PHYS 305 - INTRODUCTION TO MODERN PHYSICS Thermal expansion Question Bank Set 2 | PHYS 305

PHYS 305 - INTRODUCTION TO

MODERN PHYSICS - Thermal

expansion

Question Bank - Set 2

Liberty University

Question 1

Question

A steel rod of length 2 m is heated from an initial temperature of 20

C to a final

temperature of 120

C. If the linear expansion coefficient of steel is 1.2×10−5

per degree Celsius, determine the final length of the rod.

Solution

Step 1: Calculate the change in temperature. Given: Initial temperature, Ti=

20CFinal temperature, Tf= 120C

The change in temperature, ∆T=Tf−Ti= 120C−20C= 100C

Step 2: Calculate the change in length. The linear expansion of a material

can be given by the formula: ∆L=L0α∆Twhere, ∆L= change in length L0

= initial length α= linear expansion coefficient ∆T= change in temperature

Substitute the given values: ∆L= 2 m ×(1.2×10−5)×100C= 0.0024 m

Step 3: Determine the final length of the rod. The final length of the rod

can be calculated using the formula: Lf=L0+ ∆L

Substitute the values: Lf= 2 m + 0.0024 m = 2.0024 m

Therefore, the final length of the steel rod is 2.0024 meters.

Question 2

Question

A steel rod has a length of 2 meters at 20◦C. If the rod is heated to 100◦C, what

will be its new length? Given that the coefficient of linear expansion for steel is

12 ×10−6/◦C.

Solution

Step 1: Calculate the change in temperature. Step 2: Use the formula for linear

expansion to find the change in length. Step 3: Add the change in length to the

original length to find the new length.

Step 1: Calculate the change in temperature. The change in temperature

is given by:

∆T=Tf−Ti= 100◦C−20◦C = 80◦C

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 αis the coefficient of linear expansion, Lis the original length, and ∆T

is the change in temperature. Substitute the values to find ∆L:

∆L= 12 ×10−6/◦C·2 m ·80◦= 0.00192 m = 1.92 mm

Step 3: Add the change in length to the original length to find the new

length. The new length (Lf) can be found by:

Lf=L+ ∆L= 2 m + 0.00192 m = 2.00192 m = 2.00192 ×103mm

Therefore, when the steel rod is heated to 100◦C, its new length will be

2.00192 meters or 2001.92 millimeters.

Question 3

Question

A solid aluminum rod is initially 2 meters long at a temperature of 20◦C. If the

rod is heated to 90◦C, calculate the final length of the rod. The linear expansion

coefficient of aluminum is 23 ×10−6/◦C.

Solution

Step 1: Calculate the change in temperature. Given that the initial temperature

Ti= 20◦C and the final temperature Tf= 90◦C, the change in temperature is

∆T=Tf−Ti= 90◦C−20◦C = 70◦C.

Step 2: Calculate the change in length using the formula:

∆L=α·L·∆T,

where Lis the initial length, αis the linear expansion coefficient, and ∆Tis the

change in temperature.

Substitute L= 2 m, α= 23 ×10−6/

C, and ∆T= 70

C into the formula:

∆L= 23 ×10−6/C·2 m ·70

Step 3: Calculate the change in length:

∆L= 23 ×10−6×2×70 = 0.00322 m.

Step 4: Calculate the final length of the rod: The final length Lfof the rod

can be calculated by adding the change in length ∆Lto the initial length L:

Lf=L+ ∆L= 2 m + 0.00322 m = 2.00322 m.

Therefore, the final length of the aluminum rod when heated to 90◦C is

2.00322 meters.

Question 4

Question

A 2-meter long steel rod is heated from 20◦C to 120◦C. The rod is fixed at one

end and free to expand at the other end. If the coefficient of linear expansion

for steel is 1.2×10−5◦C−1, find the increase in length of the rod.

Solution

Let’s denote: - Initial length of the rod Li= 2 m - Initial temperature Ti= 20◦C

- Final temperature Tf= 120◦C - Coefficient of linear expansion α= 1.2×10−5

◦C−1

We need to find the increase in length. By definition, the increase in length

is given by ∆L=Lf−Li, where Lfis the final length.

Step 1: Calculate the final length of the rod, Lf.

The change in length ∆Lcan be calculated using the formula:

∆L=α·Li·∆T

where ∆T=Tf−Ti.

Substitute the given values:

∆L= (1.2×10−5◦C−1)×(2 m) ×(120◦C−20◦C)

∆L= 1.2×10−5×2×100 = 0.0024 m = 2.4 mm

Therefore, the increase in length of the rod is 2.4 mm.

Question 5

Question

A uniform steel rod is 2 meters long at 20 degrees Celsius. If the coefficient of

linear expansion for steel is 1.2×10−5/degC, find the increase in length of the

rod when its temperature is raised to 80 degrees Celsius.

Solution

Let L0be the original length of the steel rod, αbe the coefficient of linear

expansion, and ∆Tbe the change in temperature.

Step 1: Calculate the original length of the steel rod at 20 degrees Celsius.

L0= 2 m

Step 2: Calculate the change in temperature.

∆T= 80◦C−20◦C = 60◦C

Step 3: Calculate the increase in length of the steel rod. The increase in

length, ∆L, is given by the formula:

∆L=L0·α·∆T

Substitute L0= 2 m, α= 1.2×10−5/degC, and ∆T= 60 C into the formula:

∆L= 2 ·1.2×10−5·60

∆L= 1.44 ×10−4·60

∆L= 8.64 ×10−3m

So, the increase in length of the steel rod when its temperature is raised to

80 degrees Celsius is 8.64 mm.

Question 6

Question

A copper rod and an aluminum rod are both 1.0 m in length at 20◦C. If the

rods are heated to 120◦C, which rod will have a greater increase in length? The

linear expansion coefficients for copper and aluminum are 16.5×10−6◦C−1and

23 ×10−6◦C−1, respectively.

Solution

Step 1: Calculate the change in length for each rod using the formula ∆L=

L0·α·∆T, where ∆Lis the change in length, L0is the original length, αis the

coefficient of linear expansion, and ∆Tis the change in temperature.

For the copper rod: ∆Lcopper = 1.0 m ·16.5×10−6◦C−1·(120 −20)◦C =

0.0165 m

For the aluminum rod: ∆Laluminum = 1.0 m·23 ×10−6◦C−1·(120−20)◦C =

0.022 m

Step 2: Compare the change in length for both rods. Since ∆Laluminum >

∆Lcopper, the aluminum rod will have a greater increase in length when heated

to 120◦C.

Question 7

Question

A steel rod of length 2.00 m at 20.0

C is heated to a temperature of 80.0

If the linear expansion coefficient of steel is 1.20 ×10−5K−1, what is the final

length of the rod?

Solution

Step 1: First, we calculate the change in temperature: Given: Initial tempera-

ture, T1= 20.0CFinal temperature, T2= 80.0C

The change in temperature, ∆T=T2−T1= 80.0C−20.0C= 60.0C

Step 2: Next, we calculate the change in length using the linear expansion

formula:

∆L=α·L·∆T

where: αis the linear expansion coefficient, 1.20 ×10−5K−1Lis the initial

length of the rod, 2.00 m ∆Tis the change in temperature, 60.0

Substitute the values into the formula:

∆L= (1.20 ×10−5K−1)·(2.00 m) ·(60.0C)

∆L= 0.000024 m ·K−1·m·

C·m

∆L= 0.000024 m2·

Step 3: Finally, we find the final length of the rod:

Lf=Li+ ∆L

Lf= 2.00 m + 0.000024 m2·

Lf= 2.000024 m

Therefore, the final length of the steel rod after heating it to 80.0

C is

2.000024 meters.

Question 8

Question

A solid metal rod is 2 meters long at 20

C. If the coefficient of linear expansion

of the metal is 2 ×10−5per degree Celsius, by how many millimeters will the

rod expand when the temperature is increased to 100

Solution

Step 1: Calculate the change in temperature. Given that the initial temperature

is 20

C and the final temperature is 100

C, the change in temperature is ∆T=

100 −20 = 80

Step 2: Use the formula for linear expansion. The change in length ∆Lof

the rod is given by:

∆L=L0α∆T

where: - L0is the initial length of the rod, - αis the coefficient of linear

expansion, - ∆Tis the change in temperature.

Step 3: Substitute the known values into the formula. Given that L0= 2

meters and α= 2 ×10−5/◦C, we can substitute these values into the formula:

∆L= 2 ×2×10−5×80

Step 4: Calculate the change in length. Now, calculate the change in length:

∆L= 2 ×2×10−5×80 = 3.2×10−3meters

Step 5: Convert the change in length to millimeters. To convert the change

in length to millimeters, we multiply by 1000:

∆L= 3.2×10−3×1000 = 3.2 millimeters

Therefore, the rod will expand by 3.2 millimeters when the temperature is

increased to 100

Question 9

Question

A metal rod has an initial length of 2.00 m at 25

C. If the coefficient of linear

expansion for the metal is 2.5×10−5

C−1, how much will the length of the rod

change when its temperature is raised to 250

Solution

Step 1: First, we can calculate the change in temperature: Given initial tem-

perature, Ti= 25C, and final temperature, Tf= 250C.

∆T=Tf−Ti= 250C−25C= 225C

Step 2: Next, we can use the formula for linear thermal expansion:

∆L=α·L·∆T

where: ∆Lis the change in length, αis the coefficient of linear expansion, Lis

the initial length, and ∆Tis the change in temperature.

Step 3: Substitute the given values into the formula:

∆L= (2.5×10−5

C−1)·(2.00 m) ·(225C)

Step 4: Calculate the change in length:

∆L= 2.25 ×10−4m

Therefore, when the temperature of the metal rod is raised to 250

C, the

length of the rod will increase by 0.000225 m.

Question 10

Question

A metal bar has an original length of 2 meters at 20 degrees Celsius. If the

coefficient of linear expansion for the metal is 2.5×10−5per degree Celsius, by

how many millimeters will the length of the bar increase when it is heated to

100 degrees Celsius?

Solution

Step 1: Calculate the change in temperature. Given that the original tempera-

ture is 20 degrees Celsius and the final temperature is 100 degrees Celsius, the

change in temperature is:

∆T= 100◦C−20◦C= 80◦C

Step 2: Calculate the change in length. The change in length (∆L) of the

metal bar can be calculated 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.

Given that α= 2.5×10−5per degree Celsius, L= 2 meters, and ∆T= 80

degrees Celsius, we can calculate the change in length as follows:

∆L= (2.5×10−5)×2×80 = 0.00004 meters

Step 3: Convert the change in length to millimeters. To convert the change

in length from meters to millimeters, we need to multiply by 1000 (since 1 meter

= 1000 millimeters). Thus,

0.00004 meters ×1000 = 0.04 mm

Therefore, the length of the metal bar will increase by 0.04 millimeters when

heated to 100 degrees Celsius.

Question 11

Question

A steel rod is initially 2 meters long at a temperature of 20◦C. If the coefficient

of linear expansion of steel is 1.2×10−5K−1, by how many millimeters does the

rod lengthen when heated to 120◦C?

Solution

Step 1: Calculate the change in temperature. Given: Initial length of the steel

rod, L0= 2 m, Coefficient of linear expansion of steel, α= 1.2×10−5K−1,

Initial temperature, Ti= 20◦C, Final temperature, Tf= 120◦C.

The change in temperature is given by:

∆T=Tf−Ti= 120◦C−20◦C = 100◦C = 100 K

Step 2: Calculate the change in length. The change in length of the rod can

be calculated using the formula:

∆L=L0α∆T

Substitute the given values into the formula:

∆L= 2 ×1.2×10−5×100

∆L= 2.4×10−3m=2.4 mm

The steel rod lengthens by 2.4 millimeters when heated to 120◦C.

Question 12

Question

A steel rod of length 1 m is heated from 20◦C to 120◦C. If the linear expansion

coefficient of steel is 1.2×10−5/K, calculate the increase in length of the rod.

Solution

Step 1: Calculate the change in temperature Given: Initial temperature, Ti=

20◦C Final temperature, Tf= 120◦C

The change in temperature, ∆T=Tf−Ti= 120 −20 = 100K

Step 2: Calculate the increase in length The formula for linear expansion is

given by:

∆L=αL∆T

where: ∆L= increase in length α= linear expansion coefficient L= initial

length ∆T= change in temperature

Plugging in the values, we get:

∆L= (1.2×10−5)×1×100

∆L= 1.2×10−3m=1.2 mm

Therefore, the increase in length of the steel rod is 1.2 mm.

Question 13

Question

A brass ring has an initial inside diameter of 10 cm at 20

C. If the coefficient

of linear expansion for brass is 2.0×10−5K−1, find the inside diameter of the

ring when its temperature is raised to 120

Solution

Let’s denote the initial inside diameter of the brass ring as d0= 10 cm at 20C,

the final temperature as Tf= 120C, and the coefficient of linear expansion for

brass as α= 2.0×10−5K−1.

