Thermal Expansion | Physics Optional Notes for UPSC PDF Download

 Page 1


 
THERMAL EXPANSION 
When you make something hot, it usually gets larger but stays the same thing. In the 
atomic theory, the reason for this expansion is because of how the energy changes 
when things get hotter.  
 
Think of atoms as tiny things shaking. When you increase the temperature from one 
level to another, the atoms shake more, and their energy goes up. This makes the 
average space between the atoms increase, which makes the material get bigger. 
Potential energy V/s distance curve :- 
 
 
T
2
>T
1
 
Page 2


 
THERMAL EXPANSION 
When you make something hot, it usually gets larger but stays the same thing. In the 
atomic theory, the reason for this expansion is because of how the energy changes 
when things get hotter.  
 
Think of atoms as tiny things shaking. When you increase the temperature from one 
level to another, the atoms shake more, and their energy goes up. This makes the 
average space between the atoms increase, which makes the material get bigger. 
Potential energy V/s distance curve :- 
 
 
T
2
>T
1
 
E
2
>E
1
 
?? 2 avg. 
>?? 1 avg. 
 (due to asymmetric nature of curve) 
When atoms move farther apart, the whole material expands, and we call this thermal 
expansion. If the potential energy curve was symmetrical, heating wouldn't cause any 
expansion. 
Coefficient of linear expansion (?? ) : 
Most things get bigger when they are heated and smaller when they are cooled, though 
some change more than others. When a solid expands in just one direction, like a line, 
we call it "linear expansion." Imagine a rod that is a certain length when it's a certain 
temperature, say L_0 when the temperature is T_0. If the temperature goes up by a 
certain amount, let's call it ?T, the length becomes L_0 plus a bit extra, ?L. Similarly, if 
the temperature goes down by ?T, the length decreases to L_0 minus ?L. 
 
Experiments have shown that for small changes in temperature, the amount the length 
changes (?L) is directly related to the change in temperature (?T). Also, the change in 
length is linked to the starting length of the rod. 
?L
L
??T 
?L
L
=?? ?T 
?L=L?? ?T 
Equation ?L=?? L
0
?T expresses the fact that ?L is proportional to both L
0
 and 
?T(?L?L
0
?T) by using a proportionality constant ?? , which is called the coefficient of 
linear expansion. 
L
2
-L
1
=L
1
?? (T
2
-T
1
) 
 
L
2
=L
1
[1+?? (T
2
-T
1
)] 
Common unit for the coefficient of linear expansion : 
1
C
°
=(C
°
)
-1
  L
2
=L
1
(1+?? ?T) 
Page 3


 
THERMAL EXPANSION 
When you make something hot, it usually gets larger but stays the same thing. In the 
atomic theory, the reason for this expansion is because of how the energy changes 
when things get hotter.  
 
Think of atoms as tiny things shaking. When you increase the temperature from one 
level to another, the atoms shake more, and their energy goes up. This makes the 
average space between the atoms increase, which makes the material get bigger. 
Potential energy V/s distance curve :- 
 
 
T
2
>T
1
 
E
2
>E
1
 
?? 2 avg. 
>?? 1 avg. 
 (due to asymmetric nature of curve) 
When atoms move farther apart, the whole material expands, and we call this thermal 
expansion. If the potential energy curve was symmetrical, heating wouldn't cause any 
expansion. 
Coefficient of linear expansion (?? ) : 
Most things get bigger when they are heated and smaller when they are cooled, though 
some change more than others. When a solid expands in just one direction, like a line, 
we call it "linear expansion." Imagine a rod that is a certain length when it's a certain 
temperature, say L_0 when the temperature is T_0. If the temperature goes up by a 
certain amount, let's call it ?T, the length becomes L_0 plus a bit extra, ?L. Similarly, if 
the temperature goes down by ?T, the length decreases to L_0 minus ?L. 
 
