PHYS 305 - INTRODUCTION TO
MODERN PHYSICS - Thermal
expansion
Question Bank - Set 1
Liberty University
Question 1
Question
A steel bridge is 200 meters long at a temperature of 20
°
C. If the temperature
rises to 40
°
C, what is the change in length of the bridge? The coefficient of
linear expansion for steel is 12 ×10−6◦C−1.
Solution
Step 1: Let’s first calculate the change in length of the steel bridge using the
formula for linear expansion:
∆L=L0α∆T
where: - ∆Lis the change in length, - L0is the original length, - αis the
coefficient of linear expansion, and - ∆Tis the change in temperature.
Given: - L0= 200 m, - α= 12×10−6◦C−1, and - ∆T= 40◦C−20◦C = 20◦C.
Substitute the values into the formula:
∆L= 200 ×12 ×10−6×20
Step 2: Calculate the change in length:
∆L= 200 ×12 ×10−6×20 = 0.048 m
Therefore, the change in length of the steel bridge when the temperature
rises from 20
°
C to 40
°
C is 0.048 meters.
Question 2
Question
A steel rod is 2 meters long at 20◦C. The rod is heated to 120◦C. If the coefficient
of linear expansion for steel is 1.2×10−5/◦C, find the length of the rod at 120◦C.
Solution
Step 1: Calculate the change in temperature. Given: Initial temperature T1=
20◦C, Final temperature T2= 120◦C.
The change in temperature is given by:
∆T=T2−T1= 120◦C−20◦C= 100◦C
Step 2: Calculate the change in length using the formula:
∆L=L0α∆T
where: ∆L= Change in length, L0= Initial length of the steel rod, α=
Coefficient of linear expansion, ∆T= Change in temperature.
Substitute the given values:
∆L= 2 ×1.2×10−5×100
∆L= 2.4×10−3meters
Step 3: Calculate the final length of the rod. The final length Lfis given
by:
Lf=L0+ ∆L
Lf= 2 + 2.4×10−3= 2.0024 meters
Therefore, the length of the steel rod at 120◦C is 2.0024 meters.
Question 3
Question
A steel rod of length 2 m at 25
°
C is heated to 125
°
C. If the coefficient of linear
expansion of steel is 1.2×10−5◦C−1, find the change in length of the steel rod.
Solution
Step 1: Calculate the change in length of the steel rod using the formula for
linear expansion:
∆L=L0α∆T
where: ∆L= change in length of the rod, L0= initial length of the rod, α=
coefficient of linear expansion, ∆T= change in temperature.
2
Given: L0= 2 m, α= 1.2×10−5◦C−1, ∆T= 125◦C−25◦C = 100◦C.
Plugging in the values:
∆L= 2 ×1.2×10−5×100
∆L= 0.0024 m
Therefore, the change in length of the steel rod is 0.0024 meters.
Question 4
Question
A brass rod measures 2 meters at 20
°
C. If the coefficient of linear expansion for
brass is 1.9×10−5per degree Celsius, what is the length of the rod at 100
°
C?
Solution
Step 1: Calculate the change in length of the brass rod from 20
°
C to 100
°
C.
Step 2: Use the formula for linear expansion to find the final length of the brass
rod at 100
°
C.
Step 1: Calculate the change in length of the brass rod from 20
°
C to 100
°
C.
The change in length (∆L) of the brass rod can be calculated using the
formula:
∆L=L0·α·∆T
where: L0= initial length of the rod = 2 meters, α= coefficient of linear
expansion for brass = 1.9×10−5per degree Celsius, ∆T= change in temperature
= 100
°
C - 20
°
C = 80
°
C.
Substitute the values into the formula:
∆L= 2 ·(1.9×10−5)·80
∆L= 0.00304 meters
Step 2: Use the formula for linear expansion to find the final length of the
brass rod at 100
°
C.
The final length of the brass rod at 100
°
C can be found by adding the change
in length to the initial length:
Lfinal =L0+ ∆L
Substitute the values into the formula:
Lfinal = 2 + 0.00304
Lfinal = 2.00304 meters
Therefore, the length of the brass rod at 100
°
C is 2.00304 meters.
3
Question 5
Question
A steel rod with an initial length of 2 meters is heated from 20
°
C to 120
°
C. The
coefficient of linear expansion for steel is 12 ×10−6
°
C−1. Calculate the final
length of the rod after heating.
Solution
Step 1: Calculate the change in temperature. Given that the initial temperature
is 20
°
C and the final temperature is 120
°
C, the change in temperature is:
∆T=Tfinal −Tinitial = 120 C−20 C= 100 C
Step 2: Calculate the change in length using the formula:
∆L=α·L·∆T
where αis the coefficient of linear expansion, Lis the initial length, and ∆Tis
the change in temperature. Plugging in the values:
∆L= 12 ×10−6
°
C−1×2 m ×100 C= 0.0024 m
Step 3: Calculate the final length of the rod. The final length is given by:
Lfinal =Linitial + ∆L
Substitute the values:
Lfinal = 2 m + 0.0024 m = 2.0024 m
Therefore, the final length of the steel rod after heating is 2.0024 meters.
Question 6
Question
A metal rod of length 2 m at 20
°
C is heated until its temperature reaches 120
°
C.
If the coefficient of linear expansion of the metal is 2.5×10−5
°
C−1, calculate
the final length of the rod.
Solution
Step 1: Calculate the change in temperature.
Given: Initial temperature, Ti= 20C
Final temperature, Tf= 120C
Change in temperature, ∆T=Tf−Ti= 120C −20C = 100C
4
Step 2: Calculate the increase in length.
The increase in length ∆Lof the rod is given by the formula:
∆L=α·L·∆T
where: α= coefficient of linear expansion = 2.5×10−5
°
C−1L= initial length
of the rod = 2 m ∆T= change in temperature = 100C
Plugging in the values, we get:
∆L= 2.5×10−5×2×100 = 5 ×10−3m
Step 3: Calculate the final length.
The final length Lfof the rod is given by:
Lf=L+ ∆L
Lf= 2 + 0.005 = 2.005 m
Therefore, the final length of the metal rod when heated to 120
°
C is 2.005
m.
Question 7
Question
A steel rod initially has a length of 2 meters at a temperature of 20
°
C. If the
coefficient of linear expansion for steel is 1.2×10−5per degree Celsius, what
will be the length of the rod when the temperature reaches 150
°
C?
Solution
Step 1: Calculate the change in temperature. Given initial temperature Ti=
20◦C and final temperature Tf= 150◦C, the change in temperature ∆T=
Tf−Ti. ∆T= 150 −20 = 130
°
C
Step 2: Calculate the change in length using the formula for linear expansion:
∆L=α·L·∆T
where α= coefficient of linear expansion = 1.2×10−5per degree Celsius L
= initial length of the rod = 2 meters ∆T= change in temperature = 130
°
C
Substitute the values into the formula: ∆L= (1.2×10−5)·2·130 ∆L=
2.6×10−3meters
Step 3: Compute the final length of the rod. The final length Lfof the
rod will be the initial length Lplus the change in length ∆L.Lf=L+ ∆L
Lf= 2 + 2.6×10−3Lf= 2.0026 meters
Therefore, when the temperature reaches 150
°
C, the length of the steel rod
will be 2.0026 meters.
5
Question 8
Question
A steel rod is initially 2 meters long at a temperature of 25
°
C. If the coefficient
of linear expansion for steel is 1.2×10−5per
°
C, determine the length of the
rod when the temperature is raised to 100
°
C.
Solution
Step 1: Calculate the change in temperature. Given that the initial temperature
is 25
°
C and the final temperature is 100
°
C, the change in temperature is:
∆T= 100C−25C= 75C
Step 2: Use the formula for linear expansion to find the change in length.
The change in length (∆L) can be calculated using the formula:
∆L=α·L·∆T
where: α= 1.2×10−5/
°
C (coefficient of linear expansion for steel),
L= 2 m (initial length of the rod), and
∆T= 75
°
C (change in temperature).
Substitute the values into the formula:
∆L= (1.2×10−5/C)·2 m ·75C
Step 3: Calculate the change in length.
∆L= 0.00024 m ·2 m ·75 = 0.036 m
Step 4: Find the final length of the rod. The final length Lfof the rod can
be found by adding the change in length to the initial length:
Lf= 2 m + 0.036 m = 2.036 m
Therefore, when the temperature is raised to 100
°
C, the length of the steel
rod will be 2.036 meters.
Question 9
Question
A metal rod of initial length 2 meters and coefficient of linear expansion 1.2×
10−5per degree Celsius is heated from 20 degrees Celsius to 100 degrees Celsius.
Calculate the final length of the rod.
6
Solution
Step 1: Calculate the change in temperature.
∆T=Tf−Ti= 100◦C−20◦C= 80◦C
Step 2: Calculate the change in length using the formula for linear expansion.
∆L=α·L·∆T
∆L= (1.2×10−5/◦C)·(2 m) ·(80◦C)
∆L= 0.000024 m = 0.024 mm
Step 3: Calculate the final length of the rod.
Lf=Li+ ∆L
Lf= 2 m + 0.024 m
Lf= 2.024 m
Therefore, the final length of the rod after heating is 2.024 meters.
