PHYS 305 - INTRODUCTION TO MODERN PHYSICS Thermal expansion Question Bank Set 3 | PHYS 305

PHYS 305 - INTRODUCTION TO

MODERN PHYSICS - Thermal

expansion

Question Bank - Set 3

Liberty University

Question 1

Question

A steel rod initially at a length of 2 meters is heated to a high temperature,

causing it to expand. If the coefficient of linear expansion of steel is 12×10−6per

degree Celsius, and the rod increases in length by 2 centimeters when heated,

what was the temperature change in degrees Celsius?

Solution

Let’s denote the original length of the steel rod as L0= 2 meters, the increase

in length as ∆L= 0.02 meters, and the coefficient of linear expansion of steel

as α= 12 ×10−6per degree Celsius. We need to find the temperature change

in degrees Celsius.

Step 1: Calculate the change in length using the formula for linear expansion:

∆L=α·L0·∆T

Substitute the given values:

0.02 = 12 ×10−6·2·∆T

0.02 = 24 ×10−6·∆T

Solve for ∆T:

∆T=0.02

24 ×10−6= 833.33 ◦C

So, the temperature change in degrees Celsius was 833.33 ◦C.

Question 2

Question

A steel rod is initially 2 meters long at 20

C. If the rod is heated to 120

C, what

is the final length of the rod if the linear coefficient of thermal expansion for

steel is 1.2×10−5K−1?

Solution

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

thermal expansion to find the final length of the rod.

Step 1: The change in temperature is given by:

∆T=Tf−Ti= 120◦C−20◦C = 100◦C = 100 K

Step 2: The change in length of the rod can be calculated using the formula

for linear thermal expansion:

∆L=α·L·∆T

where ∆L= change in length, α= linear coefficient of thermal expansion (1.2×

10−5K−1), L= original length of the rod (2 m), ∆T= change in temperature

(100 K).

Substitute the values into the formula:

∆L= (1.2×10−5K−1)·(2 m) ·(100 K)

∆L= 0.00024 m = 0.24 mm

Therefore, the final length of the steel rod at 120

C is:

Lf=Li+ ∆L= 2 m + 0.24 mm = 2.00024 m

Thus, the final length of the rod is 2.00024 meters.

Question 3

Question

A steel rod of length 2.0 m and diameter 1.0 cm is heated from 20

C to 120

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

C−1, calculate the

change in length of the rod.

Solution

Step 1: Calculate the initial length of the rod.

Given: Initial length, L0= 2.0 m

Initial temperature, T0= 20

Coefficient of linear expansion, α= 12 ×10−6

C−1

Using the formula for linear expansion:

∆L=α·L0·∆T

Substitute the given values:

∆L= 12 ×10−6·2.0·(120 −20)

∆L= 12 ×10−6·2.0·100

∆L= 12 ×10−6·200

∆L= 2.4×10−3m

∆L= 2.4 mm

The change in length of the steel rod is 2.4 mm.

Question 4

Question

A 2 m long steel rod is heated from 20

C to 120

C. Given that the linear coef-

ficient of thermal expansion for steel is 1.2×10−5per degree Celsius, calculate

the final length of the rod.

Solution

Step 1: Calculate the change in temperature ∆T. Given: Initial temperature

Ti= 20CFinal temperature Tf= 120C

The change in temperature is given by:

∆T=Tf−Ti= 120C−20C= 100C

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

∆Lof the rod is given by:

∆L=αL∆T

where: α= 1.2×10−5per degree Celsius (linear coefficient of thermal expansion

for steel) L= 2 m (initial length of the rod) ∆T= 100

Step 3: Calculate the change in length ∆L.

∆L= (1.2×10−5)(2)(100) = 0.0024 m

Step 4: Calculate the final length of the rod. The final length Lfis given

by:

Lf=L+ ∆L

Lf= 2 + 0.0024 = 2.0024 m

Therefore, the final length of the steel rod after being heated to 120

C is

2.0024 meters.

Question 5

Question

A steel rod is initially 2 meters long at a temperature of 20

C. If the rod is heated

to 100

C, what will be its new length? The coefficient of linear expansion for

steel is 1.2×10−5

C−1.

Solution

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

20Cand final temperature, Tf= 100C, we can find the change in temperature

using the equation:

∆T=Tf−Ti= 100C−20C= 80C

Step 2: Calculate 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. Substituting the values, we get:

∆L= (1.2×10−5

C−1)·2 m ·80C

∆L= 0.000024 m = 2.4×10−5m

Step 3: Calculate the final length. The final length can be found by adding

the change in length to the original length:

Lf=Li+ ∆L= 2 m + 2.4×10−5m=2.000024 m

Therefore, the final length of the steel rod when heated to 100

C is 2.000024

meters.

