Autor: zanatto | Publicado: 2014-04-06T18:21:00-07:00

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UNIDADE II - DILATAÇÃO TÉRMICA

&nbsp;2.1-Conceito:

&nbsp;Como&nbsp; já sabemos, se aumentarmos a temperatura de
um corpo, aumentamos a agitação das partículas desse corpo. Consequentemente,
as partículas se afastam umas das outras, provocando um aumento das dimensões
(comprimento, área e volume) do corpo

 





&nbsp; A esse
aumento das dimensões do corpo dá-se o nome de Dilatação térmica.





&nbsp; Se&nbsp; a temperatura diminuir e as dimensões se
reduzirem, dizemos que houve uma Contração
Térmica.

&nbsp; Embora o
aumento ocorra em todas as dimensões do corpo simultaneamente costuma-se
dividir o estudo da dilatação térmica em três partes:

·&nbsp;&nbsp;&nbsp;&nbsp; Dilatação Linear: aquela que predomina o aumento em uma dimensão (comprimento).

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Ex.:
dilatação em fios, cabos e barras

·&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
Dilatação Superficial: aquela que predomina o
aumento em duas dimensões (área).

&nbsp;&nbsp;&nbsp; Ex.:
aumento em chapas e placas.

·&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
Dilatação volumétrica: é a dilatação em que se
considera o aumento em três dimensões.

&nbsp;&nbsp;&nbsp;&nbsp; Ex.:
aumento de volume de um corpo.

&nbsp;&nbsp; Os corpos
sólidos admitem os três tipos de dilatação, mas os líquidos e gases, por não
terem forma própria, só admitem a dilatação volumétrica.



2.2-Dilatação Linear
&nbsp;Considere uma
barra metálica de comprimento inicial Lo,
à temperatura inicial To, ao
aquecermos a barra até a temperatura T
, seu comprimento aumentará para L.&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp;&nbsp; 


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 


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<![endif]-->&nbsp;&nbsp;&nbsp; DT = Tf - Ti: variação de temperatura

&nbsp;&nbsp;&nbsp; DL = Lf - Li: variação de comprimento

&nbsp;Experimentalmente, verificou-se que a variação
do comprimento &nbsp;DL depende:

·&nbsp;&nbsp;&nbsp;&nbsp; 1º da variação de temperatura&nbsp; DT:
a variação do comprimento é diretamente proporcional a variação da temperatura.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; DL a&nbsp;&nbsp;&nbsp;&nbsp; DT

·&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
2º do comprimento inicial Li:
a variação do comprimento é diretamente proporcional ao comprimento inicial.&nbsp; 

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; DL a&nbsp;&nbsp; Li

&nbsp;Utilizando-se
barras de mesmo material, verificamos que a variação de comprimento depende
somente destes dois fatores. Será &nbsp;portanto proporcional ao produto deles.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 

&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;DL&nbsp; a&nbsp;&nbsp; Li.&nbsp;&nbsp; DT&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; DL = a. Li. DT

&nbsp;&nbsp;&nbsp; 

&nbsp;A constante
de proporcionalidade que aparece na equação, é uma característica do material
da barra e é chamada coeficiente de
dilatação linear. Há &nbsp;pois, um outro
fator que influi na dilatação linear de uma barra.

·&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
3º a variação de comprimento DL depende também do material de que é feita a barra (a ). O valor de a é uma característica de cada material mas
não é rigorosamente constante com a temperatura. Costuma-se usar um valor médio
de a entre as temperaturas usuais.

&nbsp;&nbsp;&nbsp;&nbsp; Unidades
de a : A unidade a é o inverso da unidade de
temperatura.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1/ ºC&nbsp;&nbsp;&nbsp; ou&nbsp;
ºC-1 

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1/ ºF&nbsp;&nbsp;&nbsp;&nbsp; ou&nbsp;
ºF-1

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 1/ K&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; ou&nbsp;&nbsp;
K-1





&nbsp;Significado físico de a:

O coeficiente de dilatação
linear representa a variação de comprimento unitário quando a temperatura varia
de uma unidade.

