PHYS 305 - INTRODUCTION TO
MODERN PHYSICS - Thermal
expansion
Question Bank - Set 4
Liberty University
Question 1
Question
A steel rod is 2 meters long at 20 degrees Celsius. If the coefficient of linear
expansion of steel is 1.2×10−5per degree Celsius, find the length of the rod at
100 degrees Celsius.
Solution
Let L0be the original length of the steel rod at 20 degrees Celsius, Lbe the
length of the steel rod at 100 degrees Celsius, and αbe the coefficient of linear
expansion of steel.
Step 1: Calculate the change in length of the rod from 20 degrees Celsius
to 100 degrees Celsius.
∆L=L−L0=L0α∆T
where ∆Tis the change in temperature.
∆L= 2 ×1.2×10−5×(100 −20)
∆L= 2 ×1.2×10−5×80 = 1.92 ×10−3meters
Step 2: Calculate the length of the rod at 100 degrees Celsius.
L=L0+ ∆L= 2 + 1.92 ×10−3
L= 2.00192 meters
So, the length of the steel rod at 100 degrees Celsius is 2.00192 meters.
Question 2
Question
A copper rod and an aluminum rod are both 2 meters long at 20◦C. If both
rods are heated to 120◦C, by how many millimeters will each rod elongate? The
coefficient of linear expansion for copper is 16.6×10−6/◦C and for aluminum is
23.1×10−6/◦C.
Solution
Let’s denote: - Lcopper as the initial length of the copper rod, - Laluminum as the
initial length of the aluminum rod, - αcopper as the coefficient of linear expansion
for copper, and - αaluminum as the coefficient of linear expansion for aluminum.
Step 1: Calculate the change in length of the copper rod. The change in
length of the copper rod can be found using the formula:
∆Lcopper =Lcopper ·αcopper ·∆T
where ∆Tis the change in temperature. Substitute the given values:
∆Lcopper = 2 m ·16.6×10−6/◦C·(120◦C−20◦C)
∆Lcopper = 2 ×16.6×10−6×100 m
∆Lcopper = 0.00332 m = 3.32 mm
Step 2: Calculate the change in length of the aluminum rod. Similarly, the
change in length of the aluminum rod can be found using the formula:
∆Laluminum =Laluminum ·αaluminum ·∆T
Substitute the given values:
∆Laluminum = 2 m ·23.1×10−6/◦C·(120◦C−20◦C)
∆Laluminum = 2 ×23.1×10−6×100 m
∆Laluminum = 0.00462 m = 4.62 mm
Therefore, the copper rod will elongate by 3.32 mm and the aluminum rod
will elongate by 4.62 mm when heated to 120◦C.
Question 3
Question
A brass rod is 1 m long at 20◦C. What is its length at 150◦C if the coefficient
of linear expansion of brass is 19 ×10−6K−1?
2
Solution
Step 1: Let’s denote the original length of the brass rod as L0= 1 m and the
coefficient of linear expansion as α= 19 ×10−6K−1. We want to find the final
length Lof the rod at 150◦C.
Step 2: The formula for linear expansion is given by:
∆L=α·L0·∆T
where ∆Lis the change in length, L0is the original length, αis the coefficient
of linear expansion, and ∆Tis the change in temperature.
Step 3: We can rearrange the formula to solve for the final length L:
L=L0+ ∆L=L0+α·L0·∆T
Step 4: The change in temperature is ∆T= 150◦C−20◦C = 130◦C. Sub-
stituting the values into the formula, we get:
L= 1 + 19 ×10−6×1×130 = 1 + 19 ×10−6×130
Step 5: Calculating the final length:
L= 1 + 19 ×10−6×130 = 1 + 2.47 ×10−4
Step 6: Therefore, the final length of the brass rod at 150◦C is approximately
1.00025 m.
Question 4
Question
A copper rod of length 2m is heated from 20◦C to 120◦C. If the linear expansion
coefficient of copper is 1.7×10−5◦C−1, find the change in length of the rod.
Solution
Let Lbe the original length of the copper rod, ∆Lbe the change in length, αbe
the linear expansion coefficient of copper, and ∆Tbe the change in temperature.
Step 1: Calculate the change in temperature. Given that the initial tem-
perature is 20◦C and the final temperature is 120◦C, we have:
∆T= 120◦C−20◦C = 100◦C
Step 2: Calculate the change in length. The formula for linear expansion
is given by:
∆L=α·L·∆T
Substitute the values of α,L, and ∆Tinto the formula:
∆L= (1.7×10−5◦C−1)·(2 m) ·(100 ◦C)
3
∆L= 0.000034 m
Step 3: Finalize the answer. Therefore, the change in length of the copper
rod is 0.000034 m, or 0.034 mm.
Question 5
Question
A steel beam is initially 5 m long at a temperature of 20◦C. If the temperature
of the beam increases to 50◦C, what will be the new length of the beam? The
coefficient of linear expansion for steel is 12 ×10−6
Step 1: Calculate the change in temperature. Given: Initial temperature, T1=
20◦C Final temperature, T2= 50◦C
The change in temperature, ∆T=T2−T1= 50 −20 = 30 C.
Step 2: Calculate the change in length. The coefficient of linear expansion,
α= 12 ×10−6
Question
A 2 m long rod made of steel has a hole drilled from one end towards the center.
When the temperature changes by 50
°
C, the length of the rod changes by 0.3
mm. If the diameter of the hole is 5 mm, what is the change in the length
of the hole when the temperature changes by 50
°
C? Use a coefficient of linear
expansion for steel of 12 ×10−6per degree Celsius.
Solution
Let’s denote the change in length of the hole as x. Since the rod is made of
steel and the hole is in the center, the thermal expansion of the rod will affect
the length of both the rod and the hole.
Step 1: Calculate the change in length of the rod using the coefficient of
linear expansion. The change in length of the rod can be calculated using the
formula:
Change in length of rod = α·L·∆T
where αis the coefficient of linear expansion, Lis the original length of the rod,
and ∆Tis the change in temperature.
Substitute the values:
0.3 mm = 12 ×10−6·2 m ·50
0.3×10−3m = 12 ×10−6·2·50
0.3×10−3= 1.2×10−3
x= 1.2×10−3
4
Step 2: Calculate the change in length of the hole. Since the hole is drilled
from one end towards the center, the change in length of the hole will be half
the change in length of the rod. Therefore, the change in length of the hole is:
1.2×10−3
2= 0.6×10−3
0.6 mm
Therefore, the change in the length of the hole when the temperature changes
by 50
°
C is 0.6 mm.
Question 7
Question
A steel rod has a length of 2 meters at 20◦C. If the coefficient of linear expansion
of steel is 1.2×10−5K−1, calculate the increase in length of the rod when its
temperature is raised to 200◦C.
Solution
Step 1: Calculate the change in temperature. Given: Initial temperature, T1=
20◦C Final temperature, T2= 200◦C Change in temperature, ∆T=T2−T1=
200◦C - 20◦C = 180◦C
Step 2: Calculate the increase in length of the rod. The increase in length,
∆L, can be calculated using the formula:
∆L=α·L·∆T
where: α= coefficient of linear expansion = 1.2×10−5K−1L= initial length
of the rod = 2 meters ∆T= change in temperature = 180◦C
Substitute the values into the formula:
∆L= 1.2×10−5K−1×2 m ×180
∆L= 4.32 ×10−3m=4.32 mm
Therefore, the increase in length of the rod when its temperature is raised
to 200◦C is 4.32 mm.
Question 8
Question
A steel rod measures 2 meters at a temperature of 20◦C. If the coefficient of
linear expansion for steel is 12 ×10−6/◦C, what will be the length of the rod
when its temperature is raised to 100◦C?
5
Solution
Step 1: Determine the change in temperature. Given that the initial temper-
ature is 20◦C and the final temperature is 100◦C, the change in temperature
is:
∆T= 100◦C−20◦C = 80◦C
Step 2: Calculate the change in length of the rod. The change in length of
the rod ∆Lcan be calculated using the formula:
∆L=L0α∆T
where: - L0is the initial length of the rod, - αis the coefficient of linear
expansion, - ∆Tis the change in temperature.
Given L0= 2 m and α= 12 ×10−6/◦C, we have:
∆L= 2 ×12 ×10−6×80 = 1.92 ×10−4m
Step 3: Find the final length of the rod. The final length of the rod Lfcan
be determined by adding the change in length to the initial length:
Lf=L0+ ∆L= 2 + 1.92 ×10−4= 2.000192 m
Therefore, when the temperature of the steel rod is raised to 100◦C, its
length will be 2.000192 meters.
Question 9
Question
A metal rod is 2 meters long at 0 degrees Celsius. If the coefficient of linear
expansion for the metal is 2.3×10−5/
°
C, what is the length of the rod at 100
degrees Celsius?
Solution
Step 1: First, we calculate the change in length of the rod. Given: Initial
length L0= 2 m, Coefficient of linear expansion α= 2.3×10−5/
°
C, Change in
temperature ∆T= 100
°
C.
The change in length ∆Lis given by the formula:
∆L=L0·α·∆T
Step 2: Substituting the values into the formula, we get:
∆L= 2 ·2.3×10−5·100
Step 3: Calculate the change in length:
∆L= 0.0046 m
6
Step 4: Finally, we find the length of the rod at 100 degrees Celsius. The
final length Lis given by:
L=L0+ ∆L
Step 5: Substitute the values into the formula:
L= 2 + 0.0046
Step 6: Calculate the final length of the rod:
L= 2.0046 m
Therefore, the length of the rod at 100 degrees Celsius is 2.0046 meters.
Question 10
Question
A steel bridge is constructed with a length of 100 meters at a temperature of
20
°
C. If the temperature rises to 40
°
C during a hot day, calculate the change
in length of the bridge assuming a coefficient of linear expansion of steel to be
1.2×10−5
°
C−1.
Solution
Step 1: Calculate the change in temperature: The change in temperature is
given by: ∆T=Tf−Ti, where Tfis the final temperature (40
°
C) and Tiis the
initial temperature (20
°
C). Therefore, ∆T= 40C−20C= 20C.
Step 2: Calculate the change in length using the formula for linear expansion:
The change in length (∆L) is given by: ∆L=α·L·∆T, where αis the coefficient
of linear expansion, Lis the original length (100 meters), and ∆Tis the change
in temperature. Substitute the values to find: ∆L= (1.2×10−5
°
C−1)×100 m×
20
°
C. This results in: ∆L= 0.024 m.
Therefore, the change in length of the steel bridge due to the temperature
increase is 0.024 meters (or 2.4 centimeters).
Question 11
Question
A steel rod is initially 2 meters long at a temperature of 20
°
C. If the rod is heated
to 80
°
C, what is the final length of the rod? The linear expansion coefficient of
steel is 1.2×10−5per degree Celsius.
7
Solution
Step 1: Calculate the change in temperature.
∆T= 80C−20C= 60C
Step 2: Use the formula for linear thermal expansion to find the change in
length.
∆L=α·L·∆T
∆L= (1.2×10−5)·2·60
∆L= 1.44 ×10−3m
Step 3: Find the final length of the rod.
Lf=Li+ ∆L
Lf= 2 + 1.44 ×10−3
Lf= 2.00144 m
Therefore, the final length of the steel rod when heated to 80
°
C is 2.00144
meters.
Question 12
Question
A copper rod 2 meters long at 0◦C is heated until its length increases by 2
mm. Calculate the average coefficient of linear expansion of copper in terms of
10−6/◦C.
Solution
Step 1: Recall the formula for linear expansion: The change in length (∆L) of
a material due to a change in temperature is given by:
∆L=L0α∆T
where: - L0is the original length, - αis the coefficient of linear expansion, and
- ∆Tis the change in temperature.
Step 2: Convert all the known values to SI units: Given L0= 2 m, ∆L= 2
mm = 2 ×10−3m.
Step 3: Calculate the change in temperature:
∆L=L0α∆T
2×10−3= 2 ×α×∆T
∆T=2×10−3
2×α
8
∆T=10−3
α
Step 4: Use the value of ∆Tto find the coefficient of linear expansion α: The
average coefficient of linear expansion is the change in temperature required to
cause a unit change in length, so αcan be calculated as:
α=10−3
∆T
Therefore, the average coefficient of linear expansion is α=10−3
∆T.
Question 13
Question
A steel rod is 2 meters long at 20◦C. If the coefficient of linear expansion for
steel is 12 ×10−6/K, what will be the length of the rod when its temperature
is increased to 100◦C?
Solution
Step 1: Calculate the change in temperature. Given that the initial temper-
ature is 20◦C and the final temperature is 100◦C, we can find the change in
temperature:
∆T=Tfinal −Tinitial = 100◦C−20◦C = 80◦C
Step 2: Calculate the change in length. The change in length (∆L) can be
calculated using the formula:
∆L=α·L·∆T
where α= 12 ×10−6/K is the coefficient of linear expansion for steel, L= 2 m
is the initial length of the rod, and ∆T= 80◦C is the change in temperature.
Substitute the values into the formula:
∆L= 12 ×10−6/K×2 m ×80◦C
Step 3: Calculate the change in length.