Question 6

Question

A brass cylinder with a radius of 2 cm and a height of 10 cm is heated from

C to 120

C. If the coefficient of linear expansion of brass is 1.9×10−5per

degree Celsius, by what percentage does the volume of the cylinder increase?

Solution

Step 1: Calculate the increase in height of the cylinder due to thermal expansion.

Step 2: Calculate the increase in radius of the cylinder due to thermal expansion.

Step 3: Use the formula for the volume of a cylinder to find the percentage

increase in volume.

Step 1: Given: Initial height of the cylinder, hi= 10 cm Initial temperature,

Ti= 20

C Final temperature, Tf= 120

C Coefficient of linear expansion, α=

1.9×10−5per

The increase in height, ∆h, can be calculated as:

∆h=hi·α·∆T

∆h= 10 ×1.9×10−5×(120 −20)

∆h= 0.018 cm

Step 2: The increase in radius, ∆r, can be calculated in a similar way:

∆r=ri·α·∆T

∆r= 2 ×1.9×10−5×(120 −20)

∆r= 0.038 cm

Step 3: The percentage increase in volume, %∆V, can be calculated using

the formula for the volume of a cylinder:

Vi=πr2

ihi

Vf=π(ri+ ∆r)2(hi+ ∆h)

%∆V=Vf−Vi

Vi×100

%∆V=π(2 + 0.038)2(10 + 0.018) −π×22×10

π×22×10 ×100

%∆V= 2.33%

Therefore, the volume of the cylinder increases by approximately 2.33

Question 7

Question

A steel rod is initially 2 meters long at a temperature of 20 degrees Celsius. If

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

C, what will be the

length of the rod when the temperature increases to 100 degrees Celsius?

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= 100C−20C= 80C

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

∆L= 12 ×10−6/

C×2 m ×80C

Step 3: Calculate the change in length:

∆L= 0.000192 m = 0.192 mm

Step 4: Determine the final length of the rod. The final length will be the

sum of the initial length and the change in length:

Final length = 2 m + 0.000192 m = 2.000192 m

Therefore, the length of the steel rod when the temperature increases to

100

C will be 2.000192 meters.

Question 8

Question

A steel rod has an original length of 1.5 m at 20

C. If the rod expands to 1.505

m when heated to 90

C, calculate the coefficient of linear expansion of steel.

Solution

Step 1: Identify the known values and the formula for linear expansion: Given:

Initial length, L0= 1.5 m

Final length, Lf= 1.505 m

Initial temperature, T0= 20

Final temperature, Tf= 90

The formula for linear expansion is:

Lf=L0(1 + α·∆T)

where: Lf= final length

L0= initial length

α= coefficient of linear expansion

∆T= change in temperature

Step 2: Calculate the change in temperature:

∆T=Tf−T0= 90 −20 = 70

Step 3: Substitute the known values into the formula for linear expansion:

1.505 = 1.5(1 + α·70)

Step 4: Solve for the coefficient of linear expansion, α:

1.505 = 1.5 + 105α

0.005 = 105α

α=0.005

105

α≈4.76 ×10−5

C−1

Therefore, the coefficient of linear expansion of steel is approximately 4.76 ×

10−5

C−1.

Question 9

Question

A steel rod has a length of 1.0 m at 20

C. If the rod is heated to 120

C, what will

be its new length? The coefficient of linear expansion for steel is 12 ×10−6K−1.

Solution

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

20CFinal temperature, Tf= 120CChange in temperature, ∆T=Tf−Ti=

120C−20C= 100C

Step 2: Use the formula for linear expansion: The change in length, ∆L, of

the steel rod can be calculated using the formula:

∆L=L0α∆T

where: L0= 1.0 m (initial length) α= 12 ×10−6K−1(coefficient of linear

expansion for steel) ∆T= 100C

Step 3: Calculate the change in length.

∆L= 1.0 m ×12 ×10−6K−1×100C

∆L= 0.0012 m = 1.2 mm

Step 4: Calculate the new length of the steel rod. The new length, Lf, can

be found by adding the change in length to the initial length:

Lf=L0+ ∆L= 1.0 m + 0.0012 m = 1.0012 m

Therefore, when the steel rod is heated to 120

C, its new length will be

1.0012 meters.

Question 10

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, find the change in length of the rod

when it is heated to 150◦C.

Solution

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

is 20◦C and the final temperature is 150◦C, the change in temperature is:

∆T= 150◦C−20◦C = 130◦C

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

the rod is given by:

∆L=α·L·∆T

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

the change in temperature.

Plugging in the values, we get:

∆L= (1.2×10−5)·2·130

Step 3: Calculate the change in length.

∆L= 1.56 ×10−3meters

Therefore, the change in length of the rod when heated to 150◦C is 1.56

millimeters.

Question 11

Question

A steel rod with an initial length of 2 meters undergoes a temperature change

of 100 degrees Celsius. If the coefficient of linear expansion for steel is 12×10−6

per degree Celsius, what is the final length of the rod?

