fizikis kursi

v. melaZe 

fizikis kursi 

(I nawili) 

`teqnikuri universiteti”@

saqarTvelos teqnikuri universiteti 

v. melaZe 

fizikis kursi 

(I nawili) 

damtkicebulia stu-s 

saredaqcio-sagamomcemlo sabWos 

mier. 02.07.2009, oqmi #6 

Tbilisi 

2009

fizikis kursis pirveli nawili _ meqanika, 

rxevebi da talRebi, molekuluri fizika, Termo- 

dinamika _ Sedgenilia umaRlesi teqnikuri saswavleblis 

studentebisaTvis. masSi mocemulia moqmedi 

programiT gaTvaliswinebuli yvela sakiTxi sakmao 

siRrmiT, amasTan, im varaudiT, rom mkiTxvels SeeZlos 

wignze damoukidebeli muSaoba. igi sargeb- 

lobas moutans sxva specialobis studentebsac. 

recenzenti sr. prof. k. cxakaia 

© sagamomcemlo saxli ,,teqnikuri universiteti’’, 2009 

ISBN 978-9941-14-707-4 (yvela nawili) 

ISBN 978-9941-14-708-1 (pirveli nawili) 

http://www.gtu.ge/publishinghouse/ 

yvela ufleba daculia. am wignis arc erTi nawili (iqneba es teqsti, foto, 

ilustracia Tu sxva) aranairi formiT da saSualebiT (iqneba es eleqtronuli Tu 

meqanikuri), ar SeiZleba gamoyenebul iqnas gamomcemlis werilobiTi nebarTvis 

gareSe. 

saavtoro uflebebis darRveva isjeba kanoniT.

sarCevi 

Sesavali 5 

fizikuri sidideebi. erTeulTa sistemebi 6 

1. kinematika 

1.1. aTvlis sxeuli. aTvlis sistema 11 

1.2. materialuri wertilis kinematika 11 

1.3. absoluturad myari sxeulis kinematika 17 

1.4. kuTxuri gadaadgileba, kuTxuri siCqare, kuTxuri aCqareba 18 

2. dinamika 

2.1. materialuri wertilis dinamika 22 

2.2. bunebis Zalebi 25 

2.3. muSaoba, simZlavre, energia 34 

2.4. myari sxeulis dinamika 41 

2.5. meqanikuri sistema. Senaxvis kanonebi 48 

2.6. sxeulTa wonasworobis pirobebi 52 

3. relativisturi meqanika 

3.1. fardofiTobis Teoriis safuZvlebi. ainStainis 

postulatebi 

3.2. lorencis gardaqmnebi da maTi Sedegebi 54 

3.3. relativisturi dinamika 56 

4. rxevebi da talRebi 

4.1. rxevebi. harmoniuli rxevebi 58 

4.2. siCqare da aCqareba harmoniuli rxevis 59 

4.3. harmoniuli rxevis energia 60 

4.4. harmoniuli rxevebis Sekreba 61 

4.5. milevadi rxevebi 63 

4.6. talRebi. ZiriTadi cnebebi 64 

4.7. brtyeli talRis gantoleba 65 

4.8. talRuri gantoleba 66 

4.9. talRebis interferencia 67 

3 

52

molekuluri fizika 68 

5. idealuri airebis kinetikuri Teoria 

5.1. idealuri airi. molekulur-kinetikuri 

Teoriis ZiriTadi gantoleba 71 

5.2. idealuri airis Sinagani energia 74 

5.3. molekulebis siCqareTa ganawilebis kanoni 75 

5.4. barometruli formula. 

bolcmanis ganawilebis kanoni 76 

5.5. molekulaTa dajaxebaTa ricxvis da 

saSualo Tavisufali ganarbenis gamoTvla 78 

6. Termodinamika 

6.1. Termodinamikuri sistema, damaxasiaTebeli parametrebi 78 

6.2. Termodinamikis pirveli kanoni 80 

6.3. airebis siTbotevadoba. maieris gantoleba 81 

6.4. adiabaturi procesi 83 

6.5. wriuli procesi. Seqcevadi da Seuqcevadi procesebi 84 

6.6. siTburi manqana. margi qmedebis koeficienti 85 

6.7. karnos cikli. karnos ciklis mqk 86 

6.8. Termodinamikis meore kanoni 88 

6.9. entropia. Termodinamikis gaerTianebuli kanoni 89 

7. realuri airebi 

7.1. realuri airebi. molekuluri Zalebi 90 

7.2. van-der-vaalsis gantoleba 92 

7.3. realuri airis izoTermebi 93 

damateba 95 

4

Sesavali 

fizikis sagani, misi kavSiri teqnikasTan 

fizika aris mecniereba, romelic Seiswavlis materiis moZraobis 

umartives da uzogades formebs da maT urTierTgardaqmnas. cnobilia 

materiis ori saxe: nivTiereba da veli. materiis pirvel saxes _ 

nivTierebas miekuTvnebian atomebi, molekulebi da maTgan agebuli 

sxeulebi. materiis meore saxes qmnian gravitaciuli, eleqtromagnituri 

da sxva velebi. 

adre fiqrobdnen, rom nivTierebas diskretuli agebuleba aqvs, veli 

ki uwyvetia. Semdgom aRmoCnda, rom materiis orive saxes erTdroulad 

axasiaTebs rogorc diskretuloba, ise uwyvetoba _ rac materiis 

ZiriTadi Tvisebaa. nebismieri movlena, materiis yoveli cvlileba 

warmoebs sivrcesa da droSi. sivrce da dro materiis arsebobis 

formebia. 

fizika SeiZleba gavyoT or nawilad _ klasikurad da kvanturad. 

klasikuri fizika Seiswavlis sinaTlis siCqaresTan SedarebiT mcire 

siCqareebis mqone makroskopiuli sxeulebis moZraobas. sinaTlis 

siCqaresTan maxlobeli siCqareebis mqone makroskopuli sxeulebis 

moZraobis kanonebs Seiswavlis relativisturi fizika. mikroskopiuli 

sxeulebis (atomebis da elementaruli nawilakebis) moZraobis 

Sesaswavlad klasikuri fizika (meqanika) ar gamodgeba, is icvleba 

kvanturi meqanikiT. 

fizikis aRmoCenebis safuZvelze warmoiSva da ganviTarda teqnikis 

mravali dargi. Tavis mxriv teqnika xels uwyobs fizikis ganviTarebas, 

aiaraRebs ra fizikas axali, srulyofili xelsawyoebiT da 

danadgarebiT. 

amgvarad, praqtikul moTxovnebs mivyavarT axal fizikur 

aRmoCenebamde da es aRmoCenebi xels uwyobs teqnikis winsvlas. 

fizika _ eqsperimentaluri mecnierebaa. eqsperimentaluri 

dakvirvebebi, gazomvebi da winaswar gamoTqmuli Teoriebi sabolood 

daiyvanebian fizikur sidideebsa da ricxvebs Soris 

urTierTdamokidebulebebze. amitom fizikis enaa maTematika. 

5

fizikuri sidideebi, ganzomileba, erTeulTa sistemebi 

fizikuri sididis erTeuli ewodeba pirobiT SerCeul fizikur 

sidides, romelsac gaaCnia gansaxilveli obieqtis fizikuri Tvisebebi. 

erTeulTa sistema ewodeba fizikur sidideTa erTeulebis erTobliobas, 

romlebic SeirCevian dadgenili wesiT. mocemuli sistemis ZiriTadi 

erTeulebi ewodebaT sxvadasxvagvari fizikuri sidideebis erTeulebs, 

romlebic nebismierad airCevian am sistemis Sedgenisas, erTeulTa 

sistemas ewodeba absoluturi, Tu misi ZiriTadi erTeulebia sigrZis, 

masis da drois erTeulebi. 

warmoebuli erTeulebi iseT erTeulebs ewodebaT, romlebic 

dadgenilebi arian mocemuli sistemis sxva erTeulebiT fizikuri 

kanonebis safuZvelze da warmoadgenen ZiriTadi erTeulebis 

ganzomilebaTa xarisxebis namravls. 

fizikuri sidideebis ganzomileba ewodeba gamosaxulebas, romelic 

axasiaTebs fizikuri sididis kavSirs mocemuli sistemis ZiriTad 

erTeulebTan. es gamosaxuleba warmoadgens erTwevrs ZiriTadi 

erTeulebis simboloebis namravlis saxiT Sesabamis xarisxebSi. sidides 

ewodeba uganzomilebo (ganyenebuli), Tu mis ganzomilebaSi yvela 

ZiriTadi sidideebi Sedian nulovan xarisxebSi. magaliTad, 

Tanabarwrfivi moZraobis siCqarisTvis 

s 

v = , 

t 

ganzomileba Caiwereba 

dim 

−1 

= LT , sididis erTeuli [v] = m . wm-1 . 

fizikuri sidideebis ZiriTadi erTeulebis garda SeiZleba 

gamoviyenoT jeradi da wiladi erTeulebi, romlebic miiRebian ZiriTadi 

erTeulebis 10-is romelime xarisxSi gamravlebiT an gayofiT. mamravlebi 

da TavsarTebi aTis jeradi da wilobrivi erTeulebis misaRebad (maTi 

dasaxelebebi) mocemulia cxrilSi 1. 

1960 wels zoma-wonis I generalurma konferenciam miiRo 

dadgenileba erTeulTa saerTaSoriso (SI) sistemis SemoRebis Sesaxeb., 

1971 wlidan zoma-wonis XIX konferenciaze miiRes axali ZiriTadi 

(meSvide) erTeuli moli da ori damatebiTi erTeuli radiani da 

steradiani. 

6

TavsarTi 

7 

cxrili 1. 

mamravli aRniSvna mamravli aRniSvna 

dasaxeleba dasaxeleba 

qarTuli saerTaSoriso 

qarTuli saerTSoriso 

10 18 

10 15 

10 12 

10 9 

10 6 

10 3 

10 2 

10 1 

eqsa 

peta 

tera 

giga 

mega 

kilo 

heqta 

deka 

e 

p 

t 

g 

m 

k 

h 

da 

E 

P 

T 

G 

M 

K 

h 

da 

10 -1 

10 -2 

10 -3 

10 -6 

10 -9 

10 -12 

10 -15 

10 -18 

deci 

santi 

mili 

mikro 

nano 

piko 

femto 

atto 

amgvarad, saerTaSoriso SI (Sistem International) sistemas safuZvlad 

daedo Svidi ZiriTadi erTeuli. ZiriTadi da damatebiTi sidideebi, maTi 

aRniSvnebi da erTeulebi mocemulia cxrilSi 2. 

sidide erTeuli 

d 

s 

ml 

mk 

n 

p 

f 

a 

aRniSvna 

d 

c 

m 

M 

n 

p 

f 

a 

cxrili 2. 

dasaxeleba ganzomileba dasaxeleba saerTaSoriso qarTuli 

sigrZe L metri m m 

masa M kilogrami kg kg 

dro T wami S wm 

denis Zala I amperi A a 

Termodinamikuri 

temperatura 

nivTierebis 

raodenoba 

θ kelvini K K 

NN moli mol moli 

sinaTlis Zala J kandela cd kd 

brtyeli kuTxe _ radiani rad rad 

sxeulovani 

kuTxe 

_ steradiani sr sr

ZiriTadi erTeulebis gansazRvra 

sigrZis erTeuli _ metri tolia vakuumSi brtyeli eleqtromagni- 

turi talRis gavlili manZilis wamis 1 , nawilSi. 

299792458 

masis erTeuli _ kilogrami aris kilogramis saerTaSoriso proto- 

tipis masa, damzadebuli platina-iridiumis Senadnobisgan (Pl 90%, Ir 10%) 

cilindruli sawonis formiT, diametriT da simaRliT toli 39 mm. 

drois erTeuli _ wami drois Sualedia, romelic tolia cezium- 

133-is atomis ZiriTadi mdgomareobis zenazi urTierTqmedebis or dones 

Soris gadasvlis Sesabamisi gamosxivebis 9192631770 periods. 

denis Zalis erTeuli _ amperi aris iseTi mudmivi denis Zala, 

romelic vakuumSi erTmaneTisagan 1 metriT daSorebul or paralelur, 

usasrulod grZel da mcire diametris gamtarebSi gavlisas, sigrZis 

7 

yovel metrze iwvevs 2⋅10 

niutonis tol urTierTqmedebis Zalas. 

temperaturuli erTeuli _ kelvini tolia wylis sammagi wertilis 

Termodinamikuri temperaturis 

1 nawilis. 

273, 

16 

nivTierebis raodenobis erTeuli _ moli aris sistemis nivTierebis 

raodenoba, romelic Seicavs imdenive struqturul elements (molekulas, 

ions, atoms) ramdeni atomicaa 0,012 kg C 12 naxSirbadSi. 

sinaTlis Zalis erTeulia _ kandela. kandela tolia sinaTlis 

Zalis mocemuli mimarTulebiT sinaTlis wyarodan, romelic uSvebs 

12 

monoqromatul gamosxivebas 540⋅10 hc sixSiriT da sinaTlis 

1 3t 

energetikuli Zala am mimarTulebiT Seadgens . 

683 sr 

SI sistemis damatebiTi erTeulebia: brtyeli kuTxis _ radiani, 

romelic tolia wris or radiuss Soris kuTxis, romlis momWimavi 

rkalis sigrZe radiusis tolia. 

sxeulovani kuTxis _ steradiani, romelic iseTi sivrculi kuTxea, 

romlis wvero moTavsebulia sferos centrSi da sferos zedapirze 

amokveTs misi radiusis kvadratis tol farTs. 

8

fizikuri sidide 

farTobi S 

moculoba V 

siCqare v 

simkvrive ρ 

aCqareba a 

inerciis momenti I 

warmoebuli erTeulebi 

meqanikuri sididis erTeulebi 

ganmsazRvreli 

gantoleba 

9 

sididis 

erTeuli 

sididis 

ganzomileba 

2 

S = a kvadr. metri m2 2 

L 

3 

V = a kuburi metri m3 3 

L 

s 

metri 

v = , m/wm 

t 

wm 

m 

= 

V 

ρ 3 

a 

Δv 

Δt 

−1 

LT 

kg 

L M 

m 

3 − 

m 

= −2 

2 

wm 

2 

I = mV 

kg . m2 sxeulis impulsi P P = mV 

kg 

wm 

m 

impulsis momenti L L = mV 

kg 

wm 

2 

LT 

L M 

2 

m 

−1 

LMT 

L MT 

2 −1 

Zala F F = ma niutoni (n) −2 

LMT 

Zalis momenti M M = F� 

n . m 2 −2 

L MT 

muSaoba, energia A A = FS 

jouli (j) 2 −2 

L MT 

simZlavre N 

kuTxuri aCqareba ω 

rxevis sixSire υ 

kuTxuri aCqareba ε 

A 

j 

N = vati= (vt) 

t 

wm 

ϕ 

ω = 

t 

wm 

1 

υ = 

wm 

t 

dω 

dt 

L MT 

2 −3 

rad 

−1 

T 

-1 1 − 

rad 

−2 

ε = 2 

T 

wm 

T

laTinuri anbani berZnuli anbani 

A, a _ a Α, α − alfa 

B, b _ b Β, β − beta 

C, c _ ce Γ, γ − gama 

D, d _ de Δ, δ − delta 

E, e _ e E, ε − efsilon 

F, f _ ef Ζ, ζ − Zeta 

G, g _ Je Η, η − eta 

H, h _ haS Θ, ϑ − Teta 

I, i _ i Ι, ι − iota 

J, j _ Ji Κ, κ − kapa 

K, k _ ka Λ, λ − lambada 

L, l _ el Μ, μ − miu 

M, m _ em Ν, ν − niu 

N, n _ en Ξ, ξ − qsi 

O, o _ o Ο, ο − omikroni 

P, p _ pe Π, π − pi 

Q, q _ qu Ρ, ρ − ro 

R, r _ er ∑, σ −sigma 

S, s _ es Τ, ι − tau 

T, t _ te Φ, ϕ − fi 

U, u _ u Χ, χ − xi 

V, v _ ve Υ, υ − ifsilon 

W, w _ dubl-ve Ψ, ψ − fsi 

X , x _ iqs Ω, ω − omega 

Y, y _ igrek 

Z, z _ zet 

10

meqanikis fizikuri safuZvlebi 

1. kinematika 

materiis moZraobis umartivesi saxea meqanikuri moZraoba. am 

moZraobis dros xdeba materialuri sxeulebis, an maTi nawilebis 

mdgomareobis Secvla (gadanacvleba) sivrceSi da droSi. klasikur 

(niutonis) meqanikaSi miRebulia, rom sivrces gaaCnia sami ganzomileba da 

masSi samarTliania evklides geometria. 

1.1. aTvlis sxeuli. aTvlis sistema 

sxeulis mdebareobis gansazRvra SesaZlebelia mxolod sxva 

sxeulebis mimarT. sxeuls, romlis mimarTac ganisazRvreba mocemuli 

sxeulis mdebareoba, aTvlis sxeuli ewodeba. aTvlis sxeuli, masTan 

xistad dakavSirebuli koordinatTa sistema da drois gamzomi xelsawyo 

(saaTi), Seadgenen aTvlis sistemas. fizikaSi sargebloben dekartes 

marTkuTxa koordinatTa sistemiT (nax.1), romelSic nebismieri A wertili 

xasiaTdeba x,y,z koordinatebiT an radius _ veqtoriT. r = OA. 

radius- 

veqtori aerTebs erTmaneTTan O koordinatTa saTaves A wertilTan. 

nax. 1. 

1.2. materialuri wertilis kinematika. traeqtoria 

meqanikis nawils, romelic Seiswavlis sxeulebis moZraobas 

gamomwvevi mizezebisagan damoukideblad, kinematika ewodeba. 

11

sxeulis umartivesi fizikuri modelia materialuri wertili. 

materialuri, anu nivTieri wertili ewodeba iseT sxeuls, romlis zoma 

SeiZleba ugulvebelvyoT mocemuli amocanis pirobebSi. materialuri 

wertilis moZraobisas misi koordinatebi icvlebian drois ganmavlobaSi 

da drois yovel moments Seesabameba x, y, z-is garkveuli mniSvneloba, e.i. 

koordinatebi warmoadgenen drois funqciebs, rac Semdegnairad Caiwereba: 

x=f1(t), y=f2(t), z=f3(t), (1) 

radius-veqtori agreTve drois funqciaa 

r = r(t) 

(2) 

Tu cnobilia (1) funqciebis, an veqtoruli (2) funqciis saxe, maSin 

cnobili iqneba wertilis mdebareoba drois yovel momentSi, rac 

meqanikis ZiriTadi amocanaa. (1) sam gantolebas, an maT eqvivalenturs (2) 

kinematikuri gantolebebi ewodebaT. (1) gantolebebidan SeiZleba miviRoT 

traeqtoriis gantoleba, e.i. im mrudis gantoleba, romelsac aRwers 

materialuri wertili moZraobisas. traeqtoriis formis mixedviT 

moZraoba SeiZleba iyos wrfivi an mrudwiruli. 

damoukidebel koordinatTa ricxvs, romlebic gansazRvraven 

wertilis mdebareobas sivrceSi, Tavisuflebis xarisxi ewodeba. Tu 

materialuri wertili sivrceSi Tavisuflad moZraobs, maSin mas gaaCnia 

sami Tavisuflebis xarisxi (koordinatebi x,y,z), sibrtyeze moZraobisas _ 

ori, xolo wrfis gaswvriv moZraobisas _ erTi. ganvixiloT materialuri 

wertilis moZraoba nebismier traeqtoriaze, mag. (AB) (nax. 2). 

nax. 2 

12

drois aTvla daviwyoT A mdebareobidan. ΔS _ gavlili manZili (gza) 

ewodeba traeqtoriis AB monakveTis sigrZes da warmoadgens drois 

skalarul funqcias: S = ΔS(t) 

Δ . r = r − r0 

Δ veqtors, romelic gaivleba 

wertilis sawyisi mdebareobidan sabolooSi, Δt droSi Sesrulebuli 

gadaadgildeba ewodeba. cxadia, wrfivi moZraobisas gadaadgilebis 

veqtori emTxveva Sesabamisi traeqtoriis monakveTs da misi moduli [ Δ r] 

toli iqneba Δ S gavlili gzis. 

siCqare 

materialuri wertilis moZraobis dasaxasiaTeblad ganixileba 

veqtoruli sidide _ siCqare, romelic gansazRvravs sxeulis moZraobis 

siswrafes da mis mimarTulebas drois mocemul momentSi. DdavuSvaT, 

materialuri wertili moZraobs raime mrudze (nax. 2). drois t momentSi 

igi A mdebareobaSia, xolo t + Δt 

momentSi _ B-Si. gadaadgilebis veqtori 

r r r − = Δ , sididiT is udris AB qordas da gansxvavdeba gavlili 

iqneba 0 

gzisgan (AB rkali). es gansxvaveba miT ufro mcirea, rac ufro mcirea Δ t . 

drois Sualedis _ Δ t SemcirebiT mrudwiruli moZraoba daiyvaneba 

wrfiv moZraobaze. wrfivi moZraobis saSualo siCqare tolia gavlili 

gzis Sefardebisa Sesabamis drosTan (nax. 3). 

v 

ΔS 

= , 

Δt 

13 

v 

nax. 3. 

