ro­
gori tra­eqtoriiT moZraobs
bav­­
Svi skamisa da dedamiwis
ze­dapiris mimarT (sur. 13.3)?
Q
6. sxeulis mdebareoba,
traeqtoriis saxe da gavlili manZili damokidebulia Tu ara
aTvlis sxeulis arCevaze? pasuxi
daa­sabuTeT magaliTebiT.
F
mg
sur. 13.3
13
meqanikis ZiriTadi amocanis gada­
saw­
yvetad, sivrcul koor­
dinatebTan
erTad, saWiroa kidev erTi koor­
dinatis _ drois codna.
aTvlis sxeuli, masTan dakavSirebuli koordinatTa sistema da drois
gasazomi xelsawyo erTad qmnis aTvlis sistemas, rom­lis mimarTac ga­
nixileba sxeulis moZraoba.
cxadia, sxeulis moZraobasa da mde­bareobaze saubars mxolod maSin aqvs
azri, rodesac igi ganixileba kon­kre­­tuli
aTvlis sistemis mimarT. aTv­­
lis sistema
ise unda SevarCioT, rom moZraoba masSi
martivad aRi­were­bodes.
daakvirdiT 14.1 suraTs. suraTze A da
B punqtebi erTmaneTTan dakavSi­rebulia
rogorc samanqano gziT, ise rkinigziT.
7. ramdennairi gan­s­x va­
vebuli traeq­to­riiT aris
SesaZlebeli A punqtidan
B-Si misvla?
sur. 14.1
8. gansxvavebulia Tu ara sxvadasxva
traeqtoriaze moZraobis Sedegi?
A punqtidan B-Si Casvlis usasrulod
bevri, erTmaneTisagan gansxvavebuli variantis Sedegi erTaderTia: A punqtidan
gadavaadgildiT B punqtSi. A punqti –
Cveni sawyisi mdebareoba SevaerToT swori
xazis monakveTiT B punqtTan – Cvens
sur. 14.2
saboloo mdebareobasTan. TvalsaCinoe­
bisaTvis monakveTis bolo aRv­
niSnoT isriT da am mimarTul mona­
k­
veTs
vuwodoT gadaadgileba (sur. 14.2).
wrfis mimarTul monakveTs, rome­­
lic aer­
Tebs sxeulis sawyis mde­
bareobas mis momdevno mdebareobasTan am sxeulis (nivTieri wertilis)
gadaadgileba ewodeba. gadaadgileba xasiaTdeba dadebiTi ricxviTi mniS­
vnelobiT (gadaadgilebis sigrZe) da mimar­TulebiT.
9. ra ewodeba sidides, romelic xasiaTdeba dadebiTi ricxviTi mniSvnelobiT da mimarTulebiT?
10. gadaadgilebis garda kidev romeli veqtoruli sidideebi
iciT?
11. rogor veqtorebs uwodeben tols, mopirdapires, moduliT tols?
12. rogor gamosaxaven veqtors? veqtoris moduls?
Tu S1 da S2 gadaadgilebebi tolia, maSin es ase Caiwereba S1= S2., xolo
Tu S3 da S4 gadaadgilebis veqtorebi moduliT tolia da urTierTsapirispiro mimarTuleba aqvT ase: S3=- S4.
14
13. ra mimarTule­
biT (win Tu ukan)
moZraobs velo­s i­
pedis CarCo bor­
blis qveda wertilis mi­
m arT?
borblis zeda wertilis mimarT
(sur. 15.1). pasuxi da­asabuTeT.
14. R radiusis borbalma
erTi bruni Seasrula (sur. 15.1).
sur. 15.1
gansazRvreT borblis qveda
wertilis gadaadgilebis moduli
borblis centrisa da dedamiwis zedapiris mimarT.
meqanikis ZiriTadi amocana, aTvlis sistema,
gadaadgileba
amgvarad, SeviswavloT meqanikuri moZraoba anu gadavwyvitoT meqani­
kis ZiriTadi amocana niSnavs, ganvsazRvroT sxeulis mdebareoba drois
nebismier momentSi. sxeulis mdebareobis gansasazRvravad ki aTvlis
sxeuli, koordinatTa sistema da drois gazomvaa saWiro. SemoviReT
gadaadgilebis cneba. igi veqtoruli sididea.
saSinao cda
muyaoze daxazeT 3-4 sm sigrZis AB (sur. 15.2, a). muyaosgan gamoWeriT
2-3 sm sigrZis veqtori, misi saTave
daadeT AB -is bolos da WikartiT
ise daamag­reT, rom SeiZlebodes misi
mobruneba (sur. 15.2, b). sawyis mdebareobaSi BD AB -s gagrZe­
leba iyos.
gazomeT AD -s sigrZe da CawereT. B
A
B
A
B
a
b
D
sur. 15.2
wertilis ir­
gvliv abruneT BD 10-10
AB + BD
an 20-20-gra­
dusiani kuTxiT, yoveli mobrune­
bis Semdeg gazomeT AD =veqtoris sigrZe da CawereT. rogor icvleba AD -s
ADsigrZe
= AB + BD -s sxvadasxva
mimarTulebisas?. gamo­itaneT daskvna.
15
1.5
moqmedebebi veqtorebze. veqtorebis
Sekreba, gamokleba, veqtoris namravli skalarze
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vTqva­T, A qalaqidan gamosuli
sxe­uli B qalaqSi Cavida (sur. 16.1).
1. aris Tu ara
sxeulis gada­a dgi­
leba damokide­buli
tra­
eqtoriis formasa da
drois im Sualedze, romelic
man moan­doma A-dan B-Si Casvlas?
axseniT, ratom fiqrobT ase?
sxeulis moZraoba gan­v i­x iloT
or etapad. vTqvaT, A qa­l aqidan
S1
C
S
S2
S l2
S3
D
sur. 16.1
gamosuli sxeuli C punqtSi SeCerda, am dros misi ga­
daadgileba AC = S1,
Semdeg gaag­rZela gza da Cavida B qalaqSi (sur. 16.1). gadaadgilebaa CB = S2.
samuSao rveulSi daxazeT AC = S1, CB = S2 da AB = S gadaadgilebebi.
daakvirdiT miRebul suraTs, dafiqrdiT da upasuxeT.
2. SeiZleba Tu ara vTqvaT, rom S = S1+ S2 ?.
axseniT, ratom fiqrobT ase?
(1)
S1 da S2 veqtorebis aseT Sekrebas veqtoruli Sekreba ewodeba.
vTqvaT, S1=3 km-s, S2=4 km-s.
3. Tqveni azriT, aris Tu ara S gadaadgilebis moduli
7 km? axseniT, ratom fiqrobT ase?
4. gansxvavdeba Tu ara ariTmetikuli Sekreba veqtoruli
Sekre­
bisagan? axseniT, ratom fiqrobT ase?
16.1 suraTis mixedviT SeiZleba CamovayaliboT ori veqtoris Sekrebis
wesi, romelsac samkuTxedis wesi ewodeba.
ori veqtori rom SevkriboT saWiroa, pirveli veqtoris bolos mov­
doT meore veqtoris saTave, pirveli veqtoris saTavidan meore veqto­
ris bolomde gavlebuli veqtori aris mocemuli ori veqtoris jami.
5. Seicvleba Tu ara AB = S gadaadgileba, Tu CB = S2
gadaadgilebas SevcvliT CD = S l2 da DB = S3 gadaadgilebebiT (es veqtorebi 16.1 suraTze punqtiriTaa gamosaxuli)?
6. CamoayalibeT sami (oTxi da a. S.) veqtorebis Sekrebis wesi.
16
ori, S1 da S2 , veqtori Se­iZleba Semdegnairadac SevkriboT: S1 da S2 veqtorebis saTaveebi erT A wertilSi
movdoT (sur. 17.1). Cav­
TvaloT, rom es veq­
to­
rebi para­
lelogramis gverdebia da maTze avagoT paralelogrami.
veq­
t o­
r ebis saerTo saTavidan (A werti­l i) ga­v av­l oT
diagonali da mis bolos gavukeToT isari. miRebuli S
C
B
S1
S
α
S2
A
sur. 17.1
aris S1 da S2 veqtorebis jami (sur. 17.1). veq­torTa Sekrebis aRniS­nul wess
para­le­logramis wesi ewo­
deba. cxadia, veqtorTa Sekrebis orive wesis
gamoyenebiT erTi da igive Sedegi miiReba.
17.1 suraTze gamosaxuli S1 da S2 veqtorebis jamis moduli ganisazRvreba formuliT:
S = S12+ S22+ 2S1S2cos a , (3) sadac α aris Sesakreb veqtorebs Soris kuTxe.
7. SesaZleblad migaCniaT Tu ara, rom ori veqtoris jamis
moduli naklebi iyos TiToeuli Sesakrebis modulze? pasuxi
daasabuTeT.
8. gansazRvreT erT sibrtyeSi mdebare moduliT toli sami
veqtoris jami, Tu isini erTmaneTTan 1200-ian kuTxes qmnian.
ganvsazRvroT 17.2, a suraTze gamosaxuli a da b veqtorebis sxvaoba.
aRvniSnoT igi c -Ti.
c = a - b. (4)
ori, a da b , veqtoris sxvaoba ewo­
a
deba iseT c veqtors, romelic unda
b
mivumatoT b veqtors, rom miviRoT a
a
sur. 17.2
veqtori, e. i. b + c = a. (ix. sur. 17.2, b).
17.2 b suraTis gamoyenebiT SeiZleba
CamovayaliboT ori veqtoris sxvaobis gansazRvris wesi:
a
c
b
b
ori, a da b , veqtoris sxvaoba rom ganvsazRvroT saWiroa isini
mov­
doT erT wertils. maTi boloebis Sema­
er­­
Tebeli, maklebidan sak­
lebisaken mimarTuli, veq­tori aris ori veq­toris sxva­oba (sur. 17.2, b).