Solution

Step 1: Calculate the change in length of the rod using the formula for linear

expansion:

∆L=α·L·∆T

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

length of the rod ∆T= change in temperature

Given: α= 12 ×10−6L= 2 m ∆T= 100 degrees Celsius

Substitute the given values into the formula:

∆L= 12 ×10−6·2·100

∆L= 0.0024 m

Step 2: Calculate the final length of the rod using the formula:

Lfinal =Linitial + ∆L

Lfinal = 2 + 0.0024

Lfinal = 2.0024 m

Therefore, the final length of the steel rod after a temperature change of 100

degrees Celsius is 2.0024 meters.

Question 12

Question

A 10 m long aluminum rod is heated from 20◦C to 120◦C. If the coefficient of

linear expansion of aluminum is 23 ×10−6C−1, what is the change in length of

the rod?

Solution

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

(Ti) is 20◦C and the final temperature (Tf) is 120◦C, the change in temperature

(∆T) is:

∆T=Tf−Ti= 120◦C−20◦C = 100◦C

Step 2: Use the formula for linear expansion. The change in length (∆L) of

the aluminum rod can be calculated using the formula for linear expansion:

∆L=α·L·∆T

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

and ∆Tis the change in temperature.

Step 3: Plug in the values and solve for ∆L. Substitute the known values

into the formula:

∆L= (23 ×10−6C−1)·(10 m) ·(100◦C)

Step 4: Calculate the change in length.

∆L= 23 ×10−5m·10 m ·100 = 0.023 m = 2.3 cm

Thus, the change in length of the aluminum rod is 2.3 cm.

Question 13

Question

A steel rod has an initial length of 2 meters at 20

C. If the coefficient of linear

expansion for steel is 12 ×10−6per degree Celsius, determine the length of the

rod when its temperature reaches 100

Solution

Step 1: Calculate the change in length of the steel rod due to the temperature

increase. Step 2: Determine the final length of the rod after expansion.

Step 1: Given: Initial length of the steel rod, L0= 2 meters, Change in

temperature, ∆T= 100 −20 = 80

C, Coefficient of linear expansion for steel,

α= 12 ×10−6per

The change in length of the steel rod, ∆L, can be calculated using the

formula:

∆L=α·L0·∆T

Substitute the given values to find ∆L:

∆L= 12 ×10−6·2·80

∆L= 1.92 ×10−3meters

Step 2: The final length of the steel rod after expansion is given by:

Lf=L0+ ∆L

Substitute the values for L0and ∆Lto find Lf:

Lf= 2 + 1.92 ×10−3

Lf= 2.00192 meters

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

C is

2.00192 meters.

Question 14

Question

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

of linear expansion of the metal is 2×10−5◦C−1, find the temperature at which

the rod’s length will be 2.01 meters.

Solution

Step 1: Let Lbe the final length of the metal rod, L0be the initial length of

the metal rod, Tbe the final temperature, T0be the initial temperature, and α

be the coefficient of linear expansion. Step 2: We can use the formula for linear

expansion: ∆L

L0=α∆T. Step 3: We can rearrange the formula to solve for ∆T:

∆T=∆L

αL0. Step 4: Given that L0= 2 m, ∆L= 2.01 m −2 m = 0.01 m, and

α= 2 ×10−5◦C−1, we substitute these values into the formula to find ∆T.

Step 5: ∆T=0.01 m

2×10−5m−1×2 m . Step 6: ∆T=0.01

4×10−5. Step 7: ∆T= 250 K.

Step 8: Therefore, the final temperature Tis T0+ ∆T= 20◦C + 250 K = 270 K

or 270◦C.

Question 15

Question

A brass rod is initially 1.0 m long at 20

C. If the rod is heated to 120

C, what

will be its final length? Given that the linear expansion coefficient of brass is

2.0×10−5K−1.

Solution

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

20◦C Final temperature, T2= 120◦C

The change in temperature, ∆T=T2−T1= 120◦C−20◦C = 100◦C.

Step 2: Calculate the final length of the brass rod. The linear expansion of

a material is given by the formula:

∆L=L0α∆T

where ∆L= change in length, L0= initial length, α= linear expansion coeffi-

cient, ∆T= change in temperature.

Substitute the given values into the formula:

∆L= (1.0 m) ×(2.0×10−5K−1)×(100◦C)

∆L= 2.0×10−4m

Step 3: Calculate the final length of the brass rod. The final length, Lf, is

the sum of the initial length and the change in length:

Lf=L0+ ∆L

Substitute the values:

Lf= 1.0 m + 2.0×10−4m

Lf= 1.0002 m

Therefore, the final length of the brass rod when heated to 120

C is 1.0002

meters.

Question 16

Question

A steel rod of length 2.00 m at 20.0◦C is heated until its temperature reaches

200.0◦C. If the linear expansion coefficient of steel is 12.0×10−6/◦C, what is

the final length of the rod?