Δr 

= 

Δt

es gamosaxulebebi gansazRvraven saSualo siCqaris sidides da 

mimarTulebas. Tu gadavalT zRvarze, roca Δt → 0, 

miviRebT siCqares 

drois momentSi, anu myis siCqares (saSualo da myisi siCqaris 

mimarTulebebi naCvenebia nax.3) 

Δr 

dr 

v = lim = 

Δt 

dt 

e.i. myisi, anu namdvili siCqare tolia radius-veqtoris pirveli 

rigis warmoebulisa droiT. rodesac Δt → 0 , maSin Δ S → Δr 

, amitom 

rogorc wrfivi, aseve mrudwiruli moZraobisaTvis gveqneba: 

Δr 

Δr 

ΔS 

ds 

v = v = lim = lim = lim = 

Δt→0 Δt 

Δt→0 

Δt 

Δt→0 

Δt 

dt 

e.i. myisi siCqaris ricxviTi mniSvneloba tolia gzis pirveli rigis 

warmoebulisa droiT. 

araTanabari, anu cvladi moZraobis SemTxvevaSi, radgan myisi 

siCqaris mniSvneloba droSi icvleba, ganmartaven saSualo siCqares: 

v 

ΔS 

= 

Δt 

nax.3-dan Cans, rom v > v , radgan Δ S > Δr 

, mxolod wrfivi 

moZraobisas Δ S = Δr 

sxeulis mier gavlili gza t1 -dan t 2 drois SualedSi zogadi saxiT 

gamoiTvleba integraliT: 

S = 

t 

2 

∫ 

t 

1 

v( 

t) 

dt 

Tanabari moZraobis SemTxvevaSi v = const ( t − t = Δt) 

da miviRebT: 

S = 

t 

2 

14 

, 2 1 

∫vdt = v∫dt 

= vΔ 

t 

1 

aCqareba, aCqarebis mdgenelebi 

t 

t 

2 

1 

fizikur sidides, romelic axasiaTebs siCqaris cvlilebis 

siswrafes moduliT da mimarTulebiT, aCqareba ewodeba. aCqareba 

t

araTanabari (cvladi) moZraobis damaxasiaTebeli mniSvnelovani 

veqtoruli sididea. 

davuSvaT, (1) wertilSi t drois momentSi wertilis siCqarea v 1 , 

xolo t + Δt 

momentSi (2) wertilSi v v + Δv 

saSualo aCqareba Δ t droSi 

2 = 1 (nax.4), v = v2 

− v1 

15 

nax. 4 

a 

Δ , 

Δv 

= , xolo myisi aCqareba 

Δt 

a = lim a 

Δt→0 Δv 

dv 

= lim = . 

Δt→0 

Δt 

dt 

mrudwiruli moZraobis dros aCqareba icvleba 

rogorc sididiT, aseve mimarTulebiT. imisaTvis, rom davaxasiaToT 

TiToeulis cvlileba cal-calke, gamovTvaloT Δv da warmovadginoT is 

ori mdgenelis saxiT. nax. 4-dan Cans, rom n v v v Δ + Δ = 

amitom sruli aCqareba a iqneba: 

Δv 

Δvτ 

Δvn 

a = lim = lim + lim , 

Δt→0 Δt 

Δt→0 

Δt 

Δt→0 

Δt 

dv dvτ 

dvn 

a = = + , 

dt dt dt 

Δ τ da v = v2 

− v1 

Δ , 

dvτ sadac = a τ tangencialuri (mxebi) aCqarebaa da axasiaTebs siCqaris 

dt 

dv n sididis cvlilebas, xolo = an 

_ normaluri aCqarebaa da axasiaTebs 

dt 

siCqaris mimarTulebis cvlilebas.

2 

dv 

v1 

a τ = τ , an = n 

dt 

R 

τ da n erTeulovani veqtorebia, a τ mimarTulia siCqaris gaswvriv 

traeqtoriis mxebad, xolo an _ radiusis gaswvriv centrisken. 

R_simrudis radiusia (nax. 5). amgvarad, mrudwiruli moZraobisas sruli 

aCqareba: 

a a + 

misi moduli 

a 

= τ n , an a = τ + n, 

2 2 

a = aτ 

+ an 

nax. 5. 

16 

dv 

dt 

= 

⎛ 

⎜ 

⎝ 

dv 

dt 

dv 

dt 

2 

⎞ 

⎟ 

⎠ 

2 ⎛ v ⎞ 

+ ⎜ 

⎟ 

⎝ R ⎠ 

aCqarebis tangencialuri da normaluri mdgenelebis gamoyenebiT 

SeiZleba movaxdinoT moZraobis klasificireba: 

1) a = 0, 

a = 0 _ wrfivi Tanabari moZraobaa, 

τ 

n 

2) a τ = a = const, 

a n = 0 _ wrfivi Tanabarcvladi moZraoba da 

Δv 

v2 

− v1 

aτ 

= a = = , Tu aTvla iwyeba 1 0 

Δt 

t − t 

= t momentidan, maSin t = t 

2 

1 

2 

2 da v 2 = v, 

xolo v 1 = v0 

- sawyisi siCqarea, gveqneba 

v − v0 

a = , 

t 

saidanac v = v0 

+ at, 

gavlili gza 

S = 

t t 

2 

at 

( , 

2 

∫vdt = ∫ v0 

+ at) 

dt = v0t 

+ 

0 0

3) a τ = f (t), 

a = 0 - wrfivi aCqarebuli moZraobaa, 

n 

4) a = 0, 

an = const - Tanabari moZraoba wrewirze, 

τ 

5) a = 0, 

an = f (t) 

- Tanabari mrudwiruli moZraoba, 

τ 

6) a = const, 

a ≠ 0 - Tanabarcvladi mrudwiruli moZraoba, 

n 

7) a τ = f (t), 

a n ≠ 0 - aCqarebuli mrudwiruli moZraoba. 

mxedvelobaSi unda miviRoT, rom Cvens mier ganxilul sidideebs 

Soris adgili aqvs Semdeg Tanafardobebs: 

r ( t) 

= x( 

t) 

i + y( 

t) 

j + Z( 

t) 

k, 

dr dx dy i j dzk 

v( 

t) 

= = i + j + k, 

siCqaris modulisaTvis: 

dt dt dt dt 

v = v = v + v + v 

2 

x 

dv dv x y dvz 

( t) 

= i + j + k = axi 

+ ay 

j + a k, 

dt dt dt 

a z 

2 

dv d r 

a = = aCqarebis modulisaTvis 

2 

dt dt 

2 

2 2 2 

dv d r 

a = a = ax 

+ ay 

+ az 

, magram a ≠ ≠ 2 

dt dt 

1.3. absoluturad myari sxeulis kinematika 

absoluturad myari sxeuli iseTi sxeulis fizikuri modelia, 

romelic ar ganicdis deformacias mocemul konkretul amocanaSi. myari 

sxeulis nebismieri moZraoba SeiZleba dayvanil iqnas gadataniT da 

brunviT moZraobaze. gadataniTi ewodeba iseT moZraobas, romlis drosac 

sxeulSi gavlebuli yoveli wrfe Tavisi paraleluri rCeba (nax.6). 

nax. 6 

17 

2 

4 

2 

z

unviTi moZraobis dros sxeulis yvela wertili imoZravebs wrewirze, 

romelTa centrebi ganlagdebian erT da igive wrfeze, romelsac brunvis 

RerZi ewodeba. brunvis RerZi SeiZleba moTavsebuli iyos rogorc 

sxeulSi, aseve mis gareT (nax. 7 a da b). 

a b 

nax. 7 a da b. 

sxeulis gadataniTi moZraoba aRiwereba misi erTi wertilis 

moZraobiT, romelsac masaTa centri ewodeba. 

materialuri wertilebis sistemis (sxeulis) masaTa centri _ 

wertilia romlis mdebareoba ganisazRvreba r c radius veqtoriT. 

mr1 

+ m2 

r2 

+ .... + mk 

r 

r c = 

m + m + ..... + m 

sadac m _ mTliani sxeulis masaa. 

masaTa centris siCqare 

1 

v 

c 

2 

drc 

= = 

dt 

18 

∑ 

k 

k 

mi 

r 

m 

k 

∑ 

=1 

m1ri 

i = 

m 

, 

1.4. kuTxuri gadaadgileba, kuTxuri siCqare, kuTxuri aCqareba 

wrewirze moZravi wertilis mdgomareoba ganisazRvreba 

Semobrunebis kuTxis ( Δ ϕ ) mniSvnelobiT (nax. 8). 

i

nax. 8. 

kuTxuri gadaadgileba: 

Δϕ - veqtoria, romlis sigrZe 

tolia Semobrunebis kuTxis modulis Δ ϕ , xolo mimarTuleba aiReba 

brunvis RerZis gaswvriv marjvena burRis wesiT. aseT veqtorebs _ 

fsevdoveqtorebi (aksialurebi) ewodebaT. 

kuTxuri siCqare 

Δϕ 

dϕ 

ω = lim = 

Δt→0 Δt 

dt 

kuTxuri siCqare warmoadgens RerZis garSemo mbrunavi sxeulis 

maxasiaTebels (nax.9). 

ω - fsevdoveqtoria, mimarTuli brunvis RerZis gaswvriv marjvena 

burRis wesiT. 

nax. 9. 

rad 

ω = = wm 

wm 

-1 , dimω = T-1 Tu ω = const brunva Tanabaria. brunvis periodi T – droa, romlis 

2π 

ganmavlobaSi sxeuli asrulebs erT srul bruns, T = . 

ω 

brunvis sixSire υ − drois erTeulSi Sesrulebuli brunvaTa 

ricxvia. 

19

1 ω 

υ = = 

T 2π 

1 

[ ] = 

wm 

υ =wm -1 , 

dim 

= T 

−1 

υ . 

Tu ω ≠ const − brunva araTanabaria. kuTxuri aCqareba tolia 

Δω 

dω 

ε = lim = 

Δt→0 Δt 

dt 

kuTxuri aCqareba warmoadgens siCqaris cvlilebis maxasiaTebels. ε _ 

fsevdoveqtoria, mimarTuli brunvis RerZis gaswvriv (nax.10). 

nax. 10 

aCqarebuli moZraobisas ε veqtoris mimarTuleba paraleluria ω 

veqtoris mimarTulebis (nax. 11), xolo Senelebuli moZraobisas 

mimarTulia sawinaaRmdegod (nax. 12). 

nax.11 nax. 12. 

wrewirze Tanabari moZraobis SemTxvevaSi ( ε = const) 

2 

εt 

ω = ω0 

± εt, 

ϕ ω0t 

± , 

2 

= sadac 0 

ω -sawyisi kuTxuri siCqarea. 

20

unviTi moZraobis maxasiaTeblebi, wrfivi da kuTxuri sidideebi, 

dakavSirebuli arian erTmaneTTan Semdegi TanafardobiT 

ds = Rdϕ 

, v = [ ω × r] 

, v = ωR 

, 

[ r] 

2 

2 

a τ = ε × , a τ = ε R , an = −ω 

R , an = ω R . 

aq r radius-veqtoria, gavlebuli brunvis RerZze mdebare 

koordinatTa saTavidan. is gansazRvravs materialuri wertilis 

mdebareobas. R veqtori gavlebulia gansaxilvel wertilSi da 

marTobulia brunvis RerZis. 

uZravi RerZis garSemo mbrunavi sxeulis da gadataniTi moZraobis 

Sesabamisi (analogiuri) sidideebis da gantolebebis cxrili: 

@ gadataniTi moZraoba brunviTi moZraoba 

1 masa 

2 gavlili manZili _ gza 

3 siCqare 

4 aCqareba 

m 

S 

d r 

v = 

dt 

dv 

a = 

dt 

21 

inerciis 

momenti 

Semobrunebis 

kuTxe 

5 Zala F Zalis 

6 impulsi 

7 dinamikis ZiriTadi 

gantoleba 

8 muSaoba 

9 kinetikuri energia 

p = mv 

F = ma 

d p 

F = 

dt 

dA = F ⋅ d s 

W = 

2 

mv 

2 

kuTxuri 

siCqare 

dϕ 

ω = 

dt 

kuTxuri 

aCqareba 

dω 

ε = 

dt 

momenti 

impulsis 

momenti 

dinamikis 

ZiriTadi 

gantoleba 

I 

ϕ 

M an Mz 

L = Iω 

M = Iε 

d L 

M = 

dt 

muSaoba 

M = Mdϕ 

kinetikuri 

energia 

W = 

2 

Iω 

2

2. dinamika 

2.1. materialuri wertilis dinamika 

klasikuri dinamika Seiswavlis sinaTlis siCqaresTan SedarebiT 

mcire siCqareebiT moZravi makroskopuli sxeulebis moZraobis kanonebs, 

sxeulTa urTierTqmedebis gaTvaliswinebiT. 

dinamikas safuZvlad udevs niutonis sami kanoni, romlebic 

samarTliania yvela saxis nivTieri makroskopiuli obieqtebisTvis. 

radgan movlenebs vswavlobT sivrcesa da droSi, amitom unda 

movnaxoT iseTi aTvlis sistema, romelSic sivrce da dro erTgvarovani 

da izotropulia. aseT sistemebSi Tavisufali izolirebuli sxeulis 

sivrceSi sxvadasxva mdebareoba da orientacia meqanikis TvalsazrisiT 

erTmaneTis eqvivalenturia. igive samarTliania droisaTvisac, e.i. misi 

sxvadasxva momentebi erTmaneTis eqvivalenturia. sistema, romlis mimarT 

sivrce da dro erTgvarovani da izotropulia sxva sistemebisgan 

gansxvavdeba imiT, rom, Tu am sistemaSi izolirebuli materialuri 

sxeuli drois erT momentSi aris uZravi, an moZraobs mudmivi siCqariT, 

is am mdgomareobas SeinarCunebs usasrulod didi drois ganmavlobaSi. 

aTvlis aseT sistemas ewodeba inerciuli. 

niutonis pirveli kanoni: aTvlis inerciul sistemaSi yoveli 

izolirebuli (Tavisufali) sxeuli inarCunebs uZraobis an Tanabari 

wrfivi moZraobis mdgomareobas. 

sxeulis Tvisebas, SeinarCunos uZraobis an Tanabari wrfivi 

moZraobis mdgomareoba, uwodeben inercias. aqedan warmosdgeba niutonis 

kanonis saxelwodeba _ inerciis kanoni. 

sxeulis Tvisebas, SeinarCunos Tavisi mdgomareoba, ewodeba 

inertuloba. sxeulis inertulobis zomaa _ masa. klasikur (niutonis) 

meqanikaSi, masa sididea: 

• skalaruli 

• dadebiTi (m>0), 

• aditiuri ( m = ∑ m1 

), 

• mudmivi ( m = const) 

[m]=kg 

22

Zala 

veqtoruli sidide _ Zala, warmoadgens sxeulze sxva sxeulebis an 

velebis zemoqmedebis zomas, romlis Sedegad sxeuli iZens aCqarebas an 

deformacias. 

mocemulia F Zala niSnavs, rom viciT misi moduli, mimarTuleba 

sivrceSi da modebis wertili. wrfes, romlis gaswvrivac mimarTulia 

Zala, ewodeba Zalis moqmedebis wrfe. 

sxeulebs Soris meqanikuri urTierTqmedeba SeiZleba 

ganxorcieldes rogorc manZilze (mzis mier planetebis mizidva), aseve 

uSualo kontaqtisas (xaxuni, Sexla). 

ZalTa superpoziciis principi 

materialur wertilze Zalebis erTdrouli moqmedeba eqvivalenturia 

erTi Zalis moqmedebis, romelsac tolqmedi (marezultirebeli) Zala 

ewodeba F da tolia maTi geometriuli jamis 

[ F ]= n (niutoni), dimF = LMT -2 

F = 

23 

n 

∑ Fi 

i= 

1 

materialuri wertilis impulsi 

P = m v 

sadac mi -materialuri wertilis masaa, vi - misi siCqare. 

materialur wertilTa sistemis (sxeulis) impulsi 

i 

p = 

n 

∑ 

i= 

1 

i 

pi 

materialur wertilTa sistemis impulsi tolia sistemis sruli 

masis namravlis masaTa centris siCqareze = m v 

[ p]= kg m/wm, dim =LMT-1 . 

i 

p c

niutonis meore kanoni. sxeulze momqmedi Zala tolia sxeulis impulsis 

cvlilebis siCqaris da gaaCnia Zalis moqmedebis mimarTuleba. 

d p 

F = 

dt 

Tu m = const, 

meore kanoni gamoiTqmeba: sxeulis aCqareba 

F 

pirdapirproporciulia Zalis da ukuproporciulia masis: a = , sadac: 

m 

2 

dv 

d r 

a = = , 2 

dt dt 

2 

d r 

sxeulis moZraobis gantolebaa: m = F 2 

dt 

koordinatTa RerZebze proeqciebSi: 

2 

2 

2 

d x 

d y 

d z 

m = F 2 x , m = Fy, 

m = F , 

2 

2 z 

dt 

dt 

dt 

sadac x,y,z, moZravi wertilis koordinatebia. 

niutonis mesame kanoni. sxeulebis nebismieri moqmedeba erTmaneTze 

yovelTvis urTierTqmedebis xasiaTs atarebs. Zalebi, romliTac 

urTierTmoqmedeben sxeulebi sididiT tolia, mimarTulebiT sapirispiro 

da modebulia sxvadasxva sxeulebze. mokled asec gamoiTqmeba _ qmedebis 

Zala tolia ukuqmedebis Zalis, sawinaaRmdegoa niSniT da mdebareoben 

erT wrfeze. (nax.13) 

F12 = −F 

21 , N 

p = 

nax. 13 

sadac p - sxeulis wonaa, modebulia sayrdenze, N - normaluri 

wnevis Zalaa, modebulia sxeulze, e.i. p da N qmedebis da ukuqmedebis 

Zalebia. 