Step 1: Calculate the change in temperature. We can find the change in

temperature by subtracting the initial temperature from the final temperature:

∆T=Tf−T0= 120C−20C= 100C

Step 2: Calculate the change in diameter. We can use the formula for linear

expansion:

∆L=α·L0·∆T

where ∆Lis the change in length, L0is the initial length (in this case, the initial

inside diameter), and ∆Tis the change in temperature.

Substitute the values:

∆L= (2.0×10−5K−1)·(10 cm) ·(100C)=0.02 cm

Step 3: Determine the final inside diameter. The final inside diameter is

given by:

df=d0+ ∆L= 10 cm + 0.02 cm = 10.02 cm

Therefore, when the temperature is raised to 120C, the inside diameter of

the brass ring will be 10.02 cm.

Question 14

Question

A steel beam is 10 meters long at 20

C. If the temperature changes to 80

how much longer will the beam become? The linear expansion coefficient for

steel is 12 ×10−6per Celsius degree.

Solution

Step 1: Calculate the change in temperature. Step 2: Use the linear expansion

formula to calculate the change in length.

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: Use the linear expansion formula to calculate the change in length.

The change in length (∆L) is given by:

∆L=α·L·∆T

where αis the linear expansion coefficient for steel, Lis the original length of

the beam, and ∆Tis the change in temperature.

Substitute the values into the formula:

∆L= 12 ×10−6·(10 m) ·(60)

∆L= 0.0072 m

Therefore, the beam will become 0.0072 meters longer when the temperature

changes from 20

C to 80

Question 15

Question

A steel rod has an initial length of 2 meters at 20 degrees Celsius. If the

coefficient of linear expansion for steel is 11 ×10−6◦C−1, find the length of the

rod when the temperature is increased to 100 degrees Celsius.

Solution

Step 1: Calculate the change in temperature. Let Lbe the final length of the

rod, L0be the initial length of the rod, ∆Tbe the change in temperature,

and αbe the coefficient of linear expansion. We can use the formula for linear

expansion: ∆L=α·L0·∆T. Given that L0= 2 m, α= 11 ×10−6◦C−1, and

the initial temperature is 20 degrees Celsius, and the final temperature is 100

degrees Celsius, we have:

∆T= 100 −20 = 80 ◦C

Step 2: Calculate the change in length. Substitute the known values into

the formula:

∆L= 11 ×10−6·2·80

∆L= 0.00176 m

Step 3: Calculate the final length of the rod. The final length Lis given by:

L=L0+ ∆L

. Substitute L0= 2 m and ∆L= 0.00176 m:

L= 2 + 0.00176 = 2.00176 m

Therefore, when the temperature is increased to 100 degrees Celsius, the

length of the rod will be 2.00176 meters.

Question 16

Question

A steel rod with a length of 2.5 meters at 20◦C is heated to 200◦C. If the linear

expansion coefficient of steel is 11 ×10−6/◦C, what is the final 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 200◦C, the change in temperature is:

(200 −20) = 180

Step 2: Use the linear expansion formula to find the change in length. The

change in length (∆L) of an object due to thermal expansion is given by:

∆L=α·L0·∆T

where: α= linear expansion coefficient (11 ×10−6/◦C) L0= initial length of

the rod (2.5 meters) ∆T= change in temperature (180

Substitute the given values into the formula:

∆L= (11 ×10−6/◦C) ·(2.5 m) ·(180

Step 3: Calculate the change in length.

∆L= 0.000385 m

Step 4: Determine 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=L0+ ∆L

Lf= 2.5 m + 0.000385 m

Step 5: Calculate the final length of the rod.

Lf= 2.500385 m

Therefore, the final length of the steel rod when heated to 200◦C is 2.500385

meters.

Question 17

Question

A steel rod is 2 meters long at 20◦C. If the coefficient of linear expansion for

steel is 1.2×10−5◦C−1, what will be 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 20◦C and the final temperature is 100◦C, the change in temperature is:

∆T=Tf−Ti= 100◦C−20◦C = 80◦C

Step 2: Calculate the change in length. The change in length (∆L) of the

steel rod can be calculated using the formula:

∆L=L·α·∆T

where Lis the initial length, αis the coefficient of linear expansion, and ∆T

is the change in temperature. Substitute L= 2 m, α= 1.2×10−5◦C−1, and

∆T= 80 ◦C into the formula:

∆L= 2 m ·1.2×10−5◦C−1·80 ◦C

Step 3: Calculate the final length of the rod. The final length (Lf) of the

rod can be calculated by adding the change in length to the initial length:

Lf=L+ ∆L

Substitute L= 2 m and the calculated ∆Linto the formula:

Lf= 2 m + (2 m ·1.2×10−5◦C−1·80 ◦C)

Question 18

Question

A steel rod is initially 2 meters long at 20

C. If the coefficient of linear expansion

for steel is 1.2×10−5per degree Celsius, what is the length of the rod when the

temperature reaches 100

Solution

Step 1: Calculate the change in temperature.

Given: Initial temperature, T1= 20

Final temperature, T2= 100

Change in temperature, ∆T=T2−T1= 100 −20 = 80

Step 2: Calculate the change in length using the formula for linear expansion:

∆L=αL∆T

where: ∆L= change in length

α= coefficient of linear expansion

L= initial length of the rod

∆T= change in temperature

Plugging in the values, we have:

∆L= (1.2×10−5)×2×80

Calculating,

∆L= 0.000024 ×160 = 0.00384 meters

Step 3: Find the final length of the rod. The final length, Lf, can be found

by adding the change in length to the initial length:

Lf=L+ ∆L= 2 + 0.00384 = 2.00384 meters

Therefore, the length of the rod when the temperature reaches 100

C is

2.00384 meters.

Question 19

Question

A steel rod is tightly clamped between two walls at 0◦C. If the walls are heated

to 100◦C, by how much will the rod elongate? (Given: Coefficient of linear

thermal expansion of steel = 1.2×10−5/◦Cand original length of the rod = 2

meters)

Solution

Step 1: First, we calculate the change in temperature:

∆T=Tf−Ti= 100◦C−0◦C= 100◦C

Step 2: Next, we calculate the change in length using the formula for linear

thermal expansion:

∆L=α·L·∆T

where: ∆L= change in length, α= coefficient of linear thermal expansion, L

= original length, ∆T= change in temperature.

Substitute the given values:

∆L= (1.2×10−5/◦C)·2 meters ·100◦C

Step 3: Calculate the change in length:

∆L= 0.000024 ·200 = 0.0048 meters

Step 4: Therefore, the steel rod will elongate by 0.0048 meters when the

walls are heated to 100◦C.

Question 20

Question

A steel cylinder of height 2.0 m and radius 1.0 m is initially at a temperature

of 20

C. If the temperature of the cylinder is increased to 100

C, calculate the

change in height of the cylinder. The linear expansion coefficient of steel is

1.2×10−5per

Solution

Let’s denote the initial height of the cylinder as H0and the initial radius as

R0. The change in height of the cylinder can be calculated using the formula

for linear expansion:

∆H=H0·α·∆T

where ∆H= change in height, H0= initial height of the cylinder, α= linear

expansion coefficient of steel, ∆T= change in temperature.

Given: H0= 2.0 m, α= 1.2×10−5per

C, ∆T= 100C−20C= 80C.

Step 1: Calculate the change in height

∆H= 2.0 m ·1.2×10−5·80

Step 2: Solve for ∆H

∆H= 2.0×1.2×10−5×80 = 0.00192 m

Therefore, the change in height of the cylinder is 0.00192 m.

Question 21

Question

A steel rod has a length of 1 meter at 20

C. If the coefficient of linear expansion

of steel is 1.2×10−5per degree Celsius, determine the temperature at which

the rod will have a length of 1.01 meters.

Solution

Step 1: Let’s denote the initial length of the steel rod as L0, the final length as

L, the initial temperature as T0, and the final temperature as T. We are given

that L0= 1 meter, L= 1.01 meters, and the coefficient of linear expansion

α= 1.2×10−5per degree Celsius.

Step 2: The formula for linear expansion is given by ∆L=α·L0·∆T, where

∆L=L−L0and ∆T=T−T0.

Step 3: Substituting the given values into the formula, we get ∆L= (1.2×

10−5)·1·∆T.

Step 4: Solving for ∆T, we have ∆T=∆L

1.2×10−5.

Step 5: Plugging in the values for ∆Land solving for ∆T, we get ∆T=

1.01−1

1.2×10−5.

Step 6: Calculating the value of ∆T, we have ∆T=0.01

1.2×10−5.

Step 7: Therefore, ∆T= 833.33 degrees Celsius.

Step 8: To find the final temperature T, we use the formula T=T0+ ∆T.

Step 9: Substituting the values of T0and ∆T, we get T= 20 + 833.33.

Step 10: Therefore, the temperature at which the steel rod will have a length

of 1.01 meters is 853.33 degrees Celsius.

Question 22

Question

A brass rod is 2 meters long at 20

C. If the coefficient of linear expansion of

brass is 19×10−6per

C, by how much does the length of the rod increase when

the temperature is increased to 90

Solution

Given: Initial length of brass rod, L0= 2 m

Coefficient of linear expansion, α= 19 ×10−6per

Change in temperature, ∆T= 90 −20 = 70

We can find the change in length, ∆L, using the formula:

∆L=L0α∆T

Step 1: Calculate the change in length, ∆L.

∆L= 2 ×19 ×10−6×70

= 2 ×19 ×10−6×70

= 2 ×1.33 ×10−4

= 2.66 ×10−4m

So, the length of the brass rod increases by 2.66 ×10−4meters when the

temperature is increased to 90

Question 23

Question

A brass rod is 2 meters long at 20

C. If the coefficient of linear expansion of

brass is 2.0×10−5per degree Celsius, what temperature increase is needed to

increase the length of the rod by 1 cm?

Solution

Step 1: Let’s first calculate the increase in length of the rod when the temper-

ature increases by ∆Tdegrees Celsius. The increase in length (∆L) is given by

the formula:

∆L=α·L·∆T,

where αis the coefficient of linear expansion, Lis the original length of the rod,

and ∆Tis the temperature change. Substitute the given values:

∆L= (2.0×10−5)·2·∆T.

Step 2: We are given that the increase in length is 1 cm, which is 0.01 meters.

Thus, we have:

0.01 = (2.0×10−5)·2·∆T.

Step 3: Solve for ∆T:

∆T=0.01

(2.0×10−5)·2.

Step 4: Calculating the value of ∆T:

∆T=0.01

4.0×10−5= 250 degrees Celsius.

Therefore, a temperature increase of 250

C is needed to increase the length

of the rod by 1 cm.

Question 24

Question

A steel rod of length 2.0 m is heated from 20

C to 80

C. If the coefficient of

linear expansion of steel is 1.2×10−5/

C, what is the change in length of the

rod?

Solution

Step 1: Calculate the initial length of the rod when it is at 20

C. Given: Initial

length, L0= 2.0 m Coefficient of linear expansion, α= 1.2×10−5/C Initial

temperature, Ti= 20C

Using the formula for linear expansion:

∆L=α·L0·∆T

where ∆Lis the change in length, αis the coefficient of linear expansion, L0is

the initial length, and ∆Tis the change in temperature.

Substitute the values:

∆L= (1.2×10−5/C)·2.0 m ·(80C−20C)

Step 2: Calculate the change in length of the rod.

∆L= 1.2×10−5·2·60

∆L= 1.44 ×10−4m

∆L= 0.144 mm

Therefore, the change in length of the steel rod is 0.144 mm.

Question 25

Question

A brass rod is 2.0 m long at 20

C. If the rod is heated to 100

C, what will be

its length? The linear expansion for brass is 19 ×10−6/

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=Tf−Ti= 100C−20C= 80C

Step 2: Calculate the linear expansion coefficient. The linear expansion

coefficient for brass is 19 ×10−6/

Step 3: Use the formula for linear expansion to find the change in length.

The change in length (∆L) of the brass rod can be calculated using the formula:

∆L=α·L0·∆T

where: - ∆Lis the change in length, - αis the linear expansion coefficient of

brass (19 ×10−6/

C), - L0is the initial length of the brass rod (2.0 m), - ∆T

is the change in temperature (80

C).

Substitute the values into the formula:

∆L= (19 ×10−6/

C) ·(2.0 m) ·(80C)

Step 4: Calculate the change in length.

∆L= 0.000038 m = 3.8×10−5m

Therefore, the length of the brass rod when heated to 100

C will be:

Lfinal =L0+ ∆L= 2.0 m + 3.8×10−5m=2.000038 m

So, the final length of the brass rod when heated to 100

C will be 2.000038

meters.

Question 26

Question

A copper rod is initially at a length of 1 m. When heated from 20◦C to 120◦C,

the rod expands by 2 mm. Calculate the coefficient of linear expansion of copper.

