Misalai 3 na tambayoyi kan faɗaɗa yanki
1. Takardar ƙarfe a kan zafin jiki 20oC yana da tsawon santimita 50 da faɗin santimita 30. Idan ma'aunin faɗaɗa ƙarfe mai layi shine santimita 10-5 oC-1 sannan karuwar yanki da jimillar yanki a zafin jiki na 60oC shine….
Tattaunawa
An san cewa:
Zafin farko (T)1) = 20oC
Zafin ƙarshe (T)2) = 60oC
Canjin yanayin zafi (ΔT) = 60oC - 20oC=40oC
Yankin farko (A)1) = Tsawon x faɗin = 50 cm x 30 cm = 1500 cm2
Ma'aunin faɗaɗa layin ƙarfe (α)) = 10-5 oC-1
Adadin faɗaɗa ƙarfe (β)) = 2a = 2 x 10-5 oC-1
An tambaya: Ƙara yawan yanki (ΔA)
Ƙara yawan yanki (ΔA):
ΔA = β A1 ΔT
ΔA = (2 x 10-5 oC-1)(1500 cm2)(40oC)
ΔA = (80) x 10-5)(1500 cm2)
ΔA = 120.000 x 10-5 cm2
ΔA = 1,2 x 105 x 10-5 cm2
ΔA = 1,2 cm2
Jimlar yanki (A2):
A2 = A ba1 +ΔA
A2 = 1500 cm ba2 + 1,2 cm2
A2 = 1501,2 cm2
2. Farantin aluminum mai ma'aunin faɗaɗa layi na 24 x 10-6 /oC yana da yanki na 40 cm2 a zafin jiki na 30oC. Ƙayyade zafin jiki na ƙarshe idan yankin farantin aluminum ya karu zuwa 40,2 cm2.
Tattaunawa
An san cewa:
Zafin farko (T)1) = 30oC
Ma'aunin faɗaɗa layi na aluminum (α)) = 24 x 10-6 oC-1
Adadin faɗaɗa aluminum (β)) = 2α = 2 x 24 x 10-6 oC-1 = 48 x ku 10-6 oC-1
Yankin farko (A)1= 40 cm2
Yankin ƙarshe (A)2= 40,2 cm2
Canjin yanki (ΔA) = 40,2 cm2 - 40 cm2 = 0,2 cm ba2
An tambaya: Ƙayyade zafin ƙarshe (T)2)
Amsa:
Tsarin canji a yanki (Δ)A) :
ΔA = β A1 ΔT
Zafin ƙarshe (T)2):
ΔA = β A1 (T2 - T1)
0,2 cm2 = (48 x 10-6 oC-1)(40 cm2)(T2 - 30oC)
0,2 = (1920 x 10-6)(T2 - 30)
0,2 = (1,920 x 10-3)(T2 - 30)
0,2 = (2 x 10-3)(T2 - 30)
0,2 / (2) x 10-3) = T2 - 30
0,1 x 103 = T2 - 30
1 x 102 = T ba2 - 30
100=T2 - 30
100 + 30 = T2
T2 = 130
Zafin ƙarshe = 130oC
3. Abu mai zagaye a zafin jiki na 20oC yana da radius na 20 cm. Idan a zafin jiki na 100oC radius ɗin yana ƙaruwa zuwa 20,5 cm sannan ma'aunin faɗaɗa layi na abu shine….
Tattaunawa
An san cewa:
Zafin farko (T)1) = 30oC
Zafin ƙarshe (T)2) = 100oC
Canjin yanayin zafi (ΔT) = 100oC - 30oC=70oC
Radius na farko (r)1= 20 cm
Radius na ƙarshe (r)2= 20,5 cm
An tambaya: Ma'aunin faɗaɗa yanki na abu (b)
Amsa:
Yankin farko na da'irar (A)1) = π r12 = (3,14)(20 cm)2 = (3,14)(400 cm)2= 1256 cm2
Yankin ƙarshe na da'irar (A)2) = π r22 = (3,14)(20,5 cm)2 = (3,14)(420,25 cm)2= 1319,585 cm2
Ƙara yawan yankin da'ira (ΔA) = 1319,585 cm2 - 1256 cm2 = 63,585 cm ba2
Tsarin canji a yanki (Δ)A) :
ΔA = β A1 ΔT
Ma'aunin faɗaɗa yanki:
ΔA = β A1 ΔT
63,585 cm2 = β (1256 cm2)(70 oC)
63,585 = β (87920 oC)
ba = 63,585 / 87920 oC
β = 0,00072 /oC
β = 7,2 x 10-4 /oC
Ma'aunin faɗaɗa layi (α)):
β = 2 α
α = β / 2
α = (7,2) x 10-42
α = 3,6 x 10-4 oC-1