∆L= 1.92 ×10−3m
Step 4: Find the final length of the rod. The final length of the rod can be
found by adding the change in length to the initial length:
Lfinal =Linitial + ∆L
Lfinal = 2 m + 1.92 ×10−3m
Therefore, the length of the steel rod when its temperature is increased to
100◦C is approximately 2.00192 meters.
9
Question 14
Question
A solid copper cylinder has a diameter of 10 cm at 20◦C. What will be its
diameter when it is heated to 120◦C? Given that the linear expansion coefficient
of copper is 17 ×10−6◦C−1.
Solution
Step 1: We first need to calculate the change in temperature. Step 2: Calculate
the change in diameter using the linear expansion coefficient of copper. Step 3:
Find the final diameter of the cylinder.
Step 1: The change in temperature is given by:
∆T= 120◦C−20◦C= 100◦C
Step 2: The change in length (∆L) of the cylinder can be calculated using
the formula:
∆L=αL0∆T
where αis the linear expansion coefficient, L0is the original length, and ∆Tis
the change in temperature.
The change in diameter is equal to twice the change in length, so:
∆d= 2αd0∆T
Substitute the given values:
∆d= 2 ×17 ×10−6×10 ×100 = 0.034 cm
Step 3: The final diameter (df) can be found by adding the change in
diameter to the original diameter:
df=d0+ ∆d= 10 + 0.034 = 10.034 cm
Therefore, the diameter of the copper cylinder when heated to 120◦C will
be 10.034 cm.
Question 15
Question
A brass rod has an initial length of 2.00 m at 20◦C. If the coefficient of linear
expansion of brass is 1.9×10−5◦C−1, what is the change in length of the rod
when its temperature is increased to 100◦C?
10
Solution
Step 1: Calculate the change in temperature. Let ∆Tbe the change in temper-
ature. Given: Initial temperature, Ti= 20◦C Final temperature, Tf= 100◦C
Change in temperature:
∆T=Tf−Ti= 100◦C−20◦C = 80◦C
Step 2: Calculate the change in length using the formula:
∆L=α·L·∆T
where: ∆Lis the change in length, αis the coefficient of linear expansion
(1.9×10−5◦C−1), Lis the initial length of the rod (2.00 m), ∆Tis the change
in temperature (80◦C).
Substitute the values:
∆L= (1.9×10−5◦C−1)×2.00 m ×80◦C
Step 3: Calculate the change in length.
∆L= 3.04 ×10−3m
Therefore, the change in length of the brass rod when its temperature is
increased to 100◦C is 3.04 mm.
Question 16
Question
A solid metal rod of length 2 m at 20
°
C is heated to 120
°
C. If the coefficient of
linear expansion for this metal is 1.5×10−5/
°
C, find the new length of the rod.
Solution
Step 1: Determine the change in temperature. Given that the initial temper-
ature is 20
°
C and the final temperature is 120
°
C, the change in temperature
is:
∆T= 120C−20C= 100C
Step 2: Calculate the change in length using the formula for linear expansion.
The change in length (∆L) is given by:
∆L=Lα∆T
Where: L= Initial length of the rod = 2 m, α= Coefficient of linear expansion
= 1.5×10−5/
°
C, ∆T= Change in temperature = 100
°
C.
Substitute the values into the formula:
∆L= 2 ×1.5×10−5×100
11
∆L= 2 ×1.5×10−3
∆L= 3 ×10−3m
Step 3: Find the new length of the rod. The new length of the rod can be
calculated by adding the change in length to the initial length:
New length = Initial length + ∆L
New length = 2 + 3 ×10−3
New length = 2.003 m
Therefore, the new length of the rod after heating it to 120
°
C is 2.003 meters.
Question 17
Question
A steel rod is initially 10 meters long at 20
°
C. If the coefficient of linear expan-
sion for steel is 1.2×10−5/
°
C, find the change in length of the rod when its
temperature increases to 150
°
C.
Solution
Step 1: Calculate the change in temperature.
Let Lbe the original length of the rod and ∆Tbe the change in temperature.
Given: L= 10 m (initial length) T1= 20
°
C (initial temperature) T2= 150
°
C
(final temperature) The change in temperature can be calculated as:
∆T=T2−T1= 150C −20C = 130C
Step 2: Calculate the change in length using the coefficient of linear expan-
sion formula.
The change in length (∆L) can be calculated using the formula:
∆L=L·α·∆T
where L= original length of the rod, α= coefficient of linear expansion =
1.2×10−5/
°
C, ∆T= change in temperature. Substitute the values into the
formula:
∆L= 10 ·1.2×10−5·130
∆L= 10 ×1.2×10−5×130
∆L= 0.156 m
Therefore, the change in length of the rod when its temperature increases to
150
°
C is 0.156 meters.
12
Question 18
Question
A steel rod has a length of 1.5 m at 20
°
C. If the coefficient of linear expansion
of steel is 1.2×10−5
°
C−1, calculate the change in length of the rod when it is
heated to 100
°
C.
Solution
Given: Initial length of steel rod, L0= 1.5 m
Change in temperature, ∆T= 100C−20C= 80C
Coefficient of linear expansion, α= 1.2×10−5
°
C−1
We can use the formula for linear expansion: ∆L=α·L0·∆Tto find the
change in length of the rod.
Step 1: Calculate the change in length.
∆L= 1.2×10−5
°
C−1×1.5 m ×80C
∆L= 1.2×10−5×1.5×80 = 0.00144 m = 1.44 mm
Thus, the change in length of the rod when heated to 100
°
C is 1.44 mm.
Question 19
Question
A metal rod of length 2.00 m is heated from 20
°
C to 120
°
C. If the coefficient
of linear expansion of the metal is 2.5×10−5per degree Celsius, what is the
change in length of the rod?
Solution
Step 1: Calculate the change in temperature: Given initial temperature, Tin =
20C
Given final temperature, Tfin = 120C
The change in temperature, ∆T=Tfin −Tin = 120C−20C= 100C.
Step 2: Calculate the change in length: The coefficient of linear expansion,
α= 2.5×10−5per degree Celsius.
The original length of the rod, Lin = 2.00 m.
Using the formula for linear expansion: ∆L=α·Lin ·∆T.
Substitute the known values: ∆L= (2.5×10−5)·(2.00) ·(100) = 5×10−3m.
Answer: The change in length of the rod is 5 ×10−3meters.
13
Question 20
Question
A metal cylindrical rod has a length of 2.0 m at 0
°
C. If the coefficient of linear
expansion for the metal is 2.0×10−5
°
C−1, what will be the new length of the
rod when the temperature is raised to 100
°
C?
Solution
Step 1: First, we calculate the change in length of the rod using the formula for
linear expansion:
∆L=α·L·∆T,
where: - ∆Lis the change in length, - αis the coefficient of linear expansion, -
Lis the original length, and - ∆Tis the change in temperature.
Plugging in the values, we get:
∆L= (2.0×10−5
°
C−1)·2.0 m ·(100
°
C−0
°
C) = 0.004 m.
Step 2: Next, we find the new length of the rod by adding the change in
length to the original length:
Lnew =L+ ∆L= 2.0 m + 0.004 m = 2.004 m.
Therefore, when the temperature is raised to 100
°
C, the new length of the
rod will be 2.004 m.
Question 21
Question
A brass rod is 2 meters long at 20◦C. If the coefficient of linear expansion for
brass is 1.8×10−5/◦C, what is the length of the brass rod at 200◦C?
Solution
Step 1: We can use the formula for linear expansion to find the change in length
of the brass rod:
∆L=α·L·∆T
where: - ∆Lis the change in length, - αis the coefficient of linear expansion, -
Lis the original length, - ∆Tis the change in temperature.
Step 2: Substituting the given values into the formula, we have:
∆L= (1.8×10−5/◦C) ·(2 m) ·(200 −20)◦C
Step 3: Calculate the change in length:
∆L= 1.8×10−5×2×180 = 0.00648 m
14
Step 4: The final length of the brass rod at 200◦C is the original length plus
the change in length:
Final length = 2 m + 0.00648 m = 2.00648 m
Therefore, the length of the brass rod at 200◦C is 2.00648 meters.
Question 22
Question
A steel rod of length 2 m is heated from 20
°
C to 120
°
C. If the linear expansion
coefficient of steel is 1.2×10−5K−1, calculate the change in length of the rod.
Solution
Step 1: Calculate the change in temperature. Given that the initial temperature
is 20
°
C and the final temperature is 120
°
C, we can calculate the change in
temperature:
∆T= 120C−20C= 100C
Step 2: Calculate the change in length. The change in length (∆L) of a
material due to temperature change can be calculated using the formula:
∆L=L0·α·∆T
where L0is the initial length, αis the linear expansion coefficient, and ∆Tis
the change in temperature.
Substitute the given values into the formula:
∆L= 2 m ·1.2×10−5K−1·100 K
Step 3: Calculate the change in length.
∆L= 2 ×1.2×10−3m
∆L= 2.4×10−3m
Therefore, the change in length of the steel rod is 2.4×10−3m.
Question 23
Question
A brass rod of length 1.50 m at 20
°
C supports a load that causes it to just
touch the ceiling above. If the temperature of the rod is increased to 150
°
C, by
how much will the length of the rod increase? Assume the coefficient of linear
expansion for brass is 19 ×10−6
°
C−1.
15
Solution
Step 1: We first calculate the change in temperature. Given: Initial tempera-
ture, Ti= 20
°
C Final temperature, Tf= 150
°
C
The change in temperature, ∆T=Tf−Ti= 150 −20 = 130
°
C
Step 2: Next, we can use the formula for linear expansion:
∆L=L0α∆T
where: ∆L= change in length, L0= initial length, α= coefficient of linear
expansion, ∆T= change in temperature
Substitute the given values:
∆L= (1.50 m) ×(19 ×10−6
°
C−1)×130
°
C
Step 3: Now, we can calculate the change in length:
∆L= 1.50 ×19 ×10−6×130
∆L= 1.50 ×19 ×13 ×10−6
∆L= 351 ×10−6
∆L= 0.351 m
Therefore, the length of the brass rod will increase by 0.351 m when the
temperature is increased to 150
°
C.
Question 24
Question
A steel rod is initially 2 meters long at 20
°
C. If the coefficient of linear expansion
of steel is 1.2×10−5
°
C−1, determine the length of the rod at 150
°
C.
Solution
Step 1: Calculate the change in length of the steel rod as the temperature
increases from 20
°
C to 150
°
C. Given that the coefficient of linear expansion of
steel is α= 1.2×10−5
°
C−1, the change in length ∆Lcan be calculated using
the formula:
∆L=α·L·∆T
where Lis the initial length of the rod and ∆Tis the change in temperature.
Substitute L= 2 m, α= 1.2×10−5
°
C−1, and ∆T= 150 −20 = 130
°
C into the
formula:
∆L= (1.2×10−5
°
C−1)·(2 m) ·(130
°
C)
16
Step 2: Calculate the final length of the steel rod at 150
°
C. The final length
Lfof the steel rod at 150
°
C can be determined by adding the change in length
∆Lto the initial length L:
Lf=L+ ∆L
Substitute L= 2 m and ∆Lcalculated in Step 1 into the formula:
Lf= 2 + ∆L
Question 25
Question
A brass rod of length 2 m at 20◦C consists of a copper section of length 1 m
and an iron section of length 1 m, joined end-to-end. If the coefficients of linear
expansion for brass, copper, and iron are 19 ×10−6/◦C, 17 ×10−6/◦C, and
12 ×10−6/◦C, respectively, what is the change in length of the rod when the
temperature is increased to 95◦C?
Solution
Step 1: Calculate the change in length for each metal section using the formula:
∆Li=Li·α·∆T, where irepresents the material, Liis the initial length, αis
the coefficient of linear expansion, and ∆Tis the change in temperature.
∆Lbrass = 2 m ·(19 ×10−6/◦C) ·(95 −20)◦C
∆Lcopper = 1 m ·(17 ×10−6/◦C) ·(95 −20)◦C
∆Liron = 1 m ·(12 ×10−6/◦C) ·(95 −20)◦C
Step 2: Calculate the total change in length of the rod by adding up the
changes in length for each section.
∆Ltotal = ∆Lbrass + ∆Lcopper + ∆Liron
Step 3: Substitute the calculated values and solve for the total change in
length.
∆Ltotal = 2 m·(19×10−6/◦C)·(95−20)◦C+1 m·(17×10−6/◦C)·(95−20)◦C+1 m·(12×10−6/◦C)·(95−20)◦C
Question 26
Question
A steel rod and an aluminum rod are both initially 1 meter long at 20
°
C. If the
steel rod has a coefficient of linear expansion of 1.2×10−5per degree Celsius
and the aluminum rod has a coefficient of linear expansion of 2.4×10−5per
degree Celsius, find the temperature at which the two rods will have the same
length.
17
Solution
Step 1: Let’s denote the initial length of both rods as L, the final length of the
steel rod as Ls, the final length of the aluminum rod as La, the final temperature
as T, and the initial temperature as 20
°
C.