Solution

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

20.0◦C and final temperature Tf= 200.0◦C, we have

∆T=Tf−Ti= 200.0◦C−20.0◦C = 180.0◦C

Step 2: Calculate the expansion in length. The linear expansion equation is

given by:

∆L=α·L·∆T

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

the change in temperature.

Substitute the given values into the formula:

∆L= (12.0×10−6/◦C) ·(2.00 m) ·(180.0◦C)

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

by:

Lf=L+ ∆L

Substitute the original length L= 2.00 m and the expansion ∆Lcalculated

in step 2:

Lf= 2.00 m + (12.0×10−6/◦C) ·(2.00 m) ·(180.0◦C)

After calculating, the final length of the steel rod when heated to 200.0◦C

is found to be the final length of the rod.

Question 17

Question

A steel rod has a length of 2.0 m at 20

C. If the coefficient of linear expansion

for steel is 1.2×10−5/C, what is the length of the rod at 100

Solution

Step 1: Identify the given values and the coefficient of linear expansion. The

initial length of the steel rod is L0= 2.0 m, the initial temperature is T0= 20C,

the final temperature is Tf= 100C, and the coefficient of linear expansion for

steel is α= 1.2×10−5/C.

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

The change in length of the rod can be calculated using the formula:

∆L=L0·α·∆T

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

linear expansion, and ∆Tis the change in temperature. Substitute the values

into the formula:

∆L= 2.0 m ·(1.2×10−5/C)·(100C−20C)

Step 3: Calculate the change in length.

∆L= 2.0 m ·1.2×10−5/C ·80C

∆L= 0.00192 m = 1.92 mm

Step 4: Find the final length of the rod. The final length of the rod can be

found by adding the change in length to the initial length:

Lf=L0+ ∆L

Lf= 2.0 m + 0.00192 m

Lf= 2.00192 m = 2.002 m (rounded to 3 decimal places)

Therefore, the length of the steel rod at 100

C is 2.002 meters.

Question 18

Question

A brass rod of length 1.5 m at 20

C is heated until its temperature reaches

120

C. If the linear expansion coefficient of brass is 2.0×10−5per

C, what is

the final length of the rod?

Solution

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

formula to find the final length of the rod.

Step 1: The change in temperature is given by:

∆T= (120C)−(20C) = 100C

Step 2: The linear expansion formula is given by:

∆L=α·L·∆T

where: ∆L= Change in length, α= Linear expansion coefficient, L= Original

length, and ∆T= Change in temperature.

Substitute the values into the formula:

∆L= (2.0×10−5/C)·(1.5m)·(100C)=0.0003m= 0.3mm

Therefore, the final length of the brass rod is:

1.5m+ 0.0003m= 1.5003m

Question 19

Question

A brass rod of length 2.0 m and an aluminium rod of length 1.0 m are rigidly

attached end-to-end. The rods are initially at a temperature of 20

C. If the

temperature is increased to 120

C, find the increase in the length of the rods

and the final separation between their ends assuming no external force is applied.

Given: - Brass: αBrass = 19 ×10−6

C−1- Aluminium: αAluminium = 23 ×

10−6

C−1

Solution

Let LBrass and LAluminium be the lengths of the brass and aluminum rods re-

spectively at temperature T, and LBrass0 and LAluminium0 be their lengths at

the initial temperature of 20

Step 1: Find the increase in length of the brass rod. At temperature T, the

increase in length of the brass rod is given by:

∆LBrass =LBrass −LBrass0 =LBrass0αBrass (T−20)

Substitute the values:

∆LBrass = 2.0 m ×19 ×10−6

C−1×(120 −20)

C=0.036 m

Step 2: Find the increase in length of the aluminium rod. Similarly, the

increase in length of the aluminum rod is given by:

∆LAluminium =LAluminium −LAluminium0 =LAluminium0αAluminium (T−20)

Substitute the values:

∆LAluminium = 1.0 m ×23 ×10−6

C−1×(120 −20)

C=0.023 m

Step 3: Find the final separation between their ends. Since the rods are

rigidly attached end-to-end, the final separation between their ends is:

∆LBrass −∆LAluminium = 0.036 m −0.023 m = 0.013 m

Therefore, the increase in the length of the rods is 0.036 m for brass and

0.023 m for aluminum, with a final separation of 0.013 m.

Question 20

Question

A steel rod has a length of 2 meters at 20

C. If the coefficient of linear expansion

of steel is 12 ×10−6

C−1, find the change in length of the rod when it is heated

to 200

Solution

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

T1= 20CFinal temperature, T2= 200CChange in temperature, ∆T=T2−

T1= 200C−20C= 180C

Step 2: Next, use the formula for linear expansion to find the change in

length: The formula for linear expansion is:

∆L=L0α∆T

where: ∆L= change in length L0= initial length α= coefficient of linear

expansion ∆T= change in temperature

Step 3: Substitute the given values into the formula:

∆L= 2 m ×12 ×10−6

C−1×180C

Step 4: Calculate the change in length:

∆L= 2 ×12 ×10−6×180

∆L= 4.32 ×10−3m

∆L= 4.32 mm

Therefore, the change in length of the steel rod when heated to 200

C is 4.32

mm.

