Harakat tenglamalarini integrallash va boshlang’ich shartlari. Nuqtaningistalganvaqtmomentidagiholatinitopish

r^→r^→(t,C,C ,...,C )
(9)

  1 2 6N
Integrallash doimiyliklarini boshlang’ich shartlar bilan bog’lash mumkin.Haqiqatdan(2)umumiyyechimbizgama’lumbo’lsavaboshlang’ichvaqtda

(tt₀)
bo’lgansistemanuqtaningholatlari
r^→ r^→(t)

tezliklari
0  0

v^→ v^→(t )
(10)

0  0
berilganbo’lsa,(10)nivaqtbuyichadifferensiallab

v^→v^→(t,C,C ,...,C )
(11)

  1 2 6N

Tezliklarnitopamizva(9)va(11 larda
(tt₀)
debolib,(10)asosidayozaolamiz:

0
^r→
^→
r^→(t



0
→
,C₁,C₂
,...,C_6N)
(12)

_v_0r_(t₀,C₁,C₂,...,C_6N)





r

0
^→r^→(t,t
^→,. ,r^→
→,......,v^→ )
(14)

Misol.Farazqilaylikki,elektrmaydoni
EE₀
cost
OZo’qibuyichayo’nalsin

zaryad esaOYo’qi bo’yichatushayotganbo’lsin.Uholda

E_zE₀cost
,E_xE_y 0,
v_y v₀, v_x v_z 0

Masalashartigako’ra,zaryadga
FeEcost
kuchta’siretyapti.Harakat

→

→
tenglamasiDekartkomponentalarda


m^x^0
m^y^0

Yoki
m^z^eE₀cost

e

^x^0,^y^0,^z^_mE₀ cost
(15)

8. 
   tenglamalarnivaqtbuyichabirmartaintegrallabtopamiz:
   


e
z^ _mE₀
sintC₁,
y^C₂,
x^C₃
(16

Boshlang’ichvaqtmomentitt₀
dav_y
y^0v₀,
x^0z^00
bulganiuchun(16)

dagiC₁
^eEosintm ⁰
,C₂v₀,
C₃0
buladi.

Demak(16):
e e
z^_mE₀ sint _mE₀ sint₀
y^v₀

8. 
   niyanabirmartavaqtbuyichaintegrallaymiz:
   

_zeE₀
m²
cost^eE0sint
m ⁰
C₄,

yv₀tC₅
Bundantt₀,bulgandatopamiz:


y₀0,


z₀0

ekanliginie’tiborgaolib,C₄,C₅

larni
(17)

C₄
eE₀
m²
cost₀

• 
  eE_0t
  

m⁰
sint₀

C₅v₀t₀ (18)

18. 
    larni(17)gaqo’yib,zarraningistalganvaqtmomentidagiholatinianiqlaymiz:
    

_zeE₀
m²
(cost₀

• 
  cost)^eE0(t
  

m ⁰

• 
  t)sint₀
  

yv₀(tt₀)

18. 
    datni yo’qotib,harakattenglamasinitopamiz.Buninguchun
    

(19)

tt^y
ni(19)dagizuchunifodagaquyamiz:

v
0
0

_zeE₀
(cost

• 
  cos(t
  

^y))^eE0
ysint

0

0
_m2 ⁰
Nihoyat
⁰ mv ⁰

cos(t₀

• 
  )costv₀
  

cos ^y sint

0
v₀
sin^y

v
v₀

0
Asosidatrayektoriyaningtenglamabilanifodalanishinitopamiz:

_zeE₀
((1cos^y)cost^y(1sin^y)sint)

v

v

v
m²
0 0
0 0 0

Nazoratsavollari

1. 
   Nyuton tenglamalarining Galiley almashtirishlariga nisbataninvariantliginiko’rsating.
   
2. 
   Harakat tenglamalarini integrallash va boshlang’ich shartlari haqidaayting.
   
3. 
   Nuqtaningistalganvaqtmomentidagiholatinitoping
   
4. 
   Integrallashdoimiyliklaritushuntiribbering.
   

xx(t),yyt,zzt
(1)

