Hajmni kengaytirish - muammolar va yechimlar

Chiziqli kengayish faqat qattiq jismlar tomonidan sodir bo'ladi; hajm kengayishi barcha jismlar, ham qattiq, ham suyuq va gazsimon jismlar tomonidan sodir bo'ladi. Hajm kengayishi tenglamasi chiziqli kengayish tenglamasiga o'xshaydi.

Tavsif: V_o = Boshlang'ich hajm, V = Yakuniy hajm, ΔV = V – V_o = Hajm o'zgarishi, T_o = Boshlang'ich harorat, T = Yakuniy harorat, ΔT = T – T_o = Haroratning o'zgarishi, β = hajm kengayish koeffitsienti. β o'lchov birligi = (C^o) ^-1

Yuqoridagi hajm kengayish tenglamasi faqat jismlarning (ham qattiq, ham suyuq, ham gazsimon) hajmidagi o'zgarishlar jismning asl hajmidan kichik bo'lgandagina qo'llaniladi. Agar jism hajmidagi o'zgarish jismning boshlang'ich hajmidan katta bo'lsa, hajm kengayish tenglamasi to'g'ri natijalarni bermaydi. Odatda, qattiq jismlar tomonidan sodir bo'ladigan hajm o'zgarishlari juda katta bo'lmaydi. Boshqa tomondan, suyuqlik va gazning hajm kengayish koeffitsienti katta bo'ladi. Gazsimon moddalar uchun hajm kengayish koeffitsientini harorat o'zgarganda ham o'zgartirish oson. Shuning uchun yuqoridagi formula faqat qattiq jismlarning kengayishi uchun ishlatiladi.

1. 30 da ^oC alyuminiy sharning hajmi 30 sm ga teng³ning koeffitsienti chiziqli kengayish 24 x 10 ga teng^{-6 o}C^-1Agar oxirgi hajm 30.5 sm bo'lsa³, alyuminiy sharning oxirgi harorati qanday?

Ma'lum:

Chiziqli kengayish koeffitsienti (α) = 24 x 10^{-6 o}C^-1

ning koeffitsienti hajmni kengaytirish (β) = 3 a = 3 x 24 x 10^{-6 o}C^-1 = 72 x 10^{-6 o}C^-1

boshlang'ich harorat (T₁) = 30^oC

Dastlabki hajm (V₁) = 30 sm³

Yakuniy jild (V₂) = 30.5 sm³

Ovoz balandligining o'zgarishi (ΔV) = 30.5 sm³ - 30 sm³ = 0.5 sm³

Qidirilgan: Yakuniy harorat (T₂)

yechim:

ΔV= β (V)₁)(DT)

ΔV= β (V₁)(T)₂ - T₁)

0.5 sm³ = (72 x 10^{-6 o}C^-1)(30 sm.)³)(T)₂ - 30^oC)

0.5 = (2160 x 10^-6)(T₂ - 30)

0.5 = (2.160 x 10^-3)(T2 - 30)

0.5 = (2.160 x 10^-3)(T₂ - 30)

0.5 / (2.160 x 10^-3) = T₂ - 30

0.23 x 10³ =T₂ - 30

0.23 x 1000 = T₂ - 30

230 = T₂ - 30

230 + 30 = T₂

T₂ = 260^oC

2. Metall sharning chiziqli kengayish koeffitsienti 9 x 10 ga teng^{-6 o}C^-1Metall sharning ichki diametri 20 ga teng. ^oC 2.2 sm ga teng. Agar oxirgi diametr 2.8 sm bo'lsa, oxirgi harorat qanday bo'ladi!