Step 2: We can express the final lengths of the steel and aluminum rods as
follows: Ls=L(1 + αs·(T−20)) La=L(1 + αa·(T−20))
Step 3: We want Ls=La, so we can set the two expressions equal to each
other: L(1 + αs·(T−20)) = L(1 + αa·(T−20))
Step 4: Divide both sides by Lto simplify: 1+αs·(T−20) = 1+αa·(T−20)
Step 5: Expand the equation and collect like terms: αs·T−20αs=αa·T−
20αa
Step 6: Rearrange the equation to solve for T:T(αs−αa) = 20(αa−αs)
Step 7: Plug in the given values to find the temperature at which the two
rods will have the same length: T=20(αa−αs)
αs−αa
Step 8: Substitute αa= 2.4×10−5and αs= 1.2×10−5into the equation
to find the temperature. Calculating, we get: T=20((2.4−1.2)×10−5)
(1.2−2.4)×10−5
Step 9: Simplifying further, we find: T=20×1.2×10−5
−1.2×10−5
Step 10: Finally, we get: T=−20
°
C
Therefore, the two rods will have the same length when the temperature is
-20
°
C.
Question 27
Question
A brass rod of length 2.0 m at 20
°
C is heated to 100
°
C. If the coefficient of linear
expansion for brass is 2.0×10−5
°
C−1, what is the change in length of the brass
rod?
Solution
Step 1: Identify the given values. The initial length of the brass rod, Li, is 2.0
m, the initial temperature, Ti, is 20
°
C, the final temperature, Tf, is 100
°
C, and
the coefficient of linear expansion for brass, α, is 2.0×10−5
°
C−1.
Step 2: Calculate the change in temperature. The change in temperature,
∆T, is given by
∆T=Tf−Ti= 100C−20C= 80C.
Step 3: Calculate the change in length. The change in length, ∆L, can be
calculated using the formula
∆L=α·Li·∆T.
Step 4: Substitute the known values and solve for ∆L.
∆L= 2.0×10−5
°
C−1·2.0 m ·80
°
C.
18
Step 5: Perform the calculations.
∆L= 2.0×10−5·2.0·80 = 3.2×10−3m.
Step 6: Convert the change in length to millimeters (mm). Since 1 m =
1000 mm,
∆L= 3.2×10−3m×1000 mm/m = 3.2 mm.
Therefore, the change in length of the brass rod is 3.2 mm.
Question 28
Question
A steel rod of length 2.0 m at 20◦C is heated to 120◦C. If the coefficient of linear
expansion for steel is 1.2×10−5◦C−1, what is the change in length of the rod?
Solution
Step 1: Calculate the initial length of the steel rod at 20◦C: Given: Initial length,
L0= 2.0 m Change in temperature, ∆T= 120◦C - 20◦C = 100◦C Coefficient
of linear expansion, α= 1.2×10−5◦C−1
We can use the formula for linear expansion:
∆L=L0·α·∆T
∆L= 2.0 m ·1.2×10−5◦C−1·100 ◦C
Step 2: Calculate the change in length of the steel rod:
∆L= 2.0×1.2×10−5×100
∆L= 2.4×10−3m=0.0024 m
Therefore, the change in length of the rod is 0.0024 m.
Question 29
Question
A steel rod is 2 meters long at 20◦C. If the rod is heated to 120◦C, what will
be its new length? The linear expansion coefficient for steel is 12 ×10−6K−1.
19
Solution
Step 1: Calculate the change in temperature. Step 2: Use the formula for linear
thermal expansion to calculate the new length of the rod.
Step 1: Calculate the change in temperature. The change in temperature,
∆T, is:
∆T= 120◦C−20◦C= 100◦C
Step 2: Use the formula for linear thermal expansion to calculate the new
length of the rod. The change in length, ∆L, can be calculated using the
formula:
∆L=α·Li·∆T
where: - αis the linear expansion coefficient for steel, which is 12 ×10−6K−1,
-Liis the initial length of the rod, which is 2 meters, - ∆Tis the change in
temperature, which is 100
°
C.
Substitute the values into the formula:
∆L= (12 ×10−6K−1)·(2 m) ·(100◦C)
∆L= 0.00024 m ·200◦C
∆L= 0.024 m = 2.4 cm
Therefore, the new length of the steel rod when heated to 120◦C will be the
initial length plus the change in length:
Lf=Li+ ∆L= 2 m + 0.024 m = 2.024 m
The new length of the steel rod when heated to 120◦C will be 2.024 meters.
Question 30
Question
A metal rod with a length of 2.0 m at 0◦C is heated to 100◦C. If the coefficient
of linear expansion of the metal is 2.3×10−5per degree Celsius, how much
longer is the rod after heating?
Solution
Step 1: Calculate the change in length of the rod due to the temperature
increase. Given: Initial length of the rod, L0= 2.0 m Final temperature,
Tf= 100◦C Initial temperature, Ti= 0◦C Coefficient of linear expansion,
α= 2.3×10−5per ◦C
The change in length, ∆L, is given by the formula:
∆L=L0·α·∆T
20
where ∆Tis the change in temperature. So,
∆T=Tf−Ti= 100◦C−0◦C = 100◦C
Thus,
∆L= 2.0×2.3×10−5×100 = 0.0046 m
Therefore, the rod will be 0.0046 meters longer after heating.
Question 31
Question
A brass rod of initial length 2 m and cross-sectional area 4 cm2is heated from
20
°
C to 120
°
C. If the linear expansion coefficient of brass is 2 ×10−5per degree
Celsius, calculate the change in length of the rod.
Solution
Given: Initial length of brass rod, L0= 2 m
Initial cross-sectional area of rod, A= 4 cm2= 4 ×10−4m2
Change in temperature, ∆T= 120C−20C= 100C
Linear expansion coefficient of brass, α= 2 ×10−5per degree Celsius
Step 1: Calculate the change in length using the formula
∆L=L0·α·∆T
where ∆Lis the change in length, L0is the initial length, αis the linear expan-
sion coefficient, and ∆Tis the change in temperature.
∆L= 2 ·2×10−5·100
= 2 ×2×10−3
= 4 ×10−3m
= 4 mm
Step 2: Hence, the change in length of the brass rod is 4 mm.
Question 32
Question
A copper rod of length 2.0 m at 20
°
C has a circular hole at its center. If the
temperature of the rod is raised to 80
°
C, what is the diameter of the hole?
Assume the coefficient of linear expansion for copper is 1.7×10−5
°
C−1.
21
Solution
Step 1: Calculate the change in length of the copper rod. The change in length
(∆L) of the copper rod can be calculated using the formula:
∆L=α·L·∆T
where αis the coefficient of linear expansion, Lis the original length of the rod,
and ∆Tis the change in temperature.
Given that α= 1.7×10−5
°
C−1,L= 2.0 m, and ∆T= 80 −20 = 60
°
C, we
have:
∆L= (1.7×10−5
°
C−1)·(2.0 m) ·(60
°
C)
∆L= 0.00204 m
Step 2: Determine the change in radius of the hole. Since the hole is at the
center of the rod, the change in half of the diameter of the hole will be equal
to the change in length of the rod. Thus, the change in radius (∆r) can be
determined as:
∆r=∆L
2
∆r=0.00204 m
2
∆r= 0.00102 m
Step 3: Calculate the new diameter of the hole. The new diameter of the hole
(D′) can be calculated by adding the change in radius to the original diameter
(D):
D′=D+ 2 ·∆r
Since the original diameter is equal to the original diameter of the rod, we have:
D′= 2 ·∆r
D′= 2 ·0.00102 m
D′= 0.00204 m
Therefore, the diameter of the hole when the temperature is raised to 80
°
C
is 0.00204 m.
Question 33
Question
A solid brass rod with a length of 2.0 m and a cross-sectional area of 0.02 m2
is heated from 20◦C to 120◦C. If the linear expansion coefficient of brass is
19 ×10−6◦C−1, what is the increase in length of the rod?
22
Solution
Step 1: Calculate the original length of the rod after heating.
Original length, L0= 2.0 m
Step 2: Calculate the change in temperature.
∆T= (120◦C) −(20◦C) = 100◦C
Step 3: Calculate the increase in length using the formula for linear expansion:
∆L=L0α∆T
∆L= (2.0 m) ×(19 ×10−6◦C−1)×(100◦C)
∆L= 0.0038 m
The increase in length of the rod is 0.0038 m.
Question 34
Question
A steel rod of length 2.0 m at an initial temperature of 20
°
C is heated to a
final temperature of 120
°
C. If the coefficient of linear expansion of steel is 1.2×
10−5/
°
C, what is the final length of the rod?
Solution
Step 1: Calculate the change in temperature Given that the initial temperature
(Ti) is 20
°
C and the final temperature (Tf) is 120
°
C, we can find the change in
temperature (∆T) using the formula:
∆T=Tf−Ti= 120
°
C−20
°
C = 100
°
C
Step 2: Calculate the change in length The change in length (∆L) of the
steel rod can be calculated using the formula:
∆L=α·L·∆T
where αis the coefficient of linear expansion of steel (given as 1.2×10−5/
°
C),
Lis the initial length of the rod (2.0 m), and ∆Tis the change in temperature
(100
°
C). Substitute the values into the formula:
∆L= (1.2×10−5/
°
C) ·2.0 m ·100
°
C=0.0024 m = 2.4 mm
Step 3: Calculate the final length of the rod The final length (Lf) of the rod
can be found by adding the change in length to the initial length:
Lf=L+ ∆L= 2.0 m + 0.0024 m = 2.0024 m = 2.4 m
Therefore, the final length of the steel rod when heated to 120
°
C is 2.4 m.
23
Question 2
Question
A copper rod and an aluminum rod are both 2 meters long at 20◦C. If both
rods are heated to 120◦C, by how many millimeters will each rod elongate? The
coefficient of linear expansion for copper is 16.6×10−6/◦C and for aluminum is
23.1×10−6/◦C.
Solution
Let’s denote: - Lcopper as the initial length of the copper rod, - Laluminum as the
initial length of the aluminum rod, - αcopper as the coefficient of linear expansion
for copper, and - αaluminum as the coefficient of linear expansion for aluminum.
Step 1: Calculate the change in length of the copper rod. The change in
length of the copper rod can be found using the formula:
∆Lcopper =Lcopper ·αcopper ·∆T
where ∆Tis the change in temperature. Substitute the given values:
∆Lcopper = 2 m ·16.6×10−6/◦C·(120◦C−20◦C)
∆Lcopper = 2 ×16.6×10−6×100 m
∆Lcopper = 0.00332 m = 3.32 mm
Step 2: Calculate the change in length of the aluminum rod. Similarly, the
change in length of the aluminum rod can be found using the formula:
∆Laluminum =Laluminum ·αaluminum ·∆T
Substitute the given values:
∆Laluminum = 2 m ·23.1×10−6/◦C·(120◦C−20◦C)
∆Laluminum = 2 ×23.1×10−6×100 m
∆Laluminum = 0.00462 m = 4.62 mm
Therefore, the copper rod will elongate by 3.32 mm and the aluminum rod
will elongate by 4.62 mm when heated to 120◦C.
Question 3
Question
A brass rod is 1 m long at 20◦C. What is its length at 150◦C if the coefficient
of linear expansion of brass is 19 ×10−6K−1?
2
Solution
Step 1: Let’s denote the original length of the brass rod as L0= 1 m and the
coefficient of linear expansion as α= 19 ×10−6K−1. We want to find the final
length Lof the rod at 150◦C.
Step 2: The formula for linear expansion is given by:
∆L=α·L0·∆T
where ∆Lis the change in length, L0is the original length, αis the coefficient
of linear expansion, and ∆Tis the change in temperature.
Step 3: We can rearrange the formula to solve for the final length L:
L=L0+ ∆L=L0+α·L0·∆T
Step 4: The change in temperature is ∆T= 150◦C−20◦C = 130◦C. Sub-
stituting the values into the formula, we get:
L= 1 + 19 ×10−6×1×130 = 1 + 19 ×10−6×130
Step 5: Calculating the final length:
L= 1 + 19 ×10−6×130 = 1 + 2.47 ×10−4
Step 6: Therefore, the final length of the brass rod at 150◦C is approximately
1.00025 m.
Question 4
Question
A copper rod of length 2m is heated from 20◦C to 120◦C. If the linear expansion
coefficient of copper is 1.7×10−5◦C−1, find the change in length of the rod.
Solution
Let Lbe the original length of the copper rod, ∆Lbe the change in length, αbe
the linear expansion coefficient of copper, and ∆Tbe the change in temperature.
Step 1: Calculate the change in temperature. Given that the initial tem-
perature is 20◦C and the final temperature is 120◦C, we have:
∆T= 120◦C−20◦C = 100◦C
Step 2: Calculate the change in length. The formula for linear expansion
is given by:
∆L=α·L·∆T
Substitute the values of α,L, and ∆Tinto the formula:
∆L= (1.7×10−5◦C−1)·(2 m) ·(100 ◦C)
3
∆L= 0.000034 m
Step 3: Finalize the answer. Therefore, the change in length of the copper
rod is 0.000034 m, or 0.034 mm.
Question 5
Question
A steel beam is initially 5 m long at a temperature of 20◦C. If the temperature
of the beam increases to 50◦C, what will be the new length of the beam? The
coefficient of linear expansion for steel is 12 ×10−6
Step 1: Calculate the change in temperature. Given: Initial temperature, T1=
20◦C Final temperature, T2= 50◦C
The change in temperature, ∆T=T2−T1= 50 −20 = 30 C.