24

niutonis meore da mesame kanonebis Sedegebi: 

1. nebismier meqanikur sistemaSi Sinagani Zalebis geometriuli jami 

nulis tolia 

n 

n 

∑∑ 

i= 

1 k= 

1 

25 

F 

sadac n-sistemaSi Semavali urTierTmomqmedi materialuri wertilebis 

in 

= 0 

gare 

raodenobaa. gare Zalebis mTavari veqtori ( F ) 

sadac 

tolqmedia. 

gare 

F =∑ gare 

i 

F 

gare 

F - i-ur materialur wertilze modebuli gare Zalebis 

2. impulsis warmoebuli droiT tolia sistemaze modebuli gare 

Zalebis mTavari veqtoris (impulsis cvlilebis kanoni). 

d p 

= 

dt 

gare 

F 

3. meqanikuri sistemis masaTa centris moZraobis kanoni: 

d 

dt 

( mv 

c ) = 

an mac 

= 

gare 

F , 

gare 

F 

Tu meqanikuri sistema _ myari sxeulia, maSin gadataniTi moZraobis 

ZiriTadi gantoleba iqneba: 

ma 

= 

gare 

F ( a = ac 

) 

2.2. bunebis Zalebi 

Tanamedrove fizikaSi miRebulia urTierTqmedebis 4 saxe: 

1. Zlieri anu birTvuli; 

2. eleqtromagnituri; 

3. susti; 

4. gravitaciuli.

klasikur (niutonis) meqanikaSi ganixilaven fundamentalur- 

eleqtromagnitur da gravitaciul Zalebs. drekadi da xaxunis Zalebi 

Tavisi bunebis gamo, miekuTvnebian eleqtromagnitur Zalebs. 

gravitaciuli Zalebi 

msoflio mizidulobis kanoni. gravitaciuli Zala, romliTac 

miezidebian erTmaneTs materialuri wertilebi (an ori sxeuli), 

pirdapirproporciulia maTi masebis namravlisa da ukuproporciulia 

maT Soris manZilis kvadratis (nax. 14). 

m m2 

F = G ⋅ 2 

R 

26 

R 

1 , 

sadac m 1 da m 2 - materialuri wertilebis masebia, 

R _ maT Soris manZili, 

2 

−11 n ⋅m 

G _ gravitaciuli mudmiva toli 6,67 ⋅10 

, 2 

kg 

R - erTeulovani veqtori, romelic gviCvenebs gravitaciuli Zalis 

R 

mimarTulebas. 

na.x 14 

pirvel miaxloebaSi dedamiwa sferuli formisaa da masa 

ganawilebulia masSi sferuli simetriiT. dedamiwis mizidulobis Zala m 

masis sxeulze mimarTulia dedamiwis centrisaken, misi moduli ki 

tolia: 

mM 

F = 

G 

2 

R 

R

gravitaciuli Zalebi gansazRvrven planetebis, varskvlavebis da sxva 

makroobieqtebis moZraobas. mikroobieqtebis moZraobaSi isini 

praqtikulad ar monawileoben. 

simZimis Zala da wona. 

uwonadoba 

simZimis Zala aris msoflio mizidulobis gravitaciuli Zalis kerZo 

gamovlineba. igi aris Zala, romliTac dedamiwa izidavs sxeulebs, amitom 

P simZimis Zala moqmedi m masis sxeulze 

mM 

P = mg = F = G , 2 

R 

sadac M – dedamiwis masaa. 

simZimis Zalis aCqareba g gamoiTvleba niutonis meore kanoniT: 

F M 

g = = G , 2 

m R 

e.i. g simZimis Zalis aCqareba ar aris damokidebuli sxeulis masaze da 

yvela sxeulisaTvis erTnairia. mas xSirad Tavisufali vardnis aCqarebas 

uwodeben. simZimis Zala da Sesabamisad simZimis Zalis aCqareba icvleba 

geografiuli ganedis mixedviT, rasac ganapirobebs ori faqtori _ erTia 

dedamiwis dReRamuri brunva, meore ki _ dedamiwis elifsuri forma. 

ekvatorze simZimis Zalis aCqarebis sidide g ( ϕ = 0) 

=9,78 m/wm2 . xolo 

polusze 0 

( ϕ = 90 ) 

g = 9,83 m/wm 

g aCqareba mimarTulia simZimis Zalis veqtoris mimarTulebiT, amitom 

p = mg 

sxeulis wona ewodeba Zalas, romliTac sxeuli gravitaciuli Zalis 

gavleniT moqmedebs sayrdenze. 

sxeulis mdgomareobas, rodesac is moZraobs mxolod simZimis Zalis 

gavleniT, uwonadoba ewodeba. 

27

inerciuli da arainerciuli sistemebi 

iseT sistemas, romelSic sruldeba niutonis pirveli kanoni, 

inerciuli sistema ewodeba. nebismieri sistema iqneba inerciuli, Tu is 

inerciuli sistemis mimarT uZravia, an moZraobs Tanabrad da wrfivad. 

sistema arainerciulia, Tu is moZraobs aCqarebulad nebismieri 

inerciuli sistemis mimarT (nax. 15). vTqvaT, sistema K (koordinatebiT 

x,y,z) inerciulia, xolo sistema K 1 (koordinatebiT x 1 y 1 z 1 ) moZraobs K-s 

mimarT v 0 siCqariT, ise rom x 1 y 1 z 1 RerZebi K sistemis Sesabamisi RerZebis 

paraleluria. 

nebismieri Tavisufali A wertilis radius-veqtori K sistemaSi 

aRvniSnoT r , xolo 

sistemaSi aRvniSnoT 0 

1) Tu 

nax. 15 

1 

K sistemaSi 1 

r -iT. 01 wertilis radius-veqtori K 

1 

r -iT. naxazidan miviRebT: r r + r0 

28 

= (1). 

1 

K sistema moZraobs K-s mimarT mudmivi v siCqariT, is xdeba 

inerciuli, radgan misi moZraoba K-s mimarT iqneba Tanabari da wrfivi. 

(1)-is droiT gawarmoeba mogvcems: 

1 

dr dr 

dr 

0 

1 

= + , anu v = v + v 0 

dt dt dt 

rac warmoadgens siCqareTa Sekrebis (galileis) kanons klasikur 

meqanikaSi. Tu miRebuls kvlav gavawarmoebT v = const) 

, miviRebT: 

1 

a = a 

( 0

aCqarebis tolobebi am sistemebSi niSnavs A wertilze moqmedi 

Zalebis tolobas: 

1 

F = F , rac koordinatTa sistemis arCevaze ar aris 

damokidebuli, anu K da K 1 sistemebi tolfasia. e.i. yvela inerciul 

sistemaSi meqanikis kanonebs erTnairi saxe aqvs. maSasadame, meqanikuri 

movlenebi inerciul sistemebSi erTnairad mimdinareoben. galileis mier 

miRebul am debulebebs fardobiTobis meqanikuri principi ewodeba. 

2) Tu v ≠ const, 

K1 sistema arainerciulia, amitom (1)-is meore 

warmoebuli mogvcems: 

anu aCqareba 

0 

d 

dt 

2 2 2 

1 

r d r d r 

1 

0 

= + , anu a = a + a 

2 2 2 

0 

dt 

dt 

1 

1 

a − K sistemaSi gansxvavebulia K-Si a aCqarebisagan 

1 

= a a0 

miRebulis gamravleba A wertilis masaze mogvcems: 

a − 

1 

1 

= ma 

ma 

0 , anu F = F − ma 

0 

ma − 

e.i. K 1 sistemaSi Zalac gansxvavebulia, amitom niutonis meore kanonic 

samarTliani ar iqneba. unda davuSvaT, rom K 1 sistemaSi garda 

garemomcveli sxeulTa moqmedebisa (F 1 Zala), A materialur wertilze 

moqmedebs agreTve Zala: 

F i = m − 

29 

( a 0 

gamowveuli K 1 sistemis aCqarebuli moZraobiT. F i Zalas inerciis Zala 

ewodeba. im damatebiT Zalas, romelic sxeulze moqmedebs arainerciul 

sistemaSi myofi damkvirveblis TvalsazrisiT, inerciis Zala ewodeba. 

drekadobis Zalebi 

drekadobis Zalebi warmoiqmnebian myar sxeulSi deformaciis dros da 

eqvemdebarebian hukis kanons. 

hukis kanoni: mcire deformaciis dros, deformaciis gamomwvevi Zalis 

sidide pirdapirproporciulia deformaciis sididis da gamoisaxeba 

formuliT (nax 16) F = −kx 

,sadac x- deformaciaa (wanacvleba), 

)

nax. 16. 

k- drekadobis koeficientia, damokidebuli sxeulis gvarobaze, 

sxeulis deformacia ganisazRvreba misi geometriuli zomebis 

cvlilebiT. grZivi deformaciis dros 

sadac 

Δ � 

- fardobiTi wagrZelebaa; 

� 0 

0 

Δ� 

ε = , 

� 

Δ � = � − � - absoluturi wagrZelebaa. 

� 0 - sxeulis sawyisi sigrZea. F 1 = F2 

= Fn 

nax.17 

30 

0 

.

cda gviCvenebs, rom drekadi deformaciis dros Reros wagrZeleba 

Δ � gamWimavi F normaluri Zalis da Reros sawyisi sigrZis 

pirdapirproporciulia da ukuproporciulia misi ganivkveTis S farTisa. 

sadac α drekadobis, anu 

F� 

Δ � = α , 

S 

grZivi gaWimvis koeficientia da 

damokidebulia nivTierebis gvarobaze. 

wina formulidan 

Δ� 

� 0 

F 

= α 

S 

, anu ε = α , 

S 

Fn _ farTobis erTeulze moqmedi gamWimavi Zala, anu normaluri 

S 

Zabvaa σ , miviRebT: 

ε = ασ 

miRebuli gantoleba saSualebas gvaZlevs hukis kanoni Semdegnairad 

CamovayaliboT: 

fardobiTi deformacia deformaciis gamomwvevi Zabvis 

pirdapirproporciulia. 

grZivi gaWimvis koeficientis Sebrunebul sidides iungis moduli 

ewodeba: 

1 σ 

E = da ε = 

α E 

amgvarad, normaluri meqanikuri Zabva 

F Δ� 

σ 

σ = da = . 

S � E 

e.i. iungis moduli iseTi meqanikuri Zabvis tolia, romlis drosac 

fardobiTi deformacia udris erTs. myari sxeulis deformacia hukis 

kanons eqvemdebareba garkveul zRvrul mniSvnelobamde, rac 

dakavSirebulia masalis gvarobaze. 

[ E ] 

= 

-2 

n ⋅m 

31 

dimE 

F n 

−1 

−2 

= ML T

xaxunis Zalebi 

xaxunis Zalebi warmoiSvebian erTmaneTis mimarT raime siCqariT 

moZravi myari sxeulebis Sexebis zedapirze an sxeulebis garemoSi 

(siTxeSi, gazebSi) moZraobis Sedegad da Tavisi bunebiT gansxvavdebian 

meqanikuri Zalebisagan. winaaRmdegobis F Zalas, romelic mimarTulia 

sxeulebis moZraobis sawinaaRmdegod, xaxunis Zala ewodeba. 

maqsimaluri uZraobis xaxunis Zala (srialis xaxunis Zalac), ar aris 

damokidebuli Semxebi zedapirebis farTze da proporciulia normaluri 

wnevis Zalis 

F xax = μΝ 

, 

sadac N - normaluri wnevis Zalaa, 

μ _ xaxunis koeficienti, damokidebulia Semxebi zedapirebis 

gvarobaze da mdgomareobaze. 

srialis dros xaxunis koeficienti siCqaris funqciaa. 

uZraobis xaxuni mJRavndeba uZravi sxeulis amoZravebisas. misi 

koeficientia μ 0 

nax. 18. 

srialis xaxuni mJRavndeba sxeulis moZraobisas; μ < μ0 

. 

ganvixiloT xaxunis Zalebi sxeulebis srialis dros: 

1. Tu wevis Zala emTxveva gadaadgilebis mimarTulebas (nax. 18) 

F = μmg 

2. Tu wevis Zala gadaadgilebasTan adgens α kuTxes (nax. 19 a) 

xax 

F = 

( mg − F sinα 

) 

xax 

μ wv 

32

xolo nax. 19 b SemTxvevisTvis : 

33 

nax. 19. 

F xax = μ( mg + Fwev 

sinα 

) 

3. Tu sxeuli moZraobs daxril sibrtyeze, xaxunis Zala tolia (nax. 

20): 

F xax = μmg cosα 

am dros xaxunis koeficienti μ = tgα 

nax. 20

4. Tu sxeuli raRac ZaliT ekroba vertikalur zedapirs, xaxunis 

Zala tolia (nax. 21): 

F = μF 

xax 

k 

nax. 21 

transportis moZraobisas unda gaviTvaliswinoT xaxuni ara marto 

borblebsa da gzas Soris, aramed xaxuni burTulsakisrebSic. am dros 

orive xaxunis koeficienti gaiTvaliswineba moZraobis winaaRmdegobis 

koeficientSi. 

2.3. muSaoba, simZlavre, energia 

meqanikuri muSaoba 

fizikur sidides, romelic tolia Zalisa da gadaadgilebis 

skalaruli namravlis, muSaoba ewodeba. F Zalis elementaruli muSaoba 

ΔAmcire Δr gadaadgilebaze, ganmartebis Tanaxmad, iqneba (nax. 22): 

ΔA = ( FΔr) 

= FΔr 

cosα 

= FrΔr 

nax. 22 

sadac Fr = F cosα 

_ F Zalis proeqciaa gadaadgilebis 

mimarTulebaze. α _ kuTxea Zalisa da gadaadgilebis mimarTulebebs 

34

Soris. Δ r simciris gamo F Zala SeiZleba mudmivad CavTvaloT, xolo 

Δ r → dr 

. 

elementaruli gzis sigrZe, romelsac gadis Zalis modebis 

wertili dt drois SualedSi 

maSin elementaruli muSaoba 

ds = 

35 

d r 

dA = Fds cosα 

aq d simbolo aRniSnavs mcire sidideebs. 

meqanikuri muSaoba _ skalaruli sididea, aditiuri. Tu: 

• 90 , 

° 

α < dA > 0, 

muSaoba dadebiTia. 

• 90 , 

° 

α = dA = 0, 

muSaoba tolia nulis. 

• 90 , 

° 

α > dA < 0, 

muSaoba uaryofiTia. 

n 

n 

n 

⎛ ⎞ 

E dA = ∑ dAi 

= ⎜∑ 

F i ⎟d 

r i = ∑ F i v idt 

, 

i= 

1 ⎝ i= 

1 ⎠ i= 

1 

sadac r i da v i radius-veqtori da siCqarea Zalis modebis wertilSi. 

materialuri wertilis moZraobis dros 

sadac p - wertilis impulsia. 

dA = vd 

myari sxeulis gadatanis da moZraobis dros 

dA c 

p 

= v d p 

sadac p = mv 

c - myari sxeulis impulsia, romlis masaTa centri 

moZraobs v c siCqariT. m – myari sxeulis masaa. 

cvladi Zalis muSaoba 

davuSvaT, nawilakze (sxeulze) moqmedebs cvladi Zala, romlis 

sidide damokidebulia koordinataze kanoniT Fr = F(r 

) . aseTi Zalis 

elementaruli muSaoba Δ Ai = FrΔr 

i ricxobrivad toli iqneba naxazze 

daStrixuli zolis farTis, xolo 1 

r -dan 2 

r gzaze is tolia farTis,

omelic SemosazRvrulia Fr (r) 

mrudiT, vertikaluri wrfeebiT r 1 da r 2 

da r RerZiT (nax. 23) 

∑ 

A = 

Δr 

→0 

i 

i 

r 

2 

12 lim Fr 

Δri 

= ∫ Fr 

( r) 

dr = f( r112r2 

) 

r 

cvladi Zalis meqanikuri muSaoba 

A 

= 2 

12 

1 

∫ dA = ∫ 

r 

r 

1 

2 

1 

F ( r) 

dr 

[ A ] = n ⋅m 

= j(jouli) 

, 

r 

dim A 

nax. 23 

2 −2 

= L MT 

simZlavre 

skalarul fizikur sidides, romelic tolia elementaruli 

muSaobis dA Sefardebis dt drosTan, romlis ganmavlobaSic sruldeba 

mocemuli muSaoba dA, myisi simZlavre ewodeba 

dA 

N = 

dt 

Tu dA muSaobas asrulebs F Zala, maSin N simZlavre SeiZleba 

warmovadginoT Semdegi skalaruli namravliT: 

dA d r 

N = = F ⋅ = ( F ⋅ v) 

dt dt 

e.i. namdvili (myisi) simZlavre udris Zalisa da siCqaris veqtorebis 

skalarul namravls. Tu muSaoba Tanabrad sruldeba, e.i. drois tol 

36

SualedebSi erTnairi muSaoba sruldeba, maSin namdvili simZlavre 

saSualo simZlavris toli iqneba da miviRebT: 

amitom 

A 

N = 

Δt 

es gantoleba gamosadegi iqneba drois nebismieri SualedisaTvis, 

A 

N = 

t 

e.i. simZlavre udris drois erTeulSi Sesrulebul muSaobas. 

j 

2 −3 

[ N] = = vt(vati), 

dim N = L TM 

wm 

energia, kinetikuri energia 

energia _ universaluri raodenobrivi zomaa yvela saxis materiis 

moZraobis da urTierTqmedebis. materiis moZraobis sxvadasxva saxeebTan 

dakavSirebulia energiis sxvadasxva formebi: meqanikuri, siTburi, 

eleqtromagnituri, birTvuli da sxva. 

sxeulze (nawilakze) momqmedi tolqmedi Zalis muSaoba d r 

gadaadgilebaze SeiZleba warmovadginoT Semdegnairad: 

2 

dv 

d r 

mv 

dA = ( F ⋅ d r) 

= m d r = mdv 

= mvd 

v = d( 

) 

dt dt 

2 

frCxilebSi gamosaxulebas kinetikuri energia ewodeba. 

sxeulis kinetikuri energia warmoadgens meqanikuri moZraobis 

zomas, romelic unda SevasruloT, rom gamoviwvioT mocemuli moZraoba. 

kinetikuri energia _ energiaa, romelic gaaCnia moZrav sxeulebs, 

e.i. is aris moZravi sistemis mdgomareobis funqcia. 

k 

W = 

Teorema kinetikuri energiis Sesaxeb: tolqmedi Zalis 

Sesrulebuli muSaoba nawilakze (sxeulze) tolia misi kinetikuri 

energiis cvlilebis: 

37 

mv 

2 

2

2 2 

k mv 2 mv1 

A = ΔW 

= − 

2 2 

sadac sxvaobis pirveli wevri Seesabameba nawilakis saboloo 

mdgomareobas, xolo meore _ sawyiss. 

kavSiri energiasa da impulss Soris 

W 

38 

k 

2 

p 

= 

2m 

sadac p = mv 

nawilakis impulsia, m - misi masa. 

[ W ] = j, 

dimW 

2 −2 

= L MT 

Zaluri veli, konservatoruli Zalebi 

sxeulTa urTierTqmedeba SeiZleba ganxorcieldes maTi uSualo 

SexebiT, an Tu sivrcis yovel wertilSi masze moqmedebs garkveuli Zala, 

romelic kanonzomierad icvleba. aseT SemTxvevaSi vambobT, rom sxeuli 

(nawilaki) moTavsebulia Zalur velSi (mag. gravitaciuli veli). 

Zalur vels potencialuri ewodeba, Tu masSi raime sxeulis 

gadaadgilebaze Sesrulebuli muSaoba damokidebuli araa traeqtoriis 

formaze da srulad ganisazRvreba sawyisi da saboloo mdgomareobiT. 

potencialur velSi momqmed Zalebs konservatoruli (potencialuri) 

Zalebi ewodebaT. cxadia, potencialur velSi Caketil konturze sxeulis 

gadaadgilebaze muSaoba ar sruldeba _ A=0. aseTive Tviseba gaaCnia 

centralur vels. vels centraluri ewodeba, Tu sivrcis nebismier 

wertilSi nawilakze momqmedi Zalis mimarTuleba gadis uZrav wertilSi 

(centrSi), xolo Zalis sidide damokidebulia mxolod am centramde 

manZilze. centraluri velis magaliTia dedamiwis zedapiris 

maxloblobaSi gravitaciuli veli, uZravi wertilovani muxtebis mier 

Seqmnili eleqtruli veli. 

gravitaciuli, kulonuri, drekadi Zalebi _ konservatoruli 

Zalebia. drekadi Zalebis muSaobaa 

kx1 

Adr = 

2 

kx2 

− 

2 

, 

sadac k _ drekadobis koeficientia, x 1 da x 2 nawilakis sawyisi da 

saboloo mdebareobebia. 

2 

2

Zalur vels erTgvarovani ewodeba, Tu velis nebismier wertilebSi 

nawilakze momqmedi Zalebi erTnairia sididiT da mimarTulebiT. 

⎛ dF ⎞ 

droSi ucvlel-mudmiv vels ⎜ = 0⎟ 

⎝ dt ⎠ 

⎛ dF ⎞ 

cvlads ⎜ ≠ 0⎟ 

− 

⎝ dt ⎠ 

arastacionaruli. 

disipatiuri Zalebi 

39 

stacionaruli ewodeba, xolo 

Zalebs ewodebaT disipatiuri, Tu maTi Sesrulebuli muSaoba, 

Caketil sistemaSi nebismieri moZraobisas uaryofiTia. aseTia 

winaaRmdegobis, xaxunis Zala. aseTi Zalebis sidide damokidebulia 

rogorc sxeulebis konfiguraciaze, aseve sxeulebis fardobiT siCqareze. 

isini yovelTvis mimarTulia nawilakebis (sxeulebis) siCqaris 

sawinaaRmdegod da iwveven maT damuxruWebas. 

potenciuri energia 

sxeulebis (nawilakebis) sistemis potenciuri energia ganisazRvreba 

maTi urTierTmdebareobiT da damokidebulia sxeulebs (wertilebs) 

Soris momqmedi konservatoruli Zalebis xasiaTze. ganmartebis Tanaxmad, 

is warmoadgens wertilTa koordinatebis funqcias u = u( 

x, 

y, 

z) 

. 

