Expansio areae – problemata et solutiones

1. Ad 20 ^° C, longitudo laminae ferreae est 50 cm et latitudo 30 cm. Si coefficiens expansionis linearis ferri est 10⁻⁵ ^{° C⁻¹} , determina mutationem areae et aream finalem ad 60 ^° C^.

Notum:

Temperatura initialis (T1 ₎ = 20 ^° C

Temperatura finalis (T² ₎ = 60 ^° C

Mutatio temperaturae ( ΔT ) = 60 ^° C – 20 ^° C = 40 ^° C

Area initialis (A1 ₎ = longitudo × latitudo = 50 cm × 30 cm = 1500 ^cm²

Coefficiens expansionis linearis pro chalybe ( α ⁾ = 10⁻⁵ ^{° C⁻¹}

Coefficiens expansionis areae pro chalybe ( ^β ) = 2α = 2 x ^{10⁻⁵ °} C⁻¹

Quaesitum: Mutatio areae ( ΔA )

solution:

Mutatio areae ( ΔA ):

ΔA = βA ₁ ΔT

ΔA = ( 2 × ^{10⁻⁵ °} C⁻¹ ⁾ (1500 cm² )(40 ^{° C} )

A = (80 x 10 ^-5 ) (1500 cm ² )

^ΔA = 1 × 20 000 × ^10⁻⁵ cm²

ΔA = 1. 2 x 10 ⁵ x 10 ^-5 cm ²

ΔA = ^1.2 cm²

Area finalis ( A ₂ ):

A² = _A¹ + ΔA_​

A² ⁼ 1500 ^cm² + 1,2 cm²_​

A² = 1501.2 cm²_​^​

[irp]

2. Ad 30 ^° C, area laminae aluminii est 40 cm² ^et coefficiens expansionis linearis est 24 × 10⁻⁶ ^/ ° ^C . Determina temperaturam finalem si area finalis est 40.2 ^cm².

Notum:

Temperatura initialis (T1 ₎ = 30 ^° C

Coefficiens expansionis linearis ( ^α ) = 24 × ^{10⁻⁶ °} C⁻¹

Coefficiens expansionis areae ( ^β ) = 2a = ² × 24 × ^{10⁻⁶ °} C⁻¹ = 48 × 10⁻⁶ ^{° C⁻¹}

Area initialis (A₁ ₎ = 40 ^cm²

Area finalis (A² ₎ = 40.2 ^cm²

Mutatio areae ( ΔA) = 40.2 cm² ^– 40 cm² ⁼ 0.2 ^cm²

Quaesitum: Temperaturam finalem (T² ₎ determinare .

solution:

Formula mutationis areae ( ΔA) :

ΔA = βA1ΔT_​​

Temperatura finalis (T² ₎ :

ΔA = βA₁ ( _T₂ – _T₁ )​_​

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

0.2 = (1920 × 10⁻⁶ ) ( _T² – ³⁰ )

0.2 = (1.920 × 10⁻³ ) (T² _– ³⁰ )

0.2 = (2 × 10⁻³ ) (T² _– ³⁰ )

0.2 / (2 × ^10⁻³ ) = T² _– 30

0.1 × ^10³ = T² _– 30

1 × 10² ⁼ T² _– 30

100 = T2 _– 30

100 + 30 = _T²

T² = ₁₃₀

Temperatura finalis = 130 ^° C

[irp]

3. Radius anuli ad 20 ^° C est 20 cm. Si radius finalis ad 100 ^° C est 20.5 cm, determina coefficientem expansionis areae et coefficientem expansionis linearis...

Notum:

Temperatura initialis (T1 ₎ = 30 ^° C

Temperatura finalis (T² ₎ = 100 ^° C

Mutatio temperaturae ( ΔT ) = 100 ^° C – 30 ^° C = 70 ^° C

Radius initialis (r₁ ₎ = 20 cm

Radius finalis (r² ₎ = 20.5 cm

Quaesitum: Coefficiens expansionis areae ( β )

solution:

Area initialis (A₁ ₎ = π r₁² = ( _3.14 )(20 cm²) ^{² =} (3.14)(400 cm² ⁾ = 1256 ^cm²

Area finalis (A² ₎ = π r²² = ( _3.14 )(20.5 cm²) ^{² =} (3.14)(420.25 cm² ⁾ = 1319.585 ^cm²

Mutatio areae ( ΔA) = 1319.585 cm² ^– 1256 cm² ⁼ 63.585 ^cm²

Formula mutationis areae ( ΔA) :

ΔA = βA ₁ ΔT

Coefficiens expansionis areae:

ΔA = βA ₁ ΔT

63.585 cm² ⁼ b ( 1256 cm² ⁾ (70 ^° C)

63.585 = b ( 87,920 ^° C)

β = 63.585 / 87,920 ^° C

β = 0.00072 / ^° C

β = 7.2 × 10⁻⁴ ^/ ° ^C

^β = 7.2 × 10⁻⁴ ^{° C⁻¹}

Coefficiens expansionis linearis ( α ):

β = 2α

α = β / 2

α = (7.2 ^× 10-4 ) / 2

^α = 3.6 × 10⁻⁴ ^{° C⁻¹}

[wpdm_package id='698′]

1. Scalas temperaturae convertens
2. Dilatatio linearis
3. Expansio areae
4. Expansio voluminis
5. Calor
6. Aequivalens mechanicum caloris
7. Calor specificus et capacitas calorica
8. Calor latens, calor fusionis, calor vaporisationis
9. Conservatio energiae ad translationem caloris