Experiments have shown that for small changes in temperature, the amount the length 
changes (?L) is directly related to the change in temperature (?T). Also, the change in 
length is linked to the starting length of the rod. 
?L
L
??T 
?L
L
=?? ?T 
?L=L?? ?T 
Equation ?L=?? L
0
?T expresses the fact that ?L is proportional to both L
0
 and 
?T(?L?L
0
?T) by using a proportionality constant ?? , which is called the coefficient of 
linear expansion. 
L
2
-L
1
=L
1
?? (T
2
-T
1
) 
 
L
2
=L
1
[1+?? (T
2
-T
1
)] 
Common unit for the coefficient of linear expansion : 
1
C
°
=(C
°
)
-1
  L
2
=L
1
(1+?? ?T) 
Coefficient of area expansion (?? ) : 
?A
A
??T 
?A
A
=?? ?T 
 
?? =?? 0
[1+?? (?? 2
-?? 1
) 
Also A
0
=l
0
2
 and A=l
2
 
So l
2
=l
0
2
(1+?? ??? )=[l
0
(1+?? ??? )]
2
??? =2?? 
Coefficient of volume expansion (?? ) : 
?V
V
??T 
?V=V?? ?T 
 
V
2
=V
1
[1+?? (T
2
-T
1
)] 
[Unit of ?? ,?? ,?? is 1/C or 1/K
-1
 or K
-1
 ] 
V=V
0
(1+?? ??? ) Also V=l
3
 and V
0
=l
0
3
 so ?? =3?? ? 6?? =3?? =2?? or ?? :?? :?? =1:2:3
 
Example. How much longer or shorter will a 1-meter iron rod be if its temperature 
changes by 200 degrees Celsius? The iron has a value of 5 x 10^(-5) per degree Celsius 
for its coefficient of linear expansion. 
Solution: Percentage change in length due to temperature change 
Page 4


 
THERMAL EXPANSION 
When you make something hot, it usually gets larger but stays the same thing. In the 
atomic theory, the reason for this expansion is because of how the energy changes 
when things get hotter.  
 
Think of atoms as tiny things shaking. When you increase the temperature from one 
level to another, the atoms shake more, and their energy goes up. This makes the 
average space between the atoms increase, which makes the material get bigger. 
Potential energy V/s distance curve :- 
 
 
T
2
>T
1
 
E
2
>E
1
 
?? 2 avg. 
>?? 1 avg. 
 (due to asymmetric nature of curve) 
When atoms move farther apart, the whole material expands, and we call this thermal 
expansion. If the potential energy curve was symmetrical, heating wouldn't cause any 
expansion. 
Coefficient of linear expansion (?? ) : 
Most things get bigger when they are heated and smaller when they are cooled, though 
some change more than others. When a solid expands in just one direction, like a line, 
we call it "linear expansion." Imagine a rod that is a certain length when it's a certain 
temperature, say L_0 when the temperature is T_0. If the temperature goes up by a 
certain amount, let's call it ?T, the length becomes L_0 plus a bit extra, ?L. Similarly, if 
the temperature goes down by ?T, the length decreases to L_0 minus ?L. 
 
Experiments have shown that for small changes in temperature, the amount the length 
changes (?L) is directly related to the change in temperature (?T). Also, the change in 
length is linked to the starting length of the rod. 
?L
L
??T 
?L
L
=?? ?T 
?L=L?? ?T 
Equation ?L=?? L
0
?T expresses the fact that ?L is proportional to both L
0
 and 
?T(?L?L
0
?T) by using a proportionality constant ?? , which is called the coefficient of 
linear expansion. 
L
2
-L
1
=L
1
?? (T
2
-T
1
) 
 
L
2
=L
1
[1+?? (T
2
-T
1
)] 
Common unit for the coefficient of linear expansion : 
1
C
°
=(C
°
)
-1
  L
2
=L
1
(1+?? ?T) 
Coefficient of area expansion (?? ) : 
?A
A
??T 
?A
A
=?? ?T 
 
?? =?? 0
[1+?? (?? 2
-?? 1
) 
Also A
0
=l
0
2
 and A=l
2
 
So l
2
=l
0
2
(1+?? ??? )=[l
0
(1+?? ??? )]
2
??? =2?? 
Coefficient of volume expansion (?? ) : 
?V
V
??T 
?V=V?? ?T 
 
V
2
=V
1
[1+?? (T
2
-T
1
)] 
[Unit of ?? ,?? ,?? is 1/C or 1/K
-1
 or K
-1
 ] 
V=V
0
(1+?? ??? ) Also V=l
3
 and V
0
=l
0
3
 so ?? =3?? ? 6?? =3?? =2?? or ?? :?? :?? =1:2:3
 