Tu Cven velis yovel wertils SevusabamebT aseT funqcias u(x,y,z) = 

u(r), maSin velis Zalebis mier Sesrulebuli muSaoba toli iqneba am 

funqciis Semcirebis. aseT funqcias uwodeben da utoleben potenciur 

energias. amgvarad, sxeulis (nawilakis) gadaadgilebisas X RerZis 

gaswvriv X 1 wertilidan X 2 –Si, Sesrulebuli muSaoba Cveni ganmartebis 

Tanaxmad, iqneba 

x2 

p 

p 

A12 = ∫ F( 

x) 

dx = −( 

W ( x2) 

−W 

( x1) 

) 

x 

1 

potencialuri energia _ sxeulis (nawilakebis) urTierTqmedebis 

energiaa konservatorul velTan. is damokidebulia am velSi nawilakis 

mdebareobaze.

Zala da potenciuri energia 

davuSvaT, urTierTqmedebis konservatoruli F Zalis moqmedebiT 

nivTieri wertili asrulebs dr gadaadgilebas, F Zalis Sesrulebuli 

muSaoba Caiwereba 

dA = 

40 

( Fd 

r 

meores mxriv, potenciuri energia Semcirdeba imave sididiT. 

amitom 

p 

( Fdr) = Fdrcosα 

= Frdr 

= −dW 

, ( Fr = F cosα 

) 

p 

p 

dW 

Frdr = −dW 

da Fr = − 

dr 

e.i. Zalis gegmili dr mimarTulebaze udris potenciuri energiis 

warmoebuls imave mimarTulebiT. skalaruli W funqciis warmoebuls im 

mimarTulebiT, romliskenac warmoebuli udidesia, am funqciis gradienti 

ewodeba _ grad W. 

funqciis gradienti aris veqtori, romlis gaswvriv am funqciis 

zrdis siCqare maqsimaluria da sididiT tolia am mimarTulebiT 

funqciis cvlilebisa manZilis erTeulze, veqtorulad: 

) 

p 

F = −gradW 

niSani minusi aCvenebs, rom Zala mimarTulia potencialuri 

energiis Semcirebis mimarTulebiT. simbolo 

(ortebi). 

p 

p 

p 

p ∂W 

∂W 

∂W 

grad W = i + i + k 

∂x 

∂y 

∂z 

p 

gradW -Ti aRniSnulia jami: 

sadac i , j, 

k − koordinatTa RerZebis erTeulovani veqtorebia 

w funqciis gradientis gegmilebi RerZebze konservatoruli 

p 

∂W 

Zalebis SemTxvevaSi iqnebian , 

∂x 

dagegmilebiT miviRebT: 

F x 

∂W 

= − 

∂x 

p 

, 

F x 

∂W 

= − 

∂y 

p 

, 

F x 

∂ p 

W 

∂y 

∂W 

= − 

∂z 

p 

∂W 

, 

∂z 

p 

, 

amitom 

F −gradW 

p 

= RerZebze

• Mmaterialuri wertilis (nawilakis) 

potencialuri energia erTgvarovan Zalur velSi 

dedamiwis zedapiris maxloblobaSi simZimis Zalis veli miviRoT 

erTgvarovnad. maSin nawilakis potencialuri energia 

p 

p 

W ( z) 

= mgz + W ( 0) 

. 

sadac m nawilakis masaa (z RerZi mimarTulia vertikalurad maRla, 

p 

Fz= _ (mg). W ( 0) 

nawilakis potencialuri energiis mniSvnelobaa z=0 dros. 

nawilakis potencialuri energia centraluri Zalebis velSi _ 

W 

p 

∞ 

= ∫ 

r 

p 

( r) 

F ( r) 

dr + W ( ∞) 

41 

r 

sadac Fr − F Zalis proeqciaa r veqtoris mimarTulebaze, r _ 

radius-veqtoria, gavlebuli ZalTa centridan gansaxilvel nawilakamde, 

p 

W ( ∞ ) - potencialuri energiaa usasrulobaSi, romelic nulad miiCneva. 

2.4. myari sxeulis dinamika 

myari sxeulis moZraoba SeiZleba dayvanil iqnas or martiv 

moZraobaze: masaTa centris gadataniT moZraobaze da uZravi RerZis 

garSemo brunviT moZraobaze. 

Zalis momenti 

veqtorul sidides, romelic tolia F Zalis veqtoruli namravlis 0 

uZravi wertilidan Zalis modebis wertilSi gavlebul radius-veqtorze 

r , ewodeba F Zalis momenti 0 uZravi wertilis mimarT (nax. 24) 

nax. 24 

[ r F ] 

M = ×

Zalis momenti _ fsevdoveqtoria, misi mimarTuleba ganisazRvreba 

marjvena burRis wesiT misi mobrunebisas r -dan F -sken. Zalis momentis 

moduli 

M = r ⋅ F sin α = � ⋅ F 

aq α _ kuTxea r da F veqtorebs Soris, � = r sinα 

-Zalis mxari 0 

wertilis mimarT. 

Tu adgili aqvs ramdenime Zalis, an ZalTa sistemis zemoqmedebas, 

maSin ZalTa sistemis momenti _ mTavari an tolqmedi momenti _ 

geometriuli jamia am wertilis mimarT sistemis n Zalebis: 

M 

= 

n 

∑ 

i−1 

42 

[ r1 

× Fi 

] 

sadac ri - radius-veqtoria, gavlebuli 0 wertilidan F i Zalis 

modebis wertilamde. 

[ ] = n ⋅m 

M , 

dimM 

2 −2 

= L MT 

impulsis momenti 

materialuri wertilis impulsis momenti 0 uZravi wertilis mimarT 

ewodeba L i veqtors, romelic tolia r i . radius-veqtoris veqtoruli 

namravlis impulsis Pi � veqtorze. ri � radius veqtori gaivleba 0 

wertilidan materialur wertilamde (nax. 25) 

nax. 25 

[ r i ] 

L × 

i = pi

impulsis momenti fsevdoveqtoria. misi mimarTuleba ganisazRvreba 

marjvena burRis wesiT, i r -dan p i -sken brunvisas. impulsis momentis 

moduli tolia: 

mxaria. 

L = r ⋅ p sin α = � ⋅ p 

i 

i 

43 

i 

sadac α kuTxea r da p veqtorebs Soris, xolo � = r sinα 

Zalis 

materialur wertilTa sistemis (sxeulis) impulsis momenti uZravi 0 

wertilis mimarT, ewodeba L veqtors, romelic tolia igive 0 wertilis 

mimarT wertilebis impulsebis L i momentTa geometriuli jamis: 

L = 

n 

∑ 

i= 

1 

[ r i × p i ] 

sxeulis impulsis momenti uZravi z RerZis mimarT ewodeba 

skalarul sidides, romelic tolia amave RerZze mdebare 0 wertilis 

mimarT L impulsis momentis veqtoris gegmilisa imave RerZze: 

L 

z 

= 

n 

∑ 

i= 

1 

2 

m 

L = kg = j ⋅wm, 

wm 

n 

[ r i × p i ] = ∑ 

i−1 

i 

i 

m v r 

dim L 

inerciis momenti 

i 

i 

2 −1 

= L MT 

materialuri wertilis (m) masis namravls brunvis RerZidan 

manZilis kvadratze ewodeba inerciis momenti I igive brunvis RerZis 

mimarT (nax. 26) 

RerZamde. 

I = m 

i 

sadac mi _ materialuri wertilis masaa, ri - manZili brunvis 

nax. 26 

2 

iri 

i

mTeli sxeulis inerciis momenti I udris sxeulis Semadgeneli n 

materialuri wertilebis (an elementebis) inerciis momentebis jams: 

I 

= 

2 [ I ] = kg ⋅m 

, 

44 

n 

∑ 

i= 

1 

i i r m 

2 

2 

dim I = ML 

inerciis momenti warmoadgens sxeulis inertulobis zomas brunvis 

procesSi, iseve rogorc masa gadataniTi moZraobisas. 

masis uwyveti ganawilebis SemTxvevaSi 

Δmi 

→0 

2 2 

∑ Δmir 

= ∫ r dm = ∫ 

2 

I = lim i 

ρ r dV 

sadac ρ -sxeulis simkvrivea, dV − moculobis elementi. 

Teorema brunvis RerZis gadatanis Sesaxeb _ Steineris Teorema: Tu 

m masis sxeulis inerciis momenti 0 I 0 I RerZis mimarT I , xolo C masaTa 

centrSi gamavali da 0I0I RerZis paraleluri 00 RerZis mimarT I c , maSin 

(nax. 27) maT Soris Semdegi damokidebulebaa: 

I = Ic 

+ 

2 

ma 

sadac a manZilia 00 da 0 I 0 I RerZebs Soris 

nax. 27

zogierTi erTgvarovani m masis sxeulebis 

inerciis momentebis tabula 

sxeuli RerZebis mdebareoba inerciis momenti 

Rru Telkedliani R 

radiusiani cilindri 

mTliani cilindri an 

R radiusiani diski 

simetriis RerZi 

simetriis RerZi 

l sigrZis Txeli Rero RerZi marTobia Rerosi 

da gadis mis Sua 

wertilSi 

l sigrZis Txeli Rero RerZi marTobTa Rerosi 

da gadis mis boloSi 

R radiusiani sfero RerZi 

centrSi 

gadis sferos 

mR 2 

1 

mR 

2 

1 

ml 

12 

1 

ml 

3 

siCqariT, maSin misi inerciis momenti I z igive RerZis mimarT 

45 

2 

2 

mR 

5 

myari sxeulis brunviTi moZraobis dinamikis ZiriTadi gantoleba 

brunviTi moZraobis dros dinamikis ZiriTadi sidideebia Zalis 

momenti M , impulsis momentis L da inerciis momenti I . 

dinamikis ZiriTadi kanoni warmoadgens urTierTkavSirs maT Soris. 

• Tu materialuri wertili brunavs uZravi 0 RerZis mimarT, 

maSin impulsis momentis droiT warmoebuli 0 wertilis 

mimarT tolia amave 0 wertilis mimarT Zalis momentis: 

d L 

= 

dt 

M 

i 

• Tu myari sxeuli brunavs uZravi RerZis garSemo, maSin 

dinamikis ZiriTadi kanoni Rebulobs saxes: 

d L 

dt 

∑ 

gare 

= M i 

sadac ∑ i M -gare Zalebis momentebis jamia (mTavari momenti). 

• Tu myari sxeuli brunavs Oz uZravi RerZis garSemo ω kuTxuri 

2 

2 

2

ar icvleba droSi ( I z = const). 

impulsis momenti am RerZis 

mimarT L z = I ω 

z 

uZravi RerZis garSemo mbrunavi sxeulis myari sxeulis dinamikis 

ZiriTadi kanoni miiRebs saxes: 

I z 

an 

dω 

= 

dt 

ε = 

46 

gare 

M 

1 

M 

sadac ε − sxeulis kuTxuri aCqarebaa. 

I z 

uZravi sxeulis garSemo mbrunavi sxeulis brunvis diferencialuri 

gantolebaa: 

I z 

2 

d ϕ 

= 

dt 

gare 

gare 

M 

sadac I z - inerciis momentia igive RerZis mimarT, ϕ - kuTxuri 

gadaadgileba, 

gare 

M _ gare Zalebis mTavari momenti. 

zogad SemTxvevaSi Tavisufali myari sxeulis moZraoba 

akmayofilebs or diferencialur gantolebas: 

siCqare, 

d 

dt 

gare 

( mvc 

) = F , 

d L 

= 

dt 

gare 

M 

sadac m sxeulis masaa, v c − masaTa centris gadataniTi moZraobis 

gare 

F - sxeulze modebuli gareSe ZalTa mTavari veqtori, gare 

M _ 

gare Zalebis mTavari momenti brunvis wertilis mimarT, L − sxeulis 

impulsis momenti imave wertilis mimarT. 

mbrunavi myari sxeulis kinetikuri energia 

m i masis materialuri wertilis kinetikuri energia, romelic 

brunavs ω = const mudmivi kuTxuri siCqariT, tolia (nax. 28): 

2 

i 

k miv 

1 2 

W = 

= miω 

R 

R 2 

2 

i

nax. 28. 

sadac vi = ωRi 

materialuri wertilis wiriTi siCqarea. 

Ri − manZili m i wertilidan brunvis RerZamde. 

mbrunavi absoluturad myari sxeulis nebismieri wertilisaTvis 

ω = const, 

amitom misi kinetikuri energia Caiwereba 

mimarT. 

sadac ∑ 

2 

miRi = I 

k 

k 1 2 

2 1 2 

W = ∑Wi= ω ∑miRi= 

ω I. 

2 

2 

_ sxeulis inerciis momentia brunvis RerZis 

miRebuli gamosaxuleba 

k 1 2 

( W = ω I ) 

2 

gviCvenebs, rom brunviTi 

moZraobis dros inerciis momenti asrulebs masis magivrobas, kuTxuri 

siCqare ki _ wiriTi siCqaris. Tu sxeuli monawileobas Rebulobs 

erTdroulad rogorc gadataniT, aseve brunviT moZraobaSi, maSin misi 

sruli kinetikuri energia iqneba: 

1 

= ω 

2 

2 

W Ic 

+ 

47 

1 

mv 

2 

1 2 

sadac I cω 

_ mbrunavi sxeulis kinetikuri energiaa, Ic − inerciis 

2 

momenti masaTa centrSi gamavali RerZis mimarT 

1 2 

mv c − gadataniTi moZraobis kinetikuri energia, vc -masaTa centris 

2 

siCqare, m – sxeulis masaa. 

2 

c

dros. 

iqneba: 

am formuliT gamoiTvleba sxeulis kinetikuri energia gorvis 

muSaoba brunviTi moZraobisas 

gare Zalebis elementaruli muSaoba sxeulis brunvisas dt droSi 

dA = Mωdt 

= Mdϕ 

= Mωdϕ 

( M = M cosα) 

ω 

sadac M - Zalis momentia wertilis mimarT dϕ = ωdt 

da dϕ = ωdt 

Sesabamisad mobrunebis kuTxe da kuTxuri gadaadgilebis veqtoria dt 

droSi. 

Mω − M veqtoris gegmilia ω veqtoris mimarTulebaze. 

sruli muSaoba ϕ1 -dan ϕ2 -mde gadaadgilebisas 

A 

48 

∫ 

ϕ 

= 2 

ϕ 

1 

Mdϕ 

2.5. meqanikuri sistema. Senaxvis kanonebi 

yoveli myari sxeuli SeiZleba ganvixiloT rogorc materialuri 

wertilebis sasrulo ricxvisgan Semdgari meqanikuri sistema, e.i. 

meqanikuri sistema materialuri wertilebis erTobliobaa. 

meqanikur sistemaSi Semaval materialur wertilebs (sxeulebs) 

Soris momqmed Zalebs Sinagani Zalebi ewodebaT, xolo sistemaze momqmed 

yvela sxva Zalas _ gareSe Zala. 

Tu meqanikur sistemaze gareSe Zalebi ar moqmedeben, mas Caketili 

(izolirebuli) ewodeba. 

sistemaSi Semavali urTierTmomqmedi sxeulebis koordinatebi, 

siCqareebi da aCqarebebi ganicdian mudmiv cvlilebas. arseboben iseTi 

fizikuri sidideebi, romlebic Caketil sistemaSi ar icvlebian. aseTi 

sidideebia energia, impulsi da impulsis momenti. am sidideebis Senaxvis 

kanonebi dakavSirebulni arian sivrce _ drois ZiriTad TvisebebTan.

energiis Senaxvis kanoni da drois erTgvarovneba 

sxeulebis (nawilakebis) sruli meqanikuri energia konservatorul 

ZalTa velSi gamoisaxeba Semdegnairad: 

sadac 

W + 

k p p 

= W + W W gare , 

k 

W _ sistemis kinetikuri energiaa 

p 

W _ sxeulebis (nawilakebis) urTierTqmedebis energia erTmaneTTan, 

k 

W g _ nawilakebis (sxeulebis) urTierTqmedebis potenciuri energia 

gare sxeulebTan. 

davuSvaT, sistemaSi Semaval sxeulebze garda konservatoruli 

Zalebisa, moqmedeben agreTve arakonservatoruli Zalebic. 

arakonservatoruli Zalebis muSaoba tolia sruli energiis cvlilebis: 

dA = dW , 

sadac dA - arakonservatoruli Zalebis muSaobaa, 

dW − sruli energiis cvlileba. 

rodesac dA = 0 , dW = 0 da miviRebT: 

k p p 

W + W + W = const 

p 

Tu sistema Caketilia, W = 0 da energiis Senaxvis kanoni miiRebs 

k p 

saxes W = W + W = const 

gare 

da gamoiTqmeba: Caketilis sistemis sruli meqanikuri energia 

mudmivi sididea _ ar icvleba droSi. 

energiis mudmivobis kanoni bunebis fundamentaluri kanonia, 

dakavSirebulia drois erTgvarovnebasTan. magaliTad, Tavisufali 

vardnis dros simZimis ZalTa velSi, sxeulis siCqare da Tavisufali 

vardnis xangrZlivoba damokidebulia sxeulis sawyis siCqareze da ara im 

drois momentze, roca is iwyebs vardnas (diliT Tu saRamos). 

impulsis Senaxvis kanoni da sivrcis erTgvarovneba 

sxeulebis (nawilakebis) sistemis moZraobis gantolebas, romelzec 

moqmedeben gareSe Zalebi, aqvs saxe 

49 

gare

d p 

dt 

= 

50 

∑ 

sadac p sistemis impulsia, ∑ gare 

i 

Zalebis tolqmedi. 

∑ gare 

i 

F 

gare 

i 

F - sistemaze momqmedi gareSe 

Tu sistema Caketilia, F = 0, 

p = const , rac warmoadgens 

impulsis Senaxvis kanons, romelic Semdegnairad gamoiTqmeba: 

Caketili sistemis impulsi mudmivi sididea. impulsis mudmivobis 

kanoni bunebis fundamentaluri kanonia. is Sedegia sivrcis 

erTgvarovnebis, rac dakavSirebulia im faqtTan, rom Caketili sistemis 

fizikuri Tvisebebi da misi moZraobis kanonebi ar aris damokidebuli 

koordinatTa sistemis saTavis mdebareobis arCevaze. 

reaqtiuli moZraoba 

impulsis mudmivobis kanonis damtkicebas teqnikaSi warmoadgens 

reaqtiuli moZraoba. sxeulis moZraobas, romelic aRiZvreba misgan 

gamotyorcnili nawilakebis reaqciis xarjze, reaqtiuli moZraoba 

ewodeba. gamotyorcnili nawilakebis (gazebis) nakadis reaqciis Zala 

sxeuls (raketas) aniWebs aCqarebas, romelic mimarTulia gamotyorcnili 

gazebis moZraobis sawinaaRmdegod. 

meSCerskis mier miRebuli cvladi masis sxeulis moZraobis 

gantolebaa: 

sadac F - gareSe Zalaa, xolo 

Tu F = 0, 

vRebulobT 

m a = F + 

dv dm 

m = −u 

, 

dt dt 

saidanac reaqtiuli moZraobis siCqare 

Fn 

dm 

F n = −u 

-s reaqtiuli Zala ewodeba. 

dt 

dm 

v = −u∫ 

= −u 

ln m + c 

m 

sawyisi pirobebis gamoyenebiT C = u lnm0 

( m0 − sastarto masaa), da

miRebuls ciolkovskis 

m0 

v = u ln 

m 

formula ewodeba. is miRebulia 

ararelativisturi moZraobisTvis. 

∑ i 

impulsis momentis mudmivobis kanoni da 

sivrcis izotropiuloba 

dinamikis ZiriTadi kanonis Tanaxmad, 

d L 

= ∑ M 

dt 

sadac L- sistemis impulsis momentia. Caketili sistemisTvis 

M = 0, 

miviRebT: 

d L 

= 0, 

dt 

51 

i 

L = const, 

an I ω = const 

miRebuli warmoadgens impulsis momentis mudmivobis kanons: 

Caketili sistemis impulsis momenti mudmivi sididea, rac bunebis 

fundamentaluri kanonia. is dakavSirebulia sivrcis 

izotropiulobasTan, anu fizikuri kanonebis invariantobasTan 

koordinatTa sistemis RerZebis mimarTulebis mimarT (Caketili sistemis 

SemobrunebasTan sivrceSi nebismieri kuTxiT). 

amgvarad, Caketil sistemebSi sami fizikuri sidide _ energia, 

impulsi da impulsis momenti – mudmivi sidideebia. isini koordinatebis 

da siCqareebis funqciebia da maT moZraobis integralebi ewodebaT. 

impulsis mudmivobis kanoni 

SeiZleba SevamowmoT mbrunavi skamis 

saSualebiT, romelzec zis adamiani, 

Tu brunvis dros adamiani xelebs 

gaSlis, misi inerciis momenti 

gaizrdeba da kuTxuri siCqare 

Semcirdeba. xelebis daSvebis dros 

inerciis momenti Semcirdeba, xolo 

kuTxuri siCqare imatebs.