Question 10
Question
A steel rod is initially 2 meters long at a temperature of 20◦C. If the rod expands
by 2 mm when heated to 200◦C, what is the coefficient of linear expansion of
the steel?
Solution
Step 1: Calculate the change in length of the rod due to heating. The change
in length of the rod can be calculated using the formula:
∆L=L·α·∆T
where ∆Lis the change in length, Lis the initial length, αis the coefficient
of linear expansion, and ∆Tis the temperature change. Given that ∆L= 2
mm, L= 2 m, ∆T= 200 −20 = 180 C, we can substitute these values into the
formula:
2 mm = 2 m ·α·180 C
Step 2: Solve for the coefficient of linear expansion (α). First, convert 2 mm
to meters:
2 mm = 0.002 m
Substitute ∆L= 0.002 m into the equation and solve for α:
0.002 = 2 ·α·180
7
α=0.002
2·180
α=0.002
360
α= 5.56 ×10−6C−1
Therefore, the coefficient of linear expansion of the steel is 5.56 ×10−6C−1.
Question 11
Question
A steel bridge girder is 50 meters long at 20◦C. If the bridge is exposed to
the sun and heats up to 40◦C, what is the change in length of the girder due
to thermal expansion? (Hint: The coefficient of linear expansion for steel is
1.2×10−5K−1)
Solution
Step 1: Calculate the initial length of the girder due to thermal expansion at
40◦C: Let Lbe the initial length of the girder at 20◦C. The change in length,
∆L, of the girder due to thermal expansion is given by the formula:
∆L=Lα∆T
where Lis the initial length, αis the coefficient of linear expansion, and ∆T
is the change in temperature. Given that α= 1.2×10−5K−1,L= 50 m, and
∆T= 40◦C−20◦C = 20◦C, we can substitute these values into the formula to
find ∆L.
Step 2: Calculate the change in length of the girder:
∆L= 50 m ×1.2×10−5K−1×20◦C
∆L= 50 ×1.2×10−5×20 m
∆L= 0.012 m = 1.2 cm
Therefore, the change in length of the girder due to thermal expansion when
exposed to the sun and heated to 40◦C is 1.2 cm.
Question 12
Question
A metal rod of length 1.5 m is heated from 20
°
C to 80
°
C. If the coefficient of
linear expansion of the metal is 2.5×10−5K−1, determine the change in length
of the rod.
8
Solution
Step 1: Calculate the initial length of the metal rod at 20
°
C using the formula
for linear expansion:
L0=Li×(1 + α×∆T)
where: L0= initial length of the rod, Li= length of the rod at 20
°
C, α=
coefficient of linear expansion, and ∆T= temperature change.
Plugging in the values, we get:
L0= 1.5 m ×(1 + 2.5×10−5K−1×60 K)
L0= 1.5 m ×(1 + 0.0015)
L0= 1.5 m ×1.0015
L0= 1.50225 m
Step 2: Calculate the final length of the metal rod at 80
°
C:
Lf=Li×(1 + α×∆T)
where: Lf= final length of the rod, Li= initial length of the rod, α= coefficient
of linear expansion, and ∆T= temperature change.
Plugging in the values, we get:
Lf= 1.50225 m ×(1 + 2.5×10−5K−1×60 K)
Lf= 1.50225 m ×(1 + 0.0015)
Lf= 1.50225 m ×1.0015
Lf= 1.504
Step 3: Calculate the change in length of the rod:
∆L=Lf−L0
∆L= 1.504 m −1.50225 m
∆L= 0.00175 m
Therefore, the change in length of the rod is 0.00175 m.
Question 13
Question
A steel rod is initially 2 meters long at 20
°
C. If the coefficient of linear expansion
for steel is 12 ×10−6per degree Celsius, find the length of the rod at 100
°
C.
9
Solution
Step 1: Calculate the change in length of the steel rod from 20
°
C to 100
°
C. Step
2: Find the final length of the steel rod at 100
°
C by adding the change in length
to the initial length.
Step 1: The change in length (∆L) of the steel rod can be calculated using
the formula:
∆L=L0·α·∆T
where: L0= initial length of the rod = 2 meters, α= coefficient of linear
expansion for steel = 12×10−6per degree Celsius, ∆T= change in temperature
= 100C−20C= 80C.
Plugging in the values, we get:
∆L= 2 ·12 ×10−6·80 = 1.92 ×10−4meters
Step 2: The final length of the steel rod at 100
°
C can be found by adding
the change in length to the initial length:
Lf=L0+ ∆L= 2 + 1.92 ×10−4= 2.000192 meters
Therefore, the length of the steel rod at 100
°
C is 2.000192 meters.
Question 14
Question
A steel rod of length 2.00 m is heated from 20
°
C to 120
°
C. If the linear expansion
coefficient of steel is 1.20 ×10−5
°
C−1, find the change in length of the rod.
Solution
Step 1: We can use the formula for linear expansion to find the change in length
of the rod:
∆L=Lα∆T
where ∆Lis the change in length, Lis the original length of the rod, αis the
linear expansion coefficient, and ∆Tis the change in temperature.
Step 2: First, calculate the change in temperature:
∆T=Tf−Ti= 120C−20C= 100C
Step 3: Now, we can substitute the values into the formula to find the change
in length:
∆L= 2.00 m ×1.20 ×10−5
°
C−1×100
°
C
Step 4: Calculate the change in length:
∆L= 2.00 ×1.20 ×10−5×100
10
∆L= 0.0024 m
Step 5: Therefore, the change in length of the steel rod is 0.0024 m.
Question 15
Question
A steel rod is initially 2 meters long at 20◦C. It is heated to 100◦C. If the
coefficient of linear expansion for steel is 1.2×10−5/◦C, by how many millimeters
will the length of the rod increase?
Solution
Let’s denote the original length of the steel rod as L0and the temperature
change as ∆T. The change in length of the rod can be calculated using the
formula for linear expansion:
∆L=L0·α·∆T
where: ∆L= change in length L0= original length α= coefficient of linear
expansion ∆T= change in temperature
Step 1: Calculate the change in temperature.
∆T=Tfinal −Tinitial = 100◦C−20◦C = 80◦C
Step 2: Calculate the change in length.
∆L= 2 m ×(1.2×10−5/◦C) ×80◦C
∆L= 2 ×1.2×10−5×80 m
∆L= 1.92 ×10−3m=1.92 mm
Therefore, the length of the steel rod will increase by 1.92 millimeters.
Question 16
Question
A steel rod is 2 meters long at 20
°
C. If its coefficient of linear expansion is
1.2×10−5per degree Celsius, find the length of the rod at 200
°
C.
11
Solution
Let L0be the initial length of the steel rod at 20
°
C, Lbe the final length at
200
°
C, and αbe the coefficient of linear expansion. We can use the formula for
linear expansion:
∆L=L−L0=α·L0·∆T
where ∆Tis the change in temperature. We can rearrange this formula to solve
for L:
L=L0+α·L0·∆T
Step 1: Calculate the initial length of the rod:
L0= 2 m
Step 2: Calculate the change in temperature:
∆T= 200C−20C= 180C
Step 3: Substitute the given values into the formula to find the final length
of the rod:
L= 2 + 1.2×10−5×2×180 = 2.000432 m
Therefore, the length of the steel rod at 200
°
C is approximately 2.000432
meters.
Question 17
Question
A steel rod has an original length of 2.5 meters and a coefficient of linear ex-
pansion of 11 ×10−6◦C−1. If the rod is heated from 20◦C to 100◦C, what is
the new length of the rod?
Solution
Step 1: Calculate the change in temperature. Given initial temperature, Ti=
20◦C, and final temperature, Tf= 100◦C, the change in temperature, ∆T, is
calculated as:
∆T=Tf−Ti= 100◦C−20◦C= 80◦C
Step 2: Calculate the change in length. The change in length, ∆L, is calcu-
lated using the formula:
∆L=α·L·∆T
where αis the coefficient of linear expansion, Lis the original length, and ∆T
is the change in temperature. Substituting the given values, we have:
∆L= (11 ×10−6◦C−1)·(2.5 m) ·(80 ◦C)
12
∆L= 0.000022 m = 0.022 mm
Step 3: Calculate the new length of the rod. The new length of the rod,
Lnew, is given by:
Lnew =L+ ∆L= 2.5 m + 0.000022 m
Lnew = 2.500022 m
Therefore, the new length of the steel rod when heated from 20◦C to 100◦C is
2.500022 meters.
Question 18
Question
A steel rod of length 2.0 m and diameter 2.0 cm is heated from 20
°
C to 120
°
C.
Calculate the increase in length of the rod due to thermal expansion. Given
that the linear expansion coefficient of steel is 1.2×10−5per degree Celsius.
Solution
Step 1: Calculate the initial volume of the rod. Step 2: Calculate the final
volume of the rod. Step 3: Use the formula for linear expansion to find the
increase in length.