Solution

Step 1: Identify the relevant formula for linear expansion. The linear expansion

of a material can be calculated using the formula:

∆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.

Step 2: Given values The original length L0is 1 m, the change in length ∆L

is 2 mm (which is 0.002 m), the change in temperature ∆Tis (120◦C - 20◦C)

= 100◦C.

Step 3: Substitute the values into the formula Substitute L0= 1 m, ∆L=

0.002 m, and ∆T= 100 ◦C into the formula:

0.002 = 1 ×α×100

Step 4: Solve for the coefficient of linear expansion α

α=0.002

100 = 2 ×10−5per degree Celsius

Therefore, the coefficient of linear expansion of copper is 2×10−5per degree

Celsius.

Question 27

Question

A steel rod with an original length of 2 meters experiences a temperature in-

crease of 50

C. If the coefficient of linear expansion for steel is 12 ×10−6/

by how many millimeters does the length of the rod increase?

Solution

Step 1: First we calculate the change in length using the formula given by the

linear expansion equation:

∆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.

Step 2: Substituting the given values, we get:

∆L= 12 ×10−6/

C·2 m ·50

Step 3: Calculating ∆L:

∆L= 12 ×10−6·2·50 = 1.2×10−3m

Step 4: Converting the change in length to millimeters:

∆Lmm = 1.2×10−3m×1000 = 1.2 mm

Therefore, the length of the steel rod increases by 1.2 millimeters.

Question 28

Question

A steel rod has a length of 2 meters at 20◦C. If the coefficient of linear expansion

for steel is 12 ×10−6/

C, what is the length of the rod at 100◦C?

Solution

Step 1: Calculate the change in temperature. Given: Initial temperature, T1=

20◦C Final temperature, T2= 100◦C Change in temperature, ∆T=T2−T1=

100◦C - 20◦C = 80◦C

Step 2: Use the formula for linear expansion. The change in length of the

rod can be calculated using the formula: ∆L=α·L·∆Twhere ∆L= change

in length α= coefficient of linear expansion L= initial length of the rod

Step 3: Substitute the values into the formula. Given: α= 12 ×10−6/

L= 2 meters ∆T= 80◦C

Plugging in the values, we get: ∆L= (12 ×10−6/C)·2 m ·80

Step 4: Calculate the change in length. ∆L= 12 ×10−6·2·80 = 1.92 ×10−3

meters

Step 5: Calculate the final length of the rod. The final length of the rod is

given by: L2=L1+ ∆L L2= 2 m + 1.92 ×10−3mL2= 2.00192 meters

Therefore, the length of the steel rod at 100◦C is approximately 2.00192

meters.

Question 29

Question

A metal rod with a length of 2 meters at 20

C is heated to 100

C. If the coefficient

of linear expansion for the metal is 2 ×10−5per degree Celsius, by how many

millimeters will the length of the rod increase?

Solution

Step 1: Calculate the change in temperature. Given initial temperature T1=

20Cand final temperature T2= 100C, the change in temperature ∆T=T2−

T1= 100C−20C= 80C.

Step 2: Calculate the change in length. The change in length (∆L) of the

rod can be calculated using the formula:

∆L=L·α·∆T

where Lis the original length of the rod, αis the coefficient of linear expansion,

and ∆Tis the change in temperature.

Step 3: Substitute the given values into the formula. Given L= 2 meters

and α= 2 ×10−5per

C, and ∆T= 80

∆L= 2 ·2×10−5·80 = 3.2×10−3meters

Step 4: Convert the change in length to millimeters. To convert from meters

to millimeters, we multiply by 1000:

∆L= 3.2×10−3×1000 = 3.2 millimeters

Therefore, the length of the rod will increase by 3.2 millimeters when heated

from 20

C to 100

Question 30

Question

A steel rod with a length of 2 meters at 20◦C is heated to 180◦C. If the coefficient

of linear expansion for steel is 1.2×10−5K−1, find the final length of the steel

rod.

Solution

Step 1: Calculate the change in temperature. Given that the initial temperature

is 20◦C and the final temperature is 180◦C, the change in temperature is:

∆T=Tf−Ti= 180 −20 = 160 K

Step 2: Calculate the expansion of the steel rod. 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 for steel, Lis the initial length of

the steel rod, and ∆Tis the change in temperature.

Plugging in the values:

∆L= (1.2×10−5K−1)·(2 m) ·(160 K) = 0.00384 m

Step 3: Calculate the final length of the steel rod. The final length of the

steel rod can be found by adding the change in length to the initial length:

Lf=Li+ ∆L= 2 m + 0.00384 m = 2.00384 m

Therefore, the final length of the steel rod when heated to 180◦C is 2.00384

meters.

Question 31

Question

A copper rod is 2 meters long at 0◦C. If the coefficient of linear expansion for

copper is 1.7×10−5/◦C, find the length of the rod when the temperature is

100◦C.

Solution

Step 1: Let’s first determine the change in temperature. Given: Initial length of

the rod (L) = 2 m, Coefficient of linear expansion (α)=1.7×10−5/◦C, Change

in temperature (∆T) = 100◦C.

Step 2: To find the change in length (∆L), we can use the formula:

∆L=α×L×∆T

Step 3: Plug in the values to calculate ∆L:

∆L= 1.7×10−5/◦×2 m ×100◦C

Step 4: Calculate ∆L:

∆L= 1.7×10−5×2×100 m

Step 5: Simplify the expression:

∆L= 0.0034 m

Step 6: Finally, we can find the length of the rod at 100◦C by adding the

change in length to the initial length:

L100 =L+ ∆L

Step 7: Substitute the values to find the length of the rod at 100◦C:

L100 = 2 m + 0.0034 m

Step 8: Calculate the length of the rod at 100◦C:

L100 = 2.0034 m

Therefore, the length of the copper rod at 100◦C is 2.0034 meters.

Question 32

Question

A long steel rod is heated from 20◦C to 120◦C. If the original length of the rod is

2 meters and the coefficient of linear thermal expansion of steel is 12×10−6◦C−1,

find the final length of the rod.

Solution

Step 1: Calculate the change in temperature. Given: Initial temperature, Ti=

20◦C Final temperature, Tf= 120◦C Change in temperature, ∆T=Tf−Ti=

120◦C−20◦C = 100◦C

Step 2: Use the formula for linear thermal expansion to find the change in

length. The change in length (∆L) of the rod is given by the formula:

∆L=α·L·∆T

where: αis the coefficient of linear thermal expansion of steel (α= 12 ×

10−6◦C−1)Lis the original length of the rod (2 meters) ∆Tis the change

in temperature we calculated (100◦C)

Substitute the values into the formula:

∆L= 12 ×10−6·2·100 = 0.0024 m

Step 3: Find the final length of the rod. The final length of the rod is:

Lf=L+ ∆L= 2 + 0.0024 = 2.0024 m

Therefore, the final length of the rod after being heated from 20◦C to 120◦C

is 2.0024 meters.

Question 33

Question

A steel rod is 2 meters long at a temperature of 20◦C. If the coefficient of linear

expansion for steel is 1.2×10−5/

C, what will be the length of the rod when

the temperature is raised to 80◦C?

Solution

Step 1: Calculate the change in temperature Given: Initial temperature T1=

20◦C, final temperature T2= 80◦C

∆T=T2−T1= 80◦C−20◦C = 60◦C

Step 2: Calculate the change in length using the formula

∆L=α·L·∆T

where αis the coefficient of linear expansion for steel, Lis the original length

of the rod, and ∆Tis the change in temperature.

∆L= (1.2×10−5/

C) ·2 m ·60

∆L= 0.000072 m

Step 3: Calculate the final length of the rod The final length Lfis given by

Lf=L+ ∆L

Lf= 2 m + 0.000072 m

Lf= 2.000072 m

Therefore, when the temperature is raised to 80◦C, the length of the steel

rod will be 2.000072 meters.

Question 34

Question

A steel rod is initially 2.00 m long at 20

C. If the coefficient of linear expansion

for steel is 1.20×10−5per

C, what is the length of the rod when the temperature

is increased to 200

Solution

Step 1: Calculate the change in temperature. Step 2: Use the change in tem-

perature and the coefficient of linear expansion to find the change in length.

Step 3: Determine the final length of the rod.

Step 1: Calculate the change in temperature. The change in temperature

is given by:

∆T=Tf−Ti= 200C−20C= 180C

Step 2: Use the change in temperature and the coefficient of linear expan-

sion to find the change in length. The change in length is given by:

∆L=α·L·∆T

where: α= 1.20 ×10−5/C (coefficient of linear expansion for steel) L= 2.00 m

(initial length) ∆T= 180C

Substitute the values in:

∆L= (1.20 ×10−5/C)·2.00 m ·180C

∆L= 0.000216 m

Step 3: Determine the final length of the rod. The final length of the rod

is given by:

Lf=Li+ ∆L

Lf= 2.00 m + 0.000216 m

Lf= 2.000216 m

Therefore, when the temperature is increased to 200

C, the length of the

steel rod is 2.000216 m.

Question 35

Question

A steel rod with a length of 2 meters at 20◦C is heated to 120◦C. If the coefficient

of linear expansion of steel is 1.2×10−5/◦C, what is the change in length of the

rod?

Solution

Step 1: Calculate the change in temperature: The change in temperature is

given by:

∆T=Tf−Ti= 120◦C−20◦C = 100◦C

Step 2: Calculate the change in length using the formula for thermal expan-

sion: The change in length (∆L) is given by:

∆L=L0α∆T

Solution

Step 1: Calculate the change in temperature. Step 2: Use the formula for linear

expansion to find the change in length. Step 3: Add the change in length to the

original length to find the new length.

Step 1: Calculate the change in temperature. The change in temperature

is given by:

∆T=Tf−Ti= 100◦C−20◦C = 80◦C

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 αis the coefficient of linear expansion, Lis the original length, and ∆T

is the change in temperature. Substitute the values to find ∆L:

∆L= 12 ×10−6/◦C·2 m ·80◦= 0.00192 m = 1.92 mm

Step 3: Add the change in length to the original length to find the new

length. The new length (Lf) can be found by:

Lf=L+ ∆L= 2 m + 0.00192 m = 2.00192 m = 2.00192 ×103mm

Therefore, when the steel rod is heated to 100◦C, its new length will be

2.00192 meters or 2001.92 millimeters.

Question 3

Question

A solid aluminum rod is initially 2 meters long at a temperature of 20◦C. If the

rod is heated to 90◦C, calculate the final length of the rod. The linear expansion

coefficient of aluminum is 23 ×10−6/◦C.

Solution

Step 1: Calculate the change in temperature. Given that the initial temperature

Ti= 20◦C and the final temperature Tf= 90◦C, the change in temperature is

∆T=Tf−Ti= 90◦C−20◦C = 70◦C.

Step 2: Calculate the change in length using the formula:

∆L=α·L·∆T,

where Lis the initial length, αis the linear expansion coefficient, and ∆Tis the

change in temperature.

Substitute L= 2 m, α= 23 ×10−6/

C, and ∆T= 70

C into the formula:

∆L= 23 ×10−6/C·2 m ·70

Step 3: Calculate the change in length:

∆L= 23 ×10−6×2×70 = 0.00322 m.

Step 4: Calculate the final length of the rod: The final length Lfof the rod

can be calculated by adding the change in length ∆Lto the initial length L:

Lf=L+ ∆L= 2 m + 0.00322 m = 2.00322 m.

Therefore, the final length of the aluminum rod when heated to 90◦C is

2.00322 meters.

Question 4

Question

A 2-meter long steel rod is heated from 20◦C to 120◦C. The rod is fixed at one

end and free to expand at the other end. If the coefficient of linear expansion

for steel is 1.2×10−5◦C−1, find the increase in length of the rod.

Solution

Let’s denote: - Initial length of the rod Li= 2 m - Initial temperature Ti= 20◦C

- Final temperature Tf= 120◦C - Coefficient of linear expansion α= 1.2×10−5

◦C−1

We need to find the increase in length. By definition, the increase in length

is given by ∆L=Lf−Li, where Lfis the final length.

Step 1: Calculate the final length of the rod, Lf.

The change in length ∆Lcan be calculated using the formula:

∆L=α·Li·∆T

where ∆T=Tf−Ti.

Substitute the given values:

∆L= (1.2×10−5◦C−1)×(2 m) ×(120◦C−20◦C)

∆L= 1.2×10−5×2×100 = 0.0024 m = 2.4 mm

Therefore, the increase in length of the rod is 2.4 mm.

Question 5

Question

A uniform steel rod is 2 meters long at 20 degrees Celsius. If the coefficient of

linear expansion for steel is 1.2×10−5/degC, find the increase in length of the

rod when its temperature is raised to 80 degrees Celsius.

Solution

Let L0be the original length of the steel rod, αbe the coefficient of linear

expansion, and ∆Tbe the change in temperature.

Step 1: Calculate the original length of the steel rod at 20 degrees Celsius.

L0= 2 m

Step 2: Calculate the change in temperature.