2.6 sxeulTa wonasworobis pirobebi. 

myari sxeulis moZraoba aRiwereba Semdegi gantolebebiT: 

d � � gare dL � gare 

( mvc 

) = ∑ F , = ΣM 

dt 

dt 

sadac – m sxeulis masaa, vc � - misi masaTa centris siCqarea. 

myari sxeulis wonasworobis pirobebia: 

1. sxeulze modebuli gareSe ZalTa tolqmedi unda iyos nuli: 

gare 

Σ F = 0 

� 

2. gareSe ZalTa momentebis tolqmedi sxeulis nebismieri wertilis 

mimarT unda iyos nuli 

ΣM = 0 

� 

es veqtoruli pirobebi eqvivalenturia sami skalarulis: 

gare 

ΣM 

= 0 , ΣM 

= 0 ΣM 

= 0 

x 

y 

gare 

52 

z 

gare 

3. relativisturi meqanika 

3.1. fardobiTobis Teoriis safuZvlebi. ainStainis postulatebi 

Tanamedrove fizika Camoyalibda mas Semdeg, rac ainStainis mier 

Seiqmna fardobiTobis specialuri Teoria. es Teoria warmoadgens 

sivrce-droze Tanamedrove Sexedulebebs, romelic niutonis Teoriis 

msgavsad varaudobs, rom dro erTgvarovania, xolo sivrce erTgvarovani 

da izotropiulia. amasTan erTad is uaryofs klasikuri fizikis ZiriTad 

daSvebebs eTeris Teoriis, sivrcisa da drois absoluturobis Sesaxeb. 

fardobiTobis principi, urTierTqmedebis gavrcelebis maqsimaluri 

siCqaris debuleba da sinaTlis siCqaris invariantoba _ warmoadgens 

ainStainis fardobiTobis specialuri Teoriis, relativisturi Teoriis 

ZiriTad principebs. mas safuZvlad udevs ainStainis mier 

formulirebuli ori postulati:

1. fardobiTobis principi: bunebis kanonebi invariantulia inerciuli 

sistemebis mimarT, anu bunebis kanonebi erTnairad mimdinareoben 

sxvadasxva inerciul sistemebSi. 

2. sinaTlis siCqaris invariantobis principi: sinaTlis siCqare 

vakuumSi, erTnairia yvela inerciul sistemebSi da ar aris 

damokidebuli damkvirveblis an sinaTlis wyaros siCqareze. 

ainStainis pirveli postulati warmoadgens galileis fardobiTobis 

principis ganzogadoebas nebismier fizikur procesze. gantolebebi, 

romlebic gamosaxaven fizikur kanonebs, inarCuneben TavianT saxes yvela 

aTvlis inerciul sistemaSi. 

meore postulatis Tanaxmad, nebismieri urTierTqmedebis gavrcelebis 

siCqare sasruloa. maqsimaluri urTierTqmedebis (signalis) gavrcelebis 

dasaSvebi siCqare aris sinaTlis gavrcelebis siCqare vakuumSi. 

ainStainamde miRebuli iyo msoflio eTeris arsebobis hipoTeza, 

romelic ganWolavda msoflio sivrces da warmoadgenda 

urTierTqmedebis gadamcem garemos. warmodgenebi aseT substanciaze 

arsebobda jer kidev antikuri xanidan. magaliTad, demokrites 

koncefciis mixedviT, nivTiereba Sedgeboda nawilakebisgan, romelTa 

Soris sicarielea. aristoteles koncefciis mixedviT ki, samyaroSi ar 

aris iseTi adgili, sadac `araferia~. es hipoTeza gamoiyeneboda 

mecnierebis mier movlenebis asaxsnelad gasuli saukunis dasawyisSic, 

xolo hipoTezur garemos, romelic unda arsebobdes sxeulis nawilakebs 

Soris da ganWolavdes msoflio sivrces, ewodeba eTeri. 

gansakuTrebiT gaizarda interesi eTeris bunebis Sesaxeb gamoCenili 

Teoretikosis, maqsvelis mier eleqtromagnituri velis aRmoCenis Semdeg. 

eleqtruli da magnituri velebis erTmaneTSi TanmimdevrobiTi 

monacvleoba aucilebels xdida procesebis ganxilvas gansakuTrebul 

drekad garemoSi. mas unda uzrunveleyo gravitaciuli Zalebis 

urTierTqmedeba, sinaTlis talRebis gamtarebloba-gavrceleba da 

saerTod pasuxi unda gaeca bunebis nebismier movlenebze. yvela am 

funqciebis Sesasruleblad eTers unda hqonoda Zalian gansxvavebuli da 

xSirad urTierTsawinaaRmdego Tvisebebi. magaliTad, misi drekadoba unda 

yofiliyo foladze bevrad meti da amave dros winaaRmdegoba ar unda 

53

Seeqmna masSi sinaTlis gavrcelebisaTvis da amave dros ciuri sxeulebis 

(varskvlavebis da planetebis) moZraobisaTvis. 

ainStainis fardobiTobis TeoriiT eTeri uaryofil iqna, 

fardobiTobis Teoriis paralelurad Seqmnilma velis kvanturma Teoriam 

ki ainStainis sivrceSi aRmoaCina specifikuri materialuri garemo 

eTeris Semcvleli _ romelsac fizikuri vakuumi ewoda. 

marTalia, fizikuri vakuumis realuri arseboba dasturdeba cdebiT, 

magram misi struqtura dRemde daudgenelia. fizikuri vakuumis 

struqturis da Tvisebebis dadgena warmoadgens Tanamedrove fizikis 

erT-erT mniSvnelovan mecnierul mimarTulebas. 

3.2. lorencis gardaqmnebi 

lorencis gardaqmnebi. koordinatebis da drois gardaqmnis 

formulebi, romlebic akmayofilebs ainStainis fardobiTobis princips, 

Semdegia: 

1 

x + v t 

x = 

1− β 

t 

t = 

1 

0 

2 

54 

x 

1 

x − v0t 

= 

2 

1− β 

1 

1 

y = y 

y = y 

1 

1 

z = z 

z 

1 

x 

+ v0 

2 

c 

2 

1− β 

sadac C _ sinaTlis siCqarea vakuumSi, 

1 

v0 − k sistemis siCqarea k-s mimarT. 

t 

1 

v0 β = 

c 

z = 

x 

t − v0 

2 

= 

c 

2 

1− β 

v 0

nax. 29 

lorencis gardaqmnebis Sedegebi 

1. wrfivi zomebis fardobiToba. lorencis gardaqmnebis formulebidan 

gamomdinareobs, rom inerciuli sistemis mimarT moZravi sxeulis wrfivi 

zoma mcirdeba moZraobis mimarTulebis gaswvriv (lorencis Semokleba): 

= � � 

0 

55 

2 

1 β − 

sadac � 0 sxeulis wrfivi zomaa im inerciul sistemaSi, romlis 

mimarTac is uZravia, xolo � zomaa moZravi inerciuli sistemis mimarT. 

sxeulis wrfivi zomebi udidesia uZravi sistemis mimarT, maT sakuTari 

zomebi ewodebaT. sxeulis sxva zomebi, romlebic ar emTxveva moZraobis 

mimarTulebas, ucvlelni rCebian. 

2. movlenis xangrZlioba sxvadasxva sistemebSi. 

davuSvaT k sistemis mimarT uZrav wertilSi koordinatiT x, adgili 

1 

aqvs movlenas, romlis xangrZliobaa δ . 2 1 

xangrZlioba k 1 sistemaSi aRvniSnoT 

gardaqmnis formulebidan 

1 

δ = 

saidanac gamomdinareobs, rom 

v0x 

t1 

− 2 

1 

t 1 = c , 

2 

1− β 

1. 

δ = t − t 

t 

− t 

1 

2 

1 1 2 1 

t 2 − t1 

= = , 

2 

2 

1− β 1− 

β 

1 

δ = t − t . amave movlenis 

v0 

x 

t2 

− 2 

1 

t 2 = 

c , miviRebT: 

2 

1− β 

δ 

1 

δ < δ , anu wertilSi momxdari movlenis 

xangrZlioba umciresia im inerciul sistemaSi, romlis mimarTac 

aRniSnuli wertili uZravia.

3. siCqareTa Sekrebis kanoni. galileis siCqareTa Sekrebis 

kanonis Tanaxmad, 

v = v1 

+ v0 

fardobiTobis specialur TeoriaSi siCqareTa Sekrebis kanons aqvs 

Semdegi saxe: 

v′ 

+ v0 

v = 

v′ 

v 

1+ 

c 

mcire siCqareebis SemTxvevaSi ( v ′ 

v′ 

v0 

inerciuli m masa, romelic axasiaTebs sxeulis reaqcias gare Zalaze. 

Tu v

4. rxevebi da talRebi. 

4.1. rxeva. harmoniuli rxevebi 

rxeva ewodeba iseT moZraobas an process, romelic zustad an 

raime miaxloebiT meordeba drois erTsa da imave SualedSi. fizikuri 

procesebis xasiaTis mixedviT ansxvaveben meqanikur da eleqtromagnitur 

rxevebs da maT kombinaciebs. droze damokidebulebiT ansxvaveben 

periodul (nax. 30 a,b,g) da araperiodul rxevebs (nax. 30 d) 

periodulobis pirobaa: 

nax. 30 

x ( t + T) 

= x( 

t) 

) 

sadac T - rxevis periodia, x − droSi cvladi fizikuri sidide. 

perioduli rxevis mniSvnelovani saxea harmoniuli (sinosoidaluri) 

rxevebi (nax. 30 a). merxevi sistemis mixedviT arCeven martiv (erTi 

Tavisuflebis xarisxiT), Tavmoyrili parametrebiT (Tavisuflebis 

xarisxis sasrulo raodenobiT) da ganawilebuli parametrebiT, 

romelTac gaaCniaT sakuTari sixSireebis usasrulo raodenoba. 

periodul rxevebs, romlis drosac merxevi sidide icvleba sinusis 

an kosinusis kanoniT, harmoniuli rxevebi ewodeba (nax. 31). 

x = A ω t + ϕ), 

an x = A ω t + ϕ) 

cos( 0 

nax. 31. 

sin( 0 

58

sadac x merxevi sistemis fizikuri sididis (parametris) garkveuli 

gadaxraa (wanacvleba) wonasworobis mdebareobidan – x =0 -dan. 

A - rxevis amplitudaa, udidesi gadaxra wonasworobis 

mdebareobidan. 

ω 0 t + ϕ - rxevis fazaa, 

ϕ − rxevis sawyisi fazaa. 

rxevis periodi T – erTi sruli rxevis droa. 

ω t + T) 

+ ϕ = ( ω t + ϕ) 

+ 2π 

, saidanac 

0 ( 0 

2π 

T = 

ω 

rxevis sixSire υ -rxevaTa ricxvia drois erTeulSi 

[ ] 

υ = hc(herci ), dimυ 

= T 

−1 

ω −wriuli 

(cikluri) sixSire – rxevaTa ricxvia 2π wamSi: 

0 

2π 

ω 0 = , ω0 = 2πυ 

T 

4.2. siCqare da aCqareba harmoniuli rxevis dros 

59 

0 

harmoniuli oscilatori _ sistemaa (sxeuli), romelic asrulebs 

harmoniul rxevebs mdgradi wonasworobis maxloblobaSi ω 0 sakuTari 

rxevis sixSiriT. sistemis merxevi sididis x wanacvleba (gadaxra) iqneba 

x = A ω t + ϕ) 

, 

cos( 0 

romlis gawarmoebiTac miviRebT rxevis siCqares 

dx 

v = 

dt 

⎛ π ⎞ 

= −Aω 

0 sin( ω0t 

+ ϕ) 

= Aω0 

cos⎜ω0t 

+ ϕ + ⎟ 

⎝ 2 ⎠ 

2 

d x 

2 

rxevis aCqareba a = = −Aω 

0 cos( ω0t 

+ ϕ + π ) (nax. 32) 

2 

dt 

nax. 32.

mocemulia wanacvlebis, siCqaris da aCqarebis damokidebuleba droze. 

harmoniuli rxevebi sruldebian drekadi (an kvazidrekadi) F d = −k 

x 

Zalis gavleniT. Tu visargeblebT niutonis me-2 kanoniT da am Zalebs 

gavutolebT erTmaneTs, martivi gardaqmnebiT miviRebT harmoniuli rxevis 

diferencialur gantolebas: 

k 2 

= ω0 

m 

2 

2 

d x 

d x k 

F = m F = −kx 

2 d da + = 0 

2 

dt 

dt m 

- oscilatoris sakuTari rxevis sixSirea. gantoleba Caiwereba 

2 

d x 2 

+ ω 0 x = 0 

2 

dt 

miRebuli diferencialuri gantolebis amoxsnaa: 

x = A ω t + ϕ) 

cos( 0 

4.3. harmoniuli rxevis energia. 

merxev sxeuls wonasworobis mdebareobaSi gaaCnia mxolod 

kinetikuri energia, maqsimaluri gadaxrisas ki _ mxolod potencialuri. 

potencialuri energia gaizomeba F = −kx 

drekadi Zalis 

sawinaaRmdegod x manZilze sxeulis gadasaadgileblad Sesrulebuli 

muSaobiT. elementaruli Sesrulebuli muSaoba iqneba 

sruli muSaoba 

A = 

dA = Fdx = 

x 

∫ 

0 

x 

60 

kxdx 

kx 

Fdx = ∫ kxdx = 

2 

2 2 

kx kA 2 

e.i. potenciuri energia W ( t) 

= = cos ( ω 0t 

+ ϕ) 

2 2 

p 

kinetikuri energia wonasworobidan x manZilze 

W k 

2 

mω 

A 2 

( t) 

= sin ( ω0t 

+ 

2 

sruli energiisTvis gveqneba ( k = mω 

) 

0 

2 

0 ϕ 

2 

0 

2 

2 

2 

mω 

0 A 

[ cos ( ω t + ϕ) 

+ sin ( ω t + ϕ) 

] = const. 

2 2 

mω 

0 A 

W = 

0 

0 

= 

2 

2 

2 

)

e.i. harmoniuli rxevis dros sruli energia mudmivia, rac imis Sedegia, 

rom garda wonasworobisadmi damabrunebeli drekadi (kvazidrekadi) 

Zalisa, sxva Zala ar moqmedebs. 

4.4. harmoniuli rxevebis Sekreba 

erTnairi sixSiris da mimarTulebis rxevebis veqtoruli Sekreba 

ori harmoniuli rxevis x 1 da x 2 Sekrebisas, romlebic ganisazRvrebian 

gantolebebiT: 

x = A cos( ω t + ϕ ) da x = A ω t + ϕ ) 

1 

, 0 1 

miviRebT igive sixSiris harmoniul rxevas: 

x = x + x = A ω t + ϕ) 

jamuri rxevis amplituda iqneba (nax. 33) 

A 

2 

2 

1 

1 

2 

2 

2 

2 cos( 0 2 

x = Acosωt, 

y 

= Bcos( 

ω t + ϕ) 

61 

2 

cos( 0 

= A + A + A cos( ϕ −ϕ 

) 

xolo faza ϕ gamoisaxeba gantolebiT: 

a) Tu − ϕ = 0 

ϕ 2 1 , maSin 1 2 A A A + 

1 

2A1 2 1 2 

A1 

sinϕ1 

+ A2 

sinϕ 

2 

tgϕ 

= 

A cosϕ 

+ A cosϕ 

1 

nax. 33 

2 

= amplituda udidesia 

b) Tu ϕ 2 − ϕ1 

= ± π , maSin A = A1 

− A2 

, rxevebi sawinaaRmdego fazebSia 

da amplituda minimaluria. 

urTierTmarTobi harmoniuli rxevebis Sekreba. 

Tu materialuri wertili erTdroulad irxeva OX da OY koordinatTa 

RerZebis gaswvriv kanoniT: 

2

sadac A da B sawyisi amplitudebia, 

ω _ maTi sixSireebi, 

ϕ − erT-erTi rxevis sawyisi fazaa. 

SedegiTi rxeva warmoadgens elifss, romlis gantolebaa (nax. 34): 

a) Tu ϕ = 0 

x 

A 

2 

2 

y 

+ 

B 

2 

2 

2xy 

− cosϕ 

= sin 

AB 

nax. 34 

⎛ x y ⎞ 

⎜ − ⎟ = 0, 

⎝ A B ⎠ 

traeqtoria _ wrfea, romlis gantolebaa 

B 

Y = X 

A 

wertili gairxeva gantolebidan miRebuli wrfis gaswvriv ω sixSiriT da 

2 2 

A + B amplitudiT. 

B 

b) Tu ϕ = ± π , traeqtoria isev wrfea, gantoleba iqneba Y = − X 

A 

π 

g) Tu ϕ = ± , traeqtoria elifsia 

2 

x 

A 

2 

2 

Y 

+ 

B 

62 

2 

2 

2 

= 1 

2 

ϕ

Tu A = B , elifsi gadadis wrewirSi 

iwarmoebs saaTis isris mimarTulebiT, xolo 

sawinaaRmdegod. 

4.5. milevadi rxevebi 

63 

π 

ϕ = + SemTxvevaSi moZraoba 

2 

π 

ϕ = − SemTxvevaSi _ mis 

2 

Tavisufal rxevebs, romelTa amplituda droSi mcirdeba energiis 

kargvis gamo, milevadi rxevebi ewodeba. energiis Semcireba ori mizeziT 

aris gamowveuli: winaaRmdegobis Zalebis arsebobiT da gamosxivebiT. 

milevadi meqanikuri rxevis aRmweri diferencialuri gantolebaa: 

2 

d x dx 2 

+ 2β 

+ ω 0 x = 0, 

2 

dt dt 

sadac x -droSi cvladi fizikuri sididea 

β − milevis koeficienti, 

ω − sistemis sakuTari sixSirea 

0 

mocemuli gantolebis amonaxsnia 

x = A e 

0 

sadac 

−βt 

cos( ωt 

+ ϕ) 

ω − 

2 2 

= ω0 

β - milevadi rxevis sixSirea 

dx 

A 0 da ϕ - mudmivi sidideebia, damokidebuli t = 0 momentSi x − is da 

dt 

mniSvnelobebze _ anu sawyis pirobebze. milevadi rxevis grafiki 

mocemulia nax. 35. 

nax. 35. 

milevadi rxevis amplituda icvleba eqsponencialuri kanoniT: 

A( 

t) 

= 

Aoe 

−βt

sadac A 0 sawyisi amplitudaa 

milevis siCqare ganisazRvreba milevis koeficientiT: β = 

sadac r -xaxunis koeficientia 

m - merxevi wertilis masa. 

relaqsaciis dro δ drois Sualedia, romlis ganmavlobaSi milevadi 

rxevis amplituda mcirdeba е-jer. 

milevadi rxevis periodi 

1 

δ = 

β 

2π 

T = = 

ω 

64 

2π 

2 

ω − β 

milevis logariTmuli dekrementi _ λ ewodeba rxevis ori momdevno 

amplitudebis naturaluri logariTmebis Sefardebas 

2 

0 

A( 

t) 

T 1 

λ = � n = βT 

= = 

A( 

t + T ) δ N 

sadac Ne - rxevaTa ricxvia, romlis ganmavlobaSi amplituda mcirdeba 

е-jer. 

4.6. talRebi 

talRebi _ garemoSi (an vakuumSi) gavrcelebuli SeSfoTebaa _ 

talRebs gadaaqvs energia nivTierebis gadatanis gareSe. SeiZleba iTqvas, 

rom talRa _ garemos rxevaa. 

fizikuri bunebiT ansxvaveben meqanikur (drekad, bgeriT) talRebs, 

eleqtromagnitur talRebs talRgamtarebSi da sxvadasxva garemoSi 

(sinaTle, radiotalRebi, rentgenis gamosxiveba). 