Considere
por exemplo, o coeficiente de dilatação linear do ferro que é 1,2 .10 ºC -1 , este número significa
que, se tomarmos uma barra de ferro com um metro de comprimento e elevarmos sua
temperatura em 1ºC, seu comprimento será 1,000012m.

&nbsp;&nbsp; 

&nbsp; 









Tabela de
coeficientes de dilatação linear (a) de algumas
substâncias:

 

  
Substância

  
Coeficiente&nbsp; ºC-1

  
Substancia

  
Coeficiente ºC-1

 

  
Alumínio

  
24.10-6&nbsp;
  

  
Latão

  
18.10-6

 

  
Cobre

  
17.10-6 

  
Níquel

  
13.10-6

 

  
Chumbo

  
29.10-6

  
Zinco

  
27.10-6

 

  
Aço

  
12.10-6

  
Platina

  
9.10-6

 

  
Ferro

  
12.10-6

  
Ouro

  
14.10-6

 

  
Prata

  
19.10-6

  
Vidro Comum

  
9.10-6

 

  
Mercúrio

  
41.10-6

  
Vidro Pirex

  
3.10-6

 




Nota: Lembrando que &nbsp;&nbsp;DL = Lf- Li ,podemos também escrever

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
Lf - Li = a Li .&nbsp; DT, isolando
L, temos:

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
Lf = Li + a.Li. DT , colocando Li, em evidência, obtem-se:

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
Lf = Li( 1 + a. DT )

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Lf = Li [ 1 + a( Tf- Ti )]



Lembre
que:&nbsp;&nbsp;&nbsp; Ti - temperatura inicial

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Tf - temperatura final

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; DT - variação
de temperatura

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; DT = Tf - Ti

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Li - comprimento inicial

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Lf - comprimento final

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; DL -
variação de comprimento



Exemplos:

1º)
Uma viga de concreto (a = 12.10-6 ºC -1)&nbsp; tem 100m de comprimento a 20ºC. Que
comprimento terá a viga a 30ºC?





&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;



Exercícios de aplicação:



1)Qual
será a dilatação sofrida por uma barra de ouro ( a = 15.10-6&nbsp; ºC-1&nbsp; ) inicialmente&nbsp; de&nbsp;
comprimento 40 cm
, quando a temperatura passa de 15ºC&nbsp;
para 35ºC ?



2)Um&nbsp; sarrafo de madeira (a = 5.10-6&nbsp; ºC-1 ) tem comprimento de&nbsp; 10m a 20ºC. Que comprimento terá a&nbsp; 70ºC.



3)Uma
ponte de aço tem 1000m de comprimento. O coeficiente de dilatação linear do aço
é de 11.10 -6&nbsp;&nbsp;&nbsp; ºC -1
. A expansão da ponte, quando a temperatura sobe de 0ºC para 30ºC, é de:

a)33cm&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; b)37cm&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; c) 41cm&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; d) 52cm&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; e) 99cm



4)Duas
barras metálicas são aquecidas sofrendo a mesma variação de temperatura. Qual
delas apresentará a maior dilatação linear considerando que:

a)tenham
o mesmo comprimento inicial.

b)sejam
feitas do mesmo material.

c)sejam
de materiais diferentes e tenham comprimentos iniciais diferentes.



5)Por
que se costuma deixar um espaçamento entre dois trilhos; consecutivos de uma
ferrovia?



6)O
que significa fisicamente dizer que o coeficiente de dilatação linear de certo
material vale 17.10&nbsp; -6&nbsp; ºC-1&nbsp;&nbsp; ?



7)A
10ºC, uma régua de aço mede 15cm. Qual será seu comprimento na temperatura de
80ºC?