Question 21

Question

A steel rod has an initial length of 2.0 m at 20

C. If the rod is heated to 120

what will be its final length? Assume the linear expansion coefficient for steel

is 1.2×10−5per degree Celsius.

Solution

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

formula to find the final length.

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

T1= 20

C Final temperature, T2= 120

C Change in temperature, ∆T=

T2−T1= 120 −20 = 100

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

expansion formula is given by:

∆L=α·L·∆T

where: ∆L= change in length α= linear expansion coefficient = 1.2×10−5L

= initial length = 2.0 m ∆T= change in temperature = 100

Substitute the values into the formula to find the change in length:

∆L= (1.2×10−5)·2.0·100

∆L= 0.000024 m

Now, the final length Lfcan be found by adding the change in length to the

initial length:

Lf= 2.0+0.000024

Lf= 2.000024 m

Therefore, the final length of the steel rod when heated to 120

C will be

2.000024 meters.

Question 22

Question

A copper rod, initially at 20◦C, has a length of 2 meters. If the rod is heated

to 120◦C, determine the final length of the rod given that the linear coefficient

of thermal expansion for copper is 16.8×10−6◦C−1.

Solution

Step 1: We can use the equation for linear thermal expansion:

∆L=α·L·∆T

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

expansion, - Lis the original length, and - ∆Tis the change in temperature.

Step 2: First, calculate the change in temperature:

∆T= 120◦C−20◦C = 100◦C

Step 3: Now, substitute the known values into the formula for thermal ex-

pansion to find the change in length:

∆L= (16.8×10−6◦C−1)·(2 m) ·(100 ◦C)

Step 4: Calculating the change in length:

∆L= 16.8×10−6×2×100 m = 0.00336 m = 3.36 mm

Step 5: Finally, determine the final length of the rod by adding the change

in length to the original length:

Final length = 2 m + 0.00336 m = 2.00336 m

Therefore, the final length of the copper rod when heated to 120◦C is 2.00336

meters.

Question 23

Question

A steel rod of length 1.5 m is heated from 20

C to 120

C. If the coefficient of

linear expansion for steel is 11 ×10−6

C−1, by how much does the length of the

rod increase?

Solution

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

C Final temperature, Tf= 120

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

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

The formula for linear expansion is given by:

∆L=L·α·∆T

where: ∆L= Change in length L= Initial length α= Coefficient of linear

expansion ∆T= Change in temperature

Given that: Initial length, L= 1.5 m Coefficient of linear expansion, α=

11 ×10−6

C−1Change in temperature, ∆T= 100

Substitute these values into the formula:

∆L= 1.5·11 ×10−6·100

∆L= 1.65 ×10−4m

Therefore, the length of the steel rod increases by 1.65 ×10−4meters.

Question 24

Question

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

is 1.2×10−5/◦C, find the length of the rod at 100◦C.

Solution

Step 1: Let’s denote the original length of the steel rod as L0and the final

temperature as Tf. We also know the coefficient of linear expansion, α, is

1.2×10−5/◦C.

Step 2: We can use the formula for linear expansion:

∆L=αL0∆T

where ∆Lis the change in length of the rod, L0is the original length, αis

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

Step 3: We are given the original length L0= 2 meters, the coefficient of

linear expansion α= 1.2×10−5/◦C, and the change in temperature ∆T=

100◦C.

Step 4: Substituting the values into the formula, we get:

∆L= (1.2×10−5/◦·2m) ·100◦= 0.0024m

Step 5: The final length of the steel rod at 100◦C is obtained by adding the

change in length to the original length:

Lf=L0+ ∆L= 2m + 0.0024m = 2.0024m

Step 6: Therefore, the length of the steel rod at 100◦C is 2.0024 meters.

Question 25

Question

A steel rod of length 2.5 m at 20◦C is heated to 120◦C. If the coefficient of linear

expansion of steel is 1.2×10−5/◦C, find the change in length of the rod.

Solution

Step 1: Calculate the initial length change due to the temperature increase.

Given the coefficient of linear expansion α= 1.2×10−5/◦C, the initial length

of the steel rod L0= 2.5 m, and the temperature change ∆T= 120◦C - 20◦C

= 100◦C, we can use the formula for linear expansion:

∆L=αL0∆T

∆L= (1.2×10−5/◦C)(2.5 m)(100◦C)

∆L= 0.003 m

Therefore, the initial change in length due to temperature increase is 0.003

Step 2: Calculate the final length of the rod. The final length of the rod Lf

is given by:

Lf=L0+ ∆L

Lf= 2.5 m + 0.003 m

Lf= 2.503 m

Step 3: Calculate the change in length of the rod. The change in length is

given by:

∆L=Lf−L0

∆L= 2.503 m −2.5 m

∆L= 0.003 m

Therefore, the change in length of the steel rod when heated to 120◦C is

0.003 m.