SeSfoTebis orientaciis mixedviT ansxvaveben ganiv da grZiv 

talRebs. iseT talRas, romelSic rxeva warmoebs gavrcelebis 

mimarTulebis marTobulad, ganivi talRa ewodeba. grZivi talRa ewodeba 

iseT talRas, romelSic rxeva warmoebs talRis gavrcelebis 

mimarTulebiT. 

e 

r 

m

drekadi talRebi. ZiriTadi cnebebi. drekadi talRebi drekad 

garemoSi gavrcelebuli meqanikuri SeSfoTebaa (deformaciebi) talRis 

wyaroebad gvevlinebian sxeulebi, romlebic garemoze zemoqmedebiT masSi 

aRZraven SeSfoTebebs. 

talRebia. 

bgeriTi (akustikuri) talRebi mcire intensivobis drekadi 

adamianis yuri aRiqvams bgerebs, romelTa sixSire moTavsebulia 20 

hc-dan 20 khc-mde. 20 hc dabali sixSiris rxevebs infrabgera ewodeba, 20 

khc maRals ki ultrabgera. 

garemo erTgvarovania, Tu misi fizikuri Tvisebebi ucvlelia 

wertilidan wertilSi, xolo izotropiulia Tu fizikuri Tvisebebi 

erTnairia yvela mimarTulebiT (sawinaaRmdegoa anizropuli). 

msrboli talRa _ talRaa, romelsac gadaaqvs energia sivrceSi. 

talRis fronti _ im wertilebis geometriuli adgilia, romlebsac 

talRa garkveul momentSi miaRwevs. 

talRis zedapiri _ im wertilebis geometriuli adgilia, romlebic 

erTnair fazebSi irxevian. 

talRis gavrcelebis mimarTulebas sxivi ewodeba, xolo manZils, 

romelzedac sinosoidaluri talRuri moZraoba vrceldeba erT 

periodSi, talRis sigrZe ewodeba. 

4.7. brtyeli talRis gantoleba 

talRuri zedapirebis mixedviT ganixilebian brtyeli, sferuli da 

cilindruli talRebi. talRuri zedapirebis raodenoba talRaSi 

usasruloa, gansxvavebiT talRuri frontisgan. isini gadaadgilebas ar 

ganicdian. talRuri fronti ganicdis mudmiv gadaadgilebas da drois 

yoveli momentisTvis erTaderTia. 

brtyeli sinosoidaluri, x RerZis dadebiTi mimarTulebis gaswvriv 

gavrcelebuli talRis gantolebaa 

⎡ ⎛ x ⎞ ⎤ 

ξ ( x, t) 

= Acos⎢ω⎜ 

t − ⎟ + α ⎥ 

⎣ ⎝ v ⎠ ⎦ 

sadac A − talRis amplitudaa; 

ω − talRis cikluri (wriuli) sixSire; 

65

α − talRis sawyisi faza 

⎛ x ⎞ 

ω⎜t 

− ⎟ + α − talRis fazaa 

⎝ v ⎠ 

v − talRis gavrcelebis siCqare 

x t droSi talRis gavlili gzaa. 

2π 

ω 

talRuri ricxvi k = = , 

λ v 

sadac λ talRis sigrZea, ω − wriuli sixSire, v − talRis siCqare. 

talRuri veqtoria k , romlis moduli tolia talRuri ricxvis, 

mimarTulia talRuri zedapiris marTobulad. 

brtyeli talRis gantoleba, romelic vrceldeba k veqtoriT 

gansazRvruli mimarTulebiT, iqneba: 

ξ ( r , t) 

= Acos( 

ωt 

− kr 

+ α). 

sadac r − dakvirvebis wertilis radius veqtoria; 

α − rxevis faza r = 0 wertilSi, rodesac t = 0, 

anu rxevis faza 

koordinatTa saTaveSi. 

4.8. talRuri gantoleba 

talRuri gantoleba _ meore rigis diferencialuri gantolebaa da 

aRwers talRebis gavrcelebas erTgvarovan da izotropiul garemoSi: 

an 

2 2 2 

2 

∂ ξ ∂ ξ ∂ ξ 1 ∂ ξ 

+ + = 

2 2 2 2 2 

∂x 

∂y 

∂z 

v ∂t 

Δξ 

= 

66 

1 

2 

∂ ξ 

2 2 

v ∂t 

sadac ξ -fizikuri sididea, axasiaTebs SeSfoTebis gavrcelebas 

garemoSi v siCqariT, Δ- laplasis operatori 

∂ 

Δ = 

∂ 

2 

∂ 

+ 

∂ 

2 

2 

∂ 

+ 

z 

2 2 

x y ∂ 

brtyeli talRisTvis talRuri gantoleba Caiwereba: 

2 

2 

∂ ξ 

1 ∂ ξ 

= 2 2 2 

∂x 

v ∂t 

2

sadac x − talRis gavrcelebis mimarTulebaa, 

v - gavrcelebis siCqare 

talRuri gantolebis amoxsnaa: 

sadac k - talRuri veqtoria, 

α − rxevis sawyisi faza 

ξ ( r , t) 

= Acos( 

ωt 

− kr 

+ α) 

4.9. talRebis interferencia 

koherentuli talRebi erTnairi sixSiris talRebia, romelTa 

fazaTa sxvaoba drois mixedviT mudmivia ( Δϕ = const) 

koherentuli talRebis zeddebas interferencia ewodeba. 

interferenciis Sedegad rxevebi sivrcis zogierT wertilebSi 

Zlierdebian, zogierTSi ki _ asusteben erTmaneTs. 

ori erTnairi sixSiris da amplitudis Semxvedri sinusoidaluri 

talRebis zeddebis Sedegs mdgari talRa ewodeba (nax. 36) 

mdgari talRis gantolebaa 

amplituda 

burcobebi 

kvanZebi 

nax. 36. 

⎛ x ⎞ 

ξ = ⎜2 

Acos 2π 

⎟ ⋅ cosωt 

⎝ λ ⎠ 

x 

Amd = 2 Acos 

2π 

= 2Acos 

kx 

λ 

wertilebisTvis, romlebSic kx = 0, π... Ak 

= 0 − warmoadgenen mdgari 

talRis kvanZebs, xolo wertilebisaTvis, romlebSic kx = π , 

2 

67

3π A = 2A 

2... 

md amplituda maqsimaluria _ mdgari talRis burcobebia 

(nax. 36). 

umokles manZils mezobel kvanZebs (an burcobebs Soris mdgari 

talRis sigrZe ewodeba 

sadac λ − msrboli talRis sigrZea. 

λ = 

md 

mdgar talRaSi nawilakebis siCqare ganisazRvreba gantolebiT: 

68 

λ 

, 

2 

� ∂ξ 

⎛ 

x ⎞ 

ξ = = −⎜2 

Aω 

cos 2π 

⎟sinωt 

= −( 

2Aω 

cos kx) 

⋅sinωt 

∂t 

⎝ 

λ ⎠ 

garemos deformacia ki toli iqneba: 

∂ξ 

⎛ 2π 

x ⎞ 

ε = = −⎜2 

A cos 2π 

⎟ ⋅cosωt 

= −( 

2Ak 

cos kx) 

cosωt 

∂x 

⎝ λ λ ⎠ 

mdgari talRis SemTxvevaSi adgili ar aqvs energiis gadatanas. 

energia periodulad (2ω sixSiriT) burcobebidan, sadac kinetikuri 

energia 

k 

W maqsimaluria, gadaiqaCeba kvanZebSi potenciur 

molekuluri fizika 

p 

W energiad. 

molekuluri fizika (molekulur-kinetikuri Teoria) fizikis 

nawilia, romelic Seiswavlis sxeulTa fizikur Tvisebebs sxvadasxva 

agregatul mdgomareobaSi maTi molekuluri aRnagobis ganxilvis 

safuZvelze. molekulur-kinetikuri Teoria Camoyalibda XIX saukuneSi 

daltonis, klauziusis, maqsvelis da bolcmanis Sromebis ganviTarebiT. 

nivTierebis molekulur-kinetikur Teorias safuZvlad udevs Semdegi 

debulebebi: 

1. nivTierebas gaaCnia diskretuli agebuleba, is Sedgeba 

molekulebisagan (atomebisagan). nivTierebis erTi moli agregatuli 

mdgomareobisgan damoukideblad Seicavs molekulas ( -s 

avogadros ricxvi ewodeba). 

2. molekulebi nivTierebaSi imyofebian ganuwyvetel siTbur-qaosur 

moZraobaSi.

3. molekulebis siTburi moZraobis xasiaTi damokidebulia 

molekulebs Soris urTierTqmedebis bunebaze da icvlebian nivTierebis 

erTi agregatuli mdgomareobidan meoreSi gadasvlisas. 

4. siTburi moZraobis intensivoba damokidebulia sxeulis gaTbobis 

xarisxze, rac xasiaTdeba absoluturi (Termodinamikuri) temperaturiT 

. 

5. molekulur-kinetikuri TeoriiT sxeulis sruli energia 

warmoadgens Semdeg SesakrebTa jams: 

sadac da warmoadgenen sxeulis, rogorc mTlianis kinetikur 

da potencialur energiebs, xolo warmoadgens sxeulis Semadgeneli 

molekulebis (atomebis) energias. mas Sinagani energia ewodeba. 

debulebebis irib mtkicebulebebs warmoadgenen brounis moZraoba da 

difuziis movlena, xolo pirdapirs _ atomebze dakvirveba eleqtronuli 

da ionuri mikroskopebiT. 

makroskopiuli sistema. mdgomareobis parametrebi. sistemas (sxeuls) 

ewodeba makroskopiuli, Tu is Sedgeba molekulebis didi 

raodenobisagan. makroskopiuli sistemis Tvisebebis damaxasiaTebel 

sidideebs _ parametrebi ewodebaT. ZiriTad parametrebs warmoadgenen 

sistemis wneva , moculoba da temperatura . 

wneva _ fizikuri sididea, is ricxobrivad tolia farTobis 

erTeulze moqmedi marTobi Zalis: 

sadac _ marTobi (normaluri) Zalis ricxviTi mniSvnelobaa, 

moqmedi sxeulis zedapiris mcire farTobze. 

pa (paskali), 

temperatura _ fizikuri sididea, axasiaTebs sxeulis gaTbobis 

xarisxs. Termodinamikuri wonasworobis pirobebSi sistemis Semadgeneli 

sxeulebis temperaturebi erTnairia. 

_ kelvinis skaliT, _ celsiusis skaliT. 

wneva da temperatura statistikuri sidideebia, amitom am cnebebs 

erTi an ramdenime molekulisTvis azri ara aqvs. 

69 

,

sxeulis simkvrive _ ewodeba sxeulis masis Sefardebas mis 

moculobasTan. 

xvedriTi wona _ ewodeba sxeulis wonis Sefardebas mis 

moculobasTan. 

avogadros ricxvi warmoadgens nivTierebis erT molSi 

nawilakebis (molekulebis) raodenobas. 

moli -1 . 

molis masa _ molaruli masaa, xolo erTi molis moculobas 

molaruli moculoba ewodeba. 

molekuluri fizikis meTodebi. 

makrosistemebis Tvisebebis Seswavlis ori meTodi arsebobs: 

statistikuri da Termodinamikuri. 

mTeli sxeulis mdgomareobis damaxasiaTebeli sidideebi miiReba 

calkeuli molekulebis damaxasiaTebel sidideTa gasaSualoebiT. 

magaliTad, sxeulis temperatura warmoadgens molekulebis saSualo 

kinetikuri energiis proporciul sidides; gazis wneva gaizomeba im 

saSualo impulsiT, romelsac molekulebi gadascemen kedlis erTeul 

farTobs drois erTeulSi; idealuri gazis Sinagani energia warmoadgens 

misi Semadgeneli molekulebis saSualo kinetikuri energiebis jams. 

sxeulis makroskopiuli Tvisebebis Sesaswavlad saWiroa molekuluri 

warmodgenebidan gamomdinare, gamoviyvanoT is zogadi kanonzomierebani, 

romlebic mTel sxeuls axasiaTebs. aseTi Seswavla xorcieldeba e.w. 

statistikuri meTodis saSualebiT. statistikuri meTodi emyareba 

albaTobis Teorias. mas gamoyavs is zogadi kanonzomierebani, romlebsac 

emorCileba SemTxveviTi movlenaTa erToblioba. 

Termodinamikuri meTodi _ aqsiomaturi meTodia. aqsiomebad 

gamoiyenebian is fundamentaluri kanonebi, romelTa samarTlianoba 

70

emyareba eqsperimentebs. maT Termodinamikis principebi ewodebaT. am 

principebze dayrdnobiT, Termodinamika Seiswavlis energiis gardaqmnis 

movlenebs sistemebSi maTi molekuluri aRnagobis gaTvaliswinebis 

gareSe. 

Termometruli xelsawyoebis dasagraduireblad gamoiyeneba e.w. 

`reperuli wertilebi~ _ zogierTi nivTierebis dnobis da duRilis 

temperaturebi, romlebic mudmivi moculobis pirobebSi ar icvlebian. 

5. idealuri airebis kinetikuri Teoria 

airebis kinetikuri Teoria _ Seiswavlis airebis aRnagobas, maT 

fizikur Tvisebebs da efuZneba kvlevis statistikur meTods. 

5.1. idealuri airi. molekulur-kinetikuri Teoriis 

ZiriTadi gantoleba 

iseT airs, romlis molekulebs Soris ugulvebelyofilia 

urTierTqmedebis energia da romelSiac molekulebis zoma SeiZleba 

ugulebelvyoT, idealuri airi ewodeba. es fizikuri modelia. 

urTierTqmedeba idealuri airis molekulebs Soris warmoebs mxolod 

dajaxebisas. molekulebis uamravi dajaxebebis Sedegia airis warmoebuli 

wneva WurWlis kedlebze, romelSic is moTavsebulia. 

airebis molekulur-kinetikuri Teoriis ZiriTadi gantolebaa: 

airis wneva tolia erTeul moculobaSi moTavsebuli molekulebis 

gadataniTi moZraobis kinetikuri energiis -is, 

sadac _ airis molekulebis gadataniTi moZraobis saSualo 

kinetikuri energiaa, _ siCqaris kvadratis saSualo mniSvnelobaa. 

71

_ molekulebis saSualo kvadratuli siCqarea, 

kv _ molekulebis koncentraciaa, _ airis 

moculobaa, _ molekulebis raodenoba mocemul moculobaSi. 

kavSiri temperaturasa da molekulebis 

saSualo kinetikur energias Soris 

siTburi movlenebi gvarwmunebs, rom molekulebis saSualo 

kinetikuri energia izrdeba temperaturis gazrdiT. rodesac gazs 

vaTbobT, misi wneva izrdeba, rac aisaxeba imiT, rom gaTbobis dros 

izrdeba molekulebis kinetikuri energia, amitom izrdeba dajaxebisas 

molekulebis mier kedlisadmi gadacemuli impulsi. uamravi 

dakvirvebidan gamomdinare, airis temperatura proporciulia 

molekulebis saSualo kinetikuri energiis (warmoadgens energiis zomas) 

. molekulis gadataniTi moZraobis saSualo kinetikuri energia 

gad gad 

sadac gadamyvani koeficientia energiis erTeulebidan 

temperaturis erTeulebSi. _ bolcmanis universaluri 

mudmivaa, absoluturi temperaturaa, is aiTvleba kelvinis skalaze, 

sadac nulad aRebulia , romelsac absoluturi nuli ewodeba. 

es is temperaturaa, romelzec molekulaTa gadataniTi moZraoba wydeba. 

igi dakavSirebulia celsiusis skalaze aTvlil temperaturasTan 

formuliT . 

Tu gaviTvaliswinebT wina formulebs, miviRebT, rom idealuri airis 

wneva dakavSirebulia temperaturasTan TanafardobiT, rac agreTve 

ZiriTad gantolebas warmoadgens _ . 

Tu , wneva damokidebulia mxolod koncentraciaze da ar 

aris damokidebuli molekulis (atomis) gvarobaze. 

idealuri airis gantoleba. 

airebis kinetikuri Teoriis ZiriTadi gantolebidan vRebulobT: 

72

es gantoleba amyarebs kavSirs airis mdgomareobis da 

parametrebs Soris, amitom faqtiurad warmoadgens idealuri airis 

gantolebas. am gantolebiT sargebloba praqtikulad mouxerxebelia, 

radgan is Seicavs molekulebis saerTo ricxvs airSi _ . 

aRvniSnoT airSi molebis raodenoba -iT, maSin , amitom 

, miviRebT da , rac warmoadgens 

klapeironis gantolebas idealuri airisaTvis, sadac airis masaa, _ 

molaruli masa, xolo warmoadgens airis universalur 

mudmivas. 

idealuri airis kanonebi. 

idealuri airis gantolebidan SesaZlebelia miviRoT idealuri 

airebis yvela kanoni. 

1. izoTermuli procesi _ boil-mariotis kanoni: 

izoTermis gantolebaa. 

nax. 37 

nax. 1. naCvenebia -s damokidebulebis grafiki -ze. 

2. , izobaruli procesi _ gei-lusakis kanoni: _ 

izobaris gantolebaa. 

-s damokidebuleba -ze mocemulia nax. 2. . 

3. , izoqoruli procesi _ Sarlis kanoni: _ 

izoqoris gantolebaa. 

73

-s damokidebuleba -ze mocemulia nax. 3. . 

nax. 38 nax. 39 

4. avogadros kanoni. erTnairi wnevisa da temperaturis pirobebSi 

nebismieri airis tol moculobebSi molekulebis erTnairi ricxvi 

imyofeba, radgan 

maSin 

sadac _ molekulebis raodenobaa airis nebismier tol 

moculobebSi. 

5. daltonis kanoni. airebis narevis wneva tolia narevSi Semavali 

airebis parcialuri wnevebis jamis: 

parcialuri wneva ewodeba narevis erT-erTi komponentis wnevas, Tu 

marto is Seavsebda narevis mTel moculobas. 

5.2. idealuri airis Sinagani energia 

Sinagani energia moicavs airis Semadgeneli nawilakebis Sinagani 

moZraobis da urTierTqmedebis energiis yvela saxeobas. mravalatomiani 

airis Sinagani energiis Semadgenlebia: 

1. molekulebis siTburi da brunviTi moZraobebis kinetikuri 

energiebi. 

74

2. molekulebSi atomebis rxeviTi moZraobis kinetikuri da 

potencialuri energiebi. 

3. molekulebSorisi urTierTqmedebis potencialuri energia. 

4. atomebis da ionebis garsebis energia. 

5. atomebis birTvebSi nuklonebis kinetikuri da urTierTqmedebis 

potencialuri energiebi. 

energiis Tanabari ganawilebis kanoni 

Tavisuflebis xarisxTa ricxvi ewodeba im damoukidebel 

koordinatTa ricxvs, romlebic gansazRvraven sxeulis mdebareobas 

sivrceSi. magaliTisTvis, wrfeze moZrav materialur wertils gaaCnia 

Tavisuflebis xarisxi, sibrtyeze xolo sivrceSi rac 

Seesabameba moZraobis ganmsazRvrel koordinatebs. 

energiis Tanabari ganawilebis kanoni Tavisuflebis xarisxebis 

mixedviT. nebismieri saxis moZraobisas erT Tavisuflebis xarisxs 

eTanadeba energiis raodenoba. Tu molekulas gaaCnia Tavisuflebis 

xarisxTa raodenoba, maSin misi saSualo energia 

masis airis Sinagani energia iqneba 

5.3. molekulaTa siCqareTa ganawilebis kanoni 

maqsvelis kanoni molekulebis siCqareebis ganawilebis mixedviT, 

gansazRvravs molekulebis raodenobas saerTo raodenobidan, 

romelTa siCqareebi moTavsebulia SualedSi mocemul 

temperaturaze 

sadac _ molekulis siCqaris absoluturi 

mniSvnelobaa, _ molekulis masaa. 

molekulebis ganawilebis funqcia , romlis maTematikuri 

gamosaxuleba miiRo maqsvelma, Semdegia: 

75

ganawilebis funqcia eqvemdebareba normirebis pirobas 

integrireba warmoebs 0-dan -mde, radgan siCqareebi moTavsebulia am 

intervalSi. 

grafikulad maqsvelis ganawilebis kanoni gamosaxulia nax. 4., 

siCqares, romelsac ganawilebis mrudis maqsimumi Seesabameba ( -s RerZze 

1), ualbaTesi siCqare ewodeba. ualbaTesi siCqaris maxlobeli siCqareebi 

gaaCnia molekulebis umetes nawils, 

nax. 40 nax. 41 

grafikze daStrixuli farTi ricxobrivad tolia molekulebis im 

fardobiTi raodenobis, romelTa siCqareebi moTavsebulia da 

intervals Soris. 

temperaturis gazrdiT mrudis maqsimumi inacvlebs didi 

siCqareebisken, magram siCqareebis absoluturi mniSvneloba klebulobs 

(nax. 5). 