Agar(1)danvaqtnichiqaribtashlasaknuqtaningtrayektoriyatenglamasitopiladi.Bu tenglamalarparametriktenglamalardeyiladi.
Koordinatalarorqaliifodalanganradius-vektor

k
r^→xi^→y^→jz^→
(2)

ninazardatutak,(1)nivaqtbo’yichato’liqdifferensialiMnuqtaningtezlikva

tezlanishvektorlariniberadi
→ → ^→

v^→r^→x^i

• 
  y^jz^k
  

(3-1)

→ →_

_→ → → ^→

wvr
^x^i
^y^j^z^k
(3-2)

Tezlikvatezlanishvektorlariningo’qlardagiproyeksiyalariniquyidagiko’rinishdayozishmumkin:

v_xx^,
v_yy^,
v_zz^;
w_xv^x^x^,
w_yv^y^y^,
w_zv^z^z^
(4)

Tezlikvatezlanishlarningmodullariniesa
w

(5)

ko’rinishdayozishmumkin.
(3-1) va (3-2) formulalardan tezlik vektori radius-vektordan vaqt bo’yicha olinganbirinchitartibli,tezlanishvektoriesaradus-vektordanvaqtbo’yichaolinganikkinchitartiblihosilagatengligikelib chiqadi.


Silindrikvaqutbkoordinatalarusuli
SilindrikkoordinatalarsistemasidaMnuqtaningholati^,,z koordinatalar

bilananiqlanadi.Nuqtaningharakatqonunlariko’rinishdabo’ladi.
(t),
(t),
zz(t)

2. 
   shakldanfoydalanibquyidagibog’lanishlarniyozishmumkin
   

→
xcos, ysin, zz
(8-1)


r^→e^→zk,r ²z², 
(8-2)

 
Silindrikkoordinatalarsistemasininge^→,e^→ortlaribilan
i^→,^→j
Dekartortlariorasidagi

bog’lanishnitopishuchunr^→radius-vektorharikkalasistemadagi(2)va(8-2)

_→ de
ifodalarinio’zarotenglashtiramizva(8-1)nie’tiborgaolsak,natijadaquyidagibog’lanishlargaegabo’lamiz:

e


^→i^→cos
^→jsin,
→

e
_ 
^ d
i^→sin
^→jcos

(9)

Silindrikkoordinatalarsistemasining
e^→,e^→
ortlariningyo’nalishivaqtgabog’liq

 
holdao’zgaradi,ularningvaqtbo’yichabirinchihosilalarinitopsak

2. 
   rasm
   

_→ de^→

→ →_

de^→ _→

_ 


e  ^^^e,

d

e  ^^^e

d

(9-1)

_ 

_→




Nuqtaning (8-2) radius-vektoridan vaqt bo’yicha hosila olib, (9-1) ni e’tiborgaolsak, tezlik vektori, uning moduli va proyeksiyalari uchun quyidagi ifodalarniolamiz


v^↼r^→^e^→
^e^→

• 
  zk, v
  

(10-1)

v_^,
v_^,
v_zz^
(10-2)

v_,v_,v_z
larmosravishda v tezlikvektoriningradial,ko’ndalangva aksial

proyeksiyalaridebyuritiladi.Tezlikvektoridanvaqtbo’yichahosilaolib w^→



→


tezlanishvektorvauningproyeksiyalariuchunquyidagilargaegabo’lamiz:


w^→(^^2)e^→
(2^^^)e^→

• 
  zk
  

(11-1)

w_^^2,
w_2^^^,
w_z^z^
(11-2)

Agar
z0,
r
desak,silindrikkoordinatalarsistemasitekislikdagi
r,
qutb

koordinatalarsistemasigao’tadi(2.b-rasm).

xrcos, yrsin
(12-1)

r
r^→re^→, r
(12-2)

Bundaharakatqonuni
rr(t)
(t)
tenglamlarbilanberiladi.Ulardan^tni

chiqarib, M nuqtaningqutbkoordinatalarsistemasidagi
rr()
trayektoriya

tenglamasi topiladi. Tekislikda_→harakatlanuvchi M nuqtaning qutb
koordinatlaridagir^→radius-vektori, ^v-chiziqliva^→-sektorialtezliklarihamdaw
tezlanishiuchun(10)-(12)munosabatlargako’ra(^z0,^r,^e→e→)

r
r^→re^→,
v^→r^e^→
r^e^→,
 r

k
^→¹r²^→,
2



r
w^→(^r^r^2)e^→^1d(r²^)e^→
r rdt ^

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