9. SesaZleblad migaCniaT Tu ara, ori veqtoris sxvaobis
moduli meti iyos saklebisa da maklebis modulebze? pasuxi
daasabuTeT.
a veqtoris k skalarul sidideze namravli aris veqtori, romlis
dulze, aqvs a veqtoris mimarTu­
mo­duli k-jer metia a veqtoris mo­
le­ba, rodesac k dadebiTia da a veqto­ris sa­pirispirodaa mimarTuli,
ro­desac k uaryofiTia. a veqtoris k skalarul sidideze namravli
tolia nulis, Tu k=0.
17
18.1 a da b suraTebze gamosa­
xulia a
veqtori da misi namravli k1=2 da k2=–0,5
ricxvebze.
a
a
-0,5 a
2a
a
b
sur. 18.1
10. gansazRvreT ori 3 S 1 da 2 S 1 veqtorebis a. jami; b.
pirveli da meore veqtorebis sxvaoba; g. meore da pirveli
veqtorebis sxvaoba.
veqtoris Sekreba, gamokleba, veqtoris namravli
skalarze
SeviswavleT moqmedebebi veqtorebze _ Sekreba, gamokleba, veqtoris
namravli skalarze.
saSinao cda
muyaosagan gamoWeriT AB veqtori (isari)
da mis gaswvriv gaatareT X RerZi (sur. 18.2).
isari A wertiliT WikartiT ise daamagreT,
A
sur. 18.2
B
X
rom A wertilis irgvliv misi mobruneba
SeiZlebodes. saaTis isris brunvis sapiris­
pirod abruneT AB veqtori
0
≈20-20 -iani kuTxiT da yoveli mobrunebis Semdeg B wertilidan dauSviT
marTobi X RerZze. RerZTan gadakveTis wertilebi aRniSneT, Sesabamisad,
B1, B2, B3-iT. daakvirdiT AB1, AB2, ... monakveTebs.
rogor icvleba X RerZze B wertilis koordinatis ricxviTi mniSvneloba
da niSani veqtoris mobrunebisas 0-dan 900-mde, 900-dan 1800-mde, 1800-dan
2700-mde, 2700-dan 3600-mde? dakvirvebis Sedegebi SeitaneT samuSao rveulSi
cxrilSi. gamoitaneT daskvna: A wertilis irgvliv 360 0-iani kuTxiT
veqtoris Semobrunebisas rogor icvleba misi bolos koordinatis ricxviTi
mniSvneloba da niSani?
18
1.6
veqtoris gegmili
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1. ra informacias iRebT 19.1
suraTidan?
Y
y
Sy
y0
B
βα
A
gadaadgilebis veqtoris gegmili RerZze
ewo­deba am RerZze veqtoris bolo da saw­
yisi wer­ti­­lebis koor­dinatebis sxvaobas.
x0
Sx
li X
gadaadgilebis S veqtoris gegmi­
sur. 19.1
Zze _ Sy-iT
RerZze aRvniSnoT Sx-iT, Y Rer­
(sur. 19.1).
gadaadgilebis veqtoris gegmilis ganmartebis Tanaxmad,
Sx=x–x0, ∆ABC-dan
(1)
C
x
X
Sy=y–y0. (2)
(sur. 19.1)
Sx=Scosα, (3)
Sy=Scosβ=Ssinα.(4)
(3) da (4) formulebidan gamomdi­
nareobs, rom
veqtoris gegmili RerZze tolia veqtoris modulisa da im kuTxis
kosinusis namravlis, romelsac veq­
tori qmnis RerZis dadebiT mimar­
TulebasTan.
veqtoris gegmilis es ganmarteba marTebulia yvela saxis veqtori­saTvis.
(3) da (4) formulebidan gamomdinareobs: imis mixedviT, Tu ra kuTxes
qmnis veq­
tori RerZis dadebiT mimarTule­
basTan, mi­
si gegmili SeiZleba
iyos an dadebiTi, an uaryofiTi, an nulis toli.
piTagoras Teoremis Tanaxmad, ∆ABC-dan SegviZlia davweroT
S = S 2x + S 2y . (5)
veqtoris moduli tolia kvad­ratuli fesvisa or urTierTmar­Tobul
RerZze veqtoris gegmilebis kvadr­
atebis jamidan.
S
(6)
tga = y .
Sx
veqtoris mier X RerZTan Sed­genili kuTxis tangensi tolia Y da X
RerZebze veqtoris gegmilebis Se­
far­
debisa.
imave samkuTxedidan
2. saSinao cdaSi miRebuli SedegebiT da cxrilis monacemebis mixedviT samuSao rveulSi XOY koordinatTa sistemaSi
daxazeT veqtorebi.
veqtori
S1
S2
gegmili
X RerZze
dadebiTi
gegmili Y
RerZze
dadebiTi
dadebiTi
uaryofiTi
veqtori
S3
S4
gegmili
X RerZze
uaryofiTi
gegmili
Y RerZze
uaryofiTi
uaryofiTi
dadebiTi
19
Y
20.1 suraTze gamosaxulia S1, S2 veqtorebi, maTi jami S da am veqtorebis gegmilebi
OX da OY RerZebze, Sesabamisad, S1x, S2x,
Sx, S1y, S2y da Sy.
C
S2y
S1y
Sy
suraTis gamoyenebiT SeiZleba davweroT
S = S1+ S2 . (7)
S1x=x2–x1. (8)
S2x=x3–x2. (9)
Sx=x3–x1. (10).
(10)∧(11) ⇒ Sx=S1x+S2x.
y2
y1
O
B
S1
S
A
x1
x2
S2x
S1x
x3 X
Sx
sur. 20.1
(8)∧(9) SekrebiT miiReba
S1x+S2x=x3–x1. (11)
S2
y3
(*)
3. daamtkiceT, rom Sy=S1y+S2y (**)
(*) da (**) formulebidan SeiZleba gamovitanoT daskvna:
nebismier RerZze veqtorebis ja­mis gegmili tolia imave RerZze Se­
sakrebi veqtorebis gegmilTa algeb­
ruli jamisa.
radgan veqtorebis gamokleba SeiZleba dayvanil iqnes Sekrebaze, amitom
es wesi veqtorTa sxvaobis gegmilebisTvisac marTebulia.
amgvarad, imisaTvis, rom ganvsazRvroT veqtorebis jamis an sxvaobis
gegmili, aucilebeli ar aris jamuri veqtoris povna da Semdeg misi gegmilis gansazRvra. sakmarisia, SevkriboT yvela veqtoris gegmili.
veqtoris skalarze namravlis gegmili veqtoris gegmilisa da ska­
laris namravlis tolia.
Tu cnobilia sxeulis sawyisi koordinatebi A(x0;y0) da gadaadgilebis
veqtori (sur. 19.1), maSin (1) da (2) formulebidan gamomdinareobs, rom
sxeulis saboloo mdebareobis koordinatebi iqneba
x=x0+Sx (12) da y=y0+Sy, (13)
(12)∧(13)⇒ meqanikis ZiriTadi amocanis gadasawyvetad sakmarisia sxe­
ulis sawyisi koordinatebisa da gadaadgilebis gegmilebis codna.
veqtoris gegmili
amgvarad, rom ganvsazRvroT sxeulis mdebareoba drois nebismier
momentSi saWiroa, vicodeT sxeulis sawyisi mdebareoba da drois am mo­
mentisaTvis Sesrulebuli gadaadgilebis gegmilebi: x=x0+Sx, y=y0+Sy.
20
1.7
amocanis amoxsnis nimuSi
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1. sxeuli A wertilidan mivida B-Si, Semdeg ki _ C-Si (sur. 21.1-21.4).
AB, BC da AC gadaadgilebebi aRniSnulia, Sesabamisad, S1 , S2 da S3 -iT.
ganvsazRvroT TiToeul SemTxvevaSi gadaadgilebis sawyisi da bolo
wertilebis koor­dinatebi, gegmili, moduli da mimarTuleba.
Y,
m
Y,
A
1 2 3
Y,
B
10 X,
7C
C
B
-4 -3 -2 -1
m
sur. 21.1
Y,
m
4
3
2
1
C
1 2 3 4 5 6 7 8 9 10111213
A
1 23 4
X,
-4
-5
m
7 X,
m
X,
m
sur. 21.2
m
B
A
10
-1 1 2 3 4 5 6 7 8 9 11
-2
B
A
m
C
sur. 21.4
sur. 21.3
amoxsna
suraTi veqtori
21.1
21.2
21.3
21.4
sawyisi koordinatebi
saboloo
gegmili
koordinatebi X RerZze
gegmili
Y RerZze
moduli
mimarTuleba
3;0
10;0
7
0
7
aRmosavleTiT
10;0
7;0
-3
0
3
dasavleTiT
3;0
3;0
7;0
-4;0
4
-7
0
0
4
7
aRmosavleTiT
dasavleTiT
-4;0
-1;0
3
0
3
aRmosavleTiT
0
4
3;0
-1;0
3;0
10;0
-4
7
0
7
dasavleTiT
aRmosavleTiT
10;0
13;4
3
4
5
aRmosavleTTan 530-iani kuTxiT
3;0
13;4
10
4
≈10,8
aRmosavleTTan 220-iani kuTxiT
3;0
10;0
7
0
7
aRmosavleTiT
5
aRmosavleTTan 2330-iani kuTxiT
≈5,7
aRmosavleTTan 3150-iani kuTxiT
10;0
7;-4
-3
-4
3;0
7;-4
4
-4
2. sxeulis gadaadgileba X RerZTan qmnis 300-ian kuTxes da moduliT
10 m-ia. gansazRvreT gadaadgilebis gegmilebi X da Y RerZebze
amoxsna
? Sx, Sy
moc:. a=30
S=10 m
Sx=Scosa. (*) Sy=Ssina. (**)
0
(*): Sx≈8,7 m.
(**) Sy=5 m.
21
3. A(-3;5) wertilidan sxeuli gadaadgilda
B(2;-2) wertilSi. gaavleT sxeulis gadaad­
gileba, gansazRvreT igi da misi gegmilebi X
da Y RerZebze.
amoxsna
sxeulis gadaadgileba gamo­
? S , Sx, Sy
saxulia 22.1 suraTze.
moc:. A(-3, 5)
AB / S gadaadgilebis geg­
B(2, -2)
milebi:
Sx=x2–x1=2–(-3)=5 (m)
Sy=y2–y1=-2–5=-7 (m).