Question 26

Question

A steel rod has an original length of 2 meters and a coefficient of linear expansion

of 1.2×10−5K−1. If the rod is heated from an initial temperature of 20

C to a

final temperature of 120

C, what is the final length of the rod?

Solution

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

20◦C Final temperature, Tf= 120◦C

The change in temperature, ∆T, is given by:

∆T=Tf−Ti= 120◦C−20◦C = 100◦C

Step 2: Calculate the change in length The change in length, ∆L, is given

by:

∆L=α·L·∆T

where: α= 1.2×10−5K−1(coefficient of linear expansion of steel) L= 2 m

(original length) ∆T= 100◦C

Therefore,

∆L= (1.2×10−5K−1)×(2 m) ×(100◦C)

∆L= 0.0024 m

Step 3: Calculate the final length The final length, Lf, is given by:

Lf=L+ ∆L= 2 m + 0.0024 m

Lf= 2.0024 m

Therefore, the final length of the rod after heating from 20

C to 120

C is

2.0024 meters.

Question 27

Question

A steel rod of length 2.0 m is heated from 20◦C to 120◦C. If the coefficient of

linear expansion for steel is 1.2×10−5/

C, calculate the change in length of the

rod.

Solution

Step 1: Calculate the initial length of the rod. Given: Original length of rod,

L0= 2.0 m

Step 2: Calculate the change in temperature. Given: Initial temperature,

T1= 20◦C Final temperature, T2= 120◦C

Change in temperature, ∆T=T2−T1= 120◦C - 20◦C = 100◦C

Step 3: Use the formula for linear expansion. The change in length, ∆L, is

given by:

∆L=L0α∆T

where: L0= initial length of the rod α= coefficient of linear expansion ∆T

= change in temperature

Step 4: Substitute the given values into the formula and calculate.

∆L= 2.0 m ×(1.2×10−5

C−1)×100

∆L= 2.0×1.2×10−3m

∆L= 2.4×10−3m

Therefore, the change in length of the steel rod is 2.4×10−3m.

Question 28

Question

A 2 m long steel rod is heated from 20

C to 120

C. If the linear coefficient of

thermal expansion for steel is 1.2×10−5per degree Celsius, what is the change

in length of the rod?

Solution

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

is 20

C and the final temperature is 120

C, we have:

∆T= 120C−20C= 100C

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

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

∆L=Liα∆T

where: Li= initial length of the rod α= linear coefficient of thermal expansion

∆T= change in temperature

Substitute the values into the formula:

∆L= 2 m ×1.2×10−5per

C×100

Step 3: Calculate the change in length.

∆L= 2 ×1.2×10−3m=2.4×10−3m

Therefore, the change in length of the steel rod when heated from 20

C to

120

C is 2.4×10−3m.

Question 29

Question

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

steel is 1.2×10−5/◦C, what will be the length of the beam when the temperature

rises to 100◦C?

Solution

Step 1: Let’s first calculate the change in length of the steel beam due to the

increase in temperature. Given: Initial length of the steel beam, L0= 20 m

Coefficient of linear expansion, α= 1.2×10−5/◦C Change in temperature,

∆T= 100◦−20◦= 80◦C

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

∆L=α·L0·∆T

Step 2: Now we can substitute the given values into the formula to find the

change in length.

∆L= (1.2×10−5/◦C) ·20 m ·80◦C

Step 3: Calculate the change in length.

∆L= 0.024 m

Step 4: Finally, we can find the final length of the steel beam at 100◦C by

adding the change in length to the initial length.

Final length = L0+ ∆L= 20 m + 0.024 m = 20.024 m

Therefore, the length of the steel beam when the temperature rises to 100◦C

will be 20.024 meters.

Question 30

Question

A steel rod has a length of 1 meter at 20◦C. If the coefficient of linear expansion

of steel is 1.2×10−5◦C−1, what will be the length of the rod at 200◦C?

Solution

Step 1: Calculate the change in temperature. Let ∆Tbe the change in temper-

ature. Given: initial temperature T1= 20◦C and final temperature T2= 200◦C.

Therefore, ∆T=T2−T1= 200◦C−20◦C = 180◦C.

Step 2: Calculate the change in length. The change in length ∆Lof the

steel rod is given by the formula:

∆L=α·L·∆T

where: α= coefficient of linear expansion = 1.2×10−5◦C−1L= initial length

of the steel rod = 1 meter ∆T= change in temperature = 180◦C

Substitute the values into the formula:

∆L= (1.2×10−5)·1·180

Step 3: Calculate the change in length.

