Xorazm ilm ziyo ma`ruza kinematika
X. Aylana bo`ylab harakat
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Kinematika Ma`ruza
- Bu sahifa navigatsiya:
- Parma (o`ng vint) qoidasi
- Aylana bo’ylab harakatda hususiy holatlar. 1.
- Esda tuting!
- Burchak ko`chishi formulalari
- Mavzuga ilova: 1.
X. Aylana bo`ylab harakat. Aylana bo`ylab harakatda jismning harakat trayektoriyasi to`liq aylana yoki aylananing ma`lum bir yoyini tashkil qilishi mumkin. Aylana bo`ylab tekis harakat. Ushbu holatda harakat tezligi: const = ϑ . Aylana bo`ylab harakat uchun tegishli bo`lgan kattaliklarni ko`rib o`tamiz:
Aylanish davri – aylanani to`liq bir marta aylanib chiqish uchun ketgan vaqt.
; bunda: T - aylanish davri; t - vaqt; N - aylanishlar soni. [T]=1 s. 2. Aylanish chastotasi – vaqt birligi ichidagi aylanishlar soni.
= ν ; bunda: ν - aylanish chastotasi;[ v]=1 ayl/s=1 s - 1 = 1 Hz (gerts). Chastota aylanish davriga teskari bo`lgan kattalikdir: T 1 = ν . 120 ayl/min=120 ayl/ 60 s=120:60=2 ayl/s. 3. Aylana haqida ma`lumot.
Bunda
R - aylana radiusi bo`lib, u aylana markazidan aylanagacha bo`lgan masofani anglatadi; d - aylana diametri bo`lib, aylana markazidan o`tuvchi vatarni anglatadi va u:
2 = ; l ⌣ - aylana yoyi uzunligini bildiradi va u α ⋅ = R l ⌣ ga teng. Aylana uzunligi quyidagicha ifodalanadi: d R l ⋅ = ⋅ ⋅ = π π 2 , 14 , 3 ) ( ≈ pi π . JUMANIYAZOV TEMUR
13 4. Burchak tezlik – jismning 2 π sekunddagi aylanishlar soni.
ϕ ω = ; ν π ω ⋅ ⋅ = 2
; T π ω 2 = ; t N ⋅ ⋅ = π ω 2 . Bunda: ω - burchak tezlik bo`lib, “amega” deb o`qiladi va [ω]=1 rad/s; ϕ - burchak ko`chishi va [φ]=1 rad; 0 180
1 = π rad va
14 , 3 ≈ π . 5. Chiziqli tezlik.
= ϑ ; T R ⋅ ⋅ = π ϑ 2 ; ν π ϑ ⋅ ⋅ ⋅ = R 2 ; t N R ⋅ ⋅ ⋅ = π ϑ 2 ;
⋅ =
ϑ . Bunda: R - aylana radiusi; l - aylana bo`ylab bosib o`tilgan yo`l. 6. Markazga intilma tezlanish (normal tezlanish).
2 ϑ = ; 2 2 4
R a n ⋅ ⋅ = π ; 2 2 4 ν π ⋅ ⋅ ⋅ = R a n ; 2 2 2 4 t N R a n ⋅ ⋅ ⋅ = π ; R a n ⋅ = 2 ω ; ϑ ω ⋅ = n a
Bunda: n a -markazga intilma tezlanish ya`ni normal tezlanish; [a n ]=1 m/s
2 .
Vektor kattaliklarning yo`nalish jihatdan tavsifi. 1. Chiziqli tezlik vektori – aylanaga urinma tarzida yo`naladi va yo`nalishi uzluksiz o`zgarib turadi.