S = S 2x + S 2y (*)
(*): S=7,8 m. a = arctg
Sy
. (**)
Sx
sur. 22.1
(**): a=3100.
4. mgzavrma 4 km gaiara jer aRmosavleTiT,
Semdeg _ CrdiloeTis mimarTulebiT, Semdeg ki
_ 1 km dasavleTiT. gansazRvreT gadaadgileba
jer geometriuli agebiT, Semdeg ki analizurad.
S3
S
S2
amoxsna
S
moc:. S1=4 km
S2=4 km
S3=1 km
S1
a
SevarCioT masStabi: 10 mm–1 km.
vTqvaT, mgzavrma moZraoba daiw­
sur. 22.2
yo O wertilidan. mgzavris
gadaadgilebebi aRmosavleTiT, CrdiloeTiT, dasavleTiT
aRv­niSnoT, Sesabamisad, S1 , S2 da S 3 -iT da gamovsaxoT 22.2
suraTze. imave suraTze gamosaxulia S = S1+ S2+ S3 (1)
rezultaturi S gadaadgileba.
saxazaviT SerCeuli masStabiT gavzomoT S -is sigrZe, transportiriT _ a.
miviRebT: S=5 km, a=520.
amocanis amoxsnis meore xerxi:
mgzavris gadaadgileba S = S1+ S2+ S3 . (1)
mgzavris gadaadgilebis modulia S =
RerZTan qmnis a kuTxes _ tga =
(1)⇒ Sx=S1x+S2x+S3x. (4)
(6)
22
S 2x + S 2y (2) mimarTuleba ki X
Sy
. (3)
Sx
_
S1x= S1, b
b
S
0
,
=
‘
Sy=S1y+S2y+S3y. (5) 2x
S3x=- S3.b
a
(4)⇒ Sx=S1–S3. (8)(7)
(5)⇒ Sy=S2. (9)
2
(8)∧(9)
(2)⇒ S = _ S1- S3i + S22 (*)
(8)∧(9)
(3)⇒ tga =
(*): S=5 km.
S2
S2
& a = arctg
. (**)
S1- S3
S1- S3
(**):a=520.
(6)
_
S1y= 0, b
b
S2y= S2,‘
S3y= 0. b
a
(7)
daamzadeT sawveTuriani urika
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1.8
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saSinao davaleba
daamzadeT 23.1 suraTze gamosa­
x uli
6
1
sawveTuriani uri­ka (igi SegiZliaT jgufma,
2
samma an oTxma erTad daamzadoT). urika
cdebisaTvis xSirad dagWirdebaT. poli­
eTi­­
lenis boTlze (1) da Tavze Semoxveulia mavTuli (2). mav­
Tu­
lis momrgvalebul
boloebSi (3) gayri­
lia RerZi (4). RerZze
3
4
mimag­r ebulia bor­b lebi (5), romleb­s ac
sur. 23.1
5
RerZTan erTad brunva SeuZliaT.
borblebad gamoiyeneT polieTi­
l­
enis boTlis erTmaneTTan Sewebebuli
ori Tavsaxuri. SeiZleba saTa­
maSo avtomobilis borblebis gamoye­
nebac.
Tavsaxuri gaxurebuli lursmniT gaxvriteT centrSi da RerZze mia­
magreT. SegiZliaT mavTu­
lis (2) bo­
lo­
ebi ar moamrgvaloT da gamo­
iyenoT
ro­
gorc RerZi. am SemTxvevaSi mas bor­
blebi ise gau­
keTeT, rom RerZze
Tavi­
suflad SeeZloT brunva.
borbalma mavTulze (RerZze) rom ar isrialos, borblis marjvniv da
marcxniv RerZze daaxvieT wvrili mavTuli an daawebeT webovana.
boTls misi fuZis paralelur xazze gaukeTeT 0,5 sm-iT daSo­
rebuli
sxva­­dasxva ganivkveTis wvrili ori an sami nasvreti. boTlSi wylis Casxmisas
maTgan wveTebi dro­
is sxvadasxva toli Sualedebis Semdeg Camovardeba.
ganieri xvre­
lidan wyali ufro mal-male Camo­
wveTavs.
umjobesia, sawveTurad gamoiyenoT daclili saweri pastis bolo _
liTonis an plastmasis wvrili `mili~. ga­xu­rebuli lursmniT polieTilenis boT­­­­
li advilad ixvriteba. mili naxvretSi CasviT da napirebi plastiliniT amoavseT. wve­
Tebis sixSiris regu­
li­
reba Semdegnairad SeiZ­
leba:
milis Tavi plas­
ti­
liniT dafareT da Sem­
deg nemsiT gaukeTeT sasurveli
sididis nasvreti.
boTls zeda nawilSi aucileblad gaukeTeT xvreli (6). masSi SeiZleba
pa­
tara mili CaamagroT (sur. 23.1).
boTli SeiZleba saTamaSo avtomobilzec daamagroT.
ratom aris aucilebeli xvrelis gakeTeba?
dafiqrdiT, iqneb, axla Tu ara, momavalSi sawveTuriani uri­kis ukeTesi
konstruqcia Semog­vTa­­vazoT da daamzadoT.
23
1.9
Tanabari da araTanabari moZraoba
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
ratom
ar moZraobs erTnairad met­ros
eskalatoris sxvadasxva nawili?
moZravi ori avtomobilidan erTis spidometris
(sur. 24.1) Cveneba ar icvleba, meorisa _ icvleba.
erTmaneTisagan rom ganasxvaoT, ras ityviT, rogor
moZraobs TiToeuli avtomobili?
am SekiTxvaze pasuxis gasacemad CavataroT cda:
sur. 24.1
sawveTurian urikaSi naxevramde CavasxaT wyali. davakvirdeT nasvretidan vardnil wveTebs. isini drois
spidometri _
toli Sualedebis Semdeg vardebian.
siCqaris gasazomi
avawyoT 24.2 su­
xelsawyo
raTze gamosa­xu­
li danadgari.
WiqaSi CavyaroT qviSa iseTi raodenobiT, rom odnavi biZ­
giT urika
amoZravdes. SevecadoT mivaRwioT, rom
sur. 24.2
nawveTurebs Soris manZilebi erTmaneTis toli iyos.
1. ivaraudeT, rogor Seicvleba nawveTurebs Soris
manZili WiqaSi qviSis CamatebiT? varaudis marTebuloba
SeamowmeT cdiT.
pirvel cdaSi drois tol SualedebSi nawveTurebs Soris manZili ar icvleba, meore cdaSi ki izrdeba. urikebis moZ­raoba erTmaneTisagan gansxvavdeba.
2. dafiqrdiT da SearCieT sityvebi, romliTac Catarebul cdebSi, qviSis Camatebamde da Camatebis Semdeg, urikis
moZ­
raobebs erTmaneTisgan ganasxvavebT.
24
traeqtoriis formisagan damouki­deblad, moZ­raoba SeiZleba iyos Tana­
bari da araTanabari.
avtomobili, romlis spidometris Cveneba ar icvleba, moZraobs Tanabrad
moduliT mudmivi siCqariT. xolo avtomobili, romlis spidometris Cveneba
icvleba _ araTanabrad.
Tanabaria moZraoba, rodesac sxe­
uli drois nebismier tol Su­
a­
ledebSi erTnair gadaadgilebas asrulebs. e. i. Tanabari moZraobisas
ar icvleba sxeulis moZraobis mimarTuleba da igi drois nebismier
tol SualedebSi tol manZils gadis.
moZraobas, romlis drosac sxe­
uli drois tol SualedebSi sxva­
dasxva gadaadgilebas asrulebs, ara­
Tanabari moZraoba ewodeba.
Tanabari moZ­
raoba umartivesi saxis moZraobaa.
imis gasarkvevad, ra rols asrulebs sityva nebismieri Tanabari moZra­
obis ganmartebaSi, upasuxeT SekiTxvebs:
3. Tanabaria Tu ara sxeulis moZraoba 1, Tu igi yovel
wuTSi 60 m-s gadis? am sxeuls, pirobiTad, pirveli vuwodoT.
varaudi CawereT samuSao rveulSi da daasabuTeT.
4. meore sxeuli yovel wuTSi 60 m-s gadis ise, rom pirvel
naxevar wuTSi gadis, vTqvaT, 20 m-s, meore naxevar wuTSi ki _ 40 m-s.
Tanabaria Tu ara misi moZraoba? varaudi CawereT samuSao rveulSi
da daasabuTeT.
waikiTxeT Tqven mier dawerili pirveli amocanis pasuxi da dafiqrdiT, xom ar SeicvaleT Sexeduleba? Tanabaria Tu ara pirveli sxeulis moZraoba? axseniT, ratom fiqrobT ase?
xom SeiZleba isic ise moZraobs, rogorc meore?
5. Tanabaria Tu ara sxeulis moZraoba, Tu is yovel naxevar wuTSi
tol manZilebs, 30-30 m-s, gadis? swori pasuxis gacemaSi meore amocanis gaazreba da dafiqreba dagexmarebaT. varaudi rveulSi CawereT
da daasabuTeT.
SeiZleba davaskvnaT: arc mesame da arc
mexuTe amocanis piroba ar iZleva imis saSualebas, rom kiTxvaze calsaxa pasuxi gavceT.
ganmartebis Tanaxmad, wrfivi moZroba
Tanabaria mxolod da mxolod maSin, rodesac
sxeuli drois nebismier tol SualedSi tol
manZils gadis. e.i. toli unda iyos yovel
wamSi, naxevar wamSi, meoTxed wamSi, meased
wamSi da a.S. gavlili manZilebi.
absoluturad Tanabari moZraoba ar arsebobs, TanabarTan miaxloebulia met­
r os
eskalatorze kibis Sua nawilis moZraoba,
uqaro amindSi daSvebisas paraSutistis (sur.