Step 1: The initial volume of the rod can be calculated using the formula
for the volume of a cylinder:
V=πr2h
Where ris the radius of the rod and his the initial length. Given that the
radius r= 1.0 cm = 0.01 m and the initial length h= 2.0 m, we can calculate
the initial volume:
Vinitial =π×(0.01)2×2.0=0.000628 m3
Step 2: When the rod is heated, the final length will increase but the radius
will remain the same. Therefore, the final volume can be calculated as:
Vfinal =π×(0.01)2×(2.0+∆L)
Step 3: The increase in length, ∆L, can be calculated using the linear ex-
pansion formula:
∆L=α×L×∆T
Where αis the linear expansion coefficient, Lis the initial length, and ∆T
is the change in temperature. Given that α= 1.2×10−5,L= 2.0 m, and
∆T= 100
°
C, we can calculate the increase in length:
∆L= 1.2×10−5×2.0×100 = 0.0024 m
Therefore, the increase in length of the steel rod due to thermal expansion
is 0.0024 meters.
13
Question 19
Question
A steel rod has a length of 2 meters at 0◦C. If the coefficient of linear expansion
of steel is 11 ×10−6K−1, find the length of the rod at 100◦C.
Solution
Step 1: Determine the change in temperature.
Given: Initial temperature, T1= 0◦C
Final temperature, T2= 100◦C
Change in temperature, ∆T=T2−T1= 100 −0 = 100 C
Step 2: Calculate the change in length.
The change in length ∆Lis given by the formula:
∆L=α·L·∆T
where: α= 11 ×10−6K−1(coefficient of linear expansion of steel) L= 2 m
(initial length) ∆T= 100 C (change in temperature)
Substitute the given values into the formula:
∆L= (11 ×10−6K−1)·2 m ·100 K
∆L= 0.0022 m
Step 3: Calculate the final length of the rod.
The final length Lfcan be found by adding the change in length to the initial
length:
Lf=L+ ∆L
Lf= 2 m + 0.0022 m
Lf= 2.0022 m
Therefore, the length of the steel rod at 100◦C is 2.0022 meters.
Question 20
Question
A steel rod of length 2 meters is heated from 20
°
C to 120
°
C. If the linear coef-
ficient of thermal expansion for steel is 11 ×10−6/C, find the change in length
of the rod.
14
Solution
Step 1: Calculate the initial length increase due to the temperature change.
The initial length of the rod is 2 meters, and the change in temperature is
120C−20C= 100C. The change in length (∆L) can be calculated using the
formula:
∆L=L0·α·∆T
where L0is the initial length, αis the coefficient of linear expansion, and ∆T
is the change in temperature. Substitute the given values into the formula:
∆L= 2 m ·11 ×10−6/C ·100C
∆L= 2 ×11 ×10−4m=0.00022m
∆L= 0.22 mm
Therefore, the initial length of the rod increases by 0.22 mm.
Question 21
Question
A steel rod has a length of 2 meters at 20
°
C. If the coefficient of linear expansion
of steel is 1.2×10−5
°
C−1, how much longer will the rod become when its
temperature is increased to 100
°
C?
Solution
Step 1: First, we calculate the change in temperature. Let ∆Lbe the change in
length of the steel rod, L0be its initial length, and T0be the initial temperature.
The final temperature Tfis 100
°
C. The change in temperature is given by the
formula:
∆T=Tf−T0= 100
°
C−20
°
C = 80
°
C
Step 2: Next, we calculate the change in length of the steel rod. The change
in length is given by:
∆L=αL0∆T
where αis the coefficient of linear expansion, L0is the initial length, and ∆Tis
the change in temperature. Substitute the given values to calculate the change
in length:
∆L= (1.2×10−5
°
C−1)(2 m)(80
°
C)
∆L= 1.92 ×10−3m=1.92 mm
Therefore, the steel rod will become 1.92 mm longer when its temperature
is increased to 100
°
C.
15
Question 22
Question
A solid steel rod of length 2.0 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 initial length change due to heating. Given that the
initial length of the steel rod is Linitial = 2.0 m, the change in temperature is
∆T= 120C−20C= 100C. The linear expansion of the rod can be calculated
using the formula
∆L=Linitialα∆T,
where αis the linear expansion coefficient of steel. Substitute the values into
the formula to find the initial length change:
∆L= 2.0 m ×1.2×10−5K−1×100C.
Step 2: Calculate the change in length of the rod. The initial length change
is:
∆L= 2.0 m ×1.2×10−5K−1×100C= 0.0024 m = 2.4 mm.
Therefore, the change in length of the rod due to heating from 20
°
C to 120
°
C
is 2.4 mm.
Question 23
Question
A steel rod is initially 2 meters long at 20
°
C. If its coefficient of linear expansion
is 1.2×10−5◦C−1, what will be its length when the temperature is raised to
100
°
C?
Solution
Let’s denote the original length of the steel rod as L0, the final temperature
as Tf, and the final length as Lf. We are given that L0= 2 m, T0= 20◦C,
Tf= 100◦C, and α= 1.2×10−5◦C−1.
Step 1: Calculate the change in temperature. The change in temperature
is given by:
∆T=Tf−T0= 100◦C−20◦C = 80◦C
Step 2: Use the formula for linear expansion. The change in length of the
rod is given by:
∆L=α·L0·∆T
16
Substitute the known values:
∆L= 1.2×10−5◦C−1·2 m ·80 ◦C
Step 3: Calculate the change in length.
∆L= 1.92 ×10−3m
Step 4: Determine the final length of the rod. The final length of the rod
is given by:
Lf=L0+ ∆L= 2 m + 1.92 ×10−3m
Step 5: Calculate the final length.
Lf= 2.00192 m
Therefore, when the temperature is raised to 100
°
C, the length of the steel
rod will be approximately 2.00192 meters.
Question 24
Question
A copper rod and an aluminum rod have lengths of 1.2 m and 0.8 m, respectively,
at a temperature of 20
°
C. If the temperature is raised to 120
°
C, calculate the
difference in lengths between the two rods. The coefficients of linear expansion
for copper and aluminum are 17 ×10−6
°
C−1and 23 ×10−6
°
C−1, respectively.
Solution
Step 1: Calculate the change in length for the copper rod. Given the coefficient
of linear expansion for copper, αcopper = 17 ×10−6
°
C−1, the change in length,
∆Lcopper, can be calculated using the formula:
∆Lcopper =Lcopper ·αcopper ·∆T,
where Lcopper is the original length of the copper rod, αcopper is the coefficient
of linear expansion for copper, and ∆Tis the change in temperature. Plugging
in the values, we get:
∆Lcopper = 1.2 m ·17 ×10−6
°
C−1·(120
°
C−20
°
C).
Step 2: Calculate the change in length for the aluminum rod. Similarly, the
change in length for the aluminum rod, ∆Laluminum, is given by:
∆Laluminum =Laluminum ·αaluminum ·∆T.
Plugging in the values, we get:
∆Laluminum = 0.8 m ·23 ×10−6
°
C−1·(120
°
C−20
°
C).
17
Step 3: Calculate the difference in lengths between the two rods. The dif-
ference in lengths, ∆Ldifference, is given by:
∆Ldifference = ∆Lcopper −∆Laluminum.
Substitute the calculated values to find the final answer.
Question 25
Question
A steel rod with an initial length of 2 meters is heated from 20
°
C to 100
°
C. If
the linear expansion coefficient of steel is 12×10−6per degree Celsius, calculate
the final length of the rod.
Solution
Step 1: Calculate the change in temperature ∆T. Given that the initial tem-
perature is 20
°
C and the final temperature is 100
°
C, we have:
∆T= 100C−20C= 80C
Step 2: Calculate the change in length ∆Lusing the formula for linear
expansion:
∆L=αL∆T
where αis the linear expansion coefficient, Lis the initial length, and ∆Tis the
change in temperature.
∆L= (12 ×10−6/C)×2m×80C
∆L= 1.92 ×10−3m
Step 3: Calculate the final length Lfof the rod.
Lf=Li+ ∆L
Lf= 2m+ 1.92 ×10−3m
Lf= 2.00192m
Therefore, the final length of the steel rod after heating it from 20
°
C to
100
°
C is 2.00192 meters.
Question 26
Question
A metal rod is initially at a temperature of 20◦C. It is heated until its temper-
ature reaches 80◦C. If the original length of the rod is 2 meters and its linear
expansion coefficient is 1.2×10−5per degree Celsius, find the final length of the
rod.
18
Solution
Step 1: Calculate the change in temperature. Step 2: Compute the change in
length using the linear expansion formula. Step 3: Find the final length of the
rod.
Step 1: Calculate the change in temperature. The change in temperature
is given by:
∆T=Tf−Ti= 80◦C−20◦C= 60◦C
Step 2: Compute the change in length using the linear expansion formula.
The change in length ∆Lis given by:
∆L=α·L·∆T
where αis the linear expansion coefficient, Lis the original length, and ∆Tis
the change in temperature.
Substitute the values:
∆L= (1.2×10−5/◦C)·(2 m)·(60/◦C)=0.0000144 m
Step 3: Find the final length of the rod. The final length Lfof the rod is:
Lf=Li+ ∆L
Lf= 2 m+ 0.0000144 m= 2.0000144 m
Therefore, the final length of the rod when heated to 80◦C is 2.0000144
meters.
Question 27
Question
A steel rod with a length of 2 meters at 20◦C is heated to 80◦C. If the coefficient
of linear expansion for steel is 1.2×10−5K−1, what will be the final length of
the rod?