Markazga intilma tezlanish vektori – radius bo`ylab aylana markaziga tomon yo`naladi va chiziqli tezlik vektori bilan o`zaro perpendikulyar (tik) bo`ladi. 3. Burchak tezlik – aylana tekisligiga perpendikulyar tarzda yo`naladi va yo`nalishi parma (o`ng vint) qoidasiga ko`ra topiladi. Parma (o`ng vint) qoidasi : parma dastasi aylana bo`ylab harakatlanuvchi jismning harakat yo`nalishiga mos ravishda kiradigan qilib buraladi va bunda parmaning ilgarilanma harkatining yo`nalishi burchak tezlik yo`nalishi bilan mos tushadi.
1. Tasmali uzatma: tasmali uzatmada ikkala g`ildirakdagi chiziqli tezliklar o`zaro teng bo`ladi.
2 1 ϑ ϑ = ekanligidan: 2 2
1 R R ⋅ = ⋅ ω ω 2. Tishli (friksion) va zanjirli uzatma.
2
ϑ ϑ = ekanligidan: 2 2 1 1
Z ⋅ = ⋅ ω ω ; 2 2 1 1
N Z N ⋅ = ⋅ . Bunda: 1 Z va
2 Z - birinchi va ikkinchi g`ildirakdagi tishlar soni.
Gorizontal tekislikda o`zgarmas tezlikda dumalayotgan g`ildirak ham aylanma, ham ilgarilanma harakatlanadi. Agarda g`ildirak sirpanish bialn aylansa, u holda A, B, C va D nuqtalarning yerga nisbatan tezligi quyidagicha bo`ladi (natijaviy):
u ϑ ϑ = + ; B u ϑ ϑ = − ; 2 2 C u ϑ ϑ = + ; 2 2 2 cos D u u ϑ ϑ ϑ α = + +
Agar g`ildirak sirpanishsiz dumalasa, u holda ilgarilanma harakat tezligi va gardishdagi chiziqlik tezliklar bir xil bo`ladi:
ϑ = :
A, B va C nuqtalardagi yerga nisbatan tezliklar (natijaviy): 2
ϑ ϑ ϑ
ϑ = + = ⋅ ; 0
ϑ ϑ ϑ = − = ; 2 2 2 C ϑ ϑ ϑ ϑ = + = ⋅ . Bunda: C nuqta markaziy nuqta bilan bir sathdagi nuqta. D nuqtaning yerga nisbatan tezligi (natijaviy): JUMANIYAZOV TEMUR
14 2 2 2 cos D ϑ ϑ ϑ ϑϑ α = + + ϑ -g`ildirak g`ardishining chiziqli tezligi; u - g`ildirakning ilgarilanma harakat tezligi. G`ildirakning barcha chekka nuqtalarining normal tezlanishlari bir xil (yerga nisbatan ham) bo`ladi: 2
ϑ = = = = 4. Tezlik vektorining o`zgarishi.
:
3 ; 4 1 T T
ϑ ϑ ⋅ = ∆ 2 ; : 2 1 T ϑ ϑ ⋅ = ∆ 2 ;
: T 0 = ∆ ϑ . Jism aylana bo`ylab biror α burchakka burilganda tezlik o`zgarishi: 2 2
cos ϑ ϑ ϑ ϑϑ α ∆ = + − 5. Jism o`z o`qi atrofida tekis aylanmoqda. Esda tuting! O`z o`qi atrofida aylanuvchi jismning barcha nuqtalarida aylanish davri, chastotasi, burchak tezligi va aylanishlar soni bir xil bo`ladi.
1
1 2 1 2 1 2 ; T ;
= ; N
T N ω ω ν ν = = =
2 2 1 1 2 2 1 1 2 2 ;
; ;
R R a R a R ϑ ω ϑ ω ω ω = = = =
ekanligidan: 1 1
2 2 2 R a R a ϑ ϑ = = . 6. Ma`lumki, har bir sayyora o`z o`qi atrofida aylanadi.