25.1) da wvimis wveTebis moZraoba, vakeze
mdinaris dineba (sur. 25.2) da sxv. yvela
saxis transporti ki moZ­
raobs araTanabrad.
sur. 25.1
sur. 25.2
6. rogor moZraobs TiToeuli avtomobili (sur. 25.3), Ta­
nabrad Tu araTanabrad?
a
sur. 25.3
b
1. CaTvaleT, rom ganxilul amocanebSi sxeuli moZraobs wrfivad, moZraobis mimarTulebis
Seucvlelad
25
7. Tanabaria Tu ara:
a. wylis moZraoba ReleSi (sur. 26.1), romlis
kalapoti xan viwrovdeba,
xan ki farTovdeba;
b. avtomobilis moZraoba mas Semdeg, rac mZRolma SuqniSanze wiTeli
feri SeniSna.
pasuxi daasabuTeT.
sur. 3.17
sur. 26.1
 8. sawveTuriani urikisaTvis SeafaseT drois Sualedi erTi wveTis
mowyvetidan momdevno wveTis mowyvetamde. varaudi Cawe­
reT rveulSi.
Tanabari, araTanabari, sawveTuriani urika,
spidometri
amgvarad, Tu
gadaadgilebebs
gadaadgilebebs
araTanabaria
wiria.
sxeuli drois nebismier tol SualedebSi erTnair
asrulebs, maSin misi moZraoba Tanabaria, Tu aratol
_ araTanabari.
im sxeulis moZraobac, romlis traeqtoria mrudi
moZraobas, romlis drosac sxeuli drois nebismier
tol SualedebSi erTnair gadaadgilebebs asrulebs, Tana­
bari moZraoba ewodeba.
am winadadebidan gamomdinareobs:
rodesac sxeuli moZraobs wrewirze ise, rom drois nebismier tol
SualedebSi toli sigrZis rkalebs Semowers, maSin
a. sxeuli moZraobs Tanabrad; b. sxeuli moZraobs araTanabrad;
g. SeuZlebelia davadginoT, rogor moZraobs sxeuli Tanabrad Tu
araTanabrad. d. zogjer Tanabrad moZraobs, zogjer araTanabrad.
saSinao cda
cdis mizani: drois Sualedis da sixSiris gansazRvra.
1. moifiqreT, rogor gansazRvravT sawveTuriani urikisaTvis erTi
wveTis mowyvetidan momdevno wve­
Tis mowyvetis momentebs Soris drois
Sualeds. gansazRvreT, CawereT da Sedegi SeadareT navaraudevs.
2. moaxdens Tu ara gav­lenas boTlSi wylis moculobis gazrda wve­Tebis
vardnis sixSireze? rogor?
hipoTezis marTebuloba SeamowmeT cdiT da axseniT SemCneuli movlena.
26
siCqare. siCqaris erTeulebi.
meqanikis ZiriTadi amocanis gadawyveta
Tanabari moZraobisas
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1.10
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
ra moZraobs yvelaze swrafad?
mgzavrobisas gvain­
teresebs, sad viq­
nebiT garkveuli drois Semdeg an
ra droSi mivalT daniSnulebis adgilas. xSirad amas praqtikuli mniS­
vnelobac aqvs.
1. drois mocemul SualedSi fexiT met manZils gaivliT,
manqaniT Tu TviTmfrinaviT?
moZraobis siswrafis maxasiaTebelia drois
erTeulSi gavlili manZili. rac metia es
manZili, miT metia misi siswrafe _ siCqare.
2. Tanabrad moZravi sxeuli
100m-s 12,5wm-Si gadis. gansazRvreT sxe­
u lis mier 1wm-Si
gavlili manZili - siCqare.
Tu Tanabari moZraobisas sxeulis mier t
sur. 27.1. sat­ran­sporto saSu­
ale­be­­bidan TviT­mfri­navi
yvela­ze swrafia.
drois SualedSi gadaadgilebas aRvniSnavT S -iT, siCqares v -Ti, maSin
v = St . (1)
(1) formulidan gamomdinareobs, rom Tanabari moZraobis siCqare ewodeba
mudmiv sidides, romelic tolia drois nebismier SualedSi sxeulis
gadaadgilebis fardobisa am Sualedis mniSvnelobasTan.
siCqare veqtoruli sididea, mas gadaadgilebis mimarTuleba aqvs.
3. axseniT, ratom aqvs sxeulis siCqares gadaadgilebis
mimarTuleba.
SI-Si siCqaris erTeulis dasaxelebaa 1 m/wm.
4. siCqaris kidev romeli erTeulebi iciT?
5. 1 km/sT gamosaxeT 1 m/wm-T.
6. mtredi mifrinavs 15 m/wm siCqariT. gauswrebs Tu ara igi
80 km/sT siCqariT moZrav avtomobils? pasuxi daasabuTeT.
7.
sur. 27.2
27
8. rogor fiqrobT, aris Tu ara damokidebuli sxeulis
siCqare aTvlis sxeulis Ser­Cevaze? pasuxi magaliTiT daasa­
bu­TeT.
v = St (1)⇒
& S = vt, (2) sadac S aris v siCqariT moZravi sxeulis gadaadgileba
drois t SualedSi.
radgan veqtori garda mimarTulebisa ricxviTi mniSvnelobiTac xasiaTdeba, amitom gamoTvlebisas viyenebT formulebs, romlebSic Sedis ara
veqtorebi, aramed maTi gegmilebi.
veqtoruli tolobis gegmilebiT Casawerad saWiroa avirCioT RerZebi.
Tanabari moZraobisas traeqtoria wrfea. amitom umjobesia, koordinatTa
X RerZi mivmarToT am wrfis gaswvriv, maSin sxeulis moZraobisas icvleba
mxolod erTi _ x koordinati. swored am RerZis gaswvriv aris mimarTuli
sxeulis siCqare da gadaadgileba. radgan S da v t tolia, amitom tolia
maTi gegmilebic, e. i.
Sx=vxt. (3)
(3)
x=x0+Sx formulaSi, miviRebT x=x0+vxt. (4)
(4) ⇒ vx =
x - x0 t .
(5)
(5) ⇒ koordinatTa RerZze siCqaris gegmili ricxobrivad drois
erTeulSi sxeulis koordinatis cvlilebis tolia an sxeulis siCqaris
gegmili aris misi koordinatis cvlilebis siswrafe.
(4) formuliT mocemulia Tanabari moZraobisas nivTieri wertilis koordinatis damokidebuleba droze.
Tanabari moZraobisas Tu viciT sxeulis sawyisi koordinati da siCqaris
gegmili, maSin, me-4 formulis meSveobiT, drois nebismier momentSi ganvsazRvravT sxeulis koordinats _ amovxsniT meqanikis ZiriTad amocanas.
9. 28.1 suraTze gamosaxulia koWi,
romelzedac daxveulia Zafi. Sea­
faseT, ra manZilze unda gadaaadgi­
loT Zafis bolo, rom koWma srialis
R
gareSe erTi bruni gaakeTos? koWis `borblis~ wrewiris sigrZea l, Zafis erTi xvias
A
sur. 28.1
sigrZe ki _ l1.
varaudi ganamtkiceT msjelobiT da CawereT rveulSi.
siCqare
amgvarad, sxeulis siCqaris gegmili misi koordinatis cvlilebis sis­
wrafea. Tanabari moZraobisas sxeulis siCqare mudmivi sididea. sxeulis
koordinati drois nebismier momentSi ganisazRvreba x=x0+vxt formuliT
me-9 SekiTxvis mixedviT CaatareT cda. gazomeT koWis
`borblis~ wrewirisa da Zafis erT-erTi xvias sigrZe.
gaswieT Zafis bolo ise, rom koWma srialis gareSe erTi
bruni gaakeTos. gazomeT Zafis bolos mier gavlili manZili da SeadareT
navaraudevs. axseniT cdis Sedegi.
saSinao cda
28
1.11
amocanis amoxsnis nimuSi
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1. wrfiv avtomagistralze Tanabrad moZraobs avtomobili
v=10 m/wm siCqariT (sur. 29.1). davweroT avtomobilis koordinatis droze
damokidebulebis gantoleba dedamiwasTan dakavSrebul koordinatTa
sistemaSi.
erT SemTxvevaSi koordi­
natTa saTaved miviRoT wer­
sur. 29.1
ti­li, sadac imyofeboda avto­
mo­
bili drois aTvlis dasaw­
yisSi da RerZis dadebiT mimarTulebad
avirCioT: a. moZraobis mimarTuleba; b. moZraobis sapirispiro
mimarTuleba.
meore SemTxvevaSi koordinatTa saTaved avirCioT drois aTvlis
dasawyisSi avtomobilis mdebareobidan marjvniv, 500 m-iT daSorebuli
wertili (sur. 29.1).
RerZis dadebiT mimarTulebad avirCioT: a. moZraobis mimarTuleba;
b. moZraobis sapirispiro mimarTuleba.
amoxsna
avtomobilis koordinatis droze damokidebulebis x=x0+vxt formulas
aqvs Semdegi saxe:
erT SemTxvevaSi
a. x=10t (m); b. x=–10t (m). meore SemTxvevaSi a. x=-500+10t (m);
b. x=500–10t (m).
2. motocikli da velosipedi moZraoben Semxvedri mimarTulebiT,
Sesabamisad, 10 da 5 m/wm siCqariT. dakvirvebis sawyisi momentisaTvis
maT Soris manZilia 450 m. CaTvaleT, X RerZi mimarTulia motociklis
moZraobis mimarTulebiT da misi mdebareoba emTxveva t=0 momentSi
aTvlis saTaves (sur. 29.2); davweroT motociklisa da velosipedis
x=x(t) gantolebebi; ramden xanSi da sad Sexvdebian isini erTmaneTs?
amoxsna
? x1=x1(t), x2=x2(t), t, l1.
moc:. v1=10 m/wm
v2=5 m/wm
l=450 m
0
l1
450
X
l
sur. 29.2
motociklis da velosipedis koordinatis droze damokidebulebis
x=x0+vxt (1) gantolebebs, Sesabamisad, eqnebaT Semdegi saxe:
x1=10t (m). (2)
x2=450–5t (m). (3)
Sexvedris t momentSi: xl1= xl2 (4)
(4)⇒ 10t=450–5t ⇒ t=30 wm. (5)
(2)∧(3)
(5)
(2) an (3)⇒: x′1=l1=300 m.
motociklis da velosipedis Sexvedra moxda X RerZis Tanxvdenili
mimarTulebiT sawyisi punqtidan 300 m-is daSorebiT.