Solution
Step 1: Calculate the change in temperature Step 2: Use the formula for linear
expansion to calculate the change in length Step 3: Add the change in length
to the initial length to find the final length
Step 1: Calculate the change in temperature The change in temperature,
∆T, is given by:
∆T= 80◦C−20◦C = 60◦C
Step 2: Use the formula for linear expansion to calculate the change in
length The change in length, ∆L, is given by:
∆L=α·L·∆T
19
where αis the coefficient of linear expansion, Lis the initial length, and ∆Tis
the change in temperature.
Substitute the values into the formula:
∆L= (1.2×10−5K−1)·2 m ·60◦C
∆L= 1.44 ×10−4m
Step 3: Add the change in length to the initial length to find the final length
The final length, Lf, is given by:
Lf=L+ ∆L
Lf=2m+1.44 ×10−4m
Lf= 2.000144 m
Therefore, the final length of the steel rod when heated to 80◦C is 2.000144m.
Question 28
Question
A steel rod has an initial length of 2.0 m and a coefficient of linear expansion of
1.2×10−5◦C−1. If the rod is heated from 20◦C to 80◦C, what will be its final
length?
Solution
Step 1: Calculate the change in temperature. Let ∆Tbe the change in temper-
ature.
∆T=Tf−Ti= 80◦C−20◦C = 60◦C
Step 2: Use the formula for linear expansion to find the change in length.
The change in length ∆Lof the steel rod can be calculated using the formula:
∆L=α·L·∆T
where αis the coefficient of linear expansion, Lis the initial length, and ∆Tis
the change in temperature. Substitute the given values into the formula:
∆L= (1.2×10−5◦C−1)·(2.0 m) ·(60 ◦C)
∆L= 0.00072 m
Step 3: Calculate the final length. The final length Lfof the steel rod can
be found by adding the change in length to the initial length:
Lf=Li+ ∆L= 2.0 m + 0.00072 m
Lf= 2.00072 m
Therefore, the final length of the steel rod after being heated from 20◦C to
80◦C will be 2.00072 m.
20
Question 29
Question
A steel rod is initially 2 meters long at 20
°
C. If the coefficient of linear expansion
for steel is 1.2×10−5/
°
C, what will be the length of the rod at 100
°
C?
Solution
Step 1: Calculate the change in temperature (∆T). Given that the initial tem-
perature is 20
°
C and the final temperature is 100
°
C, the change in temperature
can be calculated as:
∆T= 100C−20C= 80C
Step 2: Calculate the change in length (∆L) using the formula:
∆L=α·L·∆T
where: α= 1.2×10−5/
°
C (coefficient of linear expansion for steel),
L= 2 m (initial length of the rod),
∆T= 80
°
C (change in temperature).
Plugging in the values, we get:
∆L= (1.2×10−5)·2·80
∆L= 0.000192 ×160
∆L= 0.03072
m
Step 3: Calculate 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 + 0.03072
Lfinal = 2.03072
m
Therefore, the length of the steel rod at 100
°
C will be 2.03072 meters.
Question 30
Question
A steel rod with an initial length of 2 meters is heated from 20◦C to 120◦C. The
coefficient of linear expansion for steel is 12×10−6per degree Celsius. Calculate
the final length of the rod after it has been heated.
21
Solution
Step 1: Calculate the change in temperature. Since the rod is heated from 20◦C
to 120◦C, the change in temperature is 120◦C - 20◦C = 100◦C.
Step 2: Calculate the change in length using the formula for linear expansion:
∆L=L·α·∆T
where ∆Lis the change in length, Lis the initial length, αis the coefficient of
linear expansion, and ∆Tis the change in temperature.
Substitute L= 2 meters, α= 12 ×10−6per degree Celsius, and ∆T= 100
degrees Celsius into the formula:
∆L= 2 ·12 ×10−6·100
∆L= 0.0024 meters
Step 3: Calculate the final length of the rod. The final length of the rod is
the sum of the initial length and the change in length:
Final length = Initial length + ∆L
Final length = 2 + 0.0024
Final length = 2.0024 meters
Therefore, the final length of the rod after being heated is 2.0024 meters.
Question 31
Question
A rectangular copper plate measures 6 m by 4 m at room temperature. If the
coefficient of linear expansion of copper is 16 ×10−6K−1, by how much does
the area of the plate increase when its temperature is raised by 50◦C?
Solution
Step 1: Find the change in length of the plate in both dimensions due to the
temperature increase. Step 2: Calculate the new dimensions of the plate after
the temperature increase and find the new area. Step 3: Determine the increase
in area by comparing the original and new areas.
Step 1: Given: Original length, L0= 6 m Original width, W0= 4 m Coeffi-
cient of linear expansion, α= 16×10−6K−1Change in temperature, ∆T= 50◦C
The change in length of an object due to a temperature change can be
calculated using the formula:
∆L=α·L0·∆T
22
For the length:
∆Llength =α·L0·∆T= 16 ×10−6K−1·6 m ·50◦C=0.0048 m
For the width:
∆Lwidth =α·W0·∆T= 16 ×10−6K−1·4 m ·50◦C=0.0032 m
Step 2: The new lengths after the temperature increase will be:
Lnew =L0+ ∆Llength = 6 m + 0.0048 m = 6.0048 m
Wnew =W0+ ∆Lwidth = 4 m + 0.0032 m = 4.0032 m
The new area of the plate is:
Areanew =Lnew ×Wnew = 6.0048 m ×4.0032 m = 24.0481 m2
Step 3: The increase in area is:
∆Area = Areanew −Areaoriginal = 24.0481 m2−24 m2= 0.0481 m2
Therefore, the area of the plate increases by 0.0481 m2when its temperature
is raised by 50◦C.
Question 32
Question
A steel rod of length 4 m is heated from 20
°
C to 80
°
C. If the linear expansion
coefficient of steel is 12 ×10−6per
°
C, calculate the change in 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 80
°
C, the change in temperature
is:
∆T=Tf−Ti= 80C−20C= 60C
Step 2: Calculate the change in length using the formula for linear expansion
The change in length (∆L) of the steel rod can be calculated using the formula:
∆L=α·L·∆T
where: - α= 12 ×10−6per
°
C is the linear expansion coefficient of steel, -
L= 4 m is the original length of the rod, and - ∆T= 60
°
C is the change in
temperature.
Substitute the values into the formula to find the change in length:
∆L= 12 ×10−6·4·60 = 2.88 ×10−3m
Therefore, the change in length of the steel rod is 2.88 ×10−3m.
23
Question 33
Question
A steel rod of length 2.0 m at 20
°
C is heated until its temperature reaches 80
°
C.
If the linear expansion coefficient of steel is 1.2×10−5
°
C−1, what is the final
length of the rod?
Solution
Step 1: Calculate the change in temperature. Given: Initial temperature, Ti=
20
°
C Final temperature, Tf= 80
°
C Change in temperature, ∆T=Tf−Ti=
80 −20 = 60
°
C
Step 2: Calculate the linear expansion of the steel rod. The linear expansion
of the steel rod can be calculated using the formula:
∆L=α·L·∆T
where: ∆L= change in length α= linear expansion coefficient = 1.2×10−5
°
C−1L= initial length of the rod = 2.0 m ∆T= change in temperature = 60
°
C
Substitute the values into the formula:
∆L= (1.2×10−5
°
C−1)·(2.0 m) ·(60
°
C)
∆L= 1.44 ×10−4m
Step 3: Calculate the final length of the rod. The final length of the rod can
be calculated by adding the change in length to the initial length:
Lfinal =Linitial + ∆L
Lfinal = 2.0 m + 1.44 ×10−4m
Lfinal = 2.000144 m
Therefore, the final length of the steel rod when heated to 80
°
C is 2.000144
meters.
Question 34
Question
A steel rod has a length of 2.0 m at 20◦C. If the coefficient of linear expansion
for steel is 1.2×10−5per degree Celsius, what will be the length of the rod
when the temperature is raised to 200◦C?
24
Solution
Step 1: Determine the change in temperature. Step 2: Use the equation for
linear expansion to calculate the new length of the rod.
Step 1: The change in temperature is given by:
∆T=Tf−Ti= 200◦C−20◦C= 180◦C
Step 2: The change in length of the steel rod can be calculated using the
formula:
∆L=L0α∆T
where: ∆L= Change in length of the rod, L0= Initial length of the rod, α=
Coefficient of linear expansion, ∆T= Change in temperature.
Substitute the given values into the formula:
∆L= 2.0 m ×(1.2×10−5per ◦C) ×180◦C
∆L= 2.16 ×10−3m
Therefore, the final length of the steel rod when the temperature is 200◦C
will be:
Lf=L0+ ∆L= 2.0 m + 2.16 ×10−3m
Lf= 2.00216 m
So, the length of the rod when the temperature is raised to 200◦C will be
2.00216 meters.
Question 35
Question
A steel rod has an original length of 2 meters at 20
°
C. If the coefficient of linear
expansion for steel is 12 ×10−6
°
C−1, determine the final length of the rod when
it is heated to 100
°
C.