Step 2: Calculate the change in length. The coefficient of linear expansion,
α= 12 ×10−6
Question
A 2 m long rod made of steel has a hole drilled from one end towards the center.
When the temperature changes by 50
°
C, the length of the rod changes by 0.3
mm. If the diameter of the hole is 5 mm, what is the change in the length
of the hole when the temperature changes by 50
°
C? Use a coefficient of linear
expansion for steel of 12 ×10−6per degree Celsius.
Solution
Let’s denote the change in length of the hole as x. Since the rod is made of
steel and the hole is in the center, the thermal expansion of the rod will affect
the length of both the rod and the hole.
Step 1: Calculate the change in length of the rod using the coefficient of
linear expansion. The change in length of the rod can be calculated using the
formula:
Change in length of rod = α·L·∆T
where αis the coefficient of linear expansion, Lis the original length of the rod,
and ∆Tis the change in temperature.
Substitute the values:
0.3 mm = 12 ×10−6·2 m ·50
0.3×10−3m = 12 ×10−6·2·50
0.3×10−3= 1.2×10−3
x= 1.2×10−3
4
Step 2: Calculate the change in length of the hole. Since the hole is drilled
from one end towards the center, the change in length of the hole will be half
the change in length of the rod. Therefore, the change in length of the hole is:
1.2×10−3
2= 0.6×10−3
0.6 mm
Therefore, the change in the length of the hole when the temperature changes
by 50
°
C is 0.6 mm.
Question 7
Question
A steel rod has a length of 2 meters at 20◦C. If the coefficient of linear expansion
of steel is 1.2×10−5K−1, calculate the increase in length of the rod when its
temperature is raised to 200◦C.
Solution
Step 1: Calculate the change in temperature. Given: Initial temperature, T1=
20◦C Final temperature, T2= 200◦C Change in temperature, ∆T=T2−T1=
200◦C - 20◦C = 180◦C
Step 2: Calculate the increase in length of the rod. The increase in length,
∆L, can be calculated using the formula:
∆L=α·L·∆T
where: α= coefficient of linear expansion = 1.2×10−5K−1L= initial length
of the rod = 2 meters ∆T= change in temperature = 180◦C
Substitute the values into the formula:
∆L= 1.2×10−5K−1×2 m ×180
∆L= 4.32 ×10−3m=4.32 mm
Therefore, the increase in length of the rod when its temperature is raised
to 200◦C is 4.32 mm.
Question 8
Question
A steel rod measures 2 meters at a temperature of 20◦C. If the coefficient of
linear expansion for steel is 12 ×10−6/◦C, what will be the length of the rod
when its temperature is raised to 100◦C?
5
Solution
Step 1: Determine the change in temperature. Given that the initial temper-
ature is 20◦C and the final temperature is 100◦C, the change in temperature
is:
∆T= 100◦C−20◦C = 80◦C
Step 2: Calculate the change in length of the rod. The change in length of
the rod ∆Lcan be calculated using the formula:
∆L=L0α∆T
where: - L0is the initial length of the rod, - αis the coefficient of linear
expansion, - ∆Tis the change in temperature.
Given L0= 2 m and α= 12 ×10−6/◦C, we have:
∆L= 2 ×12 ×10−6×80 = 1.92 ×10−4m
Step 3: Find the final length of the rod. The final length of the rod Lfcan
be determined by adding the change in length to the initial length:
Lf=L0+ ∆L= 2 + 1.92 ×10−4= 2.000192 m
Therefore, when the temperature of the steel rod is raised to 100◦C, its
length will be 2.000192 meters.
Question 9
Question
A metal rod is 2 meters long at 0 degrees Celsius. If the coefficient of linear
expansion for the metal is 2.3×10−5/
°
C, what is the length of the rod at 100
degrees Celsius?
Solution
Step 1: First, we calculate the change in length of the rod. Given: Initial
length L0= 2 m, Coefficient of linear expansion α= 2.3×10−5/
°
C, Change in
temperature ∆T= 100
°
C.
The change in length ∆Lis given by the formula:
∆L=L0·α·∆T
Step 2: Substituting the values into the formula, we get:
∆L= 2 ·2.3×10−5·100
Step 3: Calculate the change in length:
∆L= 0.0046 m
6
Step 4: Finally, we find the length of the rod at 100 degrees Celsius. The
final length Lis given by:
L=L0+ ∆L
Step 5: Substitute the values into the formula:
L= 2 + 0.0046
Step 6: Calculate the final length of the rod:
L= 2.0046 m
Therefore, the length of the rod at 100 degrees Celsius is 2.0046 meters.
Question 10
Question
A steel bridge is constructed with a length of 100 meters at a temperature of
20
°
C. If the temperature rises to 40
°
C during a hot day, calculate the change
in length of the bridge assuming a coefficient of linear expansion of steel to be
1.2×10−5
°
C−1.
Solution
Step 1: Calculate the change in temperature: The change in temperature is
given by: ∆T=Tf−Ti, where Tfis the final temperature (40
°
C) and Tiis the
initial temperature (20
°
C). Therefore, ∆T= 40C−20C= 20C.
Step 2: Calculate the change in length using the formula for linear expansion:
The change in length (∆L) is given by: ∆L=α·L·∆T, where αis the coefficient
of linear expansion, Lis the original length (100 meters), and ∆Tis the change
in temperature. Substitute the values to find: ∆L= (1.2×10−5
°
C−1)×100 m×
20
°
C. This results in: ∆L= 0.024 m.
Therefore, the change in length of the steel bridge due to the temperature
increase is 0.024 meters (or 2.4 centimeters).
Question 11
Question
A steel rod is initially 2 meters long at a temperature of 20
°
C. If the rod is heated
to 80
°
C, what is the final length of the rod? The linear expansion coefficient of
steel is 1.2×10−5per degree Celsius.
7
Solution
Step 1: Calculate the change in temperature.
∆T= 80C−20C= 60C
Step 2: Use the formula for linear thermal expansion to find the change in
length.
∆L=α·L·∆T
∆L= (1.2×10−5)·2·60
∆L= 1.44 ×10−3m
Step 3: Find the final length of the rod.
Lf=Li+ ∆L
Lf= 2 + 1.44 ×10−3
Lf= 2.00144 m
Therefore, the final length of the steel rod when heated to 80
°
C is 2.00144
meters.
Question 12
Question
A copper rod 2 meters long at 0◦C is heated until its length increases by 2
mm. Calculate the average coefficient of linear expansion of copper in terms of
10−6/◦C.
Solution
Step 1: Recall the formula for linear expansion: The change in length (∆L) of
a material due to a change in temperature is given by:
∆L=L0α∆T
where: - L0is the original length, - αis the coefficient of linear expansion, and
- ∆Tis the change in temperature.
Step 2: Convert all the known values to SI units: Given L0= 2 m, ∆L= 2
mm = 2 ×10−3m.
Step 3: Calculate the change in temperature:
∆L=L0α∆T
2×10−3= 2 ×α×∆T
∆T=2×10−3
2×α
8
∆T=10−3
α
Step 4: Use the value of ∆Tto find the coefficient of linear expansion α: The
average coefficient of linear expansion is the change in temperature required to
cause a unit change in length, so αcan be calculated as:
α=10−3
∆T
Therefore, the average coefficient of linear expansion is α=10−3
∆T.
Question 13
Question
A steel rod is 2 meters long at 20◦C. If the coefficient of linear expansion for
steel is 12 ×10−6/K, what will be the length of the rod when its temperature
is increased to 100◦C?
Solution
Step 1: Calculate the change in temperature. Given that the initial temper-
ature is 20◦C and the final temperature is 100◦C, we can find the change in
temperature:
∆T=Tfinal −Tinitial = 100◦C−20◦C = 80◦C
Step 2: Calculate the change in length. The change in length (∆L) can be
calculated using the formula:
∆L=α·L·∆T
where α= 12 ×10−6/K is the coefficient of linear expansion for steel, L= 2 m
is the initial length of the rod, and ∆T= 80◦C is the change in temperature.
Substitute the values into the formula:
∆L= 12 ×10−6/K×2 m ×80◦C
Step 3: Calculate the change in length.
∆L= 1.92 ×10−3m
Step 4: Find the final length of the rod. The final length of the rod can be
found by adding the change in length to the initial length:
Lfinal =Linitial + ∆L
Lfinal = 2 m + 1.92 ×10−3m
Therefore, the length of the steel rod when its temperature is increased to
100◦C is approximately 2.00192 meters.
9
Question 14
Question
A solid copper cylinder has a diameter of 10 cm at 20◦C. What will be its
diameter when it is heated to 120◦C? Given that the linear expansion coefficient
of copper is 17 ×10−6◦C−1.
Solution
Step 1: We first need to calculate the change in temperature. Step 2: Calculate
the change in diameter using the linear expansion coefficient of copper. Step 3:
Find the final diameter of the cylinder.
Step 1: The change in temperature is given by:
∆T= 120◦C−20◦C= 100◦C
Step 2: The change in length (∆L) of the cylinder can be calculated using
the formula:
∆L=αL0∆T
where αis the linear expansion coefficient, L0is the original length, and ∆Tis
the change in temperature.
The change in diameter is equal to twice the change in length, so:
∆d= 2αd0∆T
Substitute the given values:
∆d= 2 ×17 ×10−6×10 ×100 = 0.034 cm
Step 3: The final diameter (df) can be found by adding the change in
diameter to the original diameter:
df=d0+ ∆d= 10 + 0.034 = 10.034 cm
Therefore, the diameter of the copper cylinder when heated to 120◦C will
be 10.034 cm.
Question 15
Question
A brass rod has an initial length of 2.00 m at 20◦C. If the coefficient of linear
expansion of brass is 1.9×10−5◦C−1, what is the change in length of the rod
when its temperature is increased to 100◦C?
10
Solution
Step 1: Calculate the change in temperature. Let ∆Tbe the change in temper-
ature. Given: Initial temperature, Ti= 20◦C Final temperature, Tf= 100◦C
Change in temperature:
∆T=Tf−Ti= 100◦C−20◦C = 80◦C
Step 2: Calculate the change in length using the formula:
∆L=α·L·∆T
where: ∆Lis the change in length, αis the coefficient of linear expansion
(1.9×10−5◦C−1), Lis the initial length of the rod (2.00 m), ∆Tis the change
in temperature (80◦C).
Substitute the values:
∆L= (1.9×10−5◦C−1)×2.00 m ×80◦C
Step 3: Calculate the change in length.
∆L= 3.04 ×10−3m
Therefore, the change in length of the brass rod when its temperature is
increased to 100◦C is 3.04 mm.
Question 16
Question
A solid metal rod of length 2 m at 20
°
C is heated to 120
°
C. If the coefficient of
linear expansion for this metal is 1.5×10−5/
°
C, find the new length of the rod.
Solution
Step 1: Determine the change in temperature. Given that the initial temper-
ature is 20
°
C and the final temperature is 120
°
C, the change in temperature
is:
∆T= 120C−20C= 100C
Step 2: Calculate the change in length using the formula for linear expansion.
The change in length (∆L) is given by:
∆L=Lα∆T
Where: L= Initial length of the rod = 2 m, α= Coefficient of linear expansion
= 1.5×10−5/
°
C, ∆T= Change in temperature = 100
°
C.
Substitute the values into the formula:
∆L= 2 ×1.5×10−5×100
11
∆L= 2 ×1.5×10−3
∆L= 3 ×10−3m
Step 3: Find the new length of the rod. The new length of the rod can be
calculated by adding the change in length to the initial length:
New length = Initial length + ∆L
New length = 2 + 3 ×10−3
New length = 2.003 m
Therefore, the new length of the rod after heating it to 120
°
C is 2.003 meters.
Question 17
Question
A steel rod is initially 10 meters long at 20
°
C. If the coefficient of linear expan-
sion for steel is 1.2×10−5/
°
C, find the change in length of the rod when its
temperature increases to 150
°
C.
Solution
Step 1: Calculate the change in temperature.
Let Lbe the original length of the rod and ∆Tbe the change in temperature.
Given: L= 10 m (initial length) T1= 20
°
C (initial temperature) T2= 150
°
C
(final temperature) The change in temperature can be calculated as:
∆T=T2−T1= 150C −20C = 130C
Step 2: Calculate the change in length using the coefficient of linear expan-
sion formula.
The change in length (∆L) can be calculated using the formula:
∆L=L·α·∆T
where L= original length of the rod, α= coefficient of linear expansion =
1.2×10−5/
°
C, ∆T= change in temperature. Substitute the values into the
formula:
∆L= 10 ·1.2×10−5·130
∆L= 10 ×1.2×10−5×130
∆L= 0.156 m
Therefore, the change in length of the rod when its temperature increases to
150
°
C is 0.156 meters.
12
Question 18
Question
A steel rod has a length of 1.5 m at 20
°
C. If the coefficient of linear expansion
of steel is 1.2×10−5
°
C−1, calculate the change in length of the rod when it is
heated to 100
°
C.
Solution
Given: Initial length of steel rod, L0= 1.5 m
Change in temperature, ∆T= 100C−20C= 80C
Coefficient of linear expansion, α= 1.2×10−5
°
C−1
We can use the formula for linear expansion: ∆L=α·L0·∆Tto find the
change in length of the rod.
Step 1: Calculate the change in length.