8)O
comprimento de um fio de alumínio é de 30m a 20ºC. Sabendo-se que o fio é
aquecido até 60ºC e que o coeficiente de dilatação linear do alumínio é de 24.
10- 6 ºC-1, determine:

a)
a dilatação do fio;

b)o
comprimento final do fio.



9)O
comprimento inicial de uma barra de alumínio é de 100 cm . Quando sofre
variação de 20ºC e sua dilatação é de 0,048 cm. Determinar o coeficiente de dilatação
linear do alumínio.



10)Um
trilho de aço tem 10m de comprimento a – 10ºC. supondo que a temperatura suba
para 40ºC e que o coeficiente de dilatação do aço seja exatamente 12 . 10-
6 ºC-1, determine:

a)o
acréscimo de comprimento do trilho;

b)o
comprimento final do trilho.









2.3-DILATAÇÃO SUPERFICIAL



&nbsp;&nbsp;&nbsp;&nbsp; Consideremos uma chapa que apresenta uma superfície
inicial So a uma temperatura inicial
Ti. Ao aquecermos a chapa até uma
temperatura final Tf, sua superfície
aumenta para &nbsp;Sf

 





 

  
  
  
  
 

  
  
  
 

  
  
 


&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&nbsp;



&nbsp;&nbsp; onde:



&nbsp;Si - superfície&nbsp; inicial&nbsp;&nbsp;
&nbsp;Sf- superfície final&nbsp; &nbsp;&nbsp;&nbsp;DT - variação
de temperatura

Ti - temperatura inicial&nbsp;&nbsp;&nbsp; Tf- temperatura
final&nbsp; &nbsp;DS - variação de superfície

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; DT = Tf - Ti&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; DS = Sf - Si



Verifica-se
experimentalmente que a variação da superfície é diretamente proporcional à
superfície inicial e à variação da temperatura. Portanto:

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;


&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
DS = b. Si. DT

&nbsp;&nbsp;&nbsp;&nbsp; 

&nbsp;&nbsp; A constante
de proporcionalidade b, característica&nbsp; do material da chapa, é chamada de coeficiente de dilatação superficial.

&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;b = 2. a



&nbsp;&nbsp; Significado físico de b : O coeficiente de dilatação
superficial b representa a variação de uma
superfície unitária quando a temperatura varia de uma unidade.

&nbsp;&nbsp; NOTA:
Lembrando que&nbsp; D S = Sf - Si, podemos também escrever:

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Sf -Si = b . Si . DT, isolando
Sf temos:

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Sf = Soi+ b.Si.DT,
colocando Si em evidencia, obtem-se

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Sf = Si( 1+ b.DT )&nbsp; mas&nbsp; DT = Tf - Ti

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; Sf
= Si [ 1 +b.( Tf - Ti) ]





EXEMPLO:

&nbsp; Uma placa de ferro apresenta, a 10ºC, uma
área de 100cm². Calcule a área&nbsp; da placa
a 90ºC . Dado :aFe&nbsp; = 1,2.10-6&nbsp;&nbsp; ºC. -1





Exercícios de aplicação:

11)Uma
placa metálica tem área de 500cm² a 30ºC. Qual será a área da placa a 50ºC?
Dado: a = 2.10- 6
ºC-1 



12)Aquecendo-se
uma chapa metálica com um furo no meio

a)a
chapa aumenta e o furo diminui

b)a
chapa e o furo diminuem

c)a
chapa diminui e o furo aumenta

d)a
chapa e o furo aumentam



13)Uma
chapa de aço tem área de 36 m²
a 30ºC. Calcule sua área a 50ºC, sabendo que o coeficiente de dilatação
superficial do aço é 22 . 10- 6 ºC-1 .



14)
Determine a temperatura na qual uma chapa de cobre de área 10 m² a 20ºC assume o valor
de 10,0056 m².
Considere o coeficiente de dilatação superficial do cobre igual&nbsp;&nbsp;&nbsp; a 34. 10- 6 ºC-1



15)Uma
chapa apresenta a 0ºC, 2 m²
de área. Ao ser aquecida a 50ºC, sua&nbsp;
área&nbsp; aumenta&nbsp; 10 cm² . Determine o coeficiente de dilatação
superficial e linear do material de que é feita a chapa.