∆T= 80◦C−20◦C = 60◦C

Step 3: Calculate the increase in length of the steel rod. The increase in

length, ∆L, is given by the formula:

∆L=L0·α·∆T

Substitute L0= 2 m, α= 1.2×10−5/degC, and ∆T= 60 C into the formula:

∆L= 2 ·1.2×10−5·60

∆L= 1.44 ×10−4·60

∆L= 8.64 ×10−3m

So, the increase in length of the steel rod when its temperature is raised to

80 degrees Celsius is 8.64 mm.

Question 6

Question

A copper rod and an aluminum rod are both 1.0 m in length at 20◦C. If the

rods are heated to 120◦C, which rod will have a greater increase in length? The

linear expansion coefficients for copper and aluminum are 16.5×10−6◦C−1and

23 ×10−6◦C−1, respectively.

Solution

Step 1: Calculate the change in length for each rod using the formula ∆L=

L0·α·∆T, where ∆Lis the change in length, L0is the original length, αis the

coefficient of linear expansion, and ∆Tis the change in temperature.

For the copper rod: ∆Lcopper = 1.0 m ·16.5×10−6◦C−1·(120 −20)◦C =

0.0165 m

For the aluminum rod: ∆Laluminum = 1.0 m·23 ×10−6◦C−1·(120−20)◦C =

0.022 m

Step 2: Compare the change in length for both rods. Since ∆Laluminum >

∆Lcopper, the aluminum rod will have a greater increase in length when heated

to 120◦C.

Question 7

Question

A steel rod of length 2.00 m at 20.0

C is heated to a temperature of 80.0

If the linear expansion coefficient of steel is 1.20 ×10−5K−1, what is the final

length of the rod?

Solution

Step 1: First, we calculate the change in temperature: Given: Initial tempera-

ture, T1= 20.0CFinal temperature, T2= 80.0C

The change in temperature, ∆T=T2−T1= 80.0C−20.0C= 60.0C

Step 2: Next, we calculate the change in length using the linear expansion

formula:

∆L=α·L·∆T

where: αis the linear expansion coefficient, 1.20 ×10−5K−1Lis the initial

length of the rod, 2.00 m ∆Tis the change in temperature, 60.0

Substitute the values into the formula:

∆L= (1.20 ×10−5K−1)·(2.00 m) ·(60.0C)

∆L= 0.000024 m ·K−1·m·

C·m

∆L= 0.000024 m2·

Step 3: Finally, we find the final length of the rod:

Lf=Li+ ∆L

Lf= 2.00 m + 0.000024 m2·

Lf= 2.000024 m

Therefore, the final length of the steel rod after heating it to 80.0

C is

2.000024 meters.

Question 8

Question

A solid metal rod is 2 meters long at 20

C. If the coefficient of linear expansion

of the metal is 2 ×10−5per degree Celsius, by how many millimeters will the

rod expand when the temperature is increased to 100

Solution

Step 1: Calculate the change in temperature. Given that the initial temperature

is 20

C and the final temperature is 100

C, the change in temperature is ∆T=

100 −20 = 80

Step 2: Use the formula for linear expansion. The change in length ∆Lof

the rod is given by:

∆L=L0α∆T

where: - L0is the initial length of the rod, - αis the coefficient of linear

expansion, - ∆Tis the change in temperature.

Step 3: Substitute the known values into the formula. Given that L0= 2

meters and α= 2 ×10−5/◦C, we can substitute these values into the formula:

∆L= 2 ×2×10−5×80

Step 4: Calculate the change in length. Now, calculate the change in length:

∆L= 2 ×2×10−5×80 = 3.2×10−3meters

Step 5: Convert the change in length to millimeters. To convert the change

in length to millimeters, we multiply by 1000:

∆L= 3.2×10−3×1000 = 3.2 millimeters

Therefore, the rod will expand by 3.2 millimeters when the temperature is

increased to 100

Question 9

Question

A metal rod has an initial length of 2.00 m at 25

C. If the coefficient of linear

expansion for the metal is 2.5×10−5

C−1, how much will the length of the rod

change when its temperature is raised to 250

Solution

Step 1: First, we can calculate the change in temperature: Given initial tem-

perature, Ti= 25C, and final temperature, Tf= 250C.

∆T=Tf−Ti= 250C−25C= 225C

Step 2: Next, we can use the formula for linear thermal expansion:

∆L=α·L·∆T

where: ∆Lis the change in length, αis the coefficient of linear expansion, Lis

the initial length, and ∆Tis the change in temperature.

Step 3: Substitute the given values into the formula:

∆L= (2.5×10−5

C−1)·(2.00 m) ·(225C)

Step 4: Calculate the change in length:

∆L= 2.25 ×10−4m

Therefore, when the temperature of the metal rod is raised to 250

C, the

length of the rod will increase by 0.000225 m.

Question 10

Question

A metal bar has an original length of 2 meters at 20 degrees Celsius. If the

coefficient of linear expansion for the metal is 2.5×10−5per degree Celsius, by

how many millimeters will the length of the bar increase when it is heated to

100 degrees Celsius?

Solution

Step 1: Calculate the change in temperature. Given that the original tempera-

ture is 20 degrees Celsius and the final temperature is 100 degrees Celsius, the

change in temperature is:

∆T= 100◦C−20◦C= 80◦C

Step 2: Calculate the change in length. The change in length (∆L) of the

metal bar can be calculated 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.

Given that α= 2.5×10−5per degree Celsius, L= 2 meters, and ∆T= 80

degrees Celsius, we can calculate the change in length as follows:

∆L= (2.5×10−5)×2×80 = 0.00004 meters

Step 3: Convert the change in length to millimeters. To convert the change

in length from meters to millimeters, we need to multiply by 1000 (since 1 meter

= 1000 millimeters). Thus,

0.00004 meters ×1000 = 0.04 mm

Therefore, the length of the metal bar will increase by 0.04 millimeters when

heated to 100 degrees Celsius.

Question 11

Question

A steel rod is initially 2 meters long at a temperature of 20◦C. If the coefficient

of linear expansion of steel is 1.2×10−5K−1, by how many millimeters does the

rod lengthen when heated to 120◦C?

Solution

Step 1: Calculate the change in temperature. Given: Initial length of the steel

rod, L0= 2 m, Coefficient of linear expansion of steel, α= 1.2×10−5K−1,

Initial temperature, Ti= 20◦C, Final temperature, Tf= 120◦C.

The change in temperature is given by:

∆T=Tf−Ti= 120◦C−20◦C = 100◦C = 100 K

Step 2: Calculate the change in length. The change in length of the rod can

be calculated using the formula:

∆L=L0α∆T

Substitute the given values into the formula:

∆L= 2 ×1.2×10−5×100

∆L= 2.4×10−3m=2.4 mm

The steel rod lengthens by 2.4 millimeters when heated to 120◦C.

Question 12

Question

A steel rod of length 1 m is heated from 20◦C to 120◦C. If the linear expansion

coefficient of steel is 1.2×10−5/K, calculate the increase in length of the rod.

Solution

Step 1: Calculate the change in temperature Given: Initial temperature, Ti=

20◦C Final temperature, Tf= 120◦C

The change in temperature, ∆T=Tf−Ti= 120 −20 = 100K

Step 2: Calculate the increase in length The formula for linear expansion is

given by:

∆L=αL∆T

where: ∆L= increase in length α= linear expansion coefficient L= initial

length ∆T= change in temperature

Plugging in the values, we get:

∆L= (1.2×10−5)×1×100

∆L= 1.2×10−3m=1.2 mm

Therefore, the increase in length of the steel rod is 1.2 mm.

Question 13

Question

A brass ring has an initial inside diameter of 10 cm at 20

C. If the coefficient

of linear expansion for brass is 2.0×10−5K−1, find the inside diameter of the

ring when its temperature is raised to 120

Solution

Let’s denote the initial inside diameter of the brass ring as d0= 10 cm at 20C,

the final temperature as Tf= 120C, and the coefficient of linear expansion for

brass as α= 2.0×10−5K−1.

Step 1: Calculate the change in temperature. We can find the change in

temperature by subtracting the initial temperature from the final temperature:

∆T=Tf−T0= 120C−20C= 100C

Step 2: Calculate the change in diameter. We can use the formula for linear

expansion:

∆L=α·L0·∆T

where ∆Lis the change in length, L0is the initial length (in this case, the initial

inside diameter), and ∆Tis the change in temperature.

Substitute the values:

∆L= (2.0×10−5K−1)·(10 cm) ·(100C)=0.02 cm

Step 3: Determine the final inside diameter. The final inside diameter is

given by:

df=d0+ ∆L= 10 cm + 0.02 cm = 10.02 cm

Therefore, when the temperature is raised to 120C, the inside diameter of

the brass ring will be 10.02 cm.

Question 14

Question

A steel beam is 10 meters long at 20

C. If the temperature changes to 80

how much longer will the beam become? The linear expansion coefficient for

steel is 12 ×10−6per Celsius degree.

Solution

Step 1: Calculate the change in temperature. Step 2: Use the linear expansion

formula to calculate the change in length.

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: Use the linear expansion formula to calculate the change in length.

The change in length (∆L) is given by:

∆L=α·L·∆T

where αis the linear expansion coefficient for steel, Lis the original length of

the beam, and ∆Tis the change in temperature.

Substitute the values into the formula:

∆L= 12 ×10−6·(10 m) ·(60)

∆L= 0.0072 m

Therefore, the beam will become 0.0072 meters longer when the temperature

changes from 20

C to 80

Question 15

Question

A steel rod has an initial length of 2 meters at 20 degrees Celsius. If the

coefficient of linear expansion for steel is 11 ×10−6◦C−1, find the length of the

rod when the temperature is increased to 100 degrees Celsius.

Solution

Step 1: Calculate the change in temperature. Let Lbe the final length of the

rod, L0be the initial length of the rod, ∆Tbe the change in temperature,

and αbe the coefficient of linear expansion. We can use the formula for linear

expansion: ∆L=α·L0·∆T. Given that L0= 2 m, α= 11 ×10−6◦C−1, and

the initial temperature is 20 degrees Celsius, and the final temperature is 100

degrees Celsius, we have:

∆T= 100 −20 = 80 ◦C

Step 2: Calculate the change in length. Substitute the known values into

the formula:

∆L= 11 ×10−6·2·80

∆L= 0.00176 m

Step 3: Calculate the final length of the rod. The final length Lis given by:

L=L0+ ∆L

. Substitute L0= 2 m and ∆L= 0.00176 m:

L= 2 + 0.00176 = 2.00176 m

Therefore, when the temperature is increased to 100 degrees Celsius, the

length of the rod will be 2.00176 meters.

Question 16

Question

A steel rod with a length of 2.5 meters at 20◦C is heated to 200◦C. If the linear

expansion coefficient of steel is 11 ×10−6/◦C, what is the final 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 200◦C, the change in temperature is:

(200 −20) = 180

Step 2: Use the linear expansion formula to find the change in length. The

change in length (∆L) of an object due to thermal expansion is given by:

∆L=α·L0·∆T

where: α= linear expansion coefficient (11 ×10−6/◦C) L0= initial length of

the rod (2.5 meters) ∆T= change in temperature (180

Substitute the given values into the formula:

∆L= (11 ×10−6/◦C) ·(2.5 m) ·(180

Step 3: Calculate the change in length.

∆L= 0.000385 m

Step 4: Determine 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=L0+ ∆L

Lf= 2.5 m + 0.000385 m

Step 5: Calculate the final length of the rod.

Lf= 2.500385 m

Therefore, the final length of the steel rod when heated to 200◦C is 2.500385

meters.

Question 17

Question

A steel rod is 2 meters long at 20◦C. If the coefficient of linear expansion for

steel is 1.2×10−5◦C−1, what will be 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 20◦C and the final temperature is 100◦C, the change in temperature is:

∆T=Tf−Ti= 100◦C−20◦C = 80◦C

Step 2: Calculate the change in length. The change in length (∆L) of the

steel rod can be calculated using the formula:

∆L=L·α·∆T

where Lis the initial length, αis the coefficient of linear expansion, and ∆T

is the change in temperature. Substitute L= 2 m, α= 1.2×10−5◦C−1, and

∆T= 80 ◦C into the formula:

∆L= 2 m ·1.2×10−5◦C−1·80 ◦C

Step 3: Calculate the final length of the rod. The final length (Lf) of the

rod can be calculated by adding the change in length to the initial length:

Lf=L+ ∆L

Substitute L= 2 m and the calculated ∆Linto the formula:

Lf= 2 m + (2 m ·1.2×10−5◦C−1·80 ◦C)

Question 18

Question

A steel rod is initially 2 meters long at 20

C. If the coefficient of linear expansion

for steel is 1.2×10−5per degree Celsius, what is the length of the rod when the

temperature reaches 100

Solution

Step 1: Calculate the change in temperature.