∆L= 1.2×10−5
°
C−1×1.5 m ×80C
∆L= 1.2×10−5×1.5×80 = 0.00144 m = 1.44 mm
Thus, the change in length of the rod when heated to 100
°
C is 1.44 mm.
Question 19
Question
A metal rod of length 2.00 m is heated from 20
°
C to 120
°
C. If the coefficient
of linear expansion of the metal is 2.5×10−5per degree Celsius, what is the
change in length of the rod?
Solution
Step 1: Calculate the change in temperature: Given initial temperature, Tin =
20C
Given final temperature, Tfin = 120C
The change in temperature, ∆T=Tfin −Tin = 120C−20C= 100C.
Step 2: Calculate the change in length: The coefficient of linear expansion,
α= 2.5×10−5per degree Celsius.
The original length of the rod, Lin = 2.00 m.
Using the formula for linear expansion: ∆L=α·Lin ·∆T.
Substitute the known values: ∆L= (2.5×10−5)·(2.00) ·(100) = 5×10−3m.
Answer: The change in length of the rod is 5 ×10−3meters.
13
Question 20
Question
A metal cylindrical rod has a length of 2.0 m at 0
°
C. If the coefficient of linear
expansion for the metal is 2.0×10−5
°
C−1, what will be the new length of the
rod when the temperature is raised to 100
°
C?
Solution
Step 1: First, we calculate the change in length of the rod using the formula for
linear expansion:
∆L=α·L·∆T,
where: - ∆Lis the change in length, - αis the coefficient of linear expansion, -
Lis the original length, and - ∆Tis the change in temperature.
Plugging in the values, we get:
∆L= (2.0×10−5
°
C−1)·2.0 m ·(100
°
C−0
°
C) = 0.004 m.
Step 2: Next, we find the new length of the rod by adding the change in
length to the original length:
Lnew =L+ ∆L= 2.0 m + 0.004 m = 2.004 m.
Therefore, when the temperature is raised to 100
°
C, the new length of the
rod will be 2.004 m.
Question 21
Question
A brass rod is 2 meters long at 20◦C. If the coefficient of linear expansion for
brass is 1.8×10−5/◦C, what is the length of the brass rod at 200◦C?
Solution
Step 1: We can use the formula for linear expansion to find the change in length
of the brass rod:
∆L=α·L·∆T
where: - ∆Lis the change in length, - αis the coefficient of linear expansion, -
Lis the original length, - ∆Tis the change in temperature.
Step 2: Substituting the given values into the formula, we have:
∆L= (1.8×10−5/◦C) ·(2 m) ·(200 −20)◦C
Step 3: Calculate the change in length:
∆L= 1.8×10−5×2×180 = 0.00648 m
14
Step 4: The final length of the brass rod at 200◦C is the original length plus
the change in length:
Final length = 2 m + 0.00648 m = 2.00648 m
Therefore, the length of the brass rod at 200◦C is 2.00648 meters.
Question 22
Question
A steel rod of length 2 m is heated from 20
°
C to 120
°
C. If the linear expansion
coefficient of steel is 1.2×10−5K−1, calculate the change in length of the rod.
Solution
Step 1: Calculate the change in temperature. Given that the initial temperature
is 20
°
C and the final temperature is 120
°
C, we can calculate the change in
temperature:
∆T= 120C−20C= 100C
Step 2: Calculate the change in length. The change in length (∆L) of a
material due to temperature change can be calculated using the formula:
∆L=L0·α·∆T
where L0is the initial length, αis the linear expansion coefficient, and ∆Tis
the change in temperature.
Substitute the given values into the formula:
∆L= 2 m ·1.2×10−5K−1·100 K
Step 3: Calculate the change in length.
∆L= 2 ×1.2×10−3m
∆L= 2.4×10−3m
Therefore, the change in length of the steel rod is 2.4×10−3m.
Question 23
Question
A brass rod of length 1.50 m at 20
°
C supports a load that causes it to just
touch the ceiling above. If the temperature of the rod is increased to 150
°
C, by
how much will the length of the rod increase? Assume the coefficient of linear
expansion for brass is 19 ×10−6
°
C−1.
15
Solution
Step 1: We first calculate the change in temperature. Given: Initial tempera-
ture, Ti= 20
°
C Final temperature, Tf= 150
°
C
The change in temperature, ∆T=Tf−Ti= 150 −20 = 130
°
C
Step 2: Next, we can use the formula for linear expansion:
∆L=L0α∆T
where: ∆L= change in length, L0= initial length, α= coefficient of linear
expansion, ∆T= change in temperature
Substitute the given values:
∆L= (1.50 m) ×(19 ×10−6
°
C−1)×130
°
C
Step 3: Now, we can calculate the change in length:
∆L= 1.50 ×19 ×10−6×130
∆L= 1.50 ×19 ×13 ×10−6
∆L= 351 ×10−6
∆L= 0.351 m
Therefore, the length of the brass rod will increase by 0.351 m when the
temperature is increased to 150
°
C.
Question 24
Question
A steel rod is initially 2 meters long at 20
°
C. If the coefficient of linear expansion
of steel is 1.2×10−5
°
C−1, determine the length of the rod at 150
°
C.
Solution
Step 1: Calculate the change in length of the steel rod as the temperature
increases from 20
°
C to 150
°
C. Given that the coefficient of linear expansion of
steel is α= 1.2×10−5
°
C−1, the change in length ∆Lcan be calculated using
the formula:
∆L=α·L·∆T
where Lis the initial length of the rod and ∆Tis the change in temperature.
Substitute L= 2 m, α= 1.2×10−5
°
C−1, and ∆T= 150 −20 = 130
°
C into the
formula:
∆L= (1.2×10−5
°
C−1)·(2 m) ·(130
°
C)
16
Step 2: Calculate the final length of the steel rod at 150
°
C. The final length
Lfof the steel rod at 150
°
C can be determined by adding the change in length
∆Lto the initial length L:
Lf=L+ ∆L
Substitute L= 2 m and ∆Lcalculated in Step 1 into the formula:
Lf= 2 + ∆L
Question 25
Question
A brass rod of length 2 m at 20◦C consists of a copper section of length 1 m
and an iron section of length 1 m, joined end-to-end. If the coefficients of linear
expansion for brass, copper, and iron are 19 ×10−6/◦C, 17 ×10−6/◦C, and
12 ×10−6/◦C, respectively, what is the change in length of the rod when the
temperature is increased to 95◦C?
Solution
Step 1: Calculate the change in length for each metal section using the formula:
∆Li=Li·α·∆T, where irepresents the material, Liis the initial length, αis
the coefficient of linear expansion, and ∆Tis the change in temperature.
∆Lbrass = 2 m ·(19 ×10−6/◦C) ·(95 −20)◦C
∆Lcopper = 1 m ·(17 ×10−6/◦C) ·(95 −20)◦C
∆Liron = 1 m ·(12 ×10−6/◦C) ·(95 −20)◦C
Step 2: Calculate the total change in length of the rod by adding up the
changes in length for each section.
∆Ltotal = ∆Lbrass + ∆Lcopper + ∆Liron
Step 3: Substitute the calculated values and solve for the total change in
length.
∆Ltotal = 2 m·(19×10−6/◦C)·(95−20)◦C+1 m·(17×10−6/◦C)·(95−20)◦C+1 m·(12×10−6/◦C)·(95−20)◦C
Question 26
Question
A steel rod and an aluminum rod are both initially 1 meter long at 20
°
C. If the
steel rod has a coefficient of linear expansion of 1.2×10−5per degree Celsius
and the aluminum rod has a coefficient of linear expansion of 2.4×10−5per
degree Celsius, find the temperature at which the two rods will have the same
length.
17
Solution
Step 1: Let’s denote the initial length of both rods as L, the final length of the
steel rod as Ls, the final length of the aluminum rod as La, the final temperature
as T, and the initial temperature as 20
°
C.
Step 2: We can express the final lengths of the steel and aluminum rods as
follows: Ls=L(1 + αs·(T−20)) La=L(1 + αa·(T−20))
Step 3: We want Ls=La, so we can set the two expressions equal to each
other: L(1 + αs·(T−20)) = L(1 + αa·(T−20))
Step 4: Divide both sides by Lto simplify: 1+αs·(T−20) = 1+αa·(T−20)
Step 5: Expand the equation and collect like terms: αs·T−20αs=αa·T−
20αa
Step 6: Rearrange the equation to solve for T:T(αs−αa) = 20(αa−αs)
Step 7: Plug in the given values to find the temperature at which the two
rods will have the same length: T=20(αa−αs)
αs−αa
Step 8: Substitute αa= 2.4×10−5and αs= 1.2×10−5into the equation
to find the temperature. Calculating, we get: T=20((2.4−1.2)×10−5)
(1.2−2.4)×10−5
Step 9: Simplifying further, we find: T=20×1.2×10−5
−1.2×10−5
Step 10: Finally, we get: T=−20
°
C
Therefore, the two rods will have the same length when the temperature is
-20
°
C.
Question 27
Question
A brass rod of length 2.0 m at 20
°
C is heated to 100
°
C. If the coefficient of linear
expansion for brass is 2.0×10−5
°
C−1, what is the change in length of the brass
rod?
Solution
Step 1: Identify the given values. The initial length of the brass rod, Li, is 2.0
m, the initial temperature, Ti, is 20
°
C, the final temperature, Tf, is 100
°
C, and
the coefficient of linear expansion for brass, α, is 2.0×10−5
°
C−1.
Step 2: Calculate the change in temperature. The change in temperature,
∆T, is given by
∆T=Tf−Ti= 100C−20C= 80C.
Step 3: Calculate the change in length. The change in length, ∆L, can be
calculated using the formula
∆L=α·Li·∆T.
Step 4: Substitute the known values and solve for ∆L.
∆L= 2.0×10−5
°
C−1·2.0 m ·80
°
C.
18
Step 5: Perform the calculations.
∆L= 2.0×10−5·2.0·80 = 3.2×10−3m.
Step 6: Convert the change in length to millimeters (mm). Since 1 m =
1000 mm,
∆L= 3.2×10−3m×1000 mm/m = 3.2 mm.
Therefore, the change in length of the brass rod is 3.2 mm.
Question 28
Question
A steel rod of length 2.0 m at 20◦C is heated to 120◦C. If the coefficient of linear
expansion for steel is 1.2×10−5◦C−1, what is the change in length of the rod?
Solution
Step 1: Calculate the initial length of the steel rod at 20◦C: Given: Initial length,
L0= 2.0 m Change in temperature, ∆T= 120◦C - 20◦C = 100◦C Coefficient
of linear expansion, α= 1.2×10−5◦C−1
We can use the formula for linear expansion:
∆L=L0·α·∆T
∆L= 2.0 m ·1.2×10−5◦C−1·100 ◦C
Step 2: Calculate the change in length of the steel rod:
∆L= 2.0×1.2×10−5×100
∆L= 2.4×10−3m=0.0024 m
Therefore, the change in length of the rod is 0.0024 m.
Question 29
Question
A steel rod is 2 meters long at 20◦C. If the rod is heated to 120◦C, what will
be its new length? The linear expansion coefficient for steel is 12 ×10−6K−1.
19
Solution
Step 1: Calculate the change in temperature. Step 2: Use the formula for linear
thermal expansion to calculate the new length of the rod.
Step 1: Calculate the change in temperature. The change in temperature,
∆T, is:
∆T= 120◦C−20◦C= 100◦C
Step 2: Use the formula for linear thermal expansion to calculate the new
length of the rod. The change in length, ∆L, can be calculated using the
formula:
∆L=α·Li·∆T
where: - αis the linear expansion coefficient for steel, which is 12 ×10−6K−1,
-Liis the initial length of the rod, which is 2 meters, - ∆Tis the change in
temperature, which is 100
°
C.
Substitute the values into the formula:
∆L= (12 ×10−6K−1)·(2 m) ·(100◦C)
∆L= 0.00024 m ·200◦C
∆L= 0.024 m = 2.4 cm
Therefore, the new length of the steel rod when heated to 120◦C will be the
initial length plus the change in length:
Lf=Li+ ∆L= 2 m + 0.024 m = 2.024 m
The new length of the steel rod when heated to 120◦C will be 2.024 meters.
Question 30
Question
A metal rod with a length of 2.0 m at 0◦C is heated to 100◦C. If the coefficient
of linear expansion of the metal is 2.3×10−5per degree Celsius, how much
longer is the rod after heating?
Solution
Step 1: Calculate the change in length of the rod due to the temperature
increase. Given: Initial length of the rod, L0= 2.0 m Final temperature,
Tf= 100◦C Initial temperature, Ti= 0◦C Coefficient of linear expansion,
α= 2.3×10−5per ◦C
The change in length, ∆L, is given by the formula:
∆L=L0·α·∆T
20
where ∆Tis the change in temperature. So,
∆T=Tf−Ti= 100◦C−0◦C = 100◦C
Thus,
∆L= 2.0×2.3×10−5×100 = 0.0046 m
Therefore, the rod will be 0.0046 meters longer after heating.
Question 31
Question
A brass rod of initial length 2 m and cross-sectional area 4 cm2is heated from
20
°
C to 120
°
C. If the linear expansion coefficient of brass is 2 ×10−5per degree
Celsius, calculate the change in length of the rod.