16)
Uma placa de ferro apresenta, a 90ºC, uma área de 100,192 cm² . Calcule a área
da placa a 10ºC











2.4 - DILATAÇÃO VOLUMÉTRICA



&nbsp;&nbsp;&nbsp;&nbsp; Da mesma forma que na dilatação linear e
superficial, na dilatação volumétrica a variação de volume é diretamente
proporcional ao volume inicial (Vi)
e à variação de temperatura.

 

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; DT = Tf - Ti 

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; &nbsp;DV = Vf - Vi&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; DV = g.Vi. DT

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 

&nbsp;&nbsp; A constante de proporcionalidade g, é
a característica do material de que é feito o corpo, chama-se coeficiente de dilatação volumétrica.&nbsp; g = 3. a

&nbsp;&nbsp; Significado físico de g: o coeficiente de dilatação
volumétrica representa a variação de volume de uma unidade de volume, quando a
temperatura varia de uma unidade.





EXEMPLO:



&nbsp;Um tubo de vidro&nbsp; tem volume de 500cm3 a 20ºC.
Determine o volume interno desse tubo quando a temperatura&nbsp; atingir 60ºC. Dado a vidro= 3.10 -6&nbsp; ºC-1 .



Exercícios de aplicação:

17)
Um cubo metálico tem volume 20
 cm 3à temperatura de 15ºC. Determine o seu
volume à temperatura de 25ºC, sendo o coeficiente de dilatação linear do metal
igual 0,000022ºC-1



18)Qual
será a dilatação volumétrica sofrida por uma esfera&nbsp;&nbsp; metálica de alumínio de volume igual a 150 cm 3cuja
temperatura aumenta de 100ºC? Dado a = 22.10-6&nbsp; ºC-1



19)Uma
esfera de aço tem um volume de 100cm3 a 0ºC. sabendo que o
coeficiente de dilatação linear do aço é de 12.10-6&nbsp; ºC -1, o acréscimo de volume
sofrido por essa esfera, quando aquecida a 500ºC, em cm é de:

a)0,6&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
b)1,2&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; c)1,8&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; d)3,6&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; e)5,0



20)
Um caminhão tanque com capacidade para 10 000 litros está
cheio de gasolina quando a temperatura é de 30ºC. Qual a redução de volume
sofrida pelo líquido ao ser descarregado numa ocasião em que a temperatura é de
10ºC? O coeficiente de dilatação volumétrica da gasolina é g = 9,6 . 10 -4 ºC-1.







































5-DILATAÇÃO
DOS LIQUIDOS



&nbsp;&nbsp;&nbsp; Como os
líquidos não apresentam forma própria, só tem significado o estudo de sua
dilatação volumétrica.

&nbsp;&nbsp;&nbsp;&nbsp; Ao se
estudar a dilatação dos líquidos, tem de se levar em conta a dilatação do
recipiente sólido que o contém.

&nbsp;&nbsp;&nbsp;&nbsp; De uma
maneira geral, os líquidos dilatam-se mais que os sólidos ao serem igualmente
aquecidos.










&nbsp;O líquido
irá, ao dilatar-se juntamente com o recipiente, ocupar parte da dilatação
sofrida pelo recipiente, além de mostrar dilatação própria, chamada dilatação aparente.



&nbsp;&nbsp; A dilatação
real do liquido é obtida pela soma da dilatação aparente apresentada
diretamente pelo líquido mais a dilatação volumétrica sofrida pelo recipiente

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; DV real = DVap +DV recip.