Given: Initial temperature, T1= 20

Final temperature, T2= 100

Change in temperature, ∆T=T2−T1= 100 −20 = 80

Step 2: Calculate the change in length using the formula for linear expansion:

∆L=αL∆T

where: ∆L= change in length

α= coefficient of linear expansion

L= initial length of the rod

∆T= change in temperature

Plugging in the values, we have:

∆L= (1.2×10−5)×2×80

Calculating,

∆L= 0.000024 ×160 = 0.00384 meters

Step 3: Find the final length of the rod. The final length, Lf, can be found

by adding the change in length to the initial length:

Lf=L+ ∆L= 2 + 0.00384 = 2.00384 meters

Therefore, the length of the rod when the temperature reaches 100

C is

2.00384 meters.

Question 19

Question

A steel rod is tightly clamped between two walls at 0◦C. If the walls are heated

to 100◦C, by how much will the rod elongate? (Given: Coefficient of linear

thermal expansion of steel = 1.2×10−5/◦Cand original length of the rod = 2

meters)

Solution

Step 1: First, we calculate the change in temperature:

∆T=Tf−Ti= 100◦C−0◦C= 100◦C

Step 2: Next, we calculate the change in length using the formula for linear

thermal expansion:

∆L=α·L·∆T

where: ∆L= change in length, α= coefficient of linear thermal expansion, L

= original length, ∆T= change in temperature.

Substitute the given values:

∆L= (1.2×10−5/◦C)·2 meters ·100◦C

Step 3: Calculate the change in length:

∆L= 0.000024 ·200 = 0.0048 meters

Step 4: Therefore, the steel rod will elongate by 0.0048 meters when the

walls are heated to 100◦C.

Question 20

Question

A steel cylinder of height 2.0 m and radius 1.0 m is initially at a temperature

of 20

C. If the temperature of the cylinder is increased to 100

C, calculate the

change in height of the cylinder. The linear expansion coefficient of steel is

1.2×10−5per

Solution

Let’s denote the initial height of the cylinder as H0and the initial radius as

R0. The change in height of the cylinder can be calculated using the formula

for linear expansion:

∆H=H0·α·∆T

where ∆H= change in height, H0= initial height of the cylinder, α= linear

expansion coefficient of steel, ∆T= change in temperature.

Given: H0= 2.0 m, α= 1.2×10−5per

C, ∆T= 100C−20C= 80C.

Step 1: Calculate the change in height

∆H= 2.0 m ·1.2×10−5·80

Step 2: Solve for ∆H

∆H= 2.0×1.2×10−5×80 = 0.00192 m

Therefore, the change in height of the cylinder is 0.00192 m.

Question 21

Question

A steel rod has a length of 1 meter at 20

C. If the coefficient of linear expansion

of steel is 1.2×10−5per degree Celsius, determine the temperature at which

the rod will have a length of 1.01 meters.

Solution

Step 1: Let’s denote the initial length of the steel rod as L0, the final length as

L, the initial temperature as T0, and the final temperature as T. We are given

that L0= 1 meter, L= 1.01 meters, and the coefficient of linear expansion

α= 1.2×10−5per degree Celsius.

Step 2: The formula for linear expansion is given by ∆L=α·L0·∆T, where

∆L=L−L0and ∆T=T−T0.

Step 3: Substituting the given values into the formula, we get ∆L= (1.2×

10−5)·1·∆T.

Step 4: Solving for ∆T, we have ∆T=∆L

1.2×10−5.

Step 5: Plugging in the values for ∆Land solving for ∆T, we get ∆T=

1.01−1

1.2×10−5.

Step 6: Calculating the value of ∆T, we have ∆T=0.01

1.2×10−5.

Step 7: Therefore, ∆T= 833.33 degrees Celsius.

Step 8: To find the final temperature T, we use the formula T=T0+ ∆T.

Step 9: Substituting the values of T0and ∆T, we get T= 20 + 833.33.

Step 10: Therefore, the temperature at which the steel rod will have a length

of 1.01 meters is 853.33 degrees Celsius.

Question 22

Question

A brass rod is 2 meters long at 20

C. If the coefficient of linear expansion of

brass is 19×10−6per

C, by how much does the length of the rod increase when

the temperature is increased to 90

Solution

Given: Initial length of brass rod, L0= 2 m

Coefficient of linear expansion, α= 19 ×10−6per

Change in temperature, ∆T= 90 −20 = 70

We can find the change in length, ∆L, using the formula:

∆L=L0α∆T

Step 1: Calculate the change in length, ∆L.

∆L= 2 ×19 ×10−6×70

= 2 ×19 ×10−6×70

= 2 ×1.33 ×10−4

= 2.66 ×10−4m

So, the length of the brass rod increases by 2.66 ×10−4meters when the

temperature is increased to 90

Question 23

Question

A brass rod is 2 meters long at 20

C. If the coefficient of linear expansion of

brass is 2.0×10−5per degree Celsius, what temperature increase is needed to

increase the length of the rod by 1 cm?

Solution

Step 1: Let’s first calculate the increase in length of the rod when the temper-

ature increases by ∆Tdegrees Celsius. The increase in length (∆L) is given by

the formula:

∆L=α·L·∆T,

where αis the coefficient of linear expansion, Lis the original length of the rod,

and ∆Tis the temperature change. Substitute the given values:

∆L= (2.0×10−5)·2·∆T.

Step 2: We are given that the increase in length is 1 cm, which is 0.01 meters.

Thus, we have:

0.01 = (2.0×10−5)·2·∆T.

Step 3: Solve for ∆T:

∆T=0.01

(2.0×10−5)·2.

Step 4: Calculating the value of ∆T:

∆T=0.01

4.0×10−5= 250 degrees Celsius.

Therefore, a temperature increase of 250

C is needed to increase the length

of the rod by 1 cm.

Question 24

Question

A steel rod of length 2.0 m is heated from 20

C to 80

C. If the coefficient of

linear expansion of steel is 1.2×10−5/

C, what is the change in length of the

rod?

Solution

Step 1: Calculate the initial length of the rod when it is at 20

C. Given: Initial

length, L0= 2.0 m Coefficient of linear expansion, α= 1.2×10−5/C Initial

temperature, Ti= 20C

Using the formula for linear expansion:

∆L=α·L0·∆T

where ∆Lis the change in length, αis the coefficient of linear expansion, L0is

the initial length, and ∆Tis the change in temperature.

Substitute the values:

∆L= (1.2×10−5/C)·2.0 m ·(80C−20C)

Step 2: Calculate the change in length of the rod.

∆L= 1.2×10−5·2·60

∆L= 1.44 ×10−4m

∆L= 0.144 mm

Therefore, the change in length of the steel rod is 0.144 mm.

Question 25

Question

A brass rod is 2.0 m long at 20

C. If the rod is heated to 100

C, what will be

its length? The linear expansion for brass is 19 ×10−6/

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=Tf−Ti= 100C−20C= 80C

Step 2: Calculate the linear expansion coefficient. The linear expansion

coefficient for brass is 19 ×10−6/

Step 3: Use the formula for linear expansion to find the change in length.

The change in length (∆L) of the brass rod can be calculated using the formula:

∆L=α·L0·∆T

where: - ∆Lis the change in length, - αis the linear expansion coefficient of

brass (19 ×10−6/

C), - L0is the initial length of the brass rod (2.0 m), - ∆T

is the change in temperature (80

C).

Substitute the values into the formula:

∆L= (19 ×10−6/

C) ·(2.0 m) ·(80C)

Step 4: Calculate the change in length.

∆L= 0.000038 m = 3.8×10−5m

Therefore, the length of the brass rod when heated to 100

C will be:

Lfinal =L0+ ∆L= 2.0 m + 3.8×10−5m=2.000038 m

So, the final length of the brass rod when heated to 100

C will be 2.000038

meters.

Question 26

Question

A copper rod is initially at a length of 1 m. When heated from 20◦C to 120◦C,

the rod expands by 2 mm. Calculate the coefficient of linear expansion of copper.

Solution

Step 1: Identify the relevant formula for linear expansion. The linear expansion

of a material can be calculated using the formula:

∆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.

Step 2: Given values The original length L0is 1 m, the change in length ∆L

is 2 mm (which is 0.002 m), the change in temperature ∆Tis (120◦C - 20◦C)

= 100◦C.

Step 3: Substitute the values into the formula Substitute L0= 1 m, ∆L=

0.002 m, and ∆T= 100 ◦C into the formula:

0.002 = 1 ×α×100

Step 4: Solve for the coefficient of linear expansion α

α=0.002

100 = 2 ×10−5per degree Celsius

Therefore, the coefficient of linear expansion of copper is 2×10−5per degree

Celsius.

Question 27

Question

A steel rod with an original length of 2 meters experiences a temperature in-

crease of 50

C. If the coefficient of linear expansion for steel is 12 ×10−6/

by how many millimeters does the length of the rod increase?

Solution

Step 1: First we calculate the change in length using the formula given by the

linear expansion equation:

∆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.

Step 2: Substituting the given values, we get:

∆L= 12 ×10−6/

C·2 m ·50

Step 3: Calculating ∆L:

∆L= 12 ×10−6·2·50 = 1.2×10−3m

Step 4: Converting the change in length to millimeters:

∆Lmm = 1.2×10−3m×1000 = 1.2 mm

Therefore, the length of the steel rod increases by 1.2 millimeters.

Question 28

Question

A steel rod has a length of 2 meters at 20◦C. If the coefficient of linear expansion

for steel is 12 ×10−6/

C, what is the length of the rod at 100◦C?

Solution

Step 1: Calculate the change in temperature. Given: Initial temperature, T1=

20◦C Final temperature, T2= 100◦C Change in temperature, ∆T=T2−T1=

100◦C - 20◦C = 80◦C

Step 2: Use the formula for linear expansion. The change in length of the

rod can be calculated using the formula: ∆L=α·L·∆Twhere ∆L= change

in length α= coefficient of linear expansion L= initial length of the rod

Step 3: Substitute the values into the formula. Given: α= 12 ×10−6/

L= 2 meters ∆T= 80◦C

Plugging in the values, we get: ∆L= (12 ×10−6/C)·2 m ·80

Step 4: Calculate the change in length. ∆L= 12 ×10−6·2·80 = 1.92 ×10−3

meters

Step 5: Calculate the final length of the rod. The final length of the rod is

given by: L2=L1+ ∆L L2= 2 m + 1.92 ×10−3mL2= 2.00192 meters

Therefore, the length of the steel rod at 100◦C is approximately 2.00192

meters.

Question 29

Question

A metal rod with a length of 2 meters at 20

C is heated to 100

C. If the coefficient

of linear expansion for the metal is 2 ×10−5per degree Celsius, by how many

millimeters will the length of the rod increase?

Solution

Step 1: Calculate the change in temperature. Given initial temperature T1=

20Cand final temperature T2= 100C, the change in temperature ∆T=T2−

T1= 100C−20C= 80C.

Step 2: Calculate the change in length. The change in length (∆L) of the

rod can be calculated using the formula:

∆L=L·α·∆T

where Lis the original length of the rod, αis the coefficient of linear expansion,

and ∆Tis the change in temperature.

Step 3: Substitute the given values into the formula. Given L= 2 meters

and α= 2 ×10−5per

C, and ∆T= 80

∆L= 2 ·2×10−5·80 = 3.2×10−3meters

Step 4: Convert the change in length to millimeters. To convert from meters

to millimeters, we multiply by 1000:

∆L= 3.2×10−3×1000 = 3.2 millimeters

Therefore, the length of the rod will increase by 3.2 millimeters when heated

from 20

C to 100

Question 30

Question

A steel rod with a length of 2 meters at 20◦C is heated to 180◦C. If the coefficient

of linear expansion for steel is 1.2×10−5K−1, find the final length of the steel

rod.

Solution

Step 1: Calculate the change in temperature. Given that the initial temperature

is 20◦C and the final temperature is 180◦C, the change in temperature is:

∆T=Tf−Ti= 180 −20 = 160 K

Step 2: Calculate the expansion of the steel rod. 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 for steel, Lis the initial length of

the steel rod, and ∆Tis the change in temperature.

Plugging in the values:

∆L= (1.2×10−5K−1)·(2 m) ·(160 K) = 0.00384 m

Step 3: Calculate the final length of the steel rod. The final length of the

steel rod can be found by adding the change in length to the initial length:

Lf=Li+ ∆L= 2 m + 0.00384 m = 2.00384 m

Therefore, the final length of the steel rod when heated to 180◦C is 2.00384

meters.

Question 31

Question

A copper rod is 2 meters long at 0◦C. If the coefficient of linear expansion for

copper is 1.7×10−5/◦C, find the length of the rod when the temperature is

100◦C.

Solution

Step 1: Let’s first determine the change in temperature. Given: Initial length of

the rod (L) = 2 m, Coefficient of linear expansion (α)=1.7×10−5/◦C, Change

in temperature (∆T) = 100◦C.