5.4. barometruli formula. bolcmanis ganawilebis kanoni 

dedamiwis mizidulobis velSi airis (haeris) molekulebi imyofebian 

ganuwyvetel siTbur-qaosur moZraobaSi da ganicdian simZimis Zalis 

moqmedebas. mizidulobis da siTburi moZraobis moqmedeba iwvevs airis 

stacionalur mdgomareobas, ris gamoc atmosferuli wneva da simkvrive 

mcirdeba dedamiwis zedapiridan manZilis gazrdiT. 

76

gantoleba, romelic gansazRvravs airis wnevis damokidebulebis 

kanons simaRlesTan, aris barometruli formula da aqvs Semdegi saxe: 

sadac _ airis wnevaa simaRleze, _ wnevaa simaRleze, 

_ molekulis masaa,M _ molaruli masaa. 

wnevis damokidebuleba simaRlesTan mocemulia nax. 6-ze. 

nax. 42 

radgan koncentraciis ganawilebis kanoni simaRlis 

cvlilebasTan, iqneba 

sadac da _ molekulaTa koncentraciaa im wertilebSi, 

romelTa Soris simaRleTa sxvaoba -is tolia. 

_ molekulis potencialuri energiaa simaRleze. 

Tu , adgili aqvs koncentraciis gaTanabrebas airis 

dakavebul moculobaSi. 

Tu dedamiwis atmosferos yvela molekula unda daeSvas 

dedamiwis zedapirze. 

bolcmanis ganawileba sruldeba nebismier Zalovan velSi 

sadac molekulis potencialuri energiaa. 

5.5. molekulaTa dajaxebaTa ricxvis da saSualo 

77

Tavisufali gadarbenis gamoTvla 

martivi cdebi (brounis moZraoba, difuzia) aCveneben, rom molekulis 

moZraoba airis erTi nawilidan meoresken ar aris Tavisufali, igi 

ganicdis mraval dajaxebas sxva molekulebTan. manZils _ , romelsac 

gaivlis molekula erTi dajaxebidan meoremde, Tavisufali ganarbeni 

ewodeba. radgan molekulaTa dajaxeba (Sexla) SemTxveviTi xasiaTisaa, 

amitom gansazRvraven Tavisufali ganarbenis saSualo mniSvnelobas _ 

sadac dajaxebaTa raodenobaa, 

molekulis efeqturi diametri _ minimaluri manZilia, 

romelzedac uaxlovdebian molekulebis centrebi Sexlisas. es xdeba 

maSin, roca maTi urTierTganzidvis potenciuri energia gadaaWarbebs 

maTi fardobiTi moZraobis kinetikur energias. 

gamoTvlebis Tanaxmad, drois erTeulSi TiToeuli molekula 

ganicdis dajaxebaTa saSualo raodenobas, rac tolia 

sadac _ molekulis saSualo ariTmetikuli siCqarea, _ 

koncentraciaa. 

mudmivi temperaturis pirobebSi airis koncentracia wnevis 

pirdapirproporciulia ( ) da molekulis saSualo Tavisufali 

ganarbeni tolia: 

formula aCvenebs, rom izrdeba temperaturis gazrdiT da wnevis 

SemcirebiT: da . 

6. Termodinamika 

6.1. Termodinamikuri sistema, damaxasiaTebeli parametrebi 

makroskopiuli sxeulebis Tvisebebis Seswavla SeiZleba 

ganxorcieldes maTi molekuluri aRnagobis gaTvaliswinebis gareSe. 

78

amisaTvis sakmarisia ganvixiloT energiis gardaqmnis movlenebi, 

romlebsac adgili aqvs fizikuri procesebis dros. 

fizikis im nawils, romelic Seiswavlis energiis gardaqmnis 

movlenebs, Termodinamika ewodeba. 

Termodinamikuri sistema _ makroskopiuli obieqtebia (sxeulebi da 

velebi), romlebsac SeuZliaT energocvla, rogorc erTmaneTTan, aseve 

gareSe sxeulebTan. 

Termodinamikuri sistemebis mdgomareoba calsaxad ganisazRvreba 

sami parametriT: wneviT P, moculobiT V da Termodinamikuri 

(absoluturi) temperaturiT T. 

Termodinamikuri sistemis mdgomareobas, romlis drosac misi 

mdgomareobis ganmsazRvreli parametrebi ucvlelia, ucvleli garemo 

pirobebisas, wonasworuli mdgomareoba ewodeba. am mdgomareomaSi 

sistemis temperatura mudmivia da warmoadgens molekulebis moZraobis 

intensivobis zomas. sistema arawonasaworul mdgomareobaSia, Tu mis 

Termodinamikur parametrebs sxvadasxva wertilebSi sxvadasxva 

mniSvnelobebi gaaCnia. 

relaqsacia iseTi procesia, rodesac sistema arawonasaworuli 

mdgomareobidan gadadis wonasworulSi. 

Termodinamikuri sistema stacionalur mdgomareobaSia, Tu misi 

mdgomareobis ganmsazRvreli parametrebi, droSi mudmivi rCebian sistemis 

nebismier wertilSi. 

process wonasworuli ewodeba, rodesac sistemis mdgomareobis 

cvlileba imdenad nela mimdinareobs, rom is uwyvetad gaivlis 

usasrulod maxlobel Termodinamikur wonasworul mdgomareobebs. 

procesebi, romlebic ar akmayofileben am moTxovnebs _ 

arawonasworulia. 

Termodinamikuri sistemis Sinagani energia 

sistemis Sinagani energia U damokidebulia mxolod mis 

Termodinamikur mdgomareobaze. is calsaxa funqciaa sistemis 

mdgomareobis: misi cvlileba dU-Ti sistemis gadasvlisas 1 

mdgomareobidan 2-Si ar aris damokidebuli procesis gvarobaze da 

tolia 

79

sadac da Sinagani energiis mniSvnelobebia 1 da 2 

mdgomareobebSi. Tu sistema asrulebs wriul process (cikls), anu 

ubrundeba Tavis sawyis mdgomareobas, maSin Sinagani energiis cvlileba 

iqneba nuli. 

muSaoba da siTbo 

nebismieri Termodinamikuri sistemis mdgomareobis cvlilebas Tan 

axlavs muSaoba, romelsac is asrulebs _ an muSaoba, romelsac masze 

asruleben gare Zalebi. cxadia, muSaoba gamoisaxeba sistemis 

parametrebiT da nebismieri maTi cvlileba dakavSirebulia Sesrulebul 

muSaobasTan. 

Tu sistemis elementarul Sesrulebul muSaobas gare sxeulebze 

aRvniSnavT -Ti, maSin niutonis me-3 kanonis Tanaxmad 

sadac gare sxeulebis Sesrulebuli muSaobaa sistemaze. 

sistemis mier Sesrulebuli muSaoba gare wnevis Zalis sawinaaRmdegod 

(gafarToebis muSaoba), gamoisaxeba formuliT 

sadac p _ gare wnevaa, dV _ moculobis cvlileba. 

sruli muSaoba gafarToebaze 

sistemis mdgomareobis cvlilebas procesi ewodeba. muSaoba da 

siTbo _ procesis funqciebia. amitom dA da dQ aRniSnaven muSaobis da 

siTbos mcire sidideebs, xolo dU warmoadgens srul diferencials. 

6.2. Termodinamikis pirveli kanoni 

80

Termodinamikuri sistemis Sinagani energiis Secvlis ori gza 

arsebobs: 

a) siTbos miReba an gacema, 

b) muSaobis Sesruleba. 

Termodinamikis I kanoni (principi), warmoadgens energiis Senaxvis 

kanons siTburi procesebisaTvis: sistemis mier miRebuli siTbos 

raodenoba (dQ) ixarjeba am sistemis Sinagani energiis gazrdaze da mis 

mier Sesrulebul muSaobaze gare Zalebis sawinaaRmdegod: . Tu 

Sinagani energiis cvlileba usasrulod mcirea, maSin formula aseT 

saxes miiRebs 

Tu sistema periodulad ubrundeba sawyiss mdgomareobas, misi 

Sinagani energiis cvlileba da Termodinamikis I kanonis Tanaxmad 

e.i. ar SeiZleba avagoT periodulad momqmedi siTburi manqana, 

romelic met muSaobas Seasrulebda, vidre Sesabamisi miwodebuli 

energiaa. sxva sityvebiT, pirveli gvaris mudmivi Zravi (perpetuum mobile) 

SeuZlebelia. 

6.3. airebis siTbotevadoba. maieris gantoleba 

sistemis (sxeulis) siTbotevadoba fizikuri sididea da tolia 

sistemisTvis gadacemuli siTbos raodenobis Sefardebis mis 

temperaturis cvlilebasTan: 

kuTri siTbotevadoba ( ). nivTierebis kuTri siTbotevadoba 

ewodeba skalarul fizikur sidides, romelic tolia sxeulis 

siTbotevadobis Sefardebis amave sxeulis masasTan: 

81

e.i. kuTri siTbotevadoba siTbos is raodenobaa, romelic saWiroa 

sxeulis erTi kilogramis gasaTbobad erTi kelvinis gradusiT. 

iqneba 

sxeulis gasaTbobad kelviniT, saWiro siTbos raodenoba , 

da kuTri siTbotevadobebia Sesabamisad mudmivi wnevis da 

mudmivi moculobis pirobebSi. 

molaruli siTbotevadoba ( ). nivTierebis molaruli 

siTbotevadoba ewodeba fizikur sidides, romelic tolia sistemis 

siTbotevadobis fardobis masSi Semaval nivTierebis raodenobasTan. 

ansxvaveben siTbotevadobebs mudmivi moculobis da wnevis pirobebSi. 

sadac molaruli siTbotevadobaa mudmivi moculobis pirobebSi, 

2. 

_Sinagani energiis cvlilebaa temperaturis cvlilebisas, 

_ molekuluri masaa, _ nivTierebis masa. 

sadac _ molaruli siTbotevadobaa mudmivi wnevis pirobebSi, 

_ sxeulisadmi gadacemuli siTbos raodenobaa temperaturis 

cvlilebisas. maieris gantoleba akavSirebs molarul siTbotevadobebs 

da erTmaneTTan: 

kuTri siTbotevadobebisTvis maieris gantolebaa: 

sadac _ airebis universaluri mudmivaa. 

82

airebis universaluri mudmiva ricxobrivad tolia im muSaobis, 

romelic sruldeba airis nivTierebis erTi molis gasaTbobad erTi 

kelviniT mudmivi wnevis pirobebSi. 

gazebis siTbotevadoba da Tavisuflebis xarisxi 

Taviseflebis xarisxTa mixedviT, Sinagani energiis ganawilebis 

kanonis da maieris gantolebis gaTvaliswinebiT, vRebulobT: 

sadac da molaruli siTbotevadobebia mudmivi moculobis da 

wnevis pirobebSi. Termodinamikuri procesebis ganxilvisas, sargebloben 

gamosaxulebiT 

erTatomiani molekulisTvis Tavisuflebis xarisxi = 3, 

oratomianisTvis = 5, xolo sam da mravalatomiani airebisTvis = 6. 

6.4. adiabaturi procesi 

process, romelic mimdinareobs garemosTan siTbocvlis gareSe, 

adibaturi procesi ewodeba _ = 0. 

imisaTvis, rom airi adiabaturad gafarTovdes, is unda movaTavsoT 

siTbos aragamtar WurWelSi. gafarToebisas airi siTbos ar miiRebs, e.i. 

muSaobas asrulebs Sinagani energiis xarjze da civdeba. adiabaturi 

kumSvisas muSaobas gareSe Zala asrulebs da, radgan airi siTbos ar 

gascems, misi Sinagani energia izrdeba, e.i. airi gaTbeba. 

maSasadame, adiabaturi gafarToebis dros airi civdeba, xolo 

adiabaturi kumSvis dros _ Tbeba. 

ewodeba: 

adiabaturi procesis gamomsaxvel gantolebas puasonis gantoleba 

83

adiabaturi gafarToebis muSaoba 

sadac _ adiabaturi gafarToebis muSaobaa, da procesis 

sawyisi da saboloo temperaturebia. 

6.5. wriuli procesi. Seqcevadi da Seuqcevadi procesebi 

wriuli procesi anu cikli _ Termodinamikuri procesebis 

erTobliobaa, romlis Sesrulebis Semdeg sxeuli ubrundeba sawyis 

mdgomareobas (nax. 7) 

nax. 43 

gamovTvaloT ciklis dros Sesrulebuli muSaoba. aRvniSnoT ACB 

procesis dros Sesrulebuli muSaoba -iT. is dadebiTia da ACB 

farTobiT gamoisaxeba. kumSvis procesis dros Sesrulebuli 

muSaoba uaryofiTia da ADB farTobiT gamoisaxeba. amitom, ciklis 

dros Sesrulebuli sruli muSaoba toli iqneba 

ACB ADB ACB 

e.i. ciklis dros Sesrulebuli muSaoba cikliT SemosazRvruli 

farTobiT gamoisaxeba. ciklis dros Sinagani energiis cvlileba nulis 

tolia: 

cikls pirdapiri ewodeba, Tu sistema asrulebs dadebiT muSaobas 

84

da Seqceulia (ukucikli), Tu sistemis Sesrulebuli muSaoba 

uaryofiTia 

Seqcevadi ewodeba iseT Termodinamikur process, romelic SeiZleba 

warmoebdes rogorc pirdapiri, ise Sebrunebuli mimarTulebiT; 

Sebrunebuli procesis dros sistema TanmimdevrobiT gaivlis yvela im 

saSualedo mdgomareobas, romlebic man gaiara pirdapiri procesis dros 

da ise dabrunda sawyis mdgomareobaSi,rom gareSe sxeulebSi araviTari 

cvlileba ar moxda. winaaRmdeg SemTxvevaSi cikli Seuqcevadia. 

saerTod, nebismieri wminda meqanikuri procesi (xaxunis gareSe 

mimdinare), Seqcevadia. praqtikulad Seqcevadi procesebi ar 

xorcieldeba. 

6.6. siTburi manqana. margi qmedebis koeficienti 

nebismieri siTburi manqanis Semadgeneli sami nawilia gamaxurebeli, 

macivari da muSa sxeuli. 

muSa sxeuli _ Termodinamikuri sistemaa, romelic asrulebs wriul 

process da siTbocvlaSia sxva sxeulebTan. Cveulebriv is airia. 

gamaxurebeli (siTbogadamcemi) _ sxeulia, romelic Termodinamikur 

sistemas gadascems energias siTbos saxiT. 

macivari (siTbos mimRebi) _ sxeulia, romelic Termodinamikuri 

sistemidan Rebulobs energias siTbos saxiT. 

siTburi manqanis Termodinamikuri margi qmedebis koeficienti 

ewodeba sasargeblo muSaobaze daxarjul siTbos raodenobis 

Sefardebas gamxureblidan miRebul siTbos raodenobasTan 

sadac _ muSa sxeulis mier gamaxureblidan miRebuli siTbos 

raodenobaa temperaturisas, 

_ macivris mier muSa sxeulisgan miRebuli siTbos raodenobaa 

temperaturisas, 

85

ciklis dros. 

_ siTburi manqanis mier Sesrulebuli sasargeblo muSaobaa erTi 

_ margi qmedebis Termodinamikuri (mqk) koeficientia. 

6.7. karnos cikli. karnos ciklis mqk 

karnos cikli pirdapiri wriuli procesia, romlis drosac sistemis 

Sesrulebuli muSaoba maqsimaluria. vTqvaT cilindrSi dguSis qveS 

moTavsebulia idealuri airis erTi moli (nax. 8). cilindris kedlebi da 

dguSi siTbos aragamtarebia, xolo fskeri atarebs siTbos. mivadgaT 

cilindris fskers gamaxurebeli , maSin airi miiRebs gamaxureblis 

temperaturas, rac iqneba airis sawyisi temperatura. aRvniSnoT sawyisi 

moculoba , sawyisi wneva _ . airis sawyisi mdgomareoba diagramaze 

gamoisaxeba 1 wertiliT (nax. 9). 

nax. 44 nax. 45 

Tu airis sawyisi wneva metia gareSe wnevaze, maSin is 

gafarTovdeba da Seasrulebs muSaobas, masTan, airis temperatura ar 

Seicvleba, radganac is iRebs siTbos gamaxureblisgan. Tu gafarToeba 

sakmaod nela iwarmoebs, procesi izoTermuli iqneba da 2 mdgomareobaSi 

airis parametrebi iqneba da . 1-2 izoTermuli gafarToebis dros 

muSa sxeulis miRebuli siTbos raodenoba aRvniSnoT -iT. 2 

86

mdgomareobaSi movaciloT cilindrs gamaxurebeli da mivadgaT siTbos 

aragamtari firfita, maSin airi gafarTovdeba da is siTbos arc 

miiRebs da arc gascems e.i. procesi adiabatauri iqneba. airi Seasrulebs 

muSaobas Sinagani energiis xarjze, misi temperatura Semcirdeba da 

gafarToebas SevwyvetT, rodesac airis temperatura macivris 

temperaturas gautoldeba. adiabaturi procesi 2-3 mrudiT gamoisaxeba da 

3 mdgomareobaSi airis parametrebi iqnebian da . am mdgomareobaSi 

firfitis nacvlad cilindrs mivadgaT macivari da daviwyoT airis 

kumSva. am dros muSaobas gareSe Zala asrulebs, airi ar Tbeba, radgan 

gamoyofili siTbo macivars gadaecema e.i. kumSva izoTermulad 

iwarmoebs, procesi gamoisaxeba 3-4 izoTermiT. aRvniSnoT am procesis 

dros macivrisadmi gadacemuli siTbos raodenoba -iT. me-4 

mdgomareobaSi airis parametrebia da . me-4 mdgomareobaSi 

movaciloT cilindrs macivari da mivadgaT siTbogaumtari firfita. 

radganac, exla gazi izolirebulia siTbos mxriv, kumSva adiabaturad 

iwarmoebs da, Tu meoTxe mdgomareoba saTanadod aris SerCeuli, airi 

daubrundeba sawyis 1 mdgomareobas. Sesruldeba erTi cikli, romelic 

Sedgeba ori izoTermuli da ori adiabaturi procesisgan. 

mTeli ciklis dros muSa sxeulis mier miRebuli siTbos raodenoba 

iqneba . radgan airi asrulebs wriul process _ cikls, 

energiis cvlileba da Termodinamikis I kanonis Tanaxmad 

A . 

karnos Teorema. gamxureblis da macivris erTi da igive 

temperaturebisas, verc erTi siTburi manqanis mqk ver gadaaWarbebs 

karnos Seqcevadi ciklis mqk-s. 

karnos ciklis mqk ar aris damokidebuli muSa sxeulis gvarobaze 

da warmoadgens mxolod gamaxureblis da macivris temperaturebis 

funqcias 

es formula gviCvenebs agreTve mqk gazrdis gzas. karnos Seqcevadi 

ciklisaTvis sruldeba Sefardeba 

87

nebismieri ciklisaTvis Termodinamikuri mqk 

sadac tolobis niSani Seqcevad cikls Seesabameba, utolobis ki _ 

Seuqcevads. 

6.8. Termodinamikis meore kanoni 

Termodinamikis I kanoni faqtiurad energiis mudmivobis kanonia 

siTburi procesebisaTvis da amitom yvela procesSi sruldeba, magram is 

ar aCvenebs procesis mimarTulebas. magaliTad, siTbo TavisTavad 

arasodes ar gadadis civi sxeulidan cxelze, Tumca am process 

Termodinamikis I kanoni ar ewinaaRmdegeba. 

Termodinamikis II kanoni (principi) gansazRvravs Termodinamikuri 

procesis mimarTulebas da amiT pasuxs cems kiTxvas, Tu romeli 

procesebi SeiZleba ganxorcieldes bunebaSi. 

ganvixiloT meore kanonis ramdenime formulireba: 

a) SeuZlebelia procesi, romlis erTaderTi Sedegi iqneba siTbos 

gadacema civi sxeulidan cxelze. 

b) SeuZlebelia iseTi perioduli procesi, romlis erTaderTi 

Sedegi iqneboda siTburi energiis gardaqmna mis eqvivalentur muSaobad. 

g) SeuZlebelia avagoT iseTi periodulad momqmedi manqana, romlis 

moqmedebis erTaderTi Sedegia muSaobis Sesruleba siTbos erTi wyaros 

gacivebis xarjze. 

davuSvaT, rom sxeuli vardeba dedamiwaze. vardnis procesSi 

meqanikuri potenciuri energia siTbod gardaiqmneba _ Tbeba sxeulic da 

miwac. Sebrunebuli procesi _ siTburi energiis xarjze (miwis gacivebis), 

sxeulis asvla imave simaRleze arasodes ganxorcieldeba. 