∆L= 2.16 ×10−3meters

Step 4: Calculate the final length. The final length L2of the steel rod at

200◦C is given by:

L2=L+ ∆L

Substitute the values into the formula:

L2= 1 + 2.16 ×10−3

Step 5: Calculate the final length.

L2= 1.00216 meters

Therefore, the length of the steel rod at 200◦C is 1.00216 meters.

Question 31

Question

A cylindrical rod made of steel with an initial length of 2 meters undergoes a

temperature change of 100

C. If the coefficient of linear expansion of steel is

11 ×10−6/

C, what is the final length of the rod?

Solution

Step 1: Identify the given values. The initial length of the rod, Li, is 2 meters;

the temperature change, ∆T, is 100

C; and the coefficient of linear expansion,

α, is 11 ×10−6/

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

rod, ∆L, can be calculated using the formula:

∆L=α·Li·∆T

Step 3: Substitute the given values into the formula.

∆L= (11 ×10−6/

C) ·(2 m) ·(100

Step 4: Calculate the change in length.

∆L= 0.0022 meters

Step 5: Determine the final length of the rod. The final length, Lf, can be

found by adding the change in length to the initial length:

Lf=Li+ ∆L

Step 6: Substitute the values to find the final length.

Lf= 2 m + 0.0022 m

Step 7: Calculate the final length of the rod.

Lf= 2.0022 meters

Therefore, the final length of the steel rod after a temperature change of

100

C is 2.0022 meters.

Question 32

Question

A steel rod is initially 2 meters long at 20

C. If the coefficient of linear expansion

for steel is 1.2×10−5/

C, find the temperature at which the length of the rod

will increase by 1 cm.

Solution

Step 1: Let’s start by determining the change in length of the steel rod. The

formula for linear expansion is given by:

∆L=L0α∆T

where: - ∆Lis the change in length, - L0is the initial length, - αis the coefficient

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

Step 2: We are given that the initial length L0= 2 m, the coefficient of linear

expansion α= 1.2×10−5/

C, and we need to find the change in temperature

∆Twhen the length increases by 1 cm (or 0.01 m).

Step 3: We can rewrite the formula as:

0.01 = 2 ×1.2×10−5×∆T

Step 4: Solve for ∆T:

∆T=0.01

2×1.2×10−5= 4166.67

Therefore, the temperature at which the length of the rod will increase by 1

cm is 4166.67

Question 33

Question

A copper rod has a length of 2.00 m at 20

C. If the rod is heated to 90

C, what

is the final length of the rod? (Coefficient of linear expansion for copper =

1.70 ×10−5per

Solution

Step 1: Calculate the change in temperature. At 20

C, the initial temperature,

and at 90

C, the final temperature, the change in temperature is given by:

∆T= 90C−20C= 70C

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

length. The change in length (∆L) of the rod can be calculated using the

formula:

∆L=αL∆T

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

the change in temperature. Substitute the values: α= 1.70 ×10−5,L= 2.00

m, and ∆T= 70

∆L= (1.70 ×10−5)×2.00 ×70 = 0.00238 m

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

rod can be calculated using:

Lf=Li+ ∆L

where Liis the initial length. Substitute Li= 2.00 m and ∆L= 0.00238 m.

Lf= 2.00 + 0.00238 = 2.00238 m

Therefore, the final length of the copper rod when heated to 90

C is 2.00238

Question 34

Question

A steel bridge is 200 meters long at a temperature of 20◦C. If the bridge expands

by 5 cm when the temperature rises to 40◦C, what is the coefficient of linear

expansion of steel?

Solution

Step 1: Let’s define the given values. Let L0= 200 m be the initial length of

the steel bridge at 20◦C, Lf=L0+ ∆L= 200 m + 5 cm = 200.05 m be the

final length of the bridge at 40◦C, ∆T= 40◦C - 20◦C = 20◦C be the change in

temperature, and αbe the coefficient of linear expansion of steel.

Step 2: Use the formula for linear expansion:

∆L=L0·α·∆T

Step 3: Plug in the known values and solve for α:

5 cm = 200 m ·α·20◦C

α=5 cm

200 m ·20 ◦C= 0.000125/◦C

Therefore, the coefficient of linear expansion of steel is 0.000125/◦C.

Question 2

Question

A steel rod is initially 2 meters long at 20

C. If the rod is heated to 120

C, what

is the final length of the rod if the linear coefficient of thermal expansion for

steel is 1.2×10−5K−1?

Solution

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

thermal expansion to find the final length of the rod.

Step 1: The change in temperature is given by:

∆T=Tf−Ti= 120◦C−20◦C = 100◦C = 100 K

Step 2: The change in length of the rod can be calculated using the formula

for linear thermal expansion:

∆L=α·L·∆T

where ∆L= change in length, α= linear coefficient of thermal expansion (1.2×

10−5K−1), L= original length of the rod (2 m), ∆T= change in temperature

(100 K).