Example. How much longer or shorter will a 1-meter iron rod be if its temperature 
changes by 200 degrees Celsius? The iron has a value of 5 x 10^(-5) per degree Celsius 
for its coefficient of linear expansion. 
Solution: Percentage change in length due to temperature change 
%l=
?l
l
×100=?? ??? ×100=5×10
-5
×200×100=1% 
Example. A concrete slab is 10 meters long when it's a chilly winter night at 0 degrees 
Celsius. How long will it be on a hot summer day when it's 45 degrees Celsius? The 
coefficient of linear expansion for concrete is 1.0 x 10^(-5) per degree Celsius. 
Solution : l
t
=10(1+1×10
-5
×45)=10.0045 m 
Example.  
Imagine a rod made of two materials: iron (50 cm) and aluminum (100 cm). When this 
rod is heated from 20°C to 100°C, we want to know: 
(a) How long the iron part will be at 100°C. 
(b) How long the aluminum part will be at 100°C. 
(c) What is the average expansion rate of this rod? 
Solution :(a) ?? 2
=?? 1
[1+?? (T
2
-T
1
)] 
l=50[1+12×10
-6
×80] 
?? im
=50[1+96×10
-5
]=50+48×10
-3
 
?? iron 
=50.048 cm 
(b) ?? alu 
=100[1+24×10
-6
×80] 
=100+0.192=100.192 cm 
(c) ??? equi 
=??? iron 
+??? alu 
 
=0.048+0.192 
=0.24 cm 
?? equi 
=100+50=150 cm 
?=6??? 
?? equ. 
=
?l
equ. 
l
equ. 
?T
?
0.24
150×80
?20×10
-6
/ 
°
C 
Example. The difference between lengths of a certain brass rod and of a steel rod is 
claimed to be constant at all temperatures. Is this possible? 
Solution: If ?? ?? and ?? ?? are the lengths of brass and steel rods respectively at a given 
temperature, then the lengths of the rods when temperature is changed by ?? °
C. 
Page 5


 
THERMAL EXPANSION 
When you make something hot, it usually gets larger but stays the same thing. In the 
atomic theory, the reason for this expansion is because of how the energy changes 
when things get hotter.  
 
Think of atoms as tiny things shaking. When you increase the temperature from one 
level to another, the atoms shake more, and their energy goes up. This makes the 
average space between the atoms increase, which makes the material get bigger. 
Potential energy V/s distance curve :- 
 
 
T
2
>T
1
 
E
2
>E
1
 
?? 2 avg. 
>?? 1 avg. 
 (due to asymmetric nature of curve) 
When atoms move farther apart, the whole material expands, and we call this thermal 
expansion. If the potential energy curve was symmetrical, heating wouldn't cause any 
expansion. 
Coefficient of linear expansion (?? ) : 
Most things get bigger when they are heated and smaller when they are cooled, though 
some change more than others. When a solid expands in just one direction, like a line, 
we call it "linear expansion." Imagine a rod that is a certain length when it's a certain 
temperature, say L_0 when the temperature is T_0. If the temperature goes up by a 
certain amount, let's call it ?T, the length becomes L_0 plus a bit extra, ?L. Similarly, if 
the temperature goes down by ?T, the length decreases to L_0 minus ?L. 
 
Experiments have shown that for small changes in temperature, the amount the length 
changes (?L) is directly related to the change in temperature (?T). Also, the change in 
length is linked to the starting length of the rod. 
?L
L
??T 
?L
L
=?? ?T 
?L=L?? ?T 
Equation ?L=?? L
0
?T expresses the fact that ?L is proportional to both L
0
 and 
?T(?L?L
0
?T) by using a proportionality constant ?? , which is called the coefficient of 
linear expansion. 
L
2
-L
1
=L
1
?? (T
2
-T
1
) 
 
L
2
=L
1
[1+?? (T
2
-T
1
)] 
Common unit for the coefficient of linear expansion : 
1
C
°
=(C
°
)
-1
  L
2
=L
1
(1+?? ?T) 
Coefficient of area expansion (?? ) : 
?A
A
??T 
?A
A
=?? ?T 
 
?? =?? 0
[1+?? (?? 2
-?? 1
) 
Also A
0
=l
0
2
 and A=l
2
 
So l
2
=l
0
2
(1+?? ??? )=[l
0
(1+?? ??? )]
2
??? =2?? 
Coefficient of volume expansion (?? ) : 
?V
V
??T 
?V=V?? ?T 
 