29
3. wrfiv avtomagistralze avtobusi da motocikli moZraoben
Tanabrad marcxnidan marjvniv, Sesabamisad, 15 da 10 m/wm siCqariT,
msubuqi avtomobili ki _marjvnidan marcxniv 20 m/wm siCqariT (sur.
30.1).
xe
v1
-600
v2
-200
0
sur. 30.1
1200
X, m
ekipaJTa sawyisi mdebareobis koordinatebi gamosaxulia suraTze.
davweroT TiToe­ulis koordinatis droze damokidebulebis gantoleba.
ganvsazRvroT:
a. msubuqi avtomobilis koordinati me-10 wm-is bolos;
b. 15 wm-Si mo­
tociklis mier gavlili manZili;
g. drois momenti, rodesac avtobusi daeweva motocikls;
d. drois momenti, rodesac msubuqma avto­
mobilma Cauqrola xes;
e. manZili motociklsa da msubuq avtomobils Soris drois am
momentSi.
amoxsna
? x1=x1(t); x2=x2(t); x3=x3(t)
x3′; l1; l2; l3; l4.
moc.: v1=15 m/wm
30
v2=10 m/wm
v3=20 m/wm
x01=-600 m
x02=-200 m
x03=1200 m
t1=10 wm
t2=15 wm
koordinatTa sistema davukavSi­
roT deda­
miwas. koordinatis aTvla da­
v iwyoT xidan.
RerZis dadebiT mimar­
T ulebad miviRoT
mimarTu­leba marc­xnidan mar­jvniv. avtobusis,
moto­ciklis, msu­buqi avto­mobilis koordi­natis
dro­ze damoki­debu­lebis x=x0+vxt ganto­lebebs,
Sesabamisad, eqnebaT Semdegi saxe:
x1= – 600+15t (m). (1) x2= – 200+10t (m). (2)
x3=1200–20t (m). (3)
a. CavsvaT t 1 =10 wm (3)-Si ⇒ x′ 3 =1200 –
200=1000 (m). msubuqi avto­
mobilis koor­
dinati
t1=10 wm momentSi x′3=1000 m-ia.
b. CavsvaT t2=15 wm (2)-Si. vipoviT motociklis koordinats am momentSi
x′2= – 200+150= – 50 (m). radgan moto­ciklis moZraobis mimarTuleba ar icvleba,
amitom mis mier gavlili manZili misi koordinatis cvlilebis modulis
tolia: l1=|x′2–x02|=| –50–( – 200)|=150 (m).
g. im momentSi, rodesac avtobusi motocikls daeweva, maTi koordinatebi
tolia, e. i. x1′=x2′. (4)
(1) (2) (4) ⇒ –600+15t3= –200+10t3 ⇒ t3=80 wm.
avtobusi motocikls 80 wm-Si daeweva.
d. rodesac msubuqi avtomobili xes Cauqrolebs, misi koordinati nuli
gaxdeba.
(3) ⇒ 0=1200–20t4 ⇒ t4=60 wm. (5)
e. t4=60 wm momentSi ganvsazRvroT motociklis koordinati, am momentSi
msubuqi avtomobilis koordinati nulia. (5) (2) ⇒ x2= –200+10.60=400 m.
manZili msubuq avtomobilsa da motocikls Soris l2=|400–0|=400 (m).
moZraobis gamosaxva grafikulad
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1.12
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1. ra sidideebi unda vicodeT, rom Tanabari moZraobisas
gadavwyvitoT meqanikis ZiriTadi amocana?
moZraoba SeiZleba aRvweroT grafikebis meSveobiT.
vTqvaT, avtomobili moZraobs gzatkecilze X RerZis gaswvriv (sur. 31.1)
suraTze gamosaxul mdebareobebSi igi imyofeba saaTze gamosaxul drois
momentebSi.
sur. 31.1
2. sad imyofeba avtomobili 1215 wuTze?
3. gansazRvreT drois momenti, rodesac avtomobilis
koordinatia 800 m.
horizontalur (abscisTa) RerZze arCeuli
masStabiT gadavzomoT drois aTvlis dawyebidan
8
gasuli drois Sualedi, xolo vertikalur
4
(ordinatTa) RerZze _ sxeulis koordinatis
mniSvneloba (agreTve, saTanado masS­
t abiT).
miviRebT tOX koordinatTa sistemas.
sur. 31.2
avtomobilis koordinatis droze damokidebu­
lebis grafikis asagebad x-is mniSvnelobebi gadavzomoT ordinatTa RerZze,
xolo drois _ abscisTa RerZze. x-isa da t-s Sesabamisi mniSvne­
lobebi
avagoT tOX sibrtyeze. am wertilebis SeerTebiT miviRebT grafiks, romelic
gamosaxavs avtomobilis koordinatis damokidebulebas droze (sur. 31.2).
moZraobis grafiki aris wrfe, es ki niSnavs, rom avtomobilis koordinati
wrfivadaa damokidebuli droze. asec unda yofiliyo.
TanmimdevrobiT upasuxeT qvemoT mocemul SekiTxvebs da amaSi Tavad
darwmundebiT.
4. romeli formuliT gamoisaxeba or, x da y cvlads Soris
wrfivi damokidebuleba?
5. Tanabari moZraobisas rogori damokidebulebaa sxeulis
koordinatsa da drois Sualeds Soris?
6. SeadareT y=kx+b (1) da x=x0+vxt (2) formulebi erTmaneTs.
7. ra asrulebs k-s, b-s, x-sa da y-is rols Tanabari
moZraobisas?
31
32.1 suraTze gamosaxulia ori sxe­
ulis
koordinatis droze damoki­d e­b ulebis
grafikebi. vTqvaT, sxeulebi moZraoben 32.2
suraTze gamosaxul wrfeze. grafikidan vadgenT: x01 da x02 pirveli da meore sxeulebis
saw­yisi koordinatebia _ koordinatebi t0=0
t4
momentSi. am mo­
men­
tSi sxeulebi moZraobis
traeq­to­riebis, vTqvaT, A da B wertilebSi
imyo­febian (sur. 32.2). grafikidan vadgenT,
pirveli sxeulis koordinati izrdeba,
meorisa _ mcir­
deba, e. i. sxeulebi moZraoben urTi­er­T­sapirispiro mi­mar­Tu­lebiT.
pirveli _ arCeuli Re­
rZis dadebiTi mimarTulebiT (misi siCqaris gegmili dadebiTia,
v1x>0), meore _ sapirispiro mimarTulebiT (misi
siCqaris gegmili uaryofiTia, v2x<0). grafikebis
gadakve­Tis C wer­
tilSi (sur. 32.1) sxe­
ulebis
ko­ordi­natebi erTi da igivea. C wertilis abscisa
Sex­vedris moments (t1) gviC­venebs, ordinati _ Sexvedris adgilis ko­ordinats (x1′).
traeqto­riaze D wer­tili maTi Sex­vedris adgi­
lia (sur. 32.2). 32.1 suraTidan Cans, rom me­
ore
sxe­uli dro­is t2 momentSi aTvlis saTa­
veSia _
koordinati nulis tolia. amis Semdeg misi koor­
dinati uaryofiTi xdeba.
X
x02
I
C
x1′
x01
O
II
t3
t2
t1
t
x2′
sur. 32.1
O
A
x01
D
x′1
B
x02 X
sur. 32.2
vx
v1x
I
t1 t2
t
v2x
II
sur. 32.3
8. 32.2 suraTze miuTiTeT, sad imyofeba meore sxeuli
drois t2 momentSi.
9. ra gansxvavebaa sxeulis koordinatis droze damokidebu­
lebis grafiksa da moZraobis traeqtorias Soris?
sxeulis koordinatis droze da­
mo­
kideulebis grafiki (sur. 32.1) ar
unda agerioT sxeulis moZraobis traeq­toriaSi _ wrfeSi, romlis yvela
wertilsac sxeuli moZraobisas gadis (sur. 32.2).
sxeulis wrfivi moZraobisas ko­
or­
dinatis droze damokidebulebis
grafikis meSveobiT srulad amo­ix­sneba meqanikis ZiriTadi amocana, vi­naidan
SegviZlia sxeulis mdebareobis gansaz­Rvra drois nebismier momentSi. maT
Soris, drois aTvlis dawye­
bamdec (Tu sxeuli manamdec imave siCqariT
moZra­
obda). magaliTad, 32.1 suraTis mi­
xed­
viT vadgenT, rom t3 momentSi
pirveli sxe­
ulis koordinati emTxveva meore sxeulis sawyis koordinats
_ x02-s, xolo meoresi tolia x′2-is. drois aTvlis dawyebamde t4 mo­mentSi
pirveli sxeuli aTvlis saTa­veSi iyo.
koordinatis droze damokidebu­
le­
bis grafikiT SeiZleba vimsjeloT
sxeulis siCqaris Sesaxebac. rac ufro metia kuTxe grafiksa da droTa
RerZs Soris, miT metia siCqaris moduli.
meqanikuri moZraobis aRwerisaTvis Se­iZ­leba gamoviyenoT siCqaris geg­
milis droze damokidebulebis grafi­
kic. 32.3 suraTze gamosaxulia im
32
sxeulTa siCqaris gegmilis droze da­
moki­
de­
bulebis grafikebi, romelTa
koor­
di­
natis droze damokidebuleba gamosa­
xulia 32.1 suraTze. 32.3 su­
raTze, abscisTa RerZze, arCeuli mas­StabiT, gadazomilia dro, ordinatTa
RerZze (agreTve saTanado masStabiT) _ siC­qaris gegmili. radgan Tanabari
moZ­raobisas siCqaris gegmili ar ic­vleba, amitom grafikebi droTa RerZis
para­leluria.
siCqaris grafikiT SesaZlebelia drois mocemul SualedSi sxeulis
gadaadgilebis gegmilis povnac. ricxvebi, romlebiTac gadaadgilebis mo­
duli da 32.3 suraTze daStri­
xuli marTkuTxedis farTobi gamoisa­
xeba,
erTmaneTis tolia. marTlac, marT­kuT­xedis farTobi misi ori mosaz­Rvre
gverdis sigrZeTa namravlis to­
lia. marTkuTxedis erTi gverdis sigrZe,
masStabiT, t drois Sualedis to­lia, meore ki _ vx siCqaris gegmilisa. maTi
namravli vxt=Sx swored gada­adgilebis gegmilis tolia. Tu sxeuli moZraobs
RerZis dadebiTi mimarTu­lebiT, misi gadaadgilebis gegmili ric­xobrivad
zeda marTkuTxedis fa­rTobis tolia, xolo Tu sapiris­pirod moZraobs _
minus niSniT aRe­buli qveda marTkuT­xedis farTobisa.