Solution
Step 1: First, we calculate the change in length of the steel rod as it is heated
from 20
°
C to 100
°
C. Step 2: The change in length (∆L) is given by the formula:
∆L=L0α∆T
where: L0= Original length of the rod = 2 m, α= Coefficient of linear expansion
for steel = 12 ×10−6
°
C−1, ∆T= Change in temperature = 100
°
C - 20
°
C =
80
°
C. Step 3: Substituting the values into the formula, we get:
∆L= 2 ×12 ×10−6×80 = 1.92 ×10−3m
25
Question 2
Question
A steel rod is 2 meters long at 20◦C. The rod is heated to 120◦C. If the coefficient
of linear expansion for steel is 1.2×10−5/◦C, find the length of the rod at 120◦C.
Solution
Step 1: Calculate the change in temperature. Given: Initial temperature T1=
20◦C, Final temperature T2= 120◦C.
The change in temperature is given by:
∆T=T2−T1= 120◦C−20◦C= 100◦C
Step 2: Calculate the change in length using the formula:
∆L=L0α∆T
where: ∆L= Change in length, L0= Initial length of the steel rod, α=
Coefficient of linear expansion, ∆T= Change in temperature.
Substitute the given values:
∆L= 2 ×1.2×10−5×100
∆L= 2.4×10−3meters
Step 3: Calculate the final length of the rod. The final length Lfis given
by:
Lf=L0+ ∆L
Lf= 2 + 2.4×10−3= 2.0024 meters
Therefore, the length of the steel rod at 120◦C is 2.0024 meters.
Question 3
Question
A steel rod of length 2 m at 25
°
C is heated to 125
°
C. If the coefficient of linear
expansion of steel is 1.2×10−5◦C−1, find the change in length of the steel rod.
Solution
Step 1: Calculate the change in length of the steel rod using the formula for
linear expansion:
∆L=L0α∆T
where: ∆L= change in length of the rod, L0= initial length of the rod, α=
coefficient of linear expansion, ∆T= change in temperature.
2
Given: L0= 2 m, α= 1.2×10−5◦C−1, ∆T= 125◦C−25◦C = 100◦C.
Plugging in the values:
∆L= 2 ×1.2×10−5×100
∆L= 0.0024 m
Therefore, the change in length of the steel rod is 0.0024 meters.
Question 4
Question
A brass rod measures 2 meters at 20
°
C. If the coefficient of linear expansion for
brass is 1.9×10−5per degree Celsius, what is the length of the rod at 100
°
C?
Solution
Step 1: Calculate the change in length of the brass rod from 20
°
C to 100
°
C.
Step 2: Use the formula for linear expansion to find the final length of the brass
rod at 100
°
C.
Step 1: Calculate the change in length of the brass rod from 20
°
C to 100
°
C.
The change in length (∆L) of the brass rod can be calculated using the
formula:
∆L=L0·α·∆T
where: L0= initial length of the rod = 2 meters, α= coefficient of linear
expansion for brass = 1.9×10−5per degree Celsius, ∆T= change in temperature
= 100
°
C - 20
°
C = 80
°
C.
Substitute the values into the formula:
∆L= 2 ·(1.9×10−5)·80
∆L= 0.00304 meters
Step 2: Use the formula for linear expansion to find the final length of the
brass rod at 100
°
C.
The final length of the brass rod at 100
°
C can be found by adding the change
in length to the initial length:
Lfinal =L0+ ∆L
Substitute the values into the formula:
Lfinal = 2 + 0.00304
Lfinal = 2.00304 meters
Therefore, the length of the brass rod at 100
°
C is 2.00304 meters.
3
Question 5
Question
A steel rod with an initial length of 2 meters is heated from 20
°
C to 120
°
C. The
coefficient of linear expansion for steel is 12 ×10−6
°
C−1. Calculate the final
length of the rod after heating.
Solution
Step 1: Calculate the change in temperature. Given that the initial temperature
is 20
°
C and the final temperature is 120
°
C, the change in temperature is:
∆T=Tfinal −Tinitial = 120 C−20 C= 100 C
Step 2: Calculate the change in length using the formula:
∆L=α·L·∆T
where αis the coefficient of linear expansion, Lis the initial length, and ∆Tis
the change in temperature. Plugging in the values:
∆L= 12 ×10−6
°
C−1×2 m ×100 C= 0.0024 m
Step 3: Calculate the final length of the rod. The final length is given by:
Lfinal =Linitial + ∆L
Substitute the values:
Lfinal = 2 m + 0.0024 m = 2.0024 m
Therefore, the final length of the steel rod after heating is 2.0024 meters.
Question 6
Question
A metal rod of length 2 m at 20
°
C is heated until its temperature reaches 120
°
C.
If the coefficient of linear expansion of the metal is 2.5×10−5
°
C−1, calculate
the final length of the rod.
Solution
Step 1: Calculate the change in temperature.
Given: Initial temperature, Ti= 20C
Final temperature, Tf= 120C
Change in temperature, ∆T=Tf−Ti= 120C −20C = 100C
4
Step 2: Calculate the increase in length.
The increase in length ∆Lof the rod is given by the formula:
∆L=α·L·∆T
where: α= coefficient of linear expansion = 2.5×10−5
°
C−1L= initial length
of the rod = 2 m ∆T= change in temperature = 100C
Plugging in the values, we get:
∆L= 2.5×10−5×2×100 = 5 ×10−3m
Step 3: Calculate the final length.
The final length Lfof the rod is given by:
Lf=L+ ∆L
Lf= 2 + 0.005 = 2.005 m
Therefore, the final length of the metal rod when heated to 120
°
C is 2.005
m.
Question 7
Question
A steel rod initially has a length of 2 meters at a temperature of 20
°
C. If the
coefficient of linear expansion for steel is 1.2×10−5per degree Celsius, what
will be the length of the rod when the temperature reaches 150
°
C?
Solution
Step 1: Calculate the change in temperature. Given initial temperature Ti=
20◦C and final temperature Tf= 150◦C, the change in temperature ∆T=
Tf−Ti. ∆T= 150 −20 = 130
°
C
Step 2: Calculate the change in length using the formula for linear expansion:
∆L=α·L·∆T
where α= coefficient of linear expansion = 1.2×10−5per degree Celsius L
= initial length of the rod = 2 meters ∆T= change in temperature = 130
°
C
Substitute the values into the formula: ∆L= (1.2×10−5)·2·130 ∆L=
2.6×10−3meters
Step 3: Compute the final length of the rod. The final length Lfof the
rod will be the initial length Lplus the change in length ∆L.Lf=L+ ∆L
Lf= 2 + 2.6×10−3Lf= 2.0026 meters
Therefore, when the temperature reaches 150
°
C, the length of the steel rod
will be 2.0026 meters.
5
Question 8
Question
A steel rod is initially 2 meters long at a temperature of 25
°
C. If the coefficient
of linear expansion for steel is 1.2×10−5per
°
C, determine the length of the
rod when the temperature is raised to 100
°
C.
Solution
Step 1: Calculate the change in temperature. Given that the initial temperature
is 25
°
C and the final temperature is 100
°
C, the change in temperature is:
∆T= 100C−25C= 75C
Step 2: Use the formula for linear expansion to find the change in length.
The change in length (∆L) can be calculated using the formula:
∆L=α·L·∆T
where: α= 1.2×10−5/
°
C (coefficient of linear expansion for steel),
L= 2 m (initial length of the rod), and
∆T= 75
°
C (change in temperature).
Substitute the values into the formula:
∆L= (1.2×10−5/C)·2 m ·75C
Step 3: Calculate the change in length.
∆L= 0.00024 m ·2 m ·75 = 0.036 m
Step 4: Find the final length of the rod. The final length Lfof the rod can
be found by adding the change in length to the initial length:
Lf= 2 m + 0.036 m = 2.036 m
Therefore, when the temperature is raised to 100
°
C, the length of the steel
rod will be 2.036 meters.
Question 9
Question
A metal rod of initial length 2 meters and coefficient of linear expansion 1.2×
10−5per degree Celsius is heated from 20 degrees Celsius to 100 degrees Celsius.
Calculate the final length of the rod.
6
Solution
Step 1: Calculate the change in temperature.
∆T=Tf−Ti= 100◦C−20◦C= 80◦C
Step 2: Calculate the change in length using the formula for linear expansion.
∆L=α·L·∆T
∆L= (1.2×10−5/◦C)·(2 m) ·(80◦C)
∆L= 0.000024 m = 0.024 mm
Step 3: Calculate the final length of the rod.
Lf=Li+ ∆L
Lf= 2 m + 0.024 m
Lf= 2.024 m
Therefore, the final length of the rod after heating is 2.024 meters.
Question 10
Question
A steel rod is initially 2 meters long at a temperature of 20◦C. If the rod expands
by 2 mm when heated to 200◦C, what is the coefficient of linear expansion of
the steel?
Solution
Step 1: Calculate the change in length of the rod due to heating. The change
in length of the rod can be calculated using the formula:
∆L=L·α·∆T
where ∆Lis the change in length, Lis the initial length, αis the coefficient
of linear expansion, and ∆Tis the temperature change. Given that ∆L= 2
mm, L= 2 m, ∆T= 200 −20 = 180 C, we can substitute these values into the
formula:
2 mm = 2 m ·α·180 C
Step 2: Solve for the coefficient of linear expansion (α). First, convert 2 mm
to meters:
2 mm = 0.002 m
Substitute ∆L= 0.002 m into the equation and solve for α:
0.002 = 2 ·α·180
7
α=0.002
2·180
α=0.002
360
α= 5.56 ×10−6C−1
Therefore, the coefficient of linear expansion of the steel is 5.56 ×10−6C−1.