Solution
Given: Initial length of brass rod, L0= 2 m
Initial cross-sectional area of rod, A= 4 cm2= 4 ×10−4m2
Change in temperature, ∆T= 120C−20C= 100C
Linear expansion coefficient of brass, α= 2 ×10−5per degree Celsius
Step 1: Calculate the change in length using the formula
∆L=L0·α·∆T
where ∆Lis the change in length, L0is the initial length, αis the linear expan-
sion coefficient, and ∆Tis the change in temperature.
∆L= 2 ·2×10−5·100
= 2 ×2×10−3
= 4 ×10−3m
= 4 mm
Step 2: Hence, the change in length of the brass rod is 4 mm.
Question 32
Question
A copper rod of length 2.0 m at 20
°
C has a circular hole at its center. If the
temperature of the rod is raised to 80
°
C, what is the diameter of the hole?
Assume the coefficient of linear expansion for copper is 1.7×10−5
°
C−1.
21
Solution
Step 1: Calculate the change in length of the copper rod. The change in length
(∆L) of the copper rod can be calculated using the formula:
∆L=α·L·∆T
where αis the coefficient of linear expansion, Lis the original length of the rod,
and ∆Tis the change in temperature.
Given that α= 1.7×10−5
°
C−1,L= 2.0 m, and ∆T= 80 −20 = 60
°
C, we
have:
∆L= (1.7×10−5
°
C−1)·(2.0 m) ·(60
°
C)
∆L= 0.00204 m
Step 2: Determine the change in radius of the hole. Since the hole is at the
center of the rod, the change in half of the diameter of the hole will be equal
to the change in length of the rod. Thus, the change in radius (∆r) can be
determined as:
∆r=∆L
2
∆r=0.00204 m
2
∆r= 0.00102 m
Step 3: Calculate the new diameter of the hole. The new diameter of the hole
(D′) can be calculated by adding the change in radius to the original diameter
(D):
D′=D+ 2 ·∆r
Since the original diameter is equal to the original diameter of the rod, we have:
D′= 2 ·∆r
D′= 2 ·0.00102 m
D′= 0.00204 m
Therefore, the diameter of the hole when the temperature is raised to 80
°
C
is 0.00204 m.
Question 33
Question
A solid brass rod with a length of 2.0 m and a cross-sectional area of 0.02 m2
is heated from 20◦C to 120◦C. If the linear expansion coefficient of brass is
19 ×10−6◦C−1, what is the increase in length of the rod?
22
Solution
Step 1: Calculate the original length of the rod after heating.
Original length, L0= 2.0 m
Step 2: Calculate the change in temperature.
∆T= (120◦C) −(20◦C) = 100◦C
Step 3: Calculate the increase in length using the formula for linear expansion:
∆L=L0α∆T
∆L= (2.0 m) ×(19 ×10−6◦C−1)×(100◦C)
∆L= 0.0038 m
The increase in length of the rod is 0.0038 m.
Question 34
Question
A steel rod of length 2.0 m at an initial temperature of 20
°
C is heated to a
final temperature of 120
°
C. If the coefficient of linear expansion of steel is 1.2×
10−5/
°
C, what is the final length of the rod?
Solution
Step 1: Calculate the change in temperature Given that the initial temperature
(Ti) is 20
°
C and the final temperature (Tf) is 120
°
C, we can find the change in
temperature (∆T) using the formula:
∆T=Tf−Ti= 120
°
C−20
°
C = 100
°
C
Step 2: Calculate the change in length The change in length (∆L) of the
steel rod can be calculated using the formula:
∆L=α·L·∆T
where αis the coefficient of linear expansion of steel (given as 1.2×10−5/
°
C),
Lis the initial length of the rod (2.0 m), and ∆Tis the change in temperature
(100
°
C). Substitute the values into the formula:
∆L= (1.2×10−5/
°
C) ·2.0 m ·100
°
C=0.0024 m = 2.4 mm
Step 3: Calculate the final length of the rod The final length (Lf) of the rod
can be found by adding the change in length to the initial length:
Lf=L+ ∆L= 2.0 m + 0.0024 m = 2.0024 m = 2.4 m
Therefore, the final length of the steel rod when heated to 120
°
C is 2.4 m.
23
Question 2
Question
A copper rod and an aluminum rod are both 2 meters long at 20◦C. If both
rods are heated to 120◦C, by how many millimeters will each rod elongate? The
coefficient of linear expansion for copper is 16.6×10−6/◦C and for aluminum is
23.1×10−6/◦C.
Solution
Let’s denote: - Lcopper as the initial length of the copper rod, - Laluminum as the
initial length of the aluminum rod, - αcopper as the coefficient of linear expansion
for copper, and - αaluminum as the coefficient of linear expansion for aluminum.
Step 1: Calculate the change in length of the copper rod. The change in
length of the copper rod can be found using the formula:
∆Lcopper =Lcopper ·αcopper ·∆T
where ∆Tis the change in temperature. Substitute the given values:
∆Lcopper = 2 m ·16.6×10−6/◦C·(120◦C−20◦C)
∆Lcopper = 2 ×16.6×10−6×100 m
∆Lcopper = 0.00332 m = 3.32 mm
Step 2: Calculate the change in length of the aluminum rod. Similarly, the
change in length of the aluminum rod can be found using the formula:
∆Laluminum =Laluminum ·αaluminum ·∆T
Substitute the given values:
∆Laluminum = 2 m ·23.1×10−6/◦C·(120◦C−20◦C)
∆Laluminum = 2 ×23.1×10−6×100 m
∆Laluminum = 0.00462 m = 4.62 mm
Therefore, the copper rod will elongate by 3.32 mm and the aluminum rod
will elongate by 4.62 mm when heated to 120◦C.
Question 3
Question
A brass rod is 1 m long at 20◦C. What is its length at 150◦C if the coefficient
of linear expansion of brass is 19 ×10−6K−1?
2
Solution
Step 1: Let’s denote the original length of the brass rod as L0= 1 m and the
coefficient of linear expansion as α= 19 ×10−6K−1. We want to find the final
length Lof the rod at 150◦C.
Step 2: The formula for linear expansion is given by:
∆L=α·L0·∆T
where ∆Lis the change in length, L0is the original length, αis the coefficient
of linear expansion, and ∆Tis the change in temperature.
Step 3: We can rearrange the formula to solve for the final length L:
L=L0+ ∆L=L0+α·L0·∆T
Step 4: The change in temperature is ∆T= 150◦C−20◦C = 130◦C. Sub-
stituting the values into the formula, we get:
L= 1 + 19 ×10−6×1×130 = 1 + 19 ×10−6×130
Step 5: Calculating the final length:
L= 1 + 19 ×10−6×130 = 1 + 2.47 ×10−4
Step 6: Therefore, the final length of the brass rod at 150◦C is approximately
1.00025 m.
Question 4
Question
A copper rod of length 2m is heated from 20◦C to 120◦C. If the linear expansion
coefficient of copper is 1.7×10−5◦C−1, find the change in length of the rod.
Solution
Let Lbe the original length of the copper rod, ∆Lbe the change in length, αbe
the linear expansion coefficient of copper, and ∆Tbe the change in temperature.
Step 1: Calculate the change in temperature. Given that the initial tem-
perature is 20◦C and the final temperature is 120◦C, we have:
∆T= 120◦C−20◦C = 100◦C
Step 2: Calculate the change in length. The formula for linear expansion
is given by:
∆L=α·L·∆T
Substitute the values of α,L, and ∆Tinto the formula:
∆L= (1.7×10−5◦C−1)·(2 m) ·(100 ◦C)
3
∆L= 0.000034 m
Step 3: Finalize the answer. Therefore, the change in length of the copper
rod is 0.000034 m, or 0.034 mm.
Question 5
Question
A steel beam is initially 5 m long at a temperature of 20◦C. If the temperature
of the beam increases to 50◦C, what will be the new length of the beam? The
coefficient of linear expansion for steel is 12 ×10−6
Step 1: Calculate the change in temperature. Given: Initial temperature, T1=
20◦C Final temperature, T2= 50◦C
The change in temperature, ∆T=T2−T1= 50 −20 = 30 C.
Step 2: Calculate the change in length. The coefficient of linear expansion,
α= 12 ×10−6
Question
A 2 m long rod made of steel has a hole drilled from one end towards the center.
When the temperature changes by 50
°
C, the length of the rod changes by 0.3
mm. If the diameter of the hole is 5 mm, what is the change in the length
of the hole when the temperature changes by 50
°
C? Use a coefficient of linear
expansion for steel of 12 ×10−6per degree Celsius.
Solution
Let’s denote the change in length of the hole as x. Since the rod is made of
steel and the hole is in the center, the thermal expansion of the rod will affect
the length of both the rod and the hole.
Step 1: Calculate the change in length of the rod using the coefficient of
linear expansion. The change in length of the rod can be calculated using the
formula:
Change in length of rod = α·L·∆T
where αis the coefficient of linear expansion, Lis the original length of the rod,
and ∆Tis the change in temperature.
Substitute the values:
0.3 mm = 12 ×10−6·2 m ·50
0.3×10−3m = 12 ×10−6·2·50
0.3×10−3= 1.2×10−3
x= 1.2×10−3
4
Step 2: Calculate the change in length of the hole. Since the hole is drilled
from one end towards the center, the change in length of the hole will be half
the change in length of the rod. Therefore, the change in length of the hole is:
1.2×10−3
2= 0.6×10−3
0.6 mm
Therefore, the change in the length of the hole when the temperature changes
by 50
°
C is 0.6 mm.
Question 7
Question
A steel rod has a length of 2 meters at 20◦C. If the coefficient of linear expansion
of steel is 1.2×10−5K−1, calculate the increase in length of the rod when its
temperature is raised to 200◦C.
Solution
Step 1: Calculate the change in temperature. Given: Initial temperature, T1=
20◦C Final temperature, T2= 200◦C Change in temperature, ∆T=T2−T1=
200◦C - 20◦C = 180◦C
Step 2: Calculate the increase in length of the rod. The increase in length,
∆L, can be calculated using the formula:
∆L=α·L·∆T
where: α= coefficient of linear expansion = 1.2×10−5K−1L= initial length
of the rod = 2 meters ∆T= change in temperature = 180◦C
Substitute the values into the formula:
∆L= 1.2×10−5K−1×2 m ×180
∆L= 4.32 ×10−3m=4.32 mm
Therefore, the increase in length of the rod when its temperature is raised
to 200◦C is 4.32 mm.
Question 8
Question
A steel rod measures 2 meters at a temperature of 20◦C. If the coefficient of
linear expansion for steel is 12 ×10−6/◦C, what will be the length of the rod
when its temperature is raised to 100◦C?
5
Solution
Step 1: Determine the change in temperature. Given that the initial temper-
ature is 20◦C and the final temperature is 100◦C, the change in temperature
is:
∆T= 100◦C−20◦C = 80◦C
Step 2: Calculate the change in length of the rod. The change in length of
the rod ∆Lcan be calculated using the formula:
∆L=L0α∆T
where: - L0is the initial length of the rod, - αis the coefficient of linear
expansion, - ∆Tis the change in temperature.
Given L0= 2 m and α= 12 ×10−6/◦C, we have:
∆L= 2 ×12 ×10−6×80 = 1.92 ×10−4m
Step 3: Find the final length of the rod. The final length of the rod Lfcan
be determined by adding the change in length to the initial length:
Lf=L0+ ∆L= 2 + 1.92 ×10−4= 2.000192 m
Therefore, when the temperature of the steel rod is raised to 100◦C, its
length will be 2.000192 meters.
Question 9
Question
A metal rod is 2 meters long at 0 degrees Celsius. If the coefficient of linear
expansion for the metal is 2.3×10−5/
°
C, what is the length of the rod at 100
degrees Celsius?
Solution
Step 1: First, we calculate the change in length of the rod. Given: Initial
length L0= 2 m, Coefficient of linear expansion α= 2.3×10−5/
°
C, Change in
temperature ∆T= 100
°
C.
The change in length ∆Lis given by the formula:
∆L=L0·α·∆T
Step 2: Substituting the values into the formula, we get:
∆L= 2 ·2.3×10−5·100
Step 3: Calculate the change in length:
∆L= 0.0046 m
6
Step 4: Finally, we find the length of the rod at 100 degrees Celsius. The
final length Lis given by:
L=L0+ ∆L
Step 5: Substitute the values into the formula:
L= 2 + 0.0046
Step 6: Calculate the final length of the rod:
L= 2.0046 m
Therefore, the length of the rod at 100 degrees Celsius is 2.0046 meters.
Question 10
Question
A steel bridge is constructed with a length of 100 meters at a temperature of
20
°
C. If the temperature rises to 40
°
C during a hot day, calculate the change
in length of the bridge assuming a coefficient of linear expansion of steel to be
1.2×10−5
°
C−1.
Solution
Step 1: Calculate the change in temperature: The change in temperature is
given by: ∆T=Tf−Ti, where Tfis the final temperature (40
°
C) and Tiis the
initial temperature (20
°
C). Therefore, ∆T= 40C−20C= 20C.