&nbsp;&nbsp; 

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
Mas:&nbsp;&nbsp;&nbsp;&nbsp; DVreal = greal.Vi.DT

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; DVap = gap. Vi . DT

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; DV recip.= grecip. Vi.DT 

&nbsp;Relação entre os coeficientes de dilatação real e
aparente do líquido

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;greal = gap + grecip&nbsp;&nbsp; 



&nbsp;6- DILATAÇÃO ANÔMALA DA ÁGUA

As substâncias em geral, solidas ou liquidas, sofrem
dilatação com o aumento da temperatura e naturalmente, sofrem contração com sua
diminuição. Porém, existem exceções a essa regra. Algumas substâncias, em
certas faixas de temperatura, diminuem o volume ao serem aquecidas. 

Dentre essas exceções destaca-se a água, que, ao ser
aquecida de 0º a 4ºC, sofre diminuição em seu volume. É devido a esse fenômeno
que as camadas mais profundas&nbsp; dos lagos
e mares podem estar a 4ºC e não se congelar como ocorre com suas superfícies.
Na superfície dos lagos a água esfria primeiro, diminuindo de volume. Com isso
torna-se mais densa, indo para o fundo. A água do fundo, mais quente, está
menos densa e sobe à superfície, onde é resfriada. Quando toda a água está a
4ºC, cessa a circulação. Se a temperatura da superfície cai abaixo de 4ºc, a
água vai tornar-se menos densa; permanecerá, então, na superfície, podendo
solidificar-se e formar camadas de gelo a 0ºC.

&nbsp;&nbsp; Se a água
não apresentasse esta irregularidade na dilatação, os rios e lagos se
congelariam totalmente, causando estragos às plantas e animais aquáticos.



EXEMPLO:



Um
recipiente contém 200 cm
3 de mercúrio a 10ºC. Sabendo-se que os coeficientes de dilatação volumétrica
do vidro e do mercúrio são respectivamente 0,00003 ºC –1 e 0,00018
ºC –1&nbsp; , determine a dilatação
real e a dilatação aparente do mercúrio, quando a temperatura for 120ºC



Exercícios de aplicação:



21)Um
frasco de vidro, cujo volume é 300
 cm 3
 a 10ºC, está completamente cheio de um certo líquido.
Quando se aquece o conjunto a uma temperatura de 140ºC, transbordam 2 cm
3 do liquido. Sendo o coeficiente de dilatação volumétrica do frasco
igual a&nbsp; 0,00027&nbsp;&nbsp; ºC –1, determine: 

a)&nbsp;&nbsp;&nbsp;&nbsp; o coeficiente de dilatação
volumétrica aparente do líquido,

b)&nbsp;&nbsp;&nbsp; o coeficiente de dilatação
volumétrica real do líquido.



22)Um
recipiente de vidro contém 400
 cm 3 de mercúrio a 20ºC. Determine a
dilatação aparente do mercúrio, quando a temperatura for 90ºC. Dados:gHg = 0,00018ºC –1&nbsp;&nbsp; &nbsp;e&nbsp;&nbsp; gvidro=0,00003&nbsp; ºC –1&nbsp;&nbsp;&nbsp;
&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;



23)O que se entende por dilatação aparente de um líquido?





24)Um frasco, completamente cheio de um líquido, é
aquecido e transborda. O volume de líquido que transborda mede:

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (&nbsp;&nbsp; )a dilatação real do líquido

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (&nbsp;&nbsp; )a dilatação aparente do líquido

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; (&nbsp;&nbsp; )a dilatação do frasco





25)Nos
países de inverno rigoroso, verifica-se o congelamento&nbsp; apenas da superfície dos lagos e rios. A água
não se congela completamente porque:

a)o máximo de densidade da água se verifica perto de 4ºC
e o gelo, razoável isolante térmico, é menos denso que a água.

b)o ar se resfria antes da água, congelando-se primeiro a
superfície dos líquidos em contato com o referido ar e, daí, propagando-se o
congelamento em profundidade.

c)a
água em movimento dificilmente se congela

d)a
água se comporta como a maioria dos líquidos em relação ás variações de
temperatura

e)n.d.a