33.1 suraTze gamosaxuli grafikis mixedviT davweroT sxeulis koor­
dinatis droze damokidebulebis gantoleba.
amoxsna
33.1 suraTze gamosaxuli x=x(t) damokide­bulebis grafiki wrfea. es imas
niSnavs, rom sxeuli moZ­raobs Tanabrad, romlis koordinati drois mixedviT icvleba kanoniT:
(1)
x=x0+vxt. X, m
davweroT (1) formula mocemuli sxeuli­saTvis
niSnavs, ganvsazRvroT x0, vx da CavsvaT (1)-Si.
2
33.1 suraTidan t0=0 momentSi x0=2 m. (2)
(1) ⇒
(3)
-4
-2 0
2
4
t, wm
sur. 33.1
vx-is sapovnelad umjobesia gamoviyenoT is ricxvebi, romlebic grafikis koordinatTa RerZebTan
gadakve­Tis wertilebs aweria. 33.1 suraTis mixedviT vadgenT: t = –4 wm momentidan t =0 wamamde sxeulis koordinati Seicvala 0-dan 2 m-mde. SevitanoT
es mniSvnelobebi (3)-Si. miviRebT:
(4)
(2)∧(4)
(1)⇒ x=2+0,5t (m).
pasuxi: sxeulis koordinatis droze damokidebulebis gantolebas aqvs
aseTi saxe: x=2+0,5t (m).
saSinao cda
Zafis erT bolos gamoabiT burTula. meore bolo daamagreT kedelze,
romelzedac gakrulia furceli. polieTilenis boTli Sveul mdebareobaSi
miadeT kedels, CG adeT masSi burTula ise, rom igi fskers exebodes da Zafi
daWimuli iyos. gadaaadgileT boTli horizontalurad iseT manZilze, rom
burTula boTlis TavTan mivides. daxazeT boTlis gadaadgileba kedlis
mimarT, aseve, burTulis gadaadgileba boTlisa da kedlis mimarT. SeecadeT daadginoT kavSiri am veqtorebs Soris.
33
1.13
moZraobis fardobiToba
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| saZiebo fraza k9t1p13
Y Y1
Y1
Y1
Y1
CavataroT wina paragrafis sa­
Sinao
cda: Zafze gamobmuli burTula polieTilenis boTlSi Cav­
kidoT. dafaze
dav­xazoT XOY koordinatTa sis­tema. OX
horizontalur RerZze yo­vel aT-aT santimetrze movniSnoT wer­tilebi A2, B2 da
X1
01
C2. mivadoT boTli dafas Sve­
ulad ise,
0
B2
C2
A2
rom misi fu­
Zis Sua wertili X RerZis
X
nulovani danayofis pirdapir iyos. Zafis
sur. 34.1
Tavisufali bo­
lo dafaze davamagroT
(sur. 34.1). boTli hori­z on­­t a­l urad
C C1
vamoZ­
raoT. yo­
veli 10 sm-is gavlis mo­
B
B1
men­
tSi boT­
lsa da dafaze werti­
lebiT
movniSnoT burTu­lis mdeba­reobebi. boTA
A1
lze A1, B1, C1-iT, xolo dafaze A, B, C-Ti.
es wer­
tilebi erTmaneTTan Se­
vaerToT
O1
ro­gorc boTlze, ise dafaze (sur. 34.2).
O
B2
A2
C2
ganvixiloT burTulis moZraoba ori
sur. 34.2
damkvirveblis, ori aTvlis sis­
t emis
mimarT. erTi _ dafasTan da­kav­Sirebuli
XOY sistema, vu­
wodoT mas `uZravi~ aTvlis sistema. meore _ `moZ­
ravi~
X1O1Y1 aTvlis sis­
tema _ davu­
kavSiroT boTls ise, rom O da O1 saTaveebi
erTmaneTs em­Txveodes, xolo X1 da Y1 RerZebi Sesabamisad X da Y RerZebis
para­leluri iyos. (sur. 34.1).
1. gansazRvreT boTlis gadaad­gileba uZravi aTvlis
sistemis _ dafis _ mimarT.
2. gansazRvreT burTulis gada­adgileba boTlisa da dafis
mi­marT.
34.2 suraTidan vadgenT S = S1+ S2. (1)
burTulis S gadaadgileba uZravi aTvlis sistemis (dafis) mimarT
tolia moZravi aTvlis sistemis (boTlis) mimarT misi S1 gadaadgilebisa
da uZravi aTvlis sistemis mimarT moZravi aTvlis sistemis S2
gadaadgilebis jamisa.
3. boTli dafis mimarT Ta­nabrad moZra­obs, iqneba Tu ara
Tanabari burTulis mo­Z­raoba boTlisa da dafis mimarT?
Tu (1) tolobis orive mxares gavyofT drois im Sualedze, romelSic
boTlma
gadaadgileba Seasrula, miviRebT:
(2)
34
sadac
aris burTulis siCqare dafis mimarT,
_ burTulis siCqare
_ boTlis siCqare dafis mimarT.
boTlis mimarT, xolo
me-(2)-dan SegviZlia davaskvnaT:
sxeulis (burTula) siCqare uZravi (dafa) aT­
vlis sistemis mimarT
tolia mis siCqares moZravi (boTli) aTvlis sistemis mimarT damate­
buli moZravi aTvlis sis­temis siCqare uZravis mimarT. me-2 formulas
siCqareTa Sekrebis kanoni ewodeba.
amgvarad, sxeulis gadaadgileba da siCqare sxvadasxva aTvlis sistemis
mimarT gansxvavebulia. gansxvavebulia sxeulis moZraobis traeqtoriebic
(C2C1 moZravi aTvlis sistemis mimarT da OC _ uZravi aTvlis sistemis
mimarT). swo­red amiT gamoixateba moZraobis fardo­biToba.
(1) da (2) formulebi mar­Tebulia boTlis nebismieri mimarTulebiT moZraobisas.
II cda. davabrunoT boTli sawyis mdebareobaSi. Sveul
mimarTulebaze, aT-aT santi­metrze, dafaze movniS­noT wer­ti­
lebi. avwioT boTli Sveu­
lad da yoveli 10 sm-is gavlis mo­
mentSi boT­
lsa da dafaze werti­
lebiT movniSnoT burTulis
mdeba­reoba. ga­mov­saxoT suraTze boTlis gadaad­gileba dafis
mimarT, burTu­lis gada­adgileba boT­lis mimarT da burTu­lis
gadaad­gi­leba dafis mi­marT. agebiT da gazom­vebiT Sevamow­moT
(1) formulis mar­Tebu­loba.
III cda. burTulaze gamobmuli Zafis bolo davamagroT boTlis Tavze da Zafi gadavdoT lursmanze (sur. 35.1). vamoZraoT
sur. 35.1
boTli Sve­ulad qveviT (gavzomoT boTlis gadaadgi­lebis mo­
duli dafis mimarT).
davxazoT gadaadgilebis veqto­re­bi: boTlisa dafis mimarT, burTu­li­sa
boTlis mimarT, burTulisa da­fis mi­marT. agebiT da gazomviT dava­dg
­ i­noT
(1) formu­lis marTebuloba.
bunebaSi arsebobs erTaderTi siC­
qare, romelic fardobiTi ar aris
(absoluturia). es aris siCqare, samasi mi­
lioni m/wm, romliTac eleq­
tromagnituri talRebi (sinaT­le, ra­diotalRebi, infra­wi­­Teli da ultra­
iis­feri sxi­vebi, rentge­nis sxivebi da sxva) vrceldeba vaku­umSi. sinaTlis
siCqa­ris ab­soluturoba mra­vali dakvirvebiTa da cdiT aris damtki­cebuli.
debuleba, rom sinaT­
lis siCqare absolutu­
ria da ara fardobiTi, sa­
fuZvelad udevs al­
bert ainStainis mier 1905 wels da­
muSavebul fardo­
biTobis Teorias. am Teoriidan mii­
Reba Za­
lian mniSvne­
lo­
vani Sedegebi.
magali­Tad is, rom manZili da dro ar aris absoluturi. isini aTvlis sis­
temaze damokidebuli far­dobiTi sidi­deebia. dro moZrav matarebelSi uf­ro
nela gadis, vidre uZravSi da miT ufro nela, rac ufro didia matareb­lis
siCqaris moduli. moZ­raoba iwvevs drois msvlelobis Senelebas.
moZrav matarebelSi or wertils Soris manZili moZraobis mimarTulebis
gaswvriv (magaliTad saxazavis sigrZis toli) naklebia uZrav mata­rebelSi
imave manZilTan Seda­rebiT.
Reros sigrZe udidesia im aTvlis sistemaSi, romlis mimarTac is uZravia.
fardobiTobis Teoriidan gamo­m­­di­­nareobs sxva, ufro zusti, wesi siCqareTa
Sekrebisa. magram es cvli­lebebi SesamCnevi xdeba im siCqaris dros, romelic
axlosaaa 3.108 m/wm siCqaresTan. siCqare ro­melTanac saqme gvaqvs garemomcel
samyaroSi mudam gacilebiT naklebia 3.108 m/wm-ze. amitom aseTi moZra­obi­
saTvis is cvli­lebebi, romelze­dac gakvriT visaubreT, SeumCnevelia.