Step 2: To find the change in length (∆L), we can use the formula:

∆L=α×L×∆T

Step 3: Plug in the values to calculate ∆L:

∆L= 1.7×10−5/◦×2 m ×100◦C

Step 4: Calculate ∆L:

∆L= 1.7×10−5×2×100 m

Step 5: Simplify the expression:

∆L= 0.0034 m

Step 6: Finally, we can find the length of the rod at 100◦C by adding the

change in length to the initial length:

L100 =L+ ∆L

Step 7: Substitute the values to find the length of the rod at 100◦C:

L100 = 2 m + 0.0034 m

Step 8: Calculate the length of the rod at 100◦C:

L100 = 2.0034 m

Therefore, the length of the copper rod at 100◦C is 2.0034 meters.

Question 32

Question

A long steel rod is heated from 20◦C to 120◦C. If the original length of the rod is

2 meters and the coefficient of linear thermal expansion of steel is 12×10−6◦C−1,

find the final length of the rod.

Solution

Step 1: Calculate the change in temperature. Given: Initial temperature, Ti=

20◦C Final temperature, Tf= 120◦C Change in temperature, ∆T=Tf−Ti=

120◦C−20◦C = 100◦C

Step 2: Use the formula for linear thermal expansion to find the change in

length. The change in length (∆L) of the rod is given by the formula:

∆L=α·L·∆T

where: αis the coefficient of linear thermal expansion of steel (α= 12 ×

10−6◦C−1)Lis the original length of the rod (2 meters) ∆Tis the change

in temperature we calculated (100◦C)

Substitute the values into the formula:

∆L= 12 ×10−6·2·100 = 0.0024 m

Step 3: Find the final length of the rod. The final length of the rod is:

Lf=L+ ∆L= 2 + 0.0024 = 2.0024 m

Therefore, the final length of the rod after being heated from 20◦C to 120◦C

is 2.0024 meters.

Question 33

Question

A steel rod is 2 meters long at a temperature of 20◦C. If the coefficient of linear

expansion for steel is 1.2×10−5/

C, what will be the length of the rod when

the temperature is raised to 80◦C?

Solution

Step 1: Calculate the change in temperature Given: Initial temperature T1=

20◦C, final temperature T2= 80◦C

∆T=T2−T1= 80◦C−20◦C = 60◦C

Step 2: Calculate the change in length using the formula

∆L=α·L·∆T

where αis the coefficient of linear expansion for steel, Lis the original length

of the rod, and ∆Tis the change in temperature.

∆L= (1.2×10−5/

C) ·2 m ·60

∆L= 0.000072 m

Step 3: Calculate the final length of the rod The final length Lfis given by

Lf=L+ ∆L

Lf= 2 m + 0.000072 m

Lf= 2.000072 m

Therefore, when the temperature is raised to 80◦C, the length of the steel

rod will be 2.000072 meters.

Question 34

Question

A steel rod is initially 2.00 m long at 20

C. If the coefficient of linear expansion

for steel is 1.20×10−5per

C, what is the length of the rod when the temperature

is increased to 200

Solution

Step 1: Calculate the change in temperature. Step 2: Use the change in tem-

perature and the coefficient of linear expansion to find the change in length.

Step 3: Determine the final length of the rod.

Step 1: Calculate the change in temperature. The change in temperature

is given by:

∆T=Tf−Ti= 200C−20C= 180C

Step 2: Use the change in temperature and the coefficient of linear expan-

sion to find the change in length. The change in length is given by:

∆L=α·L·∆T

where: α= 1.20 ×10−5/C (coefficient of linear expansion for steel) L= 2.00 m

(initial length) ∆T= 180C

Substitute the values in:

∆L= (1.20 ×10−5/C)·2.00 m ·180C

∆L= 0.000216 m

Step 3: Determine the final length of the rod. The final length of the rod

is given by:

Lf=Li+ ∆L

Lf= 2.00 m + 0.000216 m

Lf= 2.000216 m

Therefore, when the temperature is increased to 200

C, the length of the

steel rod is 2.000216 m.

Question 35

Question

A steel rod with a length of 2 meters at 20◦C is heated to 120◦C. If the coefficient

of linear expansion of steel is 1.2×10−5/◦C, what is the change in length of the

rod?

Solution

Step 1: Calculate the change in temperature: The change in temperature is

given by:

∆T=Tf−Ti= 120◦C−20◦C = 100◦C

Step 2: Calculate the change in length using the formula for thermal expan-

sion: The change in length (∆L) is given by:

∆L=L0α∆T

Solution

Step 1: Calculate the change in temperature. Step 2: Use the formula for linear

expansion to find the change in length. Step 3: Add the change in length to the

original length to find the new length.

Step 1: Calculate the change in temperature. The change in temperature

is given by:

∆T=Tf−Ti= 100◦C−20◦C = 80◦C

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 αis the coefficient of linear expansion, Lis the original length, and ∆T

is the change in temperature. Substitute the values to find ∆L:

∆L= 12 ×10−6/◦C·2 m ·80◦= 0.00192 m = 1.92 mm

Step 3: Add the change in length to the original length to find the new

length. The new length (Lf) can be found by:

Lf=L+ ∆L= 2 m + 0.00192 m = 2.00192 m = 2.00192 ×103mm

Therefore, when the steel rod is heated to 100◦C, its new length will be

2.00192 meters or 2001.92 millimeters.

Question 3

Question

A solid aluminum rod is initially 2 meters long at a temperature of 20◦C. If the

rod is heated to 90◦C, calculate the final length of the rod. The linear expansion

coefficient of aluminum is 23 ×10−6/◦C.

Solution

Step 1: Calculate the change in temperature. Given that the initial temperature

Ti= 20◦C and the final temperature Tf= 90◦C, the change in temperature is

∆T=Tf−Ti= 90◦C−20◦C = 70◦C.

Step 2: Calculate the change in length using the formula:

∆L=α·L·∆T,

where Lis the initial length, αis the linear expansion coefficient, and ∆Tis the

change in temperature.

Substitute L= 2 m, α= 23 ×10−6/

C, and ∆T= 70

C into the formula:

∆L= 23 ×10−6/C·2 m ·70

Step 3: Calculate the change in length:

∆L= 23 ×10−6×2×70 = 0.00322 m.

Step 4: Calculate the final length of the rod: The final length Lfof the rod

can be calculated by adding the change in length ∆Lto the initial length L:

Lf=L+ ∆L= 2 m + 0.00322 m = 2.00322 m.

Therefore, the final length of the aluminum rod when heated to 90◦C is

2.00322 meters.

Question 4

Question

A 2-meter long steel rod is heated from 20◦C to 120◦C. The rod is fixed at one

end and free to expand at the other end. If the coefficient of linear expansion

for steel is 1.2×10−5◦C−1, find the increase in length of the rod.

Solution

Let’s denote: - Initial length of the rod Li= 2 m - Initial temperature Ti= 20◦C

- Final temperature Tf= 120◦C - Coefficient of linear expansion α= 1.2×10−5

◦C−1

We need to find the increase in length. By definition, the increase in length

is given by ∆L=Lf−Li, where Lfis the final length.

Step 1: Calculate the final length of the rod, Lf.

The change in length ∆Lcan be calculated using the formula:

∆L=α·Li·∆T

where ∆T=Tf−Ti.

Substitute the given values:

∆L= (1.2×10−5◦C−1)×(2 m) ×(120◦C−20◦C)

∆L= 1.2×10−5×2×100 = 0.0024 m = 2.4 mm

Therefore, the increase in length of the rod is 2.4 mm.

Question 5

Question

A uniform steel rod is 2 meters long at 20 degrees Celsius. If the coefficient of

linear expansion for steel is 1.2×10−5/degC, find the increase in length of the

rod when its temperature is raised to 80 degrees Celsius.

Solution

Let L0be the original length of the steel rod, αbe the coefficient of linear

expansion, and ∆Tbe the change in temperature.

Step 1: Calculate the original length of the steel rod at 20 degrees Celsius.

L0= 2 m

Step 2: Calculate the change in temperature.

∆T= 80◦C−20◦C = 60◦C

Step 3: Calculate the increase in length of the steel rod. The increase in

length, ∆L, is given by the formula:

∆L=L0·α·∆T

Substitute L0= 2 m, α= 1.2×10−5/degC, and ∆T= 60 C into the formula:

∆L= 2 ·1.2×10−5·60

∆L= 1.44 ×10−4·60

∆L= 8.64 ×10−3m

So, the increase in length of the steel rod when its temperature is raised to

80 degrees Celsius is 8.64 mm.

Question 6

Question

A copper rod and an aluminum rod are both 1.0 m in length at 20◦C. If the

rods are heated to 120◦C, which rod will have a greater increase in length? The

linear expansion coefficients for copper and aluminum are 16.5×10−6◦C−1and

23 ×10−6◦C−1, respectively.

Solution

Step 1: Calculate the change in length for each rod using the formula ∆L=

L0·α·∆T, where ∆Lis the change in length, L0is the original length, αis the

coefficient of linear expansion, and ∆Tis the change in temperature.

For the copper rod: ∆Lcopper = 1.0 m ·16.5×10−6◦C−1·(120 −20)◦C =

0.0165 m

For the aluminum rod: ∆Laluminum = 1.0 m·23 ×10−6◦C−1·(120−20)◦C =

0.022 m

Step 2: Compare the change in length for both rods. Since ∆Laluminum >

∆Lcopper, the aluminum rod will have a greater increase in length when heated

to 120◦C.

Question 7

Question

A steel rod of length 2.00 m at 20.0

C is heated to a temperature of 80.0

If the linear expansion coefficient of steel is 1.20 ×10−5K−1, what is the final

length of the rod?

Solution

Step 1: First, we calculate the change in temperature: Given: Initial tempera-

ture, T1= 20.0CFinal temperature, T2= 80.0C

The change in temperature, ∆T=T2−T1= 80.0C−20.0C= 60.0C

Step 2: Next, we calculate the change in length using the linear expansion

formula:

∆L=α·L·∆T

where: αis the linear expansion coefficient, 1.20 ×10−5K−1Lis the initial

length of the rod, 2.00 m ∆Tis the change in temperature, 60.0

Substitute the values into the formula:

∆L= (1.20 ×10−5K−1)·(2.00 m) ·(60.0C)

∆L= 0.000024 m ·K−1·m·

C·m

∆L= 0.000024 m2·

Step 3: Finally, we find the final length of the rod:

Lf=Li+ ∆L

Lf= 2.00 m + 0.000024 m2·

Lf= 2.000024 m

Therefore, the final length of the steel rod after heating it to 80.0

C is

2.000024 meters.

Question 8

Question

A solid metal rod is 2 meters long at 20

C. If the coefficient of linear expansion

of the metal is 2 ×10−5per degree Celsius, by how many millimeters will the

rod expand when the temperature is increased to 100

Solution

Step 1: Calculate the change in temperature. Given that the initial temperature

is 20

C and the final temperature is 100

C, the change in temperature is ∆T=

100 −20 = 80

Step 2: Use the formula for linear expansion. The change in length ∆Lof

the rod is given by:

∆L=L0α∆T

where: - L0is the initial length of the rod, - αis the coefficient of linear

expansion, - ∆Tis the change in temperature.

Step 3: Substitute the known values into the formula. Given that L0= 2

meters and α= 2 ×10−5/◦C, we can substitute these values into the formula:

∆L= 2 ×2×10−5×80

Step 4: Calculate the change in length. Now, calculate the change in length:

∆L= 2 ×2×10−5×80 = 3.2×10−3meters

Step 5: Convert the change in length to millimeters. To convert the change

in length to millimeters, we multiply by 1000:

∆L= 3.2×10−3×1000 = 3.2 millimeters

Therefore, the rod will expand by 3.2 millimeters when the temperature is

increased to 100

Question 9

Question

A metal rod has an initial length of 2.00 m at 25

C. If the coefficient of linear

expansion for the metal is 2.5×10−5

C−1, how much will the length of the rod

change when its temperature is raised to 250

Solution

Step 1: First, we can calculate the change in temperature: Given initial tem-

perature, Ti= 25C, and final temperature, Tf= 250C.

∆T=Tf−Ti= 250C−25C= 225C

Step 2: Next, we can use the formula for linear thermal expansion:

∆L=α·L·∆T

where: ∆Lis the change in length, αis the coefficient of linear expansion, Lis

the initial length, and ∆Tis the change in temperature.

Step 3: Substitute the given values into the formula:

∆L= (2.5×10−5

C−1)·(2.00 m) ·(225C)

Step 4: Calculate the change in length:

∆L= 2.25 ×10−4m

Therefore, when the temperature of the metal rod is raised to 250

C, the

length of the rod will increase by 0.000225 m.

Question 10

Question

A metal bar has an original length of 2 meters at 20 degrees Celsius. If the

coefficient of linear expansion for the metal is 2.5×10−5per degree Celsius, by

how many millimeters will the length of the bar increase when it is heated to

100 degrees Celsius?