Ekvatordagi chiziqli tezlik quyidagicha topiladi: T R ⋅ ⋅ = π ϑ 2 ; bunda R-sayyora radiusi, T-sayyoraning o`z o`qi atrofida aylanish davri (sutkaning davomiyligi). Sayyora kengliklaridagi chiziqli tezlik esa quyidagicha topiladi: α π ϑ cos
2 ⋅ ⋅ ⋅ =
R ; bunda
α -kenglik burchagi. Esda tuting! Yer ekvatoridagi chiziqli tezlik taxminan
ϑ = 465 m/s ga teng. Aylana bo`ylab tekis o`zgaruvchan harakat. Ushbu harakatda markazga intilma tezlanish (normal tezlanish)
dan tashqari tangensial tezlanish τ
ham ishtirok etadi. Bunda tangensial tezlanish doimo aylanaga urinma tarzida yo`naladi. Quyidagi rasm aylana bo`ylab tezlanuvchan harakat keltirilgan (a τ >0): Quyidagi rasm aylana bo`ylab sekinlanuvchan harakat keltirilgan (a τ
Tangensial va normal tezlanishlar o`zaro perpendikulyar tarzda yo`nalgani uchun to`la (natijaviy) tezlanish Pifagor teoremasiga ko`ra topiladi: 2 2 2 τ
a a n T + = . Bunda:
t a 0 ϑ ϑ τ − = ; ϑ va 0 ϑ - oxirgi va boshlang`ich chiziqli tezliklar. R a ⋅ = ε τ . Burchak tezlanish – vaqt birligi ichida burchak tezlikning o`zgarishi.
0 ω ω ε − = ; 0 t ω ω
ε − = ± bunda: [ε]=1 rad/s 2 ;
ω va
ω - boshlang`ich va oxirgi burchak tezliklar; “ ε ” –
“epsilon”. Burchak ko`chishi formulalari: 2 2 0 t t ⋅ ± ⋅ = ε ω ϕ ; ε ω ω ϕ ⋅ ± − = 2 2 0 2 ; t ⋅ + = 2 0 ω ω ϕ . π ϕ 2 =
-aylanishlar soni. Burchak tezlik tenglamasi: t ⋅ ± = ε ω ω 0 . Burchak tezlik vektori aylana tekisligiga perpendikulyar tarzada yo`naladi va agar harakat tezlanuvchan bo`lsa, burchak tezlanish vektori burchak tezlik vektori bilan bir yo`nalishda yo`naladi. JUMANIYAZOV TEMUR
15 Agarda harakat sekinlanuvchan bo`lsa, burchak tezlanish vektori burchak tezlikka qarama-qarshi yo`naladi:
1. Jism balandlikdan gorizontal ravishda otildi. Jismning biror vaqtdan keyingi trayektoriya radiusi va normal tezlanishini hamda tangensial tezlanishini quyidagicha hisoblaymiz:
Trayektoriya egrilik radiusi: ( )
t g g R ⋅ ⋅ + = ⋅ = 0 3 2 2 2 0 0 3 ϑ ϑ ϑ ϑ .
Normal tezlanish (markazga intilma): 2 2 2 0 0 0 t g g g a n ⋅ + ⋅ = ⋅ = ϑ ϑ ϑ ϑ . Tangensial tezlanishni quyidagi ifodadan topish mumkin:
2 2 2 g a a n = + τ
Gorizontga burchak ostida otilgan jismning eng yuqori nuqtadagi trayektoriya egrilik radiusini quyidagicha topamiz:
g R x α ϑ ϑ 2 2 0 2 cos ⋅ = = ; bunda jismning markazga intilma tezlanishi g ga teng bo`ladi. 3. Jism harakat tiplari: a) agar 0 ; = = τ a const a n bo’lsa, jism aylana bo’ylab tekis harakatlanadi; b) agar
const a a n = ≠ τ ; 0 bo’lsa, jism aylana bo’ylab tekis o’zgaruvchan harakatlanadi; c) agar
= = τ ; 0 bo’lsa, jism to’g’ri chiziqli tekis o’zgaruvchan harakatlanadi; d) agar 0 ; 0 = = τ a a n bo’lsa, jism tekis harakatlanadi;
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