Question 11
Question
A steel bridge girder is 50 meters long at 20◦C. If the bridge is exposed to
the sun and heats up to 40◦C, what is the change in length of the girder due
to thermal expansion? (Hint: The coefficient of linear expansion for steel is
1.2×10−5K−1)
Solution
Step 1: Calculate the initial length of the girder due to thermal expansion at
40◦C: Let Lbe the initial length of the girder at 20◦C. The change in length,
∆L, of the girder due to thermal expansion is given by the formula:
∆L=Lα∆T
where Lis the initial length, αis the coefficient of linear expansion, and ∆T
is the change in temperature. Given that α= 1.2×10−5K−1,L= 50 m, and
∆T= 40◦C−20◦C = 20◦C, we can substitute these values into the formula to
find ∆L.
Step 2: Calculate the change in length of the girder:
∆L= 50 m ×1.2×10−5K−1×20◦C
∆L= 50 ×1.2×10−5×20 m
∆L= 0.012 m = 1.2 cm
Therefore, the change in length of the girder due to thermal expansion when
exposed to the sun and heated to 40◦C is 1.2 cm.
Question 12
Question
A metal rod of length 1.5 m is heated from 20
°
C to 80
°
C. If the coefficient of
linear expansion of the metal is 2.5×10−5K−1, determine the change in length
of the rod.
8
Solution
Step 1: Calculate the initial length of the metal rod at 20
°
C using the formula
for linear expansion:
L0=Li×(1 + α×∆T)
where: L0= initial length of the rod, Li= length of the rod at 20
°
C, α=
coefficient of linear expansion, and ∆T= temperature change.
Plugging in the values, we get:
L0= 1.5 m ×(1 + 2.5×10−5K−1×60 K)
L0= 1.5 m ×(1 + 0.0015)
L0= 1.5 m ×1.0015
L0= 1.50225 m
Step 2: Calculate the final length of the metal rod at 80
°
C:
Lf=Li×(1 + α×∆T)
where: Lf= final length of the rod, Li= initial length of the rod, α= coefficient
of linear expansion, and ∆T= temperature change.
Plugging in the values, we get:
Lf= 1.50225 m ×(1 + 2.5×10−5K−1×60 K)
Lf= 1.50225 m ×(1 + 0.0015)
Lf= 1.50225 m ×1.0015
Lf= 1.504
Step 3: Calculate the change in length of the rod:
∆L=Lf−L0
∆L= 1.504 m −1.50225 m
∆L= 0.00175 m
Therefore, the change in length of the rod is 0.00175 m.
Question 13
Question
A steel rod is initially 2 meters long at 20
°
C. If the coefficient of linear expansion
for steel is 12 ×10−6per degree Celsius, find the length of the rod at 100
°
C.
9
Solution
Step 1: Calculate the change in length of the steel rod from 20
°
C to 100
°
C. Step
2: Find the final length of the steel rod at 100
°
C by adding the change in length
to the initial length.
Step 1: The change in length (∆L) of the steel rod can be calculated using
the formula:
∆L=L0·α·∆T
where: L0= initial length of the rod = 2 meters, α= coefficient of linear
expansion for steel = 12×10−6per degree Celsius, ∆T= change in temperature
= 100C−20C= 80C.
Plugging in the values, we get:
∆L= 2 ·12 ×10−6·80 = 1.92 ×10−4meters
Step 2: The final length of the steel rod at 100
°
C can be found by adding
the change in length to the initial length:
Lf=L0+ ∆L= 2 + 1.92 ×10−4= 2.000192 meters
Therefore, the length of the steel rod at 100
°
C is 2.000192 meters.
Question 14
Question
A steel rod of length 2.00 m is heated from 20
°
C to 120
°
C. If the linear expansion
coefficient of steel is 1.20 ×10−5
°
C−1, find the change in length of the rod.
Solution
Step 1: We can use the formula for linear expansion to find the change in length
of the rod:
∆L=Lα∆T
where ∆Lis the change in length, Lis the original length of the rod, αis the
linear expansion coefficient, and ∆Tis the change in temperature.
Step 2: First, calculate the change in temperature:
∆T=Tf−Ti= 120C−20C= 100C
Step 3: Now, we can substitute the values into the formula to find the change
in length:
∆L= 2.00 m ×1.20 ×10−5
°
C−1×100
°
C
Step 4: Calculate the change in length:
∆L= 2.00 ×1.20 ×10−5×100
10
∆L= 0.0024 m
Step 5: Therefore, the change in length of the steel rod is 0.0024 m.
Question 15
Question
A steel rod is initially 2 meters long at 20◦C. It is heated to 100◦C. If the
coefficient of linear expansion for steel is 1.2×10−5/◦C, by how many millimeters
will the length of the rod increase?
Solution
Let’s denote the original length of the steel rod as L0and the temperature
change as ∆T. The change in length of the rod can be calculated using the
formula for linear expansion:
∆L=L0·α·∆T
where: ∆L= change in length L0= original length α= coefficient of linear
expansion ∆T= change in temperature
Step 1: Calculate the change in temperature.
∆T=Tfinal −Tinitial = 100◦C−20◦C = 80◦C
Step 2: Calculate the change in length.
∆L= 2 m ×(1.2×10−5/◦C) ×80◦C
∆L= 2 ×1.2×10−5×80 m
∆L= 1.92 ×10−3m=1.92 mm
Therefore, the length of the steel rod will increase by 1.92 millimeters.
Question 16
Question
A steel rod is 2 meters long at 20
°
C. If its coefficient of linear expansion is
1.2×10−5per degree Celsius, find the length of the rod at 200
°
C.
11
Solution
Let L0be the initial length of the steel rod at 20
°
C, Lbe the final length at
200
°
C, and αbe the coefficient of linear expansion. We can use the formula for
linear expansion:
∆L=L−L0=α·L0·∆T
where ∆Tis the change in temperature. We can rearrange this formula to solve
for L:
L=L0+α·L0·∆T
Step 1: Calculate the initial length of the rod:
L0= 2 m
Step 2: Calculate the change in temperature:
∆T= 200C−20C= 180C
Step 3: Substitute the given values into the formula to find the final length
of the rod:
L= 2 + 1.2×10−5×2×180 = 2.000432 m
Therefore, the length of the steel rod at 200
°
C is approximately 2.000432
meters.
Question 17
Question
A steel rod has an original length of 2.5 meters and a coefficient of linear ex-
pansion of 11 ×10−6◦C−1. If the rod is heated from 20◦C to 100◦C, what is
the new length of the rod?
Solution
Step 1: Calculate the change in temperature. Given initial temperature, Ti=
20◦C, and final temperature, Tf= 100◦C, the change in temperature, ∆T, is
calculated as:
∆T=Tf−Ti= 100◦C−20◦C= 80◦C
Step 2: Calculate the change in length. The change in length, ∆L, is calcu-
lated using the formula:
∆L=α·L·∆T
where αis the coefficient of linear expansion, Lis the original length, and ∆T
is the change in temperature. Substituting the given values, we have:
∆L= (11 ×10−6◦C−1)·(2.5 m) ·(80 ◦C)
12
∆L= 0.000022 m = 0.022 mm
Step 3: Calculate the new length of the rod. The new length of the rod,
Lnew, is given by:
Lnew =L+ ∆L= 2.5 m + 0.000022 m
Lnew = 2.500022 m
Therefore, the new length of the steel rod when heated from 20◦C to 100◦C is
2.500022 meters.
Question 18
Question
A steel rod of length 2.0 m and diameter 2.0 cm is heated from 20
°
C to 120
°
C.
Calculate the increase in length of the rod due to thermal expansion. Given
that the linear expansion coefficient of steel is 1.2×10−5per degree Celsius.
Solution
Step 1: Calculate the initial volume of the rod. Step 2: Calculate the final
volume of the rod. Step 3: Use the formula for linear expansion to find the
increase in length.
Step 1: The initial volume of the rod can be calculated using the formula
for the volume of a cylinder:
V=πr2h
Where ris the radius of the rod and his the initial length. Given that the
radius r= 1.0 cm = 0.01 m and the initial length h= 2.0 m, we can calculate
the initial volume:
Vinitial =π×(0.01)2×2.0=0.000628 m3
Step 2: When the rod is heated, the final length will increase but the radius
will remain the same. Therefore, the final volume can be calculated as:
Vfinal =π×(0.01)2×(2.0+∆L)
Step 3: The increase in length, ∆L, can be calculated using the linear ex-
pansion formula:
∆L=α×L×∆T
Where αis the linear expansion coefficient, Lis the initial length, and ∆T
is the change in temperature. Given that α= 1.2×10−5,L= 2.0 m, and
∆T= 100
°
C, we can calculate the increase in length:
∆L= 1.2×10−5×2.0×100 = 0.0024 m
Therefore, the increase in length of the steel rod due to thermal expansion
is 0.0024 meters.