Solution

Step 1: Calculate the change in temperature. Given that the original tempera-

ture is 20 degrees Celsius and the final temperature is 100 degrees Celsius, the

change in temperature is:

∆T= 100◦C−20◦C= 80◦C

Step 2: Calculate the change in length. The change in length (∆L) of the

metal bar can be calculated 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.

Given that α= 2.5×10−5per degree Celsius, L= 2 meters, and ∆T= 80

degrees Celsius, we can calculate the change in length as follows:

∆L= (2.5×10−5)×2×80 = 0.00004 meters

Step 3: Convert the change in length to millimeters. To convert the change

in length from meters to millimeters, we need to multiply by 1000 (since 1 meter

= 1000 millimeters). Thus,

0.00004 meters ×1000 = 0.04 mm

Therefore, the length of the metal bar will increase by 0.04 millimeters when

heated to 100 degrees Celsius.

Question 11

Question

A steel rod is initially 2 meters long at a temperature of 20◦C. If the coefficient

of linear expansion of steel is 1.2×10−5K−1, by how many millimeters does the

rod lengthen when heated to 120◦C?

Solution

Step 1: Calculate the change in temperature. Given: Initial length of the steel

rod, L0= 2 m, Coefficient of linear expansion of steel, α= 1.2×10−5K−1,

Initial temperature, Ti= 20◦C, Final temperature, Tf= 120◦C.

The change in temperature is given by:

∆T=Tf−Ti= 120◦C−20◦C = 100◦C = 100 K

Step 2: Calculate the change in length. The change in length of the rod can

be calculated using the formula:

∆L=L0α∆T

Substitute the given values into the formula:

∆L= 2 ×1.2×10−5×100

∆L= 2.4×10−3m=2.4 mm

The steel rod lengthens by 2.4 millimeters when heated to 120◦C.

Question 12

Question

A steel rod of length 1 m is heated from 20◦C to 120◦C. If the linear expansion

coefficient of steel is 1.2×10−5/K, calculate the increase in length of the rod.

Solution

Step 1: Calculate the change in temperature Given: Initial temperature, Ti=

20◦C Final temperature, Tf= 120◦C

The change in temperature, ∆T=Tf−Ti= 120 −20 = 100K

Step 2: Calculate the increase in length The formula for linear expansion is

given by:

∆L=αL∆T

where: ∆L= increase in length α= linear expansion coefficient L= initial

length ∆T= change in temperature

Plugging in the values, we get:

∆L= (1.2×10−5)×1×100

∆L= 1.2×10−3m=1.2 mm

Therefore, the increase in length of the steel rod is 1.2 mm.

Question 13

Question

A brass ring has an initial inside diameter of 10 cm at 20

C. If the coefficient

of linear expansion for brass is 2.0×10−5K−1, find the inside diameter of the

ring when its temperature is raised to 120

Solution

Let’s denote the initial inside diameter of the brass ring as d0= 10 cm at 20C,

the final temperature as Tf= 120C, and the coefficient of linear expansion for

brass as α= 2.0×10−5K−1.

Step 1: Calculate the change in temperature. We can find the change in

temperature by subtracting the initial temperature from the final temperature:

∆T=Tf−T0= 120C−20C= 100C

Step 2: Calculate the change in diameter. We can use the formula for linear

expansion:

∆L=α·L0·∆T

where ∆Lis the change in length, L0is the initial length (in this case, the initial

inside diameter), and ∆Tis the change in temperature.

Substitute the values:

∆L= (2.0×10−5K−1)·(10 cm) ·(100C)=0.02 cm

Step 3: Determine the final inside diameter. The final inside diameter is

given by:

df=d0+ ∆L= 10 cm + 0.02 cm = 10.02 cm

Therefore, when the temperature is raised to 120C, the inside diameter of

the brass ring will be 10.02 cm.

Question 14

Question

A steel beam is 10 meters long at 20

C. If the temperature changes to 80

how much longer will the beam become? The linear expansion coefficient for

steel is 12 ×10−6per Celsius degree.

Solution

Step 1: Calculate the change in temperature. Step 2: Use the linear expansion

formula to calculate the change in length.

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: Use the linear expansion formula to calculate the change in length.

The change in length (∆L) is given by:

∆L=α·L·∆T

where αis the linear expansion coefficient for steel, Lis the original length of

the beam, and ∆Tis the change in temperature.

Substitute the values into the formula:

∆L= 12 ×10−6·(10 m) ·(60)

∆L= 0.0072 m

Therefore, the beam will become 0.0072 meters longer when the temperature

changes from 20

C to 80

Question 15

Question

A steel rod has an initial length of 2 meters at 20 degrees Celsius. If the

coefficient of linear expansion for steel is 11 ×10−6◦C−1, find the length of the

rod when the temperature is increased to 100 degrees Celsius.

Solution

Step 1: Calculate the change in temperature. Let Lbe the final length of the

rod, L0be the initial length of the rod, ∆Tbe the change in temperature,

and αbe the coefficient of linear expansion. We can use the formula for linear

expansion: ∆L=α·L0·∆T. Given that L0= 2 m, α= 11 ×10−6◦C−1, and

the initial temperature is 20 degrees Celsius, and the final temperature is 100

degrees Celsius, we have:

∆T= 100 −20 = 80 ◦C

Step 2: Calculate the change in length. Substitute the known values into

the formula:

∆L= 11 ×10−6·2·80

∆L= 0.00176 m

Step 3: Calculate the final length of the rod. The final length Lis given by:

L=L0+ ∆L

. Substitute L0= 2 m and ∆L= 0.00176 m:

L= 2 + 0.00176 = 2.00176 m

Therefore, when the temperature is increased to 100 degrees Celsius, the

length of the rod will be 2.00176 meters.

Question 16

Question

A steel rod with a length of 2.5 meters at 20◦C is heated to 200◦C. If the linear

expansion coefficient of steel is 11 ×10−6/◦C, what is the final 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 200◦C, the change in temperature is:

(200 −20) = 180

Step 2: Use the linear expansion formula to find the change in length. The

change in length (∆L) of an object due to thermal expansion is given by:

∆L=α·L0·∆T

where: α= linear expansion coefficient (11 ×10−6/◦C) L0= initial length of

the rod (2.5 meters) ∆T= change in temperature (180

Substitute the given values into the formula:

∆L= (11 ×10−6/◦C) ·(2.5 m) ·(180

Step 3: Calculate the change in length.

∆L= 0.000385 m

Step 4: Determine 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=L0+ ∆L

Lf= 2.5 m + 0.000385 m

Step 5: Calculate the final length of the rod.

Lf= 2.500385 m

Therefore, the final length of the steel rod when heated to 200◦C is 2.500385

meters.

Question 17

Question

A steel rod is 2 meters long at 20◦C. If the coefficient of linear expansion for

steel is 1.2×10−5◦C−1, what will be 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 20◦C and the final temperature is 100◦C, the change in temperature is:

∆T=Tf−Ti= 100◦C−20◦C = 80◦C

Step 2: Calculate the change in length. The change in length (∆L) of the

steel rod can be calculated using the formula:

∆L=L·α·∆T

where Lis the initial length, αis the coefficient of linear expansion, and ∆T

is the change in temperature. Substitute L= 2 m, α= 1.2×10−5◦C−1, and

∆T= 80 ◦C into the formula:

∆L= 2 m ·1.2×10−5◦C−1·80 ◦C

Step 3: Calculate the final length of the rod. The final length (Lf) of the

rod can be calculated by adding the change in length to the initial length:

Lf=L+ ∆L

Substitute L= 2 m and the calculated ∆Linto the formula:

Lf= 2 m + (2 m ·1.2×10−5◦C−1·80 ◦C)

Question 18

Question

A steel rod is initially 2 meters long at 20

C. If the coefficient of linear expansion

for steel is 1.2×10−5per degree Celsius, what is the length of the rod when the

temperature reaches 100

Solution

Step 1: Calculate the change in temperature.

Given: Initial temperature, T1= 20

Final temperature, T2= 100

Change in temperature, ∆T=T2−T1= 100 −20 = 80

Step 2: Calculate the change in length using the formula for linear expansion:

∆L=αL∆T

where: ∆L= change in length

α= coefficient of linear expansion

L= initial length of the rod

∆T= change in temperature

Plugging in the values, we have:

∆L= (1.2×10−5)×2×80

Calculating,

∆L= 0.000024 ×160 = 0.00384 meters

Step 3: Find the final length of the rod. The final length, Lf, can be found

by adding the change in length to the initial length:

Lf=L+ ∆L= 2 + 0.00384 = 2.00384 meters

Therefore, the length of the rod when the temperature reaches 100

C is

2.00384 meters.

Question 19

Question

A steel rod is tightly clamped between two walls at 0◦C. If the walls are heated

to 100◦C, by how much will the rod elongate? (Given: Coefficient of linear

thermal expansion of steel = 1.2×10−5/◦Cand original length of the rod = 2

meters)

Solution

Step 1: First, we calculate the change in temperature:

∆T=Tf−Ti= 100◦C−0◦C= 100◦C

Step 2: Next, we calculate the change in length using the formula for linear

thermal expansion:

∆L=α·L·∆T

where: ∆L= change in length, α= coefficient of linear thermal expansion, L

= original length, ∆T= change in temperature.

Substitute the given values:

∆L= (1.2×10−5/◦C)·2 meters ·100◦C

Step 3: Calculate the change in length:

∆L= 0.000024 ·200 = 0.0048 meters

Step 4: Therefore, the steel rod will elongate by 0.0048 meters when the

walls are heated to 100◦C.

Question 20

Question

A steel cylinder of height 2.0 m and radius 1.0 m is initially at a temperature

of 20

C. If the temperature of the cylinder is increased to 100

C, calculate the

change in height of the cylinder. The linear expansion coefficient of steel is

1.2×10−5per

Solution

Let’s denote the initial height of the cylinder as H0and the initial radius as

R0. The change in height of the cylinder can be calculated using the formula

for linear expansion:

∆H=H0·α·∆T

where ∆H= change in height, H0= initial height of the cylinder, α= linear

expansion coefficient of steel, ∆T= change in temperature.

Given: H0= 2.0 m, α= 1.2×10−5per

C, ∆T= 100C−20C= 80C.

Step 1: Calculate the change in height

∆H= 2.0 m ·1.2×10−5·80

Step 2: Solve for ∆H

∆H= 2.0×1.2×10−5×80 = 0.00192 m

Therefore, the change in height of the cylinder is 0.00192 m.

Question 21

Question

A steel rod has a length of 1 meter at 20

C. If the coefficient of linear expansion

of steel is 1.2×10−5per degree Celsius, determine the temperature at which

the rod will have a length of 1.01 meters.

Solution

Step 1: Let’s denote the initial length of the steel rod as L0, the final length as

L, the initial temperature as T0, and the final temperature as T. We are given

that L0= 1 meter, L= 1.01 meters, and the coefficient of linear expansion

α= 1.2×10−5per degree Celsius.

Step 2: The formula for linear expansion is given by ∆L=α·L0·∆T, where

∆L=L−L0and ∆T=T−T0.

Step 3: Substituting the given values into the formula, we get ∆L= (1.2×

10−5)·1·∆T.

Step 4: Solving for ∆T, we have ∆T=∆L

1.2×10−5.

Step 5: Plugging in the values for ∆Land solving for ∆T, we get ∆T=

1.01−1

1.2×10−5.

Step 6: Calculating the value of ∆T, we have ∆T=0.01

1.2×10−5.

Step 7: Therefore, ∆T= 833.33 degrees Celsius.

Step 8: To find the final temperature T, we use the formula T=T0+ ∆T.

Step 9: Substituting the values of T0and ∆T, we get T= 20 + 833.33.

Step 10: Therefore, the temperature at which the steel rod will have a length

of 1.01 meters is 853.33 degrees Celsius.

Question 22

Question

A brass rod is 2 meters long at 20

C. If the coefficient of linear expansion of

brass is 19×10−6per

C, by how much does the length of the rod increase when

the temperature is increased to 90

Solution

Given: Initial length of brass rod, L0= 2 m

Coefficient of linear expansion, α= 19 ×10−6per

Change in temperature, ∆T= 90 −20 = 70

We can find the change in length, ∆L, using the formula:

∆L=L0α∆T

Step 1: Calculate the change in length, ∆L.

∆L= 2 ×19 ×10−6×70

= 2 ×19 ×10−6×70

= 2 ×1.33 ×10−4

= 2.66 ×10−4m

So, the length of the brass rod increases by 2.66 ×10−4meters when the

temperature is increased to 90

Question 23

Question

A brass rod is 2 meters long at 20

C. If the coefficient of linear expansion of

brass is 2.0×10−5per degree Celsius, what temperature increase is needed to

increase the length of the rod by 1 cm?