Step 2: Calculate the change in length using the formula for linear expansion:
The change in length (∆L) is given by: ∆L=α·L·∆T, where αis the coefficient
of linear expansion, Lis the original length (100 meters), and ∆Tis the change
in temperature. Substitute the values to find: ∆L= (1.2×10−5
°
C−1)×100 m×
20
°
C. This results in: ∆L= 0.024 m.
Therefore, the change in length of the steel bridge due to the temperature
increase is 0.024 meters (or 2.4 centimeters).
Question 11
Question
A steel rod is initially 2 meters long at a temperature of 20
°
C. If the rod is heated
to 80
°
C, what is the final length of the rod? The linear expansion coefficient of
steel is 1.2×10−5per degree Celsius.
7
Solution
Step 1: Calculate the change in temperature.
∆T= 80C−20C= 60C
Step 2: Use the formula for linear thermal expansion to find the change in
length.
∆L=α·L·∆T
∆L= (1.2×10−5)·2·60
∆L= 1.44 ×10−3m
Step 3: Find the final length of the rod.
Lf=Li+ ∆L
Lf= 2 + 1.44 ×10−3
Lf= 2.00144 m
Therefore, the final length of the steel rod when heated to 80
°
C is 2.00144
meters.
Question 12
Question
A copper rod 2 meters long at 0◦C is heated until its length increases by 2
mm. Calculate the average coefficient of linear expansion of copper in terms of
10−6/◦C.
Solution
Step 1: Recall the formula for linear expansion: The change in length (∆L) of
a material due to a change in temperature is given by:
∆L=L0α∆T
where: - L0is the original length, - αis the coefficient of linear expansion, and
- ∆Tis the change in temperature.
Step 2: Convert all the known values to SI units: Given L0= 2 m, ∆L= 2
mm = 2 ×10−3m.
Step 3: Calculate the change in temperature:
∆L=L0α∆T
2×10−3= 2 ×α×∆T
∆T=2×10−3
2×α
8
∆T=10−3
α
Step 4: Use the value of ∆Tto find the coefficient of linear expansion α: The
average coefficient of linear expansion is the change in temperature required to
cause a unit change in length, so αcan be calculated as:
α=10−3
∆T
Therefore, the average coefficient of linear expansion is α=10−3
∆T.
Question 13
Question
A steel rod is 2 meters long at 20◦C. If the coefficient of linear expansion for
steel is 12 ×10−6/K, what will be the length of the rod when its temperature
is increased to 100◦C?
Solution
Step 1: Calculate the change in temperature. Given that the initial temper-
ature is 20◦C and the final temperature is 100◦C, we can find the change in
temperature:
∆T=Tfinal −Tinitial = 100◦C−20◦C = 80◦C
Step 2: Calculate the change in length. The change in length (∆L) can be
calculated using the formula:
∆L=α·L·∆T
where α= 12 ×10−6/K is the coefficient of linear expansion for steel, L= 2 m
is the initial length of the rod, and ∆T= 80◦C is the change in temperature.
Substitute the values into the formula:
∆L= 12 ×10−6/K×2 m ×80◦C
Step 3: Calculate the change in length.
∆L= 1.92 ×10−3m
Step 4: Find the final length of the rod. The final length of the rod can be
found by adding the change in length to the initial length:
Lfinal =Linitial + ∆L
Lfinal = 2 m + 1.92 ×10−3m
Therefore, the length of the steel rod when its temperature is increased to
100◦C is approximately 2.00192 meters.
9
Question 14
Question
A solid copper cylinder has a diameter of 10 cm at 20◦C. What will be its
diameter when it is heated to 120◦C? Given that the linear expansion coefficient
of copper is 17 ×10−6◦C−1.
Solution
Step 1: We first need to calculate the change in temperature. Step 2: Calculate
the change in diameter using the linear expansion coefficient of copper. Step 3:
Find the final diameter of the cylinder.
Step 1: The change in temperature is given by:
∆T= 120◦C−20◦C= 100◦C
Step 2: The change in length (∆L) of the cylinder can be calculated using
the formula:
∆L=αL0∆T
where αis the linear expansion coefficient, L0is the original length, and ∆Tis
the change in temperature.
The change in diameter is equal to twice the change in length, so:
∆d= 2αd0∆T
Substitute the given values:
∆d= 2 ×17 ×10−6×10 ×100 = 0.034 cm
Step 3: The final diameter (df) can be found by adding the change in
diameter to the original diameter:
df=d0+ ∆d= 10 + 0.034 = 10.034 cm
Therefore, the diameter of the copper cylinder when heated to 120◦C will
be 10.034 cm.
Question 15
Question
A brass rod has an initial length of 2.00 m at 20◦C. If the coefficient of linear
expansion of brass is 1.9×10−5◦C−1, what is the change in length of the rod
when its temperature is increased to 100◦C?
10
Solution
Step 1: Calculate the change in temperature. Let ∆Tbe the change in temper-
ature. Given: Initial temperature, Ti= 20◦C Final temperature, Tf= 100◦C
Change in temperature:
∆T=Tf−Ti= 100◦C−20◦C = 80◦C
Step 2: Calculate the change in length using the formula:
∆L=α·L·∆T
where: ∆Lis the change in length, αis the coefficient of linear expansion
(1.9×10−5◦C−1), Lis the initial length of the rod (2.00 m), ∆Tis the change
in temperature (80◦C).
Substitute the values:
∆L= (1.9×10−5◦C−1)×2.00 m ×80◦C
Step 3: Calculate the change in length.
∆L= 3.04 ×10−3m
Therefore, the change in length of the brass rod when its temperature is
increased to 100◦C is 3.04 mm.
Question 16
Question
A solid metal rod of length 2 m at 20
°
C is heated to 120
°
C. If the coefficient of
linear expansion for this metal is 1.5×10−5/
°
C, find the new length of the rod.
Solution
Step 1: Determine the change in temperature. Given that the initial temper-
ature is 20
°
C and the final temperature is 120
°
C, the change in temperature
is:
∆T= 120C−20C= 100C
Step 2: Calculate the change in length using the formula for linear expansion.
The change in length (∆L) is given by:
∆L=Lα∆T
Where: L= Initial length of the rod = 2 m, α= Coefficient of linear expansion
= 1.5×10−5/
°
C, ∆T= Change in temperature = 100
°
C.
Substitute the values into the formula:
∆L= 2 ×1.5×10−5×100
11
∆L= 2 ×1.5×10−3
∆L= 3 ×10−3m
Step 3: Find the new length of the rod. The new length of the rod can be
calculated by adding the change in length to the initial length:
New length = Initial length + ∆L
New length = 2 + 3 ×10−3
New length = 2.003 m
Therefore, the new length of the rod after heating it to 120
°
C is 2.003 meters.
Question 17
Question
A steel rod is initially 10 meters long at 20
°
C. If the coefficient of linear expan-
sion for steel is 1.2×10−5/
°
C, find the change in length of the rod when its
temperature increases to 150
°
C.
Solution
Step 1: Calculate the change in temperature.
Let Lbe the original length of the rod and ∆Tbe the change in temperature.
Given: L= 10 m (initial length) T1= 20
°
C (initial temperature) T2= 150
°
C
(final temperature) The change in temperature can be calculated as:
∆T=T2−T1= 150C −20C = 130C
Step 2: Calculate the change in length using the coefficient of linear expan-
sion formula.
The change in length (∆L) can be calculated using the formula:
∆L=L·α·∆T
where L= original length of the rod, α= coefficient of linear expansion =
1.2×10−5/
°
C, ∆T= change in temperature. Substitute the values into the
formula:
∆L= 10 ·1.2×10−5·130
∆L= 10 ×1.2×10−5×130
∆L= 0.156 m
Therefore, the change in length of the rod when its temperature increases to
150
°
C is 0.156 meters.
12
Question 18
Question
A steel rod has a length of 1.5 m at 20
°
C. If the coefficient of linear expansion
of steel is 1.2×10−5
°
C−1, calculate the change in length of the rod when it is
heated to 100
°
C.
Solution
Given: Initial length of steel rod, L0= 1.5 m
Change in temperature, ∆T= 100C−20C= 80C
Coefficient of linear expansion, α= 1.2×10−5
°
C−1
We can use the formula for linear expansion: ∆L=α·L0·∆Tto find the
change in length of the rod.
Step 1: Calculate the change in length.
∆L= 1.2×10−5
°
C−1×1.5 m ×80C
∆L= 1.2×10−5×1.5×80 = 0.00144 m = 1.44 mm
Thus, the change in length of the rod when heated to 100
°
C is 1.44 mm.
Question 19
Question
A metal rod of length 2.00 m is heated from 20
°
C to 120
°
C. If the coefficient
of linear expansion of the metal is 2.5×10−5per degree Celsius, what is the
change in length of the rod?
Solution
Step 1: Calculate the change in temperature: Given initial temperature, Tin =
20C
Given final temperature, Tfin = 120C
The change in temperature, ∆T=Tfin −Tin = 120C−20C= 100C.
Step 2: Calculate the change in length: The coefficient of linear expansion,
α= 2.5×10−5per degree Celsius.
The original length of the rod, Lin = 2.00 m.
Using the formula for linear expansion: ∆L=α·Lin ·∆T.
Substitute the known values: ∆L= (2.5×10−5)·(2.00) ·(100) = 5×10−3m.
Answer: The change in length of the rod is 5 ×10−3meters.
13
Question 20
Question
A metal cylindrical rod has a length of 2.0 m at 0
°
C. If the coefficient of linear
expansion for the metal is 2.0×10−5
°
C−1, what will be the new length of the
rod when the temperature is raised to 100
°
C?
Solution
Step 1: First, we calculate the change in length of the rod using the formula for
linear expansion:
∆L=α·L·∆T,
where: - ∆Lis the change in length, - αis the coefficient of linear expansion, -
Lis the original length, and - ∆Tis the change in temperature.
Plugging in the values, we get:
∆L= (2.0×10−5
°
C−1)·2.0 m ·(100
°
C−0
°
C) = 0.004 m.
Step 2: Next, we find the new length of the rod by adding the change in
length to the original length:
Lnew =L+ ∆L= 2.0 m + 0.004 m = 2.004 m.
Therefore, when the temperature is raised to 100
°
C, the new length of the
rod will be 2.004 m.
Question 21
Question
A brass rod is 2 meters long at 20◦C. If the coefficient of linear expansion for
brass is 1.8×10−5/◦C, what is the length of the brass rod at 200◦C?
Solution
Step 1: We can use the formula for linear expansion to find the change in length
of the brass rod:
∆L=α·L·∆T
where: - ∆Lis the change in length, - αis the coefficient of linear expansion, -
Lis the original length, - ∆Tis the change in temperature.
Step 2: Substituting the given values into the formula, we have:
∆L= (1.8×10−5/◦C) ·(2 m) ·(200 −20)◦C
Step 3: Calculate the change in length:
∆L= 1.8×10−5×2×180 = 0.00648 m
14
Step 4: The final length of the brass rod at 200◦C is the original length plus
the change in length:
Final length = 2 m + 0.00648 m = 2.00648 m
Therefore, the length of the brass rod at 200◦C is 2.00648 meters.
Question 22
Question
A steel rod of length 2 m is heated from 20
°
C to 120
°
C. If the linear expansion
coefficient of steel is 1.2×10−5K−1, calculate the change in length of the rod.
Solution
Step 1: Calculate the change in temperature. Given that the initial temperature
is 20
°
C and the final temperature is 120
°
C, we can calculate the change in
temperature:
∆T= 120C−20C= 100C
Step 2: Calculate the change in length. The change in length (∆L) of a
material due to temperature change can be calculated using the formula:
∆L=L0·α·∆T
where L0is the initial length, αis the linear expansion coefficient, and ∆Tis
the change in temperature.
Substitute the given values into the formula:
∆L= 2 m ·1.2×10−5K−1·100 K
Step 3: Calculate the change in length.
∆L= 2 ×1.2×10−3m
∆L= 2.4×10−3m
Therefore, the change in length of the steel rod is 2.4×10−3m.
Question 23
Question
A brass rod of length 1.50 m at 20
°
C supports a load that causes it to just
touch the ceiling above. If the temperature of the rod is increased to 150
°
C, by
how much will the length of the rod increase? Assume the coefficient of linear
expansion for brass is 19 ×10−6
°
C−1.
15
Solution
Step 1: We first calculate the change in temperature. Given: Initial tempera-
ture, Ti= 20
°
C Final temperature, Tf= 150
°
C
The change in temperature, ∆T=Tf−Ti= 150 −20 = 130
°
C
Step 2: Next, we can use the formula for linear expansion:
∆L=L0α∆T
where: ∆L= change in length, L0= initial length, α= coefficient of linear
expansion, ∆T= change in temperature
Substitute the given values:
∆L= (1.50 m) ×(19 ×10−6
°
C−1)×130
°
C
Step 3: Now, we can calculate the change in length:
∆L= 1.50 ×19 ×10−6×130
∆L= 1.50 ×19 ×13 ×10−6
∆L= 351 ×10−6
∆L= 0.351 m
Therefore, the length of the brass rod will increase by 0.351 m when the
temperature is increased to 150
°
C.
Question 24
Question
A steel rod is initially 2 meters long at 20
°
C. If the coefficient of linear expansion
of steel is 1.2×10−5
°
C−1, determine the length of the rod at 150
°
C.