13
Question 19
Question
A steel rod has a length of 2 meters at 0◦C. If the coefficient of linear expansion
of steel is 11 ×10−6K−1, find the length of the rod at 100◦C.
Solution
Step 1: Determine the change in temperature.
Given: Initial temperature, T1= 0◦C
Final temperature, T2= 100◦C
Change in temperature, ∆T=T2−T1= 100 −0 = 100 C
Step 2: Calculate the change in length.
The change in length ∆Lis given by the formula:
∆L=α·L·∆T
where: α= 11 ×10−6K−1(coefficient of linear expansion of steel) L= 2 m
(initial length) ∆T= 100 C (change in temperature)
Substitute the given values into the formula:
∆L= (11 ×10−6K−1)·2 m ·100 K
∆L= 0.0022 m
Step 3: Calculate the final length of the rod.
The final length Lfcan be found by adding the change in length to the initial
length:
Lf=L+ ∆L
Lf= 2 m + 0.0022 m
Lf= 2.0022 m
Therefore, the length of the steel rod at 100◦C is 2.0022 meters.
Question 20
Question
A steel rod of length 2 meters is heated from 20
°
C to 120
°
C. If the linear coef-
ficient of thermal expansion for steel is 11 ×10−6/C, find the change in length
of the rod.
14
Solution
Step 1: Calculate the initial length increase due to the temperature change.
The initial length of the rod is 2 meters, and the change in temperature is
120C−20C= 100C. The change in length (∆L) can be calculated using the
formula:
∆L=L0·α·∆T
where L0is the initial length, αis the coefficient of linear expansion, and ∆T
is the change in temperature. Substitute the given values into the formula:
∆L= 2 m ·11 ×10−6/C ·100C
∆L= 2 ×11 ×10−4m=0.00022m
∆L= 0.22 mm
Therefore, the initial length of the rod increases by 0.22 mm.
Question 21
Question
A steel rod has a length of 2 meters at 20
°
C. If the coefficient of linear expansion
of steel is 1.2×10−5
°
C−1, how much longer will the rod become when its
temperature is increased to 100
°
C?
Solution
Step 1: First, we calculate the change in temperature. Let ∆Lbe the change in
length of the steel rod, L0be its initial length, and T0be the initial temperature.
The final temperature Tfis 100
°
C. The change in temperature is given by the
formula:
∆T=Tf−T0= 100
°
C−20
°
C = 80
°
C
Step 2: Next, we calculate the change in length of the steel rod. The change
in length is given by:
∆L=αL0∆T
where αis the coefficient of linear expansion, L0is the initial length, and ∆Tis
the change in temperature. Substitute the given values to calculate the change
in length:
∆L= (1.2×10−5
°
C−1)(2 m)(80
°
C)
∆L= 1.92 ×10−3m=1.92 mm
Therefore, the steel rod will become 1.92 mm longer when its temperature
is increased to 100
°
C.
15
Question 22
Question
A solid steel rod of length 2.0 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 initial length change due to heating. Given that the
initial length of the steel rod is Linitial = 2.0 m, the change in temperature is
∆T= 120C−20C= 100C. The linear expansion of the rod can be calculated
using the formula
∆L=Linitialα∆T,
where αis the linear expansion coefficient of steel. Substitute the values into
the formula to find the initial length change:
∆L= 2.0 m ×1.2×10−5K−1×100C.
Step 2: Calculate the change in length of the rod. The initial length change
is:
∆L= 2.0 m ×1.2×10−5K−1×100C= 0.0024 m = 2.4 mm.
Therefore, the change in length of the rod due to heating from 20
°
C to 120
°
C
is 2.4 mm.
Question 23
Question
A steel rod is initially 2 meters long at 20
°
C. If its coefficient of linear expansion
is 1.2×10−5◦C−1, what will be its length when the temperature is raised to
100
°
C?
Solution
Let’s denote the original length of the steel rod as L0, the final temperature
as Tf, and the final length as Lf. We are given that L0= 2 m, T0= 20◦C,
Tf= 100◦C, and α= 1.2×10−5◦C−1.
Step 1: Calculate the change in temperature. The change in temperature
is given by:
∆T=Tf−T0= 100◦C−20◦C = 80◦C
Step 2: Use the formula for linear expansion. The change in length of the
rod is given by:
∆L=α·L0·∆T
16
Substitute the known values:
∆L= 1.2×10−5◦C−1·2 m ·80 ◦C
Step 3: Calculate the change in length.
∆L= 1.92 ×10−3m
Step 4: Determine the final length of the rod. The final length of the rod
is given by:
Lf=L0+ ∆L= 2 m + 1.92 ×10−3m
Step 5: Calculate the final length.
Lf= 2.00192 m
Therefore, when the temperature is raised to 100
°
C, the length of the steel
rod will be approximately 2.00192 meters.
Question 24
Question
A copper rod and an aluminum rod have lengths of 1.2 m and 0.8 m, respectively,
at a temperature of 20
°
C. If the temperature is raised to 120
°
C, calculate the
difference in lengths between the two rods. The coefficients of linear expansion
for copper and aluminum are 17 ×10−6
°
C−1and 23 ×10−6
°
C−1, respectively.
Solution
Step 1: Calculate the change in length for the copper rod. Given the coefficient
of linear expansion for copper, αcopper = 17 ×10−6
°
C−1, the change in length,
∆Lcopper, can be calculated using the formula:
∆Lcopper =Lcopper ·αcopper ·∆T,
where Lcopper is the original length of the copper rod, αcopper is the coefficient
of linear expansion for copper, and ∆Tis the change in temperature. Plugging
in the values, we get:
∆Lcopper = 1.2 m ·17 ×10−6
°
C−1·(120
°
C−20
°
C).
Step 2: Calculate the change in length for the aluminum rod. Similarly, the
change in length for the aluminum rod, ∆Laluminum, is given by:
∆Laluminum =Laluminum ·αaluminum ·∆T.
Plugging in the values, we get:
∆Laluminum = 0.8 m ·23 ×10−6
°
C−1·(120
°
C−20
°
C).
17
Step 3: Calculate the difference in lengths between the two rods. The dif-
ference in lengths, ∆Ldifference, is given by:
∆Ldifference = ∆Lcopper −∆Laluminum.
Substitute the calculated values to find the final answer.
Question 25
Question
A steel rod with an initial length of 2 meters is heated from 20
°
C to 100
°
C. If
the linear expansion coefficient of steel is 12×10−6per degree Celsius, calculate
the final length of the rod.
Solution
Step 1: Calculate the change in temperature ∆T. Given that the initial tem-
perature is 20
°
C and the final temperature is 100
°
C, we have:
∆T= 100C−20C= 80C
Step 2: Calculate the change in length ∆Lusing the formula for linear
expansion:
∆L=αL∆T
where αis the linear expansion coefficient, Lis the initial length, and ∆Tis the
change in temperature.
∆L= (12 ×10−6/C)×2m×80C
∆L= 1.92 ×10−3m
Step 3: Calculate the final length Lfof the rod.
Lf=Li+ ∆L
Lf= 2m+ 1.92 ×10−3m
Lf= 2.00192m
Therefore, the final length of the steel rod after heating it from 20
°
C to
100
°
C is 2.00192 meters.
Question 26
Question
A metal rod is initially at a temperature of 20◦C. It is heated until its temper-
ature reaches 80◦C. If the original length of the rod is 2 meters and its linear
expansion coefficient is 1.2×10−5per degree Celsius, find the final length of the
rod.
18
Solution
Step 1: Calculate the change in temperature. Step 2: Compute the change in
length using the linear expansion formula. Step 3: Find the final length of the
rod.
Step 1: Calculate the change in temperature. The change in temperature
is given by:
∆T=Tf−Ti= 80◦C−20◦C= 60◦C
Step 2: Compute the change in length using the linear expansion formula.
The change in length ∆Lis given by:
∆L=α·L·∆T
where αis the linear expansion coefficient, Lis the original length, and ∆Tis
the change in temperature.
Substitute the values:
∆L= (1.2×10−5/◦C)·(2 m)·(60/◦C)=0.0000144 m
Step 3: Find the final length of the rod. The final length Lfof the rod is:
Lf=Li+ ∆L
Lf= 2 m+ 0.0000144 m= 2.0000144 m
Therefore, the final length of the rod when heated to 80◦C is 2.0000144
meters.
Question 27
Question
A steel rod with a length of 2 meters at 20◦C is heated to 80◦C. If the coefficient
of linear expansion for steel is 1.2×10−5K−1, what will be the final length of
the rod?
Solution
Step 1: Calculate the change in temperature Step 2: Use the formula for linear
expansion to calculate the change in length Step 3: Add the change in length
to the initial length to find the final length
Step 1: Calculate the change in temperature The change in temperature,
∆T, is given by:
∆T= 80◦C−20◦C = 60◦C
Step 2: Use the formula for linear expansion to calculate the change in
length The change in length, ∆L, is given by:
∆L=α·L·∆T
19
where αis the coefficient of linear expansion, Lis the initial length, and ∆Tis
the change in temperature.