Substitute the values into the formula:

∆L= (1.2×10−5K−1)·(2 m) ·(100 K)

∆L= 0.00024 m = 0.24 mm

Therefore, the final length of the steel rod at 120

C is:

Lf=Li+ ∆L= 2 m + 0.24 mm = 2.00024 m

Thus, the final length of the rod is 2.00024 meters.

Question 3

Question

A steel rod of length 2.0 m and diameter 1.0 cm is heated from 20

C to 120

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

C−1, calculate the

change in length of the rod.

Solution

Step 1: Calculate the initial length of the rod.

Given: Initial length, L0= 2.0 m

Initial temperature, T0= 20

Coefficient of linear expansion, α= 12 ×10−6

C−1

Using the formula for linear expansion:

∆L=α·L0·∆T

Substitute the given values:

∆L= 12 ×10−6·2.0·(120 −20)

∆L= 12 ×10−6·2.0·100

∆L= 12 ×10−6·200

∆L= 2.4×10−3m

∆L= 2.4 mm

The change in length of the steel rod is 2.4 mm.

Question 4

Question

A 2 m long steel rod is heated from 20

C to 120

C. Given that the linear coef-

ficient of thermal expansion for steel is 1.2×10−5per degree Celsius, calculate

the final length of the rod.

Solution

Step 1: Calculate the change in temperature ∆T. Given: Initial temperature

Ti= 20CFinal temperature Tf= 120C

The change in temperature is given by:

∆T=Tf−Ti= 120C−20C= 100C

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

∆Lof the rod is given by:

∆L=αL∆T

where: α= 1.2×10−5per degree Celsius (linear coefficient of thermal expansion

for steel) L= 2 m (initial length of the rod) ∆T= 100

Step 3: Calculate the change in length ∆L.

∆L= (1.2×10−5)(2)(100) = 0.0024 m

Step 4: Calculate the final length of the rod. The final length Lfis given

by:

Lf=L+ ∆L

Lf= 2 + 0.0024 = 2.0024 m

Therefore, the final length of the steel rod after being heated to 120

C is

2.0024 meters.

Question 5

Question

A steel rod is initially 2 meters long at a temperature of 20

C. If the rod is heated

to 100

C, what will be its new length? The coefficient of linear expansion for

steel is 1.2×10−5

C−1.

Solution

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

20Cand final temperature, Tf= 100C, we can find the change in temperature

using the equation:

∆T=Tf−Ti= 100C−20C= 80C

Step 2: Calculate 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. Substituting the values, we get:

∆L= (1.2×10−5

C−1)·2 m ·80C

∆L= 0.000024 m = 2.4×10−5m

Step 3: Calculate the final length. The final length can be found by adding

the change in length to the original length:

Lf=Li+ ∆L= 2 m + 2.4×10−5m=2.000024 m

Therefore, the final length of the steel rod when heated to 100

C is 2.000024

meters.

Question 6

Question

A brass cylinder with a radius of 2 cm and a height of 10 cm is heated from

C to 120

C. If the coefficient of linear expansion of brass is 1.9×10−5per

degree Celsius, by what percentage does the volume of the cylinder increase?

Solution

Step 1: Calculate the increase in height of the cylinder due to thermal expansion.

Step 2: Calculate the increase in radius of the cylinder due to thermal expansion.

Step 3: Use the formula for the volume of a cylinder to find the percentage

increase in volume.

Step 1: Given: Initial height of the cylinder, hi= 10 cm Initial temperature,

Ti= 20

C Final temperature, Tf= 120

C Coefficient of linear expansion, α=

1.9×10−5per

The increase in height, ∆h, can be calculated as:

∆h=hi·α·∆T

∆h= 10 ×1.9×10−5×(120 −20)

∆h= 0.018 cm

Step 2: The increase in radius, ∆r, can be calculated in a similar way:

∆r=ri·α·∆T

∆r= 2 ×1.9×10−5×(120 −20)

∆r= 0.038 cm

Step 3: The percentage increase in volume, %∆V, can be calculated using

the formula for the volume of a cylinder:

Vi=πr2

ihi

Vf=π(ri+ ∆r)2(hi+ ∆h)

%∆V=Vf−Vi

Vi×100

%∆V=π(2 + 0.038)2(10 + 0.018) −π×22×10

π×22×10 ×100

%∆V= 2.33%

Therefore, the volume of the cylinder increases by approximately 2.33

Question 7

Question

A steel rod is initially 2 meters long at a temperature of 20 degrees Celsius. If

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

C, what will be the

length of the rod when the temperature increases to 100 degrees Celsius?

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= 100C−20C= 80C

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

∆L= 12 ×10−6/

C×2 m ×80C

Step 3: Calculate the change in length:

∆L= 0.000192