Area expansion – problems and solutions

1. At 20 ^oC, the length of a sheet of steel is 50 cm and the width is 30 cm. If the coefficient of linear expansion for steel is 10^{-5 o}C^-1, determine the change in area and the final area at 60 ^oC.

Known :

The initial temperature (T₁) = 20^oC

The final temperature (T₂) = 60^oC

The change in temperature (ΔT) = 60^oC – 20^oC = 40^oC

The initial area (A₁) = length x width = 50 cm x 30 cm = 1500 cm²

The coefficient of linear expansion for steel (α) = 10^{-5 o}C^-1

The coefficient of area expansion for steel (β) = 2α = 2 x 10^{-5 o}C^-1

Wanted : The change in area (ΔA)

Solution :

The change in area (ΔA) :

ΔA = β A₁ ΔT

ΔA = (2 x 10^{-5 o}C^-1)(1500 cm²)(40^oC)

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

The final area (A₂) :

A₂ = A₁ + ΔA

A₂ = 1500 cm² + 1.2 cm²

A₂ = 1501.2 cm²

[irp]

2. At 30 ^oC, the area of a sheet of aluminum is 40 cm² and the coefficient of linear expansion is 24 x 10^-6 /^oC. Determine the final temperature if the final area is 40.2 cm².

Known :

The initial temperature (T₁) = 30^oC

The coefficient of linear expansion (α) = 24 x 10^{-6 o}C^-1

The coefficient of area expansion (β) = 2a = 2 x 24 x 10^{-6 o}C^-1 = 48 x 10^{-6 o}C^-1

The initial area (A₁) = 40 cm²

The final area (A₂) = 40.2 cm²

The change in area (ΔA) = 40.2 cm² – 40 cm² = 0.2 cm²

Wanted : Determine the final temperature (T₂)

Solution :

Formula of the change in area (ΔA) :

ΔA = β A₁ ΔT

The final temperature (T₂) :

ΔA = β A₁ (T₂ – T₁)

0.2 cm² = (48 x 10^{-6 o}C^-1)(40 cm²)(T₂ – 30^oC)

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 = T₂ – 30

100 = T₂ – 30

100 + 30 = T₂

T₂ = 130

The final temperature = 130^oC

[irp]

3. The radius of a ring at 20 ^oC is 20 cm. If the final radius at 100 ^oC is 20.5 cm, determine the coefficient of area expansion and the coefficient of linear expansion…

Known :

The initial temperature (T₁) = 30^oC

The final temperature (T₂) = 100^oC

The change in temperature (ΔT) = 100^oC – 30^oC = 70^oC

The initial radius (r₁) = 20 cm

The final radius (r₂) = 20.5 cm

Wanted : The coefficient of area expansion (β)

Solution :

The initial area (A₁) = π r₁² = (3.14)(20 cm)² = (3.14)(400 cm²) = 1256 cm²

The final area (A₂) = π r₂² = (3.14)(20.5 cm)² = (3.14)(420.25 cm²) = 1319.585 cm²

The change in area (ΔA) = 1319.585 cm² – 1256 cm² = 63.585 cm²

Formula of the change in area (ΔA) :

ΔA = β A₁ ΔT

The coefficient of area expansion :

ΔA = β A₁ ΔT

63.585 cm² = b (1256 cm²)(70 ^oC)

63.585 = b (87,920 ^oC)

β = 63.585 / 87,920 ^oC

β = 0.00072 /^oC

β = 7.2 x 10^-4 /^oC

β = 7.2 x 10^{-4 o}C^-1

The coefficient of linear expansion (α) :

β = 2 α

α = β / 2

α = (7.2 x 10^-4) / 2

α = 3.6 x 10^{-4 o}C^-1

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1. Converting temperature scales
2. Linear expansion
3. Area expansion
4. Volume expansion
5. Heat
6. Mechanical equivalent of heat
7. Specific heat and heat capacity
8. Latent heat, heat of fusion, heat of vaporization
9. Energy conservation for heat transfer