V
2
=V
1
[1+?? (T
2
-T
1
)] 
[Unit of ?? ,?? ,?? is 1/C or 1/K
-1
 or K
-1
 ] 
V=V
0
(1+?? ??? ) Also V=l
3
 and V
0
=l
0
3
 so ?? =3?? ? 6?? =3?? =2?? or ?? :?? :?? =1:2:3
 
Example. How much longer or shorter will a 1-meter iron rod be if its temperature 
changes by 200 degrees Celsius? The iron has a value of 5 x 10^(-5) per degree Celsius 
for its coefficient of linear expansion. 
Solution: Percentage change in length due to temperature change 
%l=
?l
l
×100=?? ??? ×100=5×10
-5
×200×100=1% 
Example. A concrete slab is 10 meters long when it's a chilly winter night at 0 degrees 
Celsius. How long will it be on a hot summer day when it's 45 degrees Celsius? The 
coefficient of linear expansion for concrete is 1.0 x 10^(-5) per degree Celsius. 
Solution : l
t
=10(1+1×10
-5
×45)=10.0045 m 
Example.  
Imagine a rod made of two materials: iron (50 cm) and aluminum (100 cm). When this 
rod is heated from 20°C to 100°C, we want to know: 
(a) How long the iron part will be at 100°C. 
(b) How long the aluminum part will be at 100°C. 
(c) What is the average expansion rate of this rod? 
Solution :(a) ?? 2
=?? 1
[1+?? (T
2
-T
1
)] 
l=50[1+12×10
-6
×80] 
?? im
=50[1+96×10
-5
]=50+48×10
-3
 
?? iron 
=50.048 cm 
(b) ?? alu 
=100[1+24×10
-6
×80] 
=100+0.192=100.192 cm 
(c) ??? equi 
=??? iron 
+??? alu 
 
=0.048+0.192 
=0.24 cm 
?? equi 
=100+50=150 cm 
?=6??? 
?? equ. 
=
?l
equ. 
l
equ. 
?T
?
0.24
150×80
?20×10
-6
/ 
°
C 
Example. The difference between lengths of a certain brass rod and of a steel rod is 
claimed to be constant at all temperatures. Is this possible? 
Solution: If ?? ?? and ?? ?? are the lengths of brass and steel rods respectively at a given 
temperature, then the lengths of the rods when temperature is changed by ?? °
C. 
L
B
'
=L
B
(1+?? B
??? ) and L
S
'
=L
S
(1+?? S
??? ) 
So that  L
B
'
=L
S
'
(L
B
-L
S
)+(L
B
?? B
-L
S
?? S
)??? 
So (L
B
'
-L
S
'
 ) will be equal to (L
B
-L
S
) at all temperatures if , L
B
?? B
-L
S
?? S
=0 [as ??? ?
0 ] 
or 
?? ?? ?? ?? =
?? ?? ?? ?? 
i.e., the difference in the lengths of the two rods will be independent of temperature if the 
lengths are in the inverse ratio of their coefficients of linear expansion. 
Calculating fractional change Or Percentage change : 
?? =?? ?? ?? ?? ?? differentiate ???? =?? [?? ?? ?? ?? ?? -1
???? +?? ?? ?? ?? ?? -1
???? ]
 
equation (2)÷(1) 
dz
z
=x
dA
A
+y
dB
B
 
For small change 
????
?? =?? ??? ?? +?? ??? ?? 
(% change in z)=x(% change in A)+y(% change in B) 
Example. r of cylinder increase by 0.2%& h increase by 0.5%. Find % increase in curved 
surface area & volume? 
Solution: Curved surface area (a)=2?? rh 
?A
A
=
?r
r
+
?h
h
?A
A
=0.2+0.5
?A
A
=0.7%?
 
Volume (V)=?? r
2
 h 
?V
V
=2
?r
r
+
?h
h
?2×0.2+0.5=0.4+0.5 
?V
V
=0.9%? 
Example. ?
?
3
 B
2
=A
3
 B
-2
 ? 
?z
z
=3
?A
A
+(-2)
?B
B
 
(a) % increase in z on increase A by 2% & increase B by 1% ?