35
1.14
amocanis amoxsnis nimuSebi
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1. motorianma navma mdinare unda gadacuros. mdinari­
s di­
nebis siCqa­r ea v1 , navis siCqare wylis mimarT v2 . gansazRvre­T
rogo­r i a kuTxiT unda mimarTon navis siC­q a­­­­­­r e napi­r i­­s mimarT,
rom mdinare gadacuros: a. minimalur drois SualedSi?
b. umok­­­­lesi manZiliT?
amoxsna
A
aTvlis uZravi XOY sistema davuavSiroT
v1
v2
napirs, koordinatis aTvlis saTaved miviRoT is
l
O wertili, saidanac navi iwyebs moZraobas, OX
v
RerZi mivmarToT mdinaris dinebis mimarTulebiT,
a b
OY RerZi napiris marTobulad (sur. 36.1).
napiris mimarT navis siCqare: v = v1+ v2. (1)
sur. 36.1
navis siCqaris gegmilebi X da Y RerZebze:
vx=v2cosa+v1;
(2)
vy=v2sina. (3)
navisTvis x=x0+vxt da y=y0+vyt gantolebebs eqnebaT saxe:
x=(v2cosa+v1)t, (4) y=(v2sina)t. (5)
vTqvaT, navi meore napirTan mivida drois t=t (6) momentSi, maSin y=l, (7)
sadac l aris mdinaris sigane.
(6)∧(7)
(5)⇒ l = ^v2sin ah x & x =
l . (8)
v2sin a
a. (8)⇒ mdinaris gadacurvisaTvis saWiro drois Sualedi t minimaluria,
roca sina=1 ⇒ a = r . (9) aqedan gamomdinareobs: mdinaris gadacurvis drois
2
Sualedi minimaluri rom iyos, navma kursi unda aiRos napiris marTobuli
mimarTulebiT.
b. navma mdinare rom gadacuros umoklesi manZiliT, e. i. O-dan A-sken,
saWiroa, ise imoZraos, rom misi x koordinati ar Seicvalos, maSasadame,
x=0. (10)
v
(10)
(4)⇒ (v2cosa+v1)t=0 ⇒ v2cosa+v1=0 ⇒ cosa =- v1 . (11)
2
aqedan gamomdinareobs: navis kursi iseTi unda iyos, rom Sesruldes
pirobebi a > r
da cos a = v1 .. amgvarad, mdinaris umoklesi gziT gada­
v2
curva SesaZlebelia, roca v2>v1 da b = r .
2
2
2. mocurave, romlis siCqaris moduli wylis mimarT 1,4 m/wm-ia, 126 m
siganis mdinareSi napiris marTobulad moZraobs wyalTan dakavSire­
bul aTvlis sistemaSi. dinebis siCqaris modulia 0,8 m/wm. ra droSi
gadacuravs mocurave mdinares? gansazRvreT mocuravis gadaadgi­
leba
da siCqare napiris mimarT. ra manZilze daacurebs mdinare mocuraves?
36
amoxsna
koordinatTa sistema davukavSiroT dedamiwas. aTvlis
saTaved avirCioT mocuravis mdebareoba drois aTvlis
dasawyisSi. X RerZi mivmarToT dinebis gaswvriv, Y RerZi
_ mis marTobulad (sur. 37.1).
mocuravis siCqare
(1)
(1)-is dagegmilebiT X da Y RerZebze miviRebT:
vx=v1x+v2x=v2, (2)
vy=v1y+v2y=v1. (3)
mocuravis Tanabari moZraobisas misi koordinatebi:
x=vxt. (4)
y=vyt. (5)
mdinaris gadacurvis momentSi y=l1. (6)
(6)
(5) ⇒ l1=v1t1 ⇒
(*)
(7)
mocuravis siCqaris moduli
(2)
(3)
Y
(7) ⇒
l1
v1
v2
(**)
mocuravis siCqaris mimarTulebas _ X RerZTan
Seqmnil α kuTxes _ ganvsazRvravT formuliT
X
sur. 37.1
(***)
mocuravis gadaadgilebis moduli
(****)
gadaadgilebis mimarTuleba emTxveva siCqaris mimarTulebas.
Tu (*)
curaves:
(2)
(4), miviRebT manZils, romliTac mdinare daacurebs mo-
(*****)
(*):
t1=90 wm.
(***):
tgα=1.75 ⇒ α≈610.
(**):
v=1,6 m/wm.
(****):
S=144 m.
(*****): l2=72 m.
3. ori avtomobili moZraobs urTierTmarTobulad. maTi siCqaris
modulebia, Sesabamisad, 3 da 4 m/wm (sur. 37.2, a). gansazRvreT meore av­
tomobilis siCqare pirvelis mimarT.
amoxsna
siCqareTa Sekrebis kanonis Tanaxmad: meore avtomobilis siCqare (v2),
`uZ­ravi~ aTvlis sistemis (de­da­miwis)
mimarT tolia mis siCqares (u) `moZ­
ravi~ aTvlis sistemis (pirveli avtomobili) mimarT damate­buli `moZ­
ravi~ aTvlis sistemis siCqare ( ) `uZravis~ mimarT, e. i.
a
(*)
maSasadame, meore avtomobilis siCqare pirvelis
mimarT tolia meorisa da pirvelis siCqareebis
sxvao­bisa (sur. 37.2, b).
b
α
sur. 37.2
37
piTagoras Teoremis Tanaxmad,
pasuxi: meore avtomobilis siCqaris moduli pirvelis mimarT 5 m/wm-ia
da
-Tan qmnis ≈370-ian kuTxes.
amocanis amoxsnis meore xerxi
aTvlis sistema davukavSiroT pirvel avtomobils, maSin igi `uZravia~,
xolo dedamiwa mis mimarT moZraobs _
siCqariT. meore avtomobilis
siCqare
uZravi aTvlis sitemis mimarT tolia mis siCqares moZravi aTvlis sistemis (dedamiwis) mimarT
-s damatebuli moZravi aTvlis sistemis
siCqare uZravis mimarT _
, e. i.
4. ori matarebeli moZraobs Semxvedri mimarTulebiT, Sesabamisad,
10 m/wm da 15 m/wm siCqariT. pirvel matarebelSi mjdomma mgzavrma
SeamCnia, rom meore matarebelma mas Cauara 6 wm-is ganmavlobaSi.
gansazRvreT meore matareblis sigrZe.
amoxsna
koordinatTa sistema davukav­
S iroT pirvel matarebels. aTvlis
? l
saTaved avirCioT O wertili, sadac
X
moc:. v1=10 m/wm imyofeba mgzavri. drois aTvla v1
O
v2=15 m/wm daviwyoT im momentidan, rodesac
v2
meore matareblis dasawyisi gaus­
t=6 wm
l
worda mgzavrs. X RerZi mivmarToT
sur. 38.1
meore matareblis moZraobis mimarTulebiT (sur. 38.1).
meore matareblis koordinatis droze damoki­
debulebis x=x0+uxt
(1)
formulaSi x0=–l. (2)
(3) aris meore matareblis siCqare.
X RerZze gegmilebiT (3) ⇒ ux=v2x-vx=v2–(-v1)=v2+v1. (4)
(2)∧(4)
(1)⇒ x=-l+(v2+v1)t. (5)
t=t momentSi, rodesac meore matareblis bolo Cauqrolebs mgzavrs
x=x′=0. (6)
(6)
(5)⇒ 0=–l+(v2+v1)t ⇒ l=(v2+v1)t.
(*)
(*): l=150 m.
ra pirobebi unda sruldebodes, rom sxeuli iyos uZravi an
moZraobdes Tanabrad? am SekiTxvebze pasuxis gacemas SevZlebT Semdegi
paragrafis ganxilvisas.
38
1.15
CaatareT, gamoikvlieT,
ivaraudeT, daadgineT
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cdebis mizani: sxeulis siCqaris cvlilebaze dakvirveba.
cdisTvis saWiroa
 horizontalu­ri
da daxrili zeda­
pirebi
 burTula
 qviSa an marili
CaatareT 39.1 suraTis Sesabamisi cdebi.
Tu yuradRebiT daakvirdebiT da ifiqrebT,
gamo­itanT daskvnas, rome­lic bunebis erT-erTi
ZiriTadi kanonia da mis marTebulobamde pirve­
lad galileo galilei mivida.
1. dadeT burTula ho­
ri­
zontalur zeda­
pirze
da daakvirdiT. SesaZleblad migaCniaT Tu ara,
burTulam Ta­
visiT Seicvalos siCqare?
2. dadeT burTula daxril sibrtyeze da
gaaTavisufleT. romel zedapirze ufro male
Cerdeba burTula xaoian (balaxian, qviSian,
marilian) Tu swor da gaprialebulze? ratom?
a
3. ra iwvevs burTulis siCqaris Semcirebas?
4. rogor imoZravebda burTula, Tu siCqaris
Semcirebis mizezi ar iarsebebda?
b
gamoitaneT daskvna.
d suraTze daxrili sibrtyidan dago­rebuli
burTula horizontalur ubanze mcire manZilis gavlis Semdeg meore daxril sib­
rtyeze
agrZelebs moZraobas. CaatareT Sesa­bamisi cda.
5. rogor icvleba pirvel daxril sibrtyeze
burTulis siCqare? meoreze?
6. horizontisadmi sibrtyis daxra aris Tu
ara burTulis siCqaris gadidebis da Semcirebis mizezi?
g
I
d
II
sur. 39.1
7. horizontalur ubanze burTulis moZraobisas arsebobs siCqaris
Secvlis aseTi mizezi?
8. Tu horizontalur ubanze burTulis moZraobas xels araferi SeuSlis,
Seicvleba Tu ara misi siCqare?
9. rogor gaagrZelebs burTula moZraobas, Tu horizontaluri zedapiri usasrulod didi iqneba?
varaudi da dakvirvebisas SemCneuli movlenebi dawvrilebiT CawereT
sa­
muSao rveulSi. gamoitaneT daskvna: rogor imoZravebs sxeuli, Tu
masze sxva sxeulebi ar imoqmedebs?
39
1.16
aristotele da galilei
MmoZraobis Sesaxeb
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
moZraobis
gamomwvevi mizezi ar arsebobs.
1. SesaZleblad migaCniaT, sxe­
uli sxva sxeulis moqmedebis gareSe amoZravdes?
_ `uZraoba sxeulis bunebrivi mdgo­­­
mareobaa~. moZraoba?
` ... sxe­
ulis moZra­
obisaTvis saWi­
roa masze sxva sxeuli moqmedebdes; ar
arse­
bobs moZraoba sxva sxeulis moqme­
debis gareSe~ _ aseTi iyo gamo­
Cenili moazrovnis aristote­
les Tvalsazrisi.
TiTqmis 2000 weli batonobda es mcdari azri. albaT, ase imitom moxda,
rom es mtkicebebi arcTu ise cudad Seesaba­
meba yoveldRiur cdebsa da
dakvir­
vebebs _ egreT wo­
debul saR azrs.
2. ra aris saWiro karadis, urikis, sazidris da sxva sxe­ulebis
erTi adgilidan meoreze gadasatanad?
arc moZravi sxe­
u­
lis siCqaris
Semci­
reba xdeba Ta­
visTavad. Seg­
viZlia warmovidginoT, ra Znelia,
Tundac gamor­
Tuli Zra­
vaTi, didi
siCqariT moZ­ravi manqanis gaCereba.
cxa­dia, es sxeulebi adre Tu gvian
Cveni Carevis gareSec gaCerdebian,
magram ara TavisTavad: gaCerebis
sur. 40 .1. dafrenisas Tanamedrove TviTmfri­
_ siCqaris Semcirebis mizezic
navebis gasaCe­reb­lad speci­alur paraSutebsac
sxva sxeulebis (hae­risa da Semxebi
ki iyeneben
zedapiris) moq­me­debaa _ winaaRmde­
goba, xaxuni.
daixsomeT! erTi sxeulis meoreze moqmedeba aris maTi siCqaris Secv­
lis da ara moZraobis gamomwvevi mizezi.
3. daasaxeleT romelime sxeuli, romelic teqstSi moyvanili
ar aris da romlis moqmedebiTac icvleba sxva sxeulis siCqare.
sidides, romliTac xasiaTdeba erTi sxeulis moqmedeba meoreze, Zala
ewodeba. Zalis moqmedebiT icvleba sxeulis siCqare.
Sin Catarebul cdebSi pirvel daxril sibrtyeze burTulis siCqaris
gadidebis da meoreze Semcirebis mizezi masze moqmedi de­
da­
miwis mizi­
dulobis Zala da zedapiris horizontisadmi daxraa. rac ufro naklebia
daxris kuTxe, miT ufro nela icvleba burTulis siCqare.
Tu horizontisadmi daxra nulis tolia, maSin absoluturad gluv
hori­zon­talur ubanze abso­lu­tu­rad gluvi burTulis siCqa­ris Sec­vlis
40
mizezi ar arsebobs, amitom man mu­
dmivi siCqariT
azrobrivi _
unda imoZ­raos _ gaagrZela az­robrivi eqsperimenti mocemul SemTxvevaSi
galileim.
fiqriT.
sxeulis siCqaris mimarTulebis Secvlac sxva
sxeulis, e.i. Zalis moqmedebiT xdeba.
dakvirvebebidan da cdebidan ga­mom­dinareobs daskvna: sxeulis siC­qaris
Secvla mxolod Zalis moqmedebiTaa SesaZ­
lebeli.
4. horizontalur zedapirze
gamorTuli ZravaTi moZravi avtomobilis siCqaris moduli ganuwyvetliv mcirdeba srul gaCe­
rebamde (sur. 41.1). axseniT, ratom
xdeba ase?
sur. 41.1
amrigad, Tu sxeulze sxva sxeuli ar moqmedebs,
maSin igi an uZravia, an Ta­nabrad moZ­raobs. am das­
kvnam­
de pirvelad mivida galileo galilei. misi
mtkicebiT, Tanabari moZraoba sxeulis ise­Tive
`bu­neb­rivi~ mdgomare­obaa, rogoric uZraoba.
miRebuli daskvna bu­
nebis erT-erTi Zi­
ri­
Tadi
kanonia. ma­s inerciis
inercia (laT.) _ uZraokanoni ewo­deba.
bas niSnavs.
genialurma ingli­
selma mecnierma isaak
niutonma ganazogada galileis daskvna da igi
meqanikis ZiriTad kanonTa ricxvSi Seitana. kanons,
romelsac niutonma Semdegi for­muli­reba misca,
niutonis pirvel kanons uwodeben:
yvela sxeuli inarCunebs uZ­
ra­
obis an Tana­ isaak niutoni (1643-1727)
bari moZraobis mdgo­mareobas manam, sanam sxva
sxe­
ulis moqmedeba gamoiyvans am mdgomareobidan.
aTvlis sistemas, romlis mimarT sxeuli inarCunebs mudmiv siCqares,
rodesac masze sxva sxeuli (Zala) ar moqmedebs, inerciuli ewodeba.
praqtikulad, inerciulia aTvlis sistema, romelSic sruldeba niutonis
kanonebi.
inerciis movlena gamovlindeba yovelTvis, rodesac sxeuli gamoh­
yavT
uZraobis mdgoma­reobidan, an aCere­ben, an cvlian misi moZ­raobis mimarTu­
lebas.
5. ra mimarTulebiT ixrebian mgzavrebi transportis daZvrisas? damuxruWebisas? moxvevisas? aris Tu ara transporti
TiToeul SemTxvevaSi aTvlis inerciuli sistema? axseniT,
ratom?
6. 41.2 suraTze gamosaxulia CaquCze taris dagebis
ori xerxi. axseniT, rogor
xdeba taris dageba TiToeul
SemTxvevaSi.
sur. 41.2
41
7. Tanabrad moZrav vagonSi Zafze kidia burTula.
a. rogori mimarTuleba aqvs Zafs?
ras SeeZlo Zafis mimarTulebis Secvla Tu Sexebis gareSe:
b. Zafi matareblis moZraobis mimarTulebiT gadaixara;
g. moZraobis sapirispiro mimarTulebiT gadaixara;
d. moZraobis marTobuli mimarTulebiT gadaixara.
inercia, Zala
amgvarad, moZraobis gamomwvevi mizezi ar arsebobs. erTi sxeulis
meoreze moqmedeba aris mxolod siCqaris Secvlis mizezi (maT Soris
amoZravebis da gaCerebis).
Tu sxeulze sxva sxeuli ar moqmedebs, maSin is an uZravia, an Ta­
nabrad moZraobs.
uZraoba da Tanabari moZraoba sxeulis bunebrivi mdgomareobaa.
sxeuli bunebriv mdgomareobas inarCunebs manam, sanam sxva sxeulis
moqmedeba gamoiyvans am mdgomareobidan.
yvela sxeuli cdilobs SeinarCunos Tavisi bunebrivi mdgomareoba.
am movlenas inercias uwodeben.
saSinao cda
sawveTurian urikaSi an polieTilenis
boTlSi naxevramde CaasxiT wyali. Tavi
mWidroT moukariT da dadeT magidaze
(sur. 42.1).
sur. 42.1
jer ivaraudeT, Semdeg daadgineT:
boTlSi wylis zedapiris forma uZrav da Tanabrad moZrav urikaSi.
ivaraudeT, rogor cvlilebebs ganicdis wylis zedapiri boTlis
swrafad amoZravebisa da damuxruWebisas?
varaudis marTebuloba SeamowmeT cdiT.
cda ramdenjerme gaimeoreT. CawereT dakvirvebisas SemCneuli
movlenebi da SeecadeT axsnaT.
42
1.17
SeamowmeT Tqveni codna
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Y
amoxseniT amocanebi
Y1 ,m
-2
5
X1 ,m
B O1 2
4
3
sur. 43.1
2
1
1
2
3
4
5
X, m
8
0,
1. gansazRvreT ­A da B wertilebis
koordinatebi XOY da X1O1Y1 koor­dinatTa
sistemebSi (sur.43.1). gansaz­
RvreT A da B
wertilebs Soris man­
Zili orive koor­
di­
natTa sis­t e­­m aSi. da­m oki­d ebulia Tu ara
wertilis ko­
ordinatebi da or wertils
Soris man­Zili koordinatTa sistemis arCe­
vaze? gamoitaneT daskvna.
A
3
2
1
O1
2. gansazRvreT gadaadgileba, TUu misi gegmi­lebi Sx=-3 m, Sy=5 m.
3. 43.2 suraTze gamosaxuli gra­
f ikis mixedviT X, m
4
dawereT sxeulis ko­ordinatis droze damokidebulebis
3
formula.
2
4. wrfiv avtomagistralze moZraobs ori avtomobili.
1
er­
Tis siCqaris modulia 80, xolo meoris _ 60 km/
1 2 3
sT. gansazRvreT meoris siCqare pirvelis mimarT, Tu
isi­ni moZraoben a. erTi mimarTulebiT; b. sapirispiro
sur. 43.2
mimarTulebiT.
t, wm
testi
1. S veqtorisa da misi mopirdapire veqtoris sxvaobis moduli tolia
a. S-is; b. nulis; g. 2S-is; d. 0,5S-is.
2. rodesac veqtoris gegmili X RerZze dadebiTia, maSin misi gegmili
Y RerZze
a. dadebiTia;
b. uaryofiTia;
g. nulia;
d. SeiZleba dadebiTi iyos, SeiZleba uaryofiTi, SeiZleba nuli.
3. avtomobilis spidometri gviCvenebs avtomobilis
a. siCqaris moduls; b. siCqaris mimarTulebas;
g. siCqaris gegmils; d. siCqares.
4. rodesac ori sxeuli moZraobs urTierTmarTobulad da maTi
siCqaris modulebi, Sesabamisad, 3 da 4 m/wm-ia, maSin meore sxeulis
siCqaris moduli pirvelis mimarT aris
a. 1 m/wm;
b. 3 m/wm;
g. 4 m/wm;
d. 5 m/wm;
e. 7 m/wm.
43