Ma'lum:

Chiziqli kengayish koeffitsienti (α) = 9 x 10^{-6 o}C^-1

Hajm kengayish koeffitsienti (β) = 3 a = 3 x 9 x 10^{-6 o}C^-1 = 27 x 10^{-6 o}C^-1

Dastlabki harorat (T₁) = 20^oC

Dastlabki diametr (D₁) = 2.2 sm

Yakuniy diametr (D₂) = 2.8 sm

Boshlang'ich radius (r₁) = D₁ / 2 = 2.2 sm³ / 2 = 1.1 sm³

Yakuniy radius (r₂) = D₂ / 2 = 2.8 sm³ / 2 = 1.4 sm³

Dastlabki hajm (V₁) = 4/3 π r₁³ = (4/3)(3.14)(1.1 sm)³ = (4/3)(3.14)(1.331 sm)³) = 5.57 sm³

Yakuniy jild (V₂) = 4/3 π r₂³ = (4/3)(3.14)(1.4 sm)³ = (4/3)(3.14)(2.744 sm)³) = 11.48 sm³

Ovoz balandligining o'zgarishi (ΔV) = 11.48 sm³ - 5.57 sm³ = 5.91 sm³

Qidirilgan: Yakuniy harorat (T₂)

yechim:

ΔV= β (V)₁)(DT)

5.91 sm³ = (27 x 10^{-6 o}C^-1)(5.57 sm³)(T)₂ - 20^oC)

5.91 = (150.39 x 10^-6)(T₂ - 20)

5.91/150.39 x 10^-6 =T₂ - 20

0.039 x 10⁶ =T₂ - 20

39 x 10³ =T₂ - 20

39,000 = T₂ - 20

39,000 + 20 = T₂

T₂ = 39,020 ^oC

3. 2000 sm³ alyuminiy idish, 0 da suv bilan to'ldirilgan^oC. Va keyin 90 darajaga qadar qizdiriladi^oC. Agar alyuminiy uchun chiziqli kengayish koeffitsienti 24 x 10 ga teng bo'lsa^-6 (^oC)^-1 va suv uchun hajm kengayish koeffitsienti 6.3 x 10 ga teng^-4 (^oC)^-1, to'kilgan suv hajmini aniqlang.

Ma'lum:

Alyuminiy idish va suvning boshlang'ich hajmi (V)_o) = 2000 sm³ = 2 x 10³ cm³

Alyuminiy idish va suvning boshlang'ich harorati (T)₁) = 0^oC

Alyuminiy idish va suvning oxirgi harorati (T)₂) = 90^oC

Alyuminiy uchun chiziqli kengayish koeffitsienti (α) = 24 x 10^-6 (^oC)^-1

Alyuminiy uchun hajm kengayish koeffitsienti (γ) = 3α = 3 (24 x 10^-6 (^oC)^-1 ) = 72 x 10^{-6 o}C^-1

Suv uchun hajm kengayish koeffitsienti (γ) = 6.3 x 10^-4 (^oC)^-1

Qidirilgan: To'kilgan suv hajmi

yechim:

Hajm kengayishining tenglamasi:

V = V_o + γ V_o DT

V – V_o = γ V_o DT

ΔV = γ V_o DT

V = yakuniy hajm, V_o = boshlang'ich hajm, ΔV = hajmning o'zgarishi, γ = hajm kengayish koeffitsienti, ΔT = harorat o'zgarishi

Alyuminiy idish hajmining o'zgarishini hisoblang:

ΔV = γ V_o ΔT = (72) x 10^-6)(2 x 10³)(90) = 12960 x 10^-3 = 12.960 sm³

Suv hajmining o'zgarishini hisoblang:

ΔV = γ V_o ΔT = (6.3 x 10^-4)(2 x 10³)(90) = 1134 x 10^-1 = 113.4 sm³

Suv hajmining o'zgarishi alyuminiy idishdagidan kattaroq, shuning uchun bir oz suv to'kilgan.

To'kilgan suv hajmini hisoblang:

113.4 sm³ - 12.960 sm³ = 100.44 sm³

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1. Harorat shkalalarini konvertatsiya qilish
2. Chiziqli kengayish
3. Hududni kengaytirish
4. Hajmni kengaytirish
5. issiqlik
6. Issiqlikning mexanik ekvivalenti
7. Maxsus issiqlik va issiqlik sig'imi
8. Yashirin issiqlik, erish issiqligi, bug'lanish issiqligi
9. Issiqlik uzatish uchun energiya tejash