Solution

Step 1: Let’s first calculate the increase in length of the rod when the temper-

ature increases by ∆Tdegrees Celsius. The increase in length (∆L) is given by

the formula:

∆L=α·L·∆T,

where αis the coefficient of linear expansion, Lis the original length of the rod,

and ∆Tis the temperature change. Substitute the given values:

∆L= (2.0×10−5)·2·∆T.

Step 2: We are given that the increase in length is 1 cm, which is 0.01 meters.

Thus, we have:

0.01 = (2.0×10−5)·2·∆T.

Step 3: Solve for ∆T:

∆T=0.01

(2.0×10−5)·2.

Step 4: Calculating the value of ∆T:

∆T=0.01

4.0×10−5= 250 degrees Celsius.

Therefore, a temperature increase of 250

C is needed to increase the length

of the rod by 1 cm.

Question 24

Question

A steel rod of length 2.0 m is heated from 20

C to 80

C. If the coefficient of

linear expansion of steel is 1.2×10−5/

C, what is the change in length of the

rod?

Solution

Step 1: Calculate the initial length of the rod when it is at 20

C. Given: Initial

length, L0= 2.0 m Coefficient of linear expansion, α= 1.2×10−5/C Initial

temperature, Ti= 20C

Using the formula for linear expansion:

∆L=α·L0·∆T

where ∆Lis the change in length, αis the coefficient of linear expansion, L0is

the initial length, and ∆Tis the change in temperature.

Substitute the values:

∆L= (1.2×10−5/C)·2.0 m ·(80C−20C)

Step 2: Calculate the change in length of the rod.

∆L= 1.2×10−5·2·60

∆L= 1.44 ×10−4m

∆L= 0.144 mm

Therefore, the change in length of the steel rod is 0.144 mm.

Question 25

Question

A brass rod is 2.0 m long at 20

C. If the rod is heated to 100

C, what will be

its length? The linear expansion for brass is 19 ×10−6/

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=Tf−Ti= 100C−20C= 80C

Step 2: Calculate the linear expansion coefficient. The linear expansion

coefficient for brass is 19 ×10−6/

Step 3: Use the formula for linear expansion to find the change in length.

The change in length (∆L) of the brass rod can be calculated using the formula:

∆L=α·L0·∆T

where: - ∆Lis the change in length, - αis the linear expansion coefficient of

brass (19 ×10−6/

C), - L0is the initial length of the brass rod (2.0 m), - ∆T

is the change in temperature (80

C).

Substitute the values into the formula:

∆L= (19 ×10−6/

C) ·(2.0 m) ·(80C)

Step 4: Calculate the change in length.

∆L= 0.000038 m = 3.8×10−5m

Therefore, the length of the brass rod when heated to 100

C will be:

Lfinal =L0+ ∆L= 2.0 m + 3.8×10−5m=2.000038 m

So, the final length of the brass rod when heated to 100

C will be 2.000038

meters.

Question 26

Question

A copper rod is initially at a length of 1 m. When heated from 20◦C to 120◦C,

the rod expands by 2 mm. Calculate the coefficient of linear expansion of copper.

Solution

Step 1: Identify the relevant formula for linear expansion. The linear expansion

of a material can be calculated using the formula:

∆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.

Step 2: Given values The original length L0is 1 m, the change in length ∆L

is 2 mm (which is 0.002 m), the change in temperature ∆Tis (120◦C - 20◦C)

= 100◦C.

Step 3: Substitute the values into the formula Substitute L0= 1 m, ∆L=

0.002 m, and ∆T= 100 ◦C into the formula:

0.002 = 1 ×α×100

Step 4: Solve for the coefficient of linear expansion α

α=0.002

100 = 2 ×10−5per degree Celsius

Therefore, the coefficient of linear expansion of copper is 2×10−5per degree

Celsius.

Question 27

Question

A steel rod with an original length of 2 meters experiences a temperature in-

crease of 50

C. If the coefficient of linear expansion for steel is 12 ×10−6/

by how many millimeters does the length of the rod increase?

Solution

Step 1: First we calculate the change in length using the formula given by the

linear expansion equation:

∆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.

Step 2: Substituting the given values, we get:

∆L= 12 ×10−6/

C·2 m ·50

Step 3: Calculating ∆L:

∆L= 12 ×10−6·2·50 = 1.2×10−3m

Step 4: Converting the change in length to millimeters:

∆Lmm = 1.2×10−3m×1000 = 1.2 mm

Therefore, the length of the steel rod increases by 1.2 millimeters.

Question 28

Question

A steel rod has a length of 2 meters at 20◦C. If the coefficient of linear expansion

for steel is 12 ×10−6/

C, what is the length of the rod at 100◦C?

Solution

Step 1: Calculate the change in temperature. Given: Initial temperature, T1=

20◦C Final temperature, T2= 100◦C Change in temperature, ∆T=T2−T1=

100◦C - 20◦C = 80◦C

Step 2: Use the formula for linear expansion. The change in length of the

rod can be calculated using the formula: ∆L=α·L·∆Twhere ∆L= change

in length α= coefficient of linear expansion L= initial length of the rod

Step 3: Substitute the values into the formula. Given: α= 12 ×10−6/

L= 2 meters ∆T= 80◦C

Plugging in the values, we get: ∆L= (12 ×10−6/C)·2 m ·80

Step 4: Calculate the change in length. ∆L= 12 ×10−6·2·80 = 1.92 ×10−3

meters

Step 5: Calculate the final length of the rod. The final length of the rod is

given by: L2=L1+ ∆L L2= 2 m + 1.92 ×10−3mL2= 2.00192 meters

Therefore, the length of the steel rod at 100◦C is approximately 2.00192

meters.

Question 29

Question

A metal rod with a length of 2 meters at 20

C is heated to 100

C. If the coefficient

of linear expansion for the metal is 2 ×10−5per degree Celsius, by how many

millimeters will the length of the rod increase?

Solution

Step 1: Calculate the change in temperature. Given initial temperature T1=

20Cand final temperature T2= 100C, the change in temperature ∆T=T2−

T1= 100C−20C= 80C.

Step 2: Calculate the change in length. The change in length (∆L) of the

rod can be calculated using the formula:

∆L=L·α·∆T

where Lis the original length of the rod, αis the coefficient of linear expansion,

and ∆Tis the change in temperature.

Step 3: Substitute the given values into the formula. Given L= 2 meters

and α= 2 ×10−5per

C, and ∆T= 80

∆L= 2 ·2×10−5·80 = 3.2×10−3meters

Step 4: Convert the change in length to millimeters. To convert from meters

to millimeters, we multiply by 1000:

∆L= 3.2×10−3×1000 = 3.2 millimeters

Therefore, the length of the rod will increase by 3.2 millimeters when heated

from 20

C to 100

Question 30

Question

A steel rod with a length of 2 meters at 20◦C is heated to 180◦C. If the coefficient

of linear expansion for steel is 1.2×10−5K−1, find the final length of the steel

rod.

Solution

Step 1: Calculate the change in temperature. Given that the initial temperature

is 20◦C and the final temperature is 180◦C, the change in temperature is:

∆T=Tf−Ti= 180 −20 = 160 K

Step 2: Calculate the expansion of the steel rod. 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 for steel, Lis the initial length of

the steel rod, and ∆Tis the change in temperature.

Plugging in the values:

∆L= (1.2×10−5K−1)·(2 m) ·(160 K) = 0.00384 m

Step 3: Calculate the final length of the steel rod. The final length of the

steel rod can be found by adding the change in length to the initial length:

Lf=Li+ ∆L= 2 m + 0.00384 m = 2.00384 m

Therefore, the final length of the steel rod when heated to 180◦C is 2.00384

meters.

Question 31

Question

A copper rod is 2 meters long at 0◦C. If the coefficient of linear expansion for

copper is 1.7×10−5/◦C, find the length of the rod when the temperature is

100◦C.

Solution

Step 1: Let’s first determine the change in temperature. Given: Initial length of

the rod (L) = 2 m, Coefficient of linear expansion (α)=1.7×10−5/◦C, Change

in temperature (∆T) = 100◦C.

Step 2: To find the change in length (∆L), we can use the formula:

∆L=α×L×∆T

Step 3: Plug in the values to calculate ∆L:

∆L= 1.7×10−5/◦×2 m ×100◦C

Step 4: Calculate ∆L:

∆L= 1.7×10−5×2×100 m

Step 5: Simplify the expression:

∆L= 0.0034 m

Step 6: Finally, we can find the length of the rod at 100◦C by adding the

change in length to the initial length:

L100 =L+ ∆L

Step 7: Substitute the values to find the length of the rod at 100◦C:

L100 = 2 m + 0.0034 m

Step 8: Calculate the length of the rod at 100◦C:

L100 = 2.0034 m

Therefore, the length of the copper rod at 100◦C is 2.0034 meters.

Question 32

Question

A long steel rod is heated from 20◦C to 120◦C. If the original length of the rod is

2 meters and the coefficient of linear thermal expansion of steel is 12×10−6◦C−1,

find the final length of the rod.

Solution

Step 1: Calculate the change in temperature. Given: Initial temperature, Ti=

20◦C Final temperature, Tf= 120◦C Change in temperature, ∆T=Tf−Ti=

120◦C−20◦C = 100◦C

Step 2: Use the formula for linear thermal expansion to find the change in

length. The change in length (∆L) of the rod is given by the formula:

∆L=α·L·∆T

where: αis the coefficient of linear thermal expansion of steel (α= 12 ×

10−6◦C−1)Lis the original length of the rod (2 meters) ∆Tis the change

in temperature we calculated (100◦C)

Substitute the values into the formula:

∆L= 12 ×10−6·2·100 = 0.0024 m

Step 3: Find the final length of the rod. The final length of the rod is:

Lf=L+ ∆L= 2 + 0.0024 = 2.0024 m

Therefore, the final length of the rod after being heated from 20◦C to 120◦C

is 2.0024 meters.

Question 33

Question

A steel rod is 2 meters long at a temperature of 20◦C. If the coefficient of linear

expansion for steel is 1.2×10−5/

C, what will be the length of the rod when

the temperature is raised to 80◦C?

Solution

Step 1: Calculate the change in temperature Given: Initial temperature T1=

20◦C, final temperature T2= 80◦C

∆T=T2−T1= 80◦C−20◦C = 60◦C

Step 2: Calculate the change in length using the formula

∆L=α·L·∆T

where αis the coefficient of linear expansion for steel, Lis the original length

of the rod, and ∆Tis the change in temperature.

∆L= (1.2×10−5/

C) ·2 m ·60

∆L= 0.000072 m

Step 3: Calculate the final length of the rod The final length Lfis given by

Lf=L+ ∆L

Lf= 2 m + 0.000072 m

Lf= 2.000072 m

Therefore, when the temperature is raised to 80◦C, the length of the steel

rod will be 2.000072 meters.

Question 34

Question

A steel rod is initially 2.00 m long at 20

C. If the coefficient of linear expansion

for steel is 1.20×10−5per

C, what is the length of the rod when the temperature

is increased to 200

Solution

Step 1: Calculate the change in temperature. Step 2: Use the change in tem-

perature and the coefficient of linear expansion to find the change in length.

Step 3: Determine the final length of the rod.

Step 1: Calculate the change in temperature. The change in temperature

is given by:

∆T=Tf−Ti= 200C−20C= 180C

Step 2: Use the change in temperature and the coefficient of linear expan-

sion to find the change in length. The change in length is given by:

∆L=α·L·∆T

where: α= 1.20 ×10−5/C (coefficient of linear expansion for steel) L= 2.00 m

(initial length) ∆T= 180C

Substitute the values in:

∆L= (1.20 ×10−5/C)·2.00 m ·180C

∆L= 0.000216 m

Step 3: Determine the final length of the rod. The final length of the rod

is given by:

Lf=Li+ ∆L

Lf= 2.00 m + 0.000216 m

Lf= 2.000216 m

Therefore, when the temperature is increased to 200

C, the length of the

steel rod is 2.000216 m.

Question 35

Question

A steel rod with a length of 2 meters at 20◦C is heated to 120◦C. If the coefficient

of linear expansion of steel is 1.2×10−5/◦C, what is the change in length of the

rod?

Solution

Step 1: Calculate the change in temperature: The change in temperature is

given by:

∆T=Tf−Ti= 120◦C−20◦C = 100◦C

Step 2: Calculate the change in length using the formula for thermal expan-

sion: The change in length (∆L) is given by:

∆L=L0α∆T

where L0is the original length of the rod, αis the coefficient of linear expansion,

and ∆Tis the change in temperature.

Substitute the given values:

∆L= 2 m ×1.2×10−5/◦C×100◦C=0.0024 m

Step 3: Convert the change in length to centimeters: Since 1 m = 100 cm,

we have:

∆L= 0.0024 m ×100 = 0.24 cm

Therefore, the change in length of the steel rod when heated from 20◦C to

120◦C is 0.24 cm.

PHYS 305 - INTRODUCTION TO MODERN PHYSICS Thermal expansion Question Bank Set 2

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