Solution
Step 1: Calculate the change in length of the steel rod as the temperature
increases from 20
°
C to 150
°
C. Given that the coefficient of linear expansion of
steel is α= 1.2×10−5
°
C−1, the change in length ∆Lcan be calculated using
the formula:
∆L=α·L·∆T
where Lis the initial length of the rod and ∆Tis the change in temperature.
Substitute L= 2 m, α= 1.2×10−5
°
C−1, and ∆T= 150 −20 = 130
°
C into the
formula:
∆L= (1.2×10−5
°
C−1)·(2 m) ·(130
°
C)
16
Step 2: Calculate the final length of the steel rod at 150
°
C. The final length
Lfof the steel rod at 150
°
C can be determined by adding the change in length
∆Lto the initial length L:
Lf=L+ ∆L
Substitute L= 2 m and ∆Lcalculated in Step 1 into the formula:
Lf= 2 + ∆L
Question 25
Question
A brass rod of length 2 m at 20◦C consists of a copper section of length 1 m
and an iron section of length 1 m, joined end-to-end. If the coefficients of linear
expansion for brass, copper, and iron are 19 ×10−6/◦C, 17 ×10−6/◦C, and
12 ×10−6/◦C, respectively, what is the change in length of the rod when the
temperature is increased to 95◦C?
Solution
Step 1: Calculate the change in length for each metal section using the formula:
∆Li=Li·α·∆T, where irepresents the material, Liis the initial length, αis
the coefficient of linear expansion, and ∆Tis the change in temperature.
∆Lbrass = 2 m ·(19 ×10−6/◦C) ·(95 −20)◦C
∆Lcopper = 1 m ·(17 ×10−6/◦C) ·(95 −20)◦C
∆Liron = 1 m ·(12 ×10−6/◦C) ·(95 −20)◦C
Step 2: Calculate the total change in length of the rod by adding up the
changes in length for each section.
∆Ltotal = ∆Lbrass + ∆Lcopper + ∆Liron
Step 3: Substitute the calculated values and solve for the total change in
length.
∆Ltotal = 2 m·(19×10−6/◦C)·(95−20)◦C+1 m·(17×10−6/◦C)·(95−20)◦C+1 m·(12×10−6/◦C)·(95−20)◦C
Question 26
Question
A steel rod and an aluminum rod are both initially 1 meter long at 20
°
C. If the
steel rod has a coefficient of linear expansion of 1.2×10−5per degree Celsius
and the aluminum rod has a coefficient of linear expansion of 2.4×10−5per
degree Celsius, find the temperature at which the two rods will have the same
length.
17
Solution
Step 1: Let’s denote the initial length of both rods as L, the final length of the
steel rod as Ls, the final length of the aluminum rod as La, the final temperature
as T, and the initial temperature as 20
°
C.
Step 2: We can express the final lengths of the steel and aluminum rods as
follows: Ls=L(1 + αs·(T−20)) La=L(1 + αa·(T−20))
Step 3: We want Ls=La, so we can set the two expressions equal to each
other: L(1 + αs·(T−20)) = L(1 + αa·(T−20))
Step 4: Divide both sides by Lto simplify: 1+αs·(T−20) = 1+αa·(T−20)
Step 5: Expand the equation and collect like terms: αs·T−20αs=αa·T−
20αa
Step 6: Rearrange the equation to solve for T:T(αs−αa) = 20(αa−αs)
Step 7: Plug in the given values to find the temperature at which the two
rods will have the same length: T=20(αa−αs)
αs−αa
Step 8: Substitute αa= 2.4×10−5and αs= 1.2×10−5into the equation
to find the temperature. Calculating, we get: T=20((2.4−1.2)×10−5)
(1.2−2.4)×10−5
Step 9: Simplifying further, we find: T=20×1.2×10−5
−1.2×10−5
Step 10: Finally, we get: T=−20
°
C
Therefore, the two rods will have the same length when the temperature is
-20
°
C.
Question 27
Question
A brass rod of length 2.0 m at 20
°
C is heated to 100
°
C. If the coefficient of linear
expansion for brass is 2.0×10−5
°
C−1, what is the change in length of the brass
rod?
Solution
Step 1: Identify the given values. The initial length of the brass rod, Li, is 2.0
m, the initial temperature, Ti, is 20
°
C, the final temperature, Tf, is 100
°
C, and
the coefficient of linear expansion for brass, α, is 2.0×10−5
°
C−1.
Step 2: Calculate the change in temperature. The change in temperature,
∆T, is given by
∆T=Tf−Ti= 100C−20C= 80C.
Step 3: Calculate the change in length. The change in length, ∆L, can be
calculated using the formula
∆L=α·Li·∆T.
Step 4: Substitute the known values and solve for ∆L.
∆L= 2.0×10−5
°
C−1·2.0 m ·80
°
C.
18
Step 5: Perform the calculations.
∆L= 2.0×10−5·2.0·80 = 3.2×10−3m.
Step 6: Convert the change in length to millimeters (mm). Since 1 m =
1000 mm,
∆L= 3.2×10−3m×1000 mm/m = 3.2 mm.
Therefore, the change in length of the brass rod is 3.2 mm.
Question 28
Question
A steel rod of length 2.0 m at 20◦C is heated to 120◦C. If the coefficient of linear
expansion for steel is 1.2×10−5◦C−1, what is the change in length of the rod?
Solution
Step 1: Calculate the initial length of the steel rod at 20◦C: Given: Initial length,
L0= 2.0 m Change in temperature, ∆T= 120◦C - 20◦C = 100◦C Coefficient
of linear expansion, α= 1.2×10−5◦C−1
We can use the formula for linear expansion:
∆L=L0·α·∆T
∆L= 2.0 m ·1.2×10−5◦C−1·100 ◦C
Step 2: Calculate the change in length of the steel rod:
∆L= 2.0×1.2×10−5×100
∆L= 2.4×10−3m=0.0024 m
Therefore, the change in length of the rod is 0.0024 m.
Question 29
Question
A steel rod is 2 meters long at 20◦C. If the rod is heated to 120◦C, what will
be its new length? The linear expansion coefficient for steel is 12 ×10−6K−1.
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Solution
Step 1: Calculate the change in temperature. Step 2: Use the formula for linear
thermal expansion to calculate the new length of the rod.
Step 1: Calculate the change in temperature. The change in temperature,
∆T, is:
∆T= 120◦C−20◦C= 100◦C
Step 2: Use the formula for linear thermal expansion to calculate the new
length of the rod. The change in length, ∆L, can be calculated using the
formula:
∆L=α·Li·∆T
where: - αis the linear expansion coefficient for steel, which is 12 ×10−6K−1,
-Liis the initial length of the rod, which is 2 meters, - ∆Tis the change in
temperature, which is 100
°
C.
Substitute the values into the formula:
∆L= (12 ×10−6K−1)·(2 m) ·(100◦C)
∆L= 0.00024 m ·200◦C
∆L= 0.024 m = 2.4 cm
Therefore, the new length of the steel rod when heated to 120◦C will be the
initial length plus the change in length:
Lf=Li+ ∆L= 2 m + 0.024 m = 2.024 m
The new length of the steel rod when heated to 120◦C will be 2.024 meters.
Question 30
Question
A metal rod with a length of 2.0 m at 0◦C is heated to 100◦C. If the coefficient
of linear expansion of the metal is 2.3×10−5per degree Celsius, how much
longer is the rod after heating?
Solution
Step 1: Calculate the change in length of the rod due to the temperature
increase. Given: Initial length of the rod, L0= 2.0 m Final temperature,
Tf= 100◦C Initial temperature, Ti= 0◦C Coefficient of linear expansion,
α= 2.3×10−5per ◦C
The change in length, ∆L, is given by the formula:
∆L=L0·α·∆T
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where ∆Tis the change in temperature. So,
∆T=Tf−Ti= 100◦C−0◦C = 100◦C
Thus,
∆L= 2.0×2.3×10−5×100 = 0.0046 m
Therefore, the rod will be 0.0046 meters longer after heating.
Question 31
Question
A brass rod of initial length 2 m and cross-sectional area 4 cm2is heated from
20
°
C to 120
°
C. If the linear expansion coefficient of brass is 2 ×10−5per degree
Celsius, calculate the change in length of the rod.
Solution
Given: Initial length of brass rod, L0= 2 m
Initial cross-sectional area of rod, A= 4 cm2= 4 ×10−4m2
Change in temperature, ∆T= 120C−20C= 100C
Linear expansion coefficient of brass, α= 2 ×10−5per degree Celsius
Step 1: Calculate the change in length using the formula
∆L=L0·α·∆T
where ∆Lis the change in length, L0is the initial length, αis the linear expan-
sion coefficient, and ∆Tis the change in temperature.
∆L= 2 ·2×10−5·100
= 2 ×2×10−3
= 4 ×10−3m
= 4 mm
Step 2: Hence, the change in length of the brass rod is 4 mm.
Question 32
Question
A copper rod of length 2.0 m at 20
°
C has a circular hole at its center. If the
temperature of the rod is raised to 80
°
C, what is the diameter of the hole?
Assume the coefficient of linear expansion for copper is 1.7×10−5
°
C−1.
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Solution
Step 1: Calculate the change in length of the copper rod. The change in length
(∆L) of the copper rod can be calculated using the formula:
∆L=α·L·∆T
where αis the coefficient of linear expansion, Lis the original length of the rod,
and ∆Tis the change in temperature.
Given that α= 1.7×10−5
°
C−1,L= 2.0 m, and ∆T= 80 −20 = 60
°
C, we
have:
∆L= (1.7×10−5
°
C−1)·(2.0 m) ·(60
°
C)
∆L= 0.00204 m
Step 2: Determine the change in radius of the hole. Since the hole is at the
center of the rod, the change in half of the diameter of the hole will be equal
to the change in length of the rod. Thus, the change in radius (∆r) can be
determined as:
∆r=∆L
2
∆r=0.00204 m
2
∆r= 0.00102 m
Step 3: Calculate the new diameter of the hole. The new diameter of the hole
(D′) can be calculated by adding the change in radius to the original diameter
(D):
D′=D+ 2 ·∆r
Since the original diameter is equal to the original diameter of the rod, we have:
D′= 2 ·∆r
D′= 2 ·0.00102 m
D′= 0.00204 m
Therefore, the diameter of the hole when the temperature is raised to 80
°
C
is 0.00204 m.
Question 33
Question
A solid brass rod with a length of 2.0 m and a cross-sectional area of 0.02 m2
is heated from 20◦C to 120◦C. If the linear expansion coefficient of brass is
19 ×10−6◦C−1, what is the increase in length of the rod?
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Solution
Step 1: Calculate the original length of the rod after heating.
Original length, L0= 2.0 m
Step 2: Calculate the change in temperature.
∆T= (120◦C) −(20◦C) = 100◦C
Step 3: Calculate the increase in length using the formula for linear expansion:
∆L=L0α∆T
∆L= (2.0 m) ×(19 ×10−6◦C−1)×(100◦C)
∆L= 0.0038 m
The increase in length of the rod is 0.0038 m.
Question 34
Question
A steel rod of length 2.0 m at an initial temperature of 20
°
C is heated to a
final temperature of 120
°
C. If the coefficient of linear expansion of steel is 1.2×
10−5/
°
C, what is the final length of the rod?
Solution
Step 1: Calculate the change in temperature Given that the initial temperature
(Ti) is 20
°
C and the final temperature (Tf) is 120
°
C, we can find the change in
temperature (∆T) using the formula:
∆T=Tf−Ti= 120
°
C−20
°
C = 100
°
C
Step 2: Calculate the change in length The change in length (∆L) of the
steel rod can be calculated using the formula:
∆L=α·L·∆T
where αis the coefficient of linear expansion of steel (given as 1.2×10−5/
°
C),
Lis the initial length of the rod (2.0 m), and ∆Tis the change in temperature
(100
°
C). Substitute the values into the formula:
∆L= (1.2×10−5/
°
C) ·2.0 m ·100
°
C=0.0024 m = 2.4 mm
Step 3: Calculate the final length of the rod The final length (Lf) of the rod
can be found by adding the change in length to the initial length:
Lf=L+ ∆L= 2.0 m + 0.0024 m = 2.0024 m = 2.4 m
Therefore, the final length of the steel rod when heated to 120
°
C is 2.4 m.
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Question 35
Question
A steel rod has a length of 1.5 m at 20
°
C. If the coefficient of linear expansion
for steel is 1.2×10−5K−1, what is the length of the rod when the temperature
rises to 100
°
C?
Solution
Step 1: Identify the given values and the unknown. Let: L0= 1.5 m be the
initial length of the rod, α= 1.2×10−5K−1be the coefficient of linear expansion,
and ∆T= 100 −20 = 80 K be the change in temperature.
We want to find the final length of the rod when the temperature rises to
100
°
C.
Step 2: Use the formula for linear expansion. The formula for linear expan-
sion is: ∆L=L0α∆T
Step 3: Substitute the given values into the formula and solve for the final
length. ∆L= 1.5×1.2×10−5×80 ∆L= 0.00144 m
The final length of the rod is: L=L0+ ∆L= 1.5+0.00144 = 1.50144 m
Therefore, when the temperature rises to 100
°
C, the length of the steel rod
is 1.50144 meters.
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