Substitute the values into the formula:
∆L= (1.2×10−5K−1)·2 m ·60◦C
∆L= 1.44 ×10−4m
Step 3: Add the change in length to the initial length to find the final length
The final length, Lf, is given by:
Lf=L+ ∆L
Lf=2m+1.44 ×10−4m
Lf= 2.000144 m
Therefore, the final length of the steel rod when heated to 80◦C is 2.000144m.
Question 28
Question
A steel rod has an initial length of 2.0 m and a coefficient of linear expansion of
1.2×10−5◦C−1. If the rod is heated from 20◦C to 80◦C, what will be its final
length?
Solution
Step 1: Calculate the change in temperature. Let ∆Tbe the change in temper-
ature.
∆T=Tf−Ti= 80◦C−20◦C = 60◦C
Step 2: Use the formula for linear expansion to find the change in length.
The change in length ∆Lof the steel rod can be calculated using the formula:
∆L=α·L·∆T
where αis the coefficient of linear expansion, Lis the initial length, and ∆Tis
the change in temperature. Substitute the given values into the formula:
∆L= (1.2×10−5◦C−1)·(2.0 m) ·(60 ◦C)
∆L= 0.00072 m
Step 3: Calculate the final length. The final length Lfof the steel rod can
be found by adding the change in length to the initial length:
Lf=Li+ ∆L= 2.0 m + 0.00072 m
Lf= 2.00072 m
Therefore, the final length of the steel rod after being heated from 20◦C to
80◦C will be 2.00072 m.
20
Question 29
Question
A steel rod is initially 2 meters long at 20
°
C. If the coefficient of linear expansion
for steel is 1.2×10−5/
°
C, what will be the length of the rod at 100
°
C?
Solution
Step 1: Calculate the change in temperature (∆T). Given that the initial tem-
perature is 20
°
C and the final temperature is 100
°
C, the change in temperature
can be calculated as:
∆T= 100C−20C= 80C
Step 2: Calculate the change in length (∆L) using the formula:
∆L=α·L·∆T
where: α= 1.2×10−5/
°
C (coefficient of linear expansion for steel),
L= 2 m (initial length of the rod),
∆T= 80
°
C (change in temperature).
Plugging in the values, we get:
∆L= (1.2×10−5)·2·80
∆L= 0.000192 ×160
∆L= 0.03072
m
Step 3: Calculate 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 + 0.03072
Lfinal = 2.03072
m
Therefore, the length of the steel rod at 100
°
C will be 2.03072 meters.
Question 30
Question
A steel rod with an initial length of 2 meters is heated from 20◦C to 120◦C. The
coefficient of linear expansion for steel is 12×10−6per degree Celsius. Calculate
the final length of the rod after it has been heated.
21
Solution
Step 1: Calculate the change in temperature. Since the rod is heated from 20◦C
to 120◦C, the change in temperature is 120◦C - 20◦C = 100◦C.
Step 2: Calculate the change in length using the formula for linear expansion:
∆L=L·α·∆T
where ∆Lis the change in length, Lis the initial length, αis the coefficient of
linear expansion, and ∆Tis the change in temperature.
Substitute L= 2 meters, α= 12 ×10−6per degree Celsius, and ∆T= 100
degrees Celsius into the formula:
∆L= 2 ·12 ×10−6·100
∆L= 0.0024 meters
Step 3: Calculate the final length of the rod. The final length of the rod is
the sum of the initial length and the change in length:
Final length = Initial length + ∆L
Final length = 2 + 0.0024
Final length = 2.0024 meters
Therefore, the final length of the rod after being heated is 2.0024 meters.
Question 31
Question
A rectangular copper plate measures 6 m by 4 m at room temperature. If the
coefficient of linear expansion of copper is 16 ×10−6K−1, by how much does
the area of the plate increase when its temperature is raised by 50◦C?
Solution
Step 1: Find the change in length of the plate in both dimensions due to the
temperature increase. Step 2: Calculate the new dimensions of the plate after
the temperature increase and find the new area. Step 3: Determine the increase
in area by comparing the original and new areas.
Step 1: Given: Original length, L0= 6 m Original width, W0= 4 m Coeffi-
cient of linear expansion, α= 16×10−6K−1Change in temperature, ∆T= 50◦C
The change in length of an object due to a temperature change can be
calculated using the formula:
∆L=α·L0·∆T
22
For the length:
∆Llength =α·L0·∆T= 16 ×10−6K−1·6 m ·50◦C=0.0048 m
For the width:
∆Lwidth =α·W0·∆T= 16 ×10−6K−1·4 m ·50◦C=0.0032 m
Step 2: The new lengths after the temperature increase will be:
Lnew =L0+ ∆Llength = 6 m + 0.0048 m = 6.0048 m
Wnew =W0+ ∆Lwidth = 4 m + 0.0032 m = 4.0032 m
The new area of the plate is:
Areanew =Lnew ×Wnew = 6.0048 m ×4.0032 m = 24.0481 m2
Step 3: The increase in area is:
∆Area = Areanew −Areaoriginal = 24.0481 m2−24 m2= 0.0481 m2
Therefore, the area of the plate increases by 0.0481 m2when its temperature
is raised by 50◦C.
Question 32
Question
A steel rod of length 4 m is heated from 20
°
C to 80
°
C. If the linear expansion
coefficient of steel is 12 ×10−6per
°
C, calculate the change in 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 80
°
C, the change in temperature
is:
∆T=Tf−Ti= 80C−20C= 60C
Step 2: Calculate the change in length using the formula for linear expansion
The change in length (∆L) of the steel rod can be calculated using the formula:
∆L=α·L·∆T
where: - α= 12 ×10−6per
°
C is the linear expansion coefficient of steel, -
L= 4 m is the original length of the rod, and - ∆T= 60
°
C is the change in
temperature.
Substitute the values into the formula to find the change in length:
∆L= 12 ×10−6·4·60 = 2.88 ×10−3m
Therefore, the change in length of the steel rod is 2.88 ×10−3m.
23
Question 33
Question
A steel rod of length 2.0 m at 20
°
C is heated until its temperature reaches 80
°
C.
If the linear expansion coefficient of steel is 1.2×10−5
°
C−1, what is the final
length of the rod?
Solution
Step 1: Calculate the change in temperature. Given: Initial temperature, Ti=
20
°
C Final temperature, Tf= 80
°
C Change in temperature, ∆T=Tf−Ti=
80 −20 = 60
°
C
Step 2: Calculate the linear expansion of the steel rod. The linear expansion
of the steel rod can be calculated using the formula:
∆L=α·L·∆T
where: ∆L= change in length α= linear expansion coefficient = 1.2×10−5
°
C−1L= initial length of the rod = 2.0 m ∆T= change in temperature = 60
°
C
Substitute the values into the formula:
∆L= (1.2×10−5
°
C−1)·(2.0 m) ·(60
°
C)
∆L= 1.44 ×10−4m
Step 3: Calculate the final length of the rod. The final length of the rod can
be calculated by adding the change in length to the initial length:
Lfinal =Linitial + ∆L
Lfinal = 2.0 m + 1.44 ×10−4m
Lfinal = 2.000144 m
Therefore, the final length of the steel rod when heated to 80
°
C is 2.000144
meters.
Question 34
Question
A steel rod has a length of 2.0 m at 20◦C. If the coefficient of linear expansion
for steel is 1.2×10−5per degree Celsius, what will be the length of the rod
when the temperature is raised to 200◦C?
24
Solution
Step 1: Determine the change in temperature. Step 2: Use the equation for
linear expansion to calculate the new length of the rod.
Step 1: The change in temperature is given by:
∆T=Tf−Ti= 200◦C−20◦C= 180◦C
Step 2: The change in length of the steel rod can be calculated using the
formula:
∆L=L0α∆T
where: ∆L= Change in length of the rod, L0= Initial length of the rod, α=
Coefficient of linear expansion, ∆T= Change in temperature.
Substitute the given values into the formula:
∆L= 2.0 m ×(1.2×10−5per ◦C) ×180◦C
∆L= 2.16 ×10−3m
Therefore, the final length of the steel rod when the temperature is 200◦C
will be:
Lf=L0+ ∆L= 2.0 m + 2.16 ×10−3m
Lf= 2.00216 m
So, the length of the rod when the temperature is raised to 200◦C will be
2.00216 meters.
Question 35
Question
A steel rod has an original length of 2 meters at 20
°
C. If the coefficient of linear
expansion for steel is 12 ×10−6
°
C−1, determine the final length of the rod when
it is heated to 100
°
C.
Solution
Step 1: First, we calculate the change in length of the steel rod as it is heated
from 20
°
C to 100
°
C. Step 2: The change in length (∆L) is given by the formula:
∆L=L0α∆T
where: L0= Original length of the rod = 2 m, α= Coefficient of linear expansion
for steel = 12 ×10−6
°
C−1, ∆T= Change in temperature = 100
°
C - 20
°
C =
80
°
C. Step 3: Substituting the values into the formula, we get:
∆L= 2 ×12 ×10−6×80 = 1.92 ×10−3m
25
Step 4: The final length of the rod (Lf) is the sum of the original length and
the change in length:
Lf=L0+ ∆L= 2 + 1.92 ×10−3= 2.00192 m
Therefore, the final length of the steel rod when heated to 100
°
C is 2.00192
meters.
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