Termodinamikis I kanonis TvalsazrisiT es procesi ar aris SeuZlebeli, 

magram igi ewinaaRmdegeba Termodinamikis meore kanons, radgan 

Sesruldeba muSaoba mxolod erTi wyaros _ dedamiwis gacivebis xarjze. 

88

6.9. entropia. Termodinamikis gaerTianebuli kanoni 

siTburi procesebis Seswavlidan gamomdinareobs, rom unda 

arsebobdes iseTi funqcia , romlis cvlileba 1-dan 2 mdgomareobaSi 

raime Seqcevadi procesis gadasvlisas, toli iqneba 

funqcias entropia ewodeba. da entropiis mniSvnelobebia 1 da 

2 mdgomareobebi. -s dayvanili siTbo ewodeba. 

amgvarad, entropia warmoadgens mdgomareobis funqcias, romlis 

cvlileba Seqcevadi procesis dros, gvaZlevs dayvanil siTbos 

raodenobaTa jams. 

Tu sistema izolirebulia, maSin , Seqcevadi procesisTvis 

gveqneba , xolo nebismierisTvis e.i. izolirebul 

sistemaSi procesebi ise warmoebs, rom sistemis entropia mudmivi rCeba, 

an izrdeba. es debuleba warmoadgens Termodinamikis meore kanonis 

uzogades formulirebas. 

Tu 1 da 2 wertilebi usasrulod axlosaa erTmaneTTan e.i. sistemis 

mdgomareoba usasrulod mcired icvleba, maSin entropiis sasrulo 

cvlilebis magivrad gveqneba usasrulod mcire nazrdi da 

Termodinamikis meore kanoni Semdeg saxes miiRebs (entropiis zrdis 

kanoni): 

xolo, Tu sistema araizolirebulia, maSin 

kavSiri entropiasa da informacias Soris mogvca bolcmanma 

sadac _ bolcmanis mudmivaa, _ Termodinamikuri albaToba. am 

formulidan gamomdinareobs entropiis fizikuri arsi kinetikuri 

TeoriiT _ entropia warmoadgens fizikuri sistemis mouwesrigeblobis 

zomas. 

89

entropiis warmodgena, rogorc damoukidebeli parametris ucnauri 

Canda, radgan ar arsebobda misi gazomvis pirdapiri meTodi. es problema 

moxsna nerstis siTburma Teoriam, romlis Tanaxmadac Tu , maSin 

, e.i. absolutur nulze entropiac nuls utoldeba. amgvarad, 

nerstis TeoremiT moxda entropiis kalibreba. 

cnobilia, rom nebismier energetikul kontaqts Seesabameba 

parametrebis wyvili. Termodinamikis meore kanonidan , e.i. 

temperatura da entropia parametrTa wyvilia. radgan siTbocvla 

proporciulia entropiis zrdis, is miCneulia eqstenciur parametrad, 

xolo temperatura intensiurad. 

Termodinamikis I da II kanonebis ( ) SerwymiT miviRebT 

Termodinamikis gaerTianebul kanons: 

es formula saSualebas gvaZlevs srulad miviRoT Termodinamikuri 

Tanafardobebi nebismieri konkretuli fizikuri sistemisaTvis. 

7. realuri airebi 

7.1. realuri airebi. molekuluri Zalebi 

realuri airebis Tvisebebis gansxvaveba idealurebTan gansakuTrebiT 

SesamCnevia didi wnevebisa da dabali temperaturebis dros. didi wnevebis 

dros airis molekulebs Soris manZili imdenac mcirdeba, rom aRar 

SeiZleba molekulebis sakuTari moculobisa da maTi urTierTqmedebis 

ugulvebelyofa, xolo temperaturis Semcirebis dros, molekulebis 

saSualo kinetikuri energia mcirdeba, amitom dabal temperaturaze 

molekulebis urTierTqmedebis energiebs veRar ugulvebelvyofT maT 

kinetikur energiebTan SedarebiT. 

realuri iseT airebs ewodebaT, romlis molekulebs Soris 

moqmedebs mizidva-ganzidvis Zalebi. es Zalebi sxvadasxvanairad arian 

damokidebulni molekulebs Soris manZilebze. atomis zomis tol 

manZilebze (10 -10 m) Warbobs ganzidvis Zalebi, xolo 10 -9 m manZilebze _ 

ki urTierTmizidvis Zalebi (nax. 10). tolqmedi Zala 

90

Tu 

Tu xolo Tu 

_ aRniSnulia molekulebSorisi moqmedebis radiusi _ es aris 

manZili molekulebs Soris wonasworuli mdgomareobisas, rodesac 

siTburi moZraoba ar warmoebs. 

eTanadeba molekulebis wonasworul mdgomareobas, rasac 

ewinaaRmdegeba molekulebis siTburi moZraoba. 

nax. 46 

nivTierebebis sxvadasxva agregatuli mdgomareobis kriteriums 

warmoadgens Tanafardoba molekulebis urTierTqmedebis minimalur 

potencialur da sidides Soris. gansazRvravs mizidvis 

Zalebis sawinaaRmdego muSaobas, raTa daaSoron molekulebi 

wonasworul mdgomareobas ( ), xolo sidide gansazRvravs 

molekulebis siTburi moZraobis intensivobis xarisxs. 

Tu , nivTiereba imyofeba airovan mdgomareobaSi, radgan 

molekulebis intensiuri siTburi moZraoba ewinaaRmdegeba maT SeerTebas. 

Tu , nivTiereba imyofeba myar mdgomareobaSi. am dros 

molekulebs Soris mizidvis Zalebs gaaCniaT didi mniSvnelobebi da 

isini asruleben mxolod rxeviT moZraobas wonasworuli mdgomareobis 

91

maxloblobaSi. Tu nivTiereba imyofeba Txevad mdgomareobaSi. 

siTburi moZraobis Sedegad molekulebi gadaadgildebian siTxis mTel 

moculobaSi, magram ar scildebian erTmaneTs -ze met manZilze. 

amgvarad, nebismieri nivTierebis agregatuli mniSvneloba 

ganisazRvreba -sididiT. magaliTad, inertuli airebis mcire 

sidideebia, xolo liTonebis didi, amitomac Cveulebriv (oTaxis) 

temperaturaze isini Sesabamisad imyofebian airovan da myar 

mdgomareobebSi. 

7.2. van der vaalsis gantoleba 

realuri airis gantolebis misaRebad holandielma mecnierma 

idealuri airis gantolebaSi ori Sesworeba Seitana: 1. Sesworeba 

molekulebis sakuTar moculobaze, 2. Sesworeba molekulebis 

urTierTmizidvaze. 

1. imis gamo, rom realuri airis molekulebs sakuTari moculoba 

gaaCniaT, maT SeuZliaT moZraoba ara mTel moculobaSi, aramed 

moculobis im nawilebSi, romelic Tavisufalia. Tu -Ti aRvniSnavT 

moculobis im nawils, romelic molekulebisTvis SeuRwevadia, maSin 

molekulebis moZraobis Tavisufali moculoba iqneba ara , aramed 

. warmoadgens Sesworebas molekulebis sakuTari moculobis 

gaTvaliswinebiT, da is tolia erT mol gazSi arsebuli molekulebis 

gaoTxkecebuli moculobis: 

sadac _ avogadros ricxvia, _ molekulis diametri. 

2. airSi molekulebis urTierTmizidvis gaTvaliswineba iwvevs 

damatebiTi wnevis warmoqmnas, romelsac Sinagani wneva ewodeba. van der 

vaalsis gamoTvlebiT Sinagani wneva tolia 

sadac _ mudmivaa, damokidebulia airis gvarobaze, _ molaruli 

moculobaa. 

92

Sesworebebis gaTvaliswinebiT realuri airis mdgomareobis 

gantoleba _ van der vaalsis gantoleba _ erTi moli airisTvis, iqneba 

saxisaa 

nebismieri masisaTvis realuri airis mdgomareobis gatoleba Semdegi 

sadac airis moculobaa. 

Tu da , van der vaalsis gantoleba gadadis 

klapeironis gantolebaSi. 

7.3. realuri airis izoTermebi 

realuri airis izoTerma warmoadgens airis molaruli moculobis 

damokidebulebas wnevaze mudmivi temperaturis pirobebSi. kritikul 

mdgomareobamde ( -mde), mrudebi warmoadgenen nivTierebis uwyvet 

gadasvlas airovani mdgomareobidan siTxeSi (nax. 11). 

monakveTi Seesabameba nivTierebis airovan mdgomareobas, _ 

airovani mdgomareobis gadasvlas siTxur mdgomareobaSi, _ siTxes. 

wertili Seesabameba mduRare siTxes, _ mSral najer orTqls (nax. 

11). 

nax. 47 

93

temperaturis gadidebiT grafiki iwevs zeviT da orTqlis 

kondensaciis dasawyisis da damTavrebis wertilebi erTmaneTs 

uaxlovdeba. rac metia temperatura, miT naklebi moculobis dros iwyeba 

da miT meti moculobis dros mTavrdeba orTqlis siTxed qceva. Tu 

avagebT izoTermebs TandaTan ufro maRal temperaturaze, miviRebT, rom 

erT garkveul temperaturaze, romelsac kritikuli ewodeba, grafikis 

nawili moispoba da am ubnis nacvlad miviRebT gadaRunvis 

wertils. -s Sesabamiss mdgomareobas kritikuli ewodeba, xolo mis 

Sesabamis parametrebs _ kritikuli wneva , kritikuli moculoba da 

kritikuli temperatura . 

kritikul temperaturaze ispoba gansxvaveba airsa da siTxes Soris. 

Tu izoTermas avagebT temperaturaze, vnaxavT, rom realuri airis 

izoTerma emTxveva idealuri airis izoTermas, e.i. aq nivTiereba SeiZleba 

imyofebodes mxolod airis mdgomareobaSi, anu -s zeviT SeuZlebelia 

airis (orTqlis) gaTxevadeba. 

van der vaalsis gantolebidan kritikuli mdgomareobisaTvis, 

SesaZlebelia kritikuli parametrebis gamoTvla. 

94

damateba 

cdomilebaTa Teoriis ZiriTadi debulebebi 

fizikuri sididis gazomva niSnavs mis Sedarebas imave gvaris 

fizikur sididesTan, specialuri gamzomi saSualebebis (xelsawyoebis) 

gamoyenebiT. 

gazomvas, romlis drosac fizikuri sididis mniSvnelobas uSualod 

vsazRvravT cdiT, pirdapiri gazomva ewodeba. aseTi gazomvis magaliTia 

temperaturis gazomva TermometriT, denis Zalis – ampermetriT da sxva. 

pirdapiri gazomvisas sidideebis mniSvnelobebi uSualod aiTvlebian 

xelsawyoebze. 

arapirdapiri gazomvebis SemTxvevaSi, saZiebeli sidideebis 

gansazRvrisas, eyrdnobian sxva fizikuri sidideebis pirdapiri gazomvis 

Sedegs, romlebTanac saZiebeli sidide funqcionalur kavSirSia. 

uxeSi. 

gazomvis cdomilebebi SeiZleba iyos sistematuri, SemTxveviTi da 

uxeSi cdomilebebi dakavSirebulia gamzomi xelsawyoebis 

dazianebasTan, eqsperimentatoris SecdomebTan anaTvlis aRebisas an 

gazomvis pirobebis mkveTr cvlilebasTan. 

sistemuri cdomileba ori saxisaa: meToduri da instrumentaluri. 

meToduri cdomileba gamowveulia movlenis fizikuri Teoriis, 

gazomvis meTodis da saZiebeli sididis saangariSo formulis 

naklovanebebiT. 

instrumentaluri cdomileba gamowveulia danadgaris konstruqciis 

da gamzomi xelsawyobis arasrulfasovnebiT. 

gazomvis SemxveviTi cdomilebebi iseTi cdomilebebia, romelTa 

absoluturi sidideebi da niSani icvlebian erTi da igive fizikuri 

sididis mravaljeradi gazomvebisas. maTi gamomwvevi faqtorebi mravalia 

da ar eqvemdebarebian aRricxvas. SemTxveviT cdomilebebi Tan sdevs 

nebismier gazomvas, maTi Tavidan acileba SeuZlebelia. 

cdomilebaTa Teoriis saSualebiT SeiZleba gamoviTval- 

oTYmosalodneli cdomileba da davadginoT sazRvrebi, romelTa Sorisac 

unda iyos moTavsebuli gasazomi sididis WeSmariti mniSvneloba. 

95

cdomilebaTa sruli Teoriis ganxilva warmoadgens sakmarisad rTul 

amocanas. Cven gavecnobiT damuSavebis ZiriTad da gavrcelebul meTodebs. 

cdomilebis Semcirebis mizniT mizanSewonilia saZiebeli sididis 

mravaljeradi gazomva. mravali gazomvis saSualo ariTmetikuli, cxadia 

yvelaze axlos iqneba gasazomi sididis WeSmarit mniSvnelobasTan. 

davuSvaT Catarda fizikuri sididis -jer gazomva da gazomvis 

Semdegad miRebuli iqna mniSvnelobebi. gasazomi fizikuri 

sididis saSualo mniSvneloba iqneba 

calkeuli gazomvebis absoluturi cdomilebebi 

sididis saSualo absoluturi cdomileba iqneba: 

gasazomi sididis WeSmariti mniSvneloba moTavsebuli iqneba 

da sidideebs Soris, cdis pasuxi Caiwereba: 

romelsac xSirad ase warmoadgenen: 

mxolod absoluturi cdomileba ver gansazRvravs gazomvis 

sizustes. erTi da igive absoluturi cdomileba SeiZleba mcired 

CaiTvalos erT SemTxvevaSi da piriqiT, Zalian uxeSad sxva 

SemTxvevisaTvis, es damokidebulia gasazomi fizikuri sididis 

mniSvnelobaze. gazomvis sizustis Sesafaseblad saWiroa ganisazRvros 

fardobiTi cdomileba. fardobiTi cdomileba aris absoluturi 

cdomilebis Sefardeba gasazomi fizikuri sididis WeSmarit 

mniSvnelobasTan, is gviCvenebs gasazomi sididis ra nawils Seadgens 

absoluturi cdomileba. Cveulebriv fardobiT cdomilebas procentebSi 

gamosaxaven. cdis sizuste ganisazRvreba saSualo fardobiTi 

cdomilebiT . 

96

Tu calkeul gazomvaTa SemTxvevaSi cdomilebebi ar aRemateba im 

Secdomas, rasac xelsawyo iZleva, sakmarisia Catardes erTjeradi 

gazomva. (magaliTad: temperaturis gazomva TermometriT). im SemTxvevaSi 

rodesac SeiZleba SemovifargloT erTi gazomviT, absoluturi 

cdomileba xelsawyos umciresi danayofis naxevris toli iqneba. 

magaliTad, Tu Termometris umciresi danayofi -ia, maSin absoluturi 

cdomileba aiReba . 

arapirdapiri gazomvebis SemTxveviTi cdomilebebis gamoTvla. 

sidideebis arapirdapiri gamoTvlebisas fizikuri sidide 

gamoiTvleba formuliT 

sadac fizikuri sididis pirdapiri 

gazomvis Sedegebia. arapirdapiri gazomvis absoluturi cdomileba 

gamoiTvleba formuliT , (1) 

sadac , funqciis kerZo warmoebulia -iT; absoluturi 

cdomilebaa sididis gazomvisas. 

arapirdapiri, (iribi) gazomvis Sedegs warmoadgenen: 

sadac - sadac funqcia -is 

mniSvneloba Seesabameba -bis saSualo mniSvnelobas. fardobiTi 

cdomilebebisTvis gveqneba 

sadac 

gamoviTvaloT cdomileba cilindruli formis sxeulis simkvrivis 

gansazRvrisas: 

sadac - sxeulis simkvrivea, - misi masa, - cilindris diamteria, 

- misi simaRlea. 

(1) formuliT gamoviTvaloT absoluturi cdomileba: 

; 

97

sadac , , - pirdapiri gazomvebis absoluturi cdomilebebia – 

masis, diametris da simaRlis. 

gamoTvlebis pasuxia 

cdomilebebis gamoTvla (2) formuliT 

gavalogariTmoT -s formula: 

gansazRvroT warmoebulebi : 

fardobiTi cdomileba: 

. 

sadac -sidideebis saSualo mniSvne- 

lobebia, -pirdapiri gazomvebis cdomilebebia, _ 

absoluturi cdomilebaa. 

gamoTvlebis pasuxia 

absolutur cdomilebas da saSualo Sedegs amrgvaleben 

Semdegnairad: 

• Tu pirveli ukusagdebi cifri metia 4-ze, maSin mis Semdeg cifrs 

emateba erTi. magaliTad, 18,6782 damrgvalebisas measedamde gveqneba 18,68. 

• Tu pirveli ukusagdebi cifri naklebia an tolia 4, maSin bolo 

cifri ar icvleba. magaliTad, Tu vamrgvalebT measemde 29,3531, gveqneba 

29,35. 

• Tu dasamrgvalebeli nawilis bolo ricxvSia 5, maSin mas 

amrgvaleben ise, rom bolo ricxvi darCes luwi. magaliTad meaTemde 

damrgvalebisas 25,65 Caiwereba 25,6, xolo 27,75 iqneba 27,8. 

saSualo ariTmetikuli meTodiT cdomilebaTa gamoTvlis wesi: 

1) erTi da igive ucvleli sididis gazomva xdeba mravaljeradad 

erTnair pirobebSi. 

2) yvela gazomva warmoebs erTnairi aTvlis cdomilebiT 

3) am meTods iyeneben im SemTxvevaSi, rodesac gansxvaveba calkeul 

gazomvebs Soris aRemateba TiToeuli gazomvis an dasaSveb 

instrumentalur cdomilebas. 

4) Tu saZiebeli sidide mniSvnelovania (didia) da damokidebulia 

gazomvebis mravaljeradobaze, maSin praqtikulad savsebiT sakmarisia 

98

gazomvebis imdenjer gameoreba, rom saSualo ariTmetikuli cdomileba 

miuaxlovdes dasaSveb instrumentalurs, an dayvanil iqnas aTvlis 

calkeuli gazomvis cdomilebaze. 

5) Tu ganmeorebiTi gazomvisas miiReba erTi da igive Sedegebi, maSin 

gazomvis cdomilebad miiReben gamzom xelsawyos, an instrumentalur 

cdomilebas. 

SeniSvnis saxiT aRvniSnavT, rom saSualo ariTmetikuli cdomileba 

dakavSirebulia saSualo kvadratul cdomilebasTan formuliT 

sadac 

Tu miviCnevT, rom maSin wina formula SeiZleba 

CavweroT gamoTvlebisaTvis iol formaSi . 

qvemoT cxrilis saxiT mogvyavs absoluturi da fardobiTi 

cdomilebebis formulebi. da miaxloebiTi ricxvebia, -Sesabamisi 

absoluturi cdomilebebis sazRvrebi, . 

N 

1 

2 

3 

4 

5 

6 

7 

8 

algebruli 

maqsimaluri cdomileba 

gamosaxuleba absoluturi fardobiTi 

rogorc formulebi aCveneben, absoluturi cdo-milebis moZebnis 

wesebi emTxveva gadiferencialebis wesebs, im gansxvavebiT rom 

absolutur cdomilebaTa formulebSi damoukidebeli cvladebis 

99

diferencialebi Secvlilia fizikur sidideTa gazomvis absoluturi 

cdomilebebiT da kerZo warmoebulebi aRebulia dadebiTi niSnebiT. 

fardobiTi cdomilebis wesebi ki emTxveva logariTmul gadife- 

rencialebis Sesabamis wesebs. 

100

ibeWdeba avtoris mier warmodgenili saxiT 

gadaeca warmoebas 03.07.2009. xelmowerilia dasabeWdad 

10.07.2009. qaRaldis zoma 60X84 1/16. pirobiTi nabeWdi Tabaxi 6. 

tiraJi 100 egz. 

sagamomcemlo saxli `teqnikuri universiteti~, Tbilisi, 

kostavas 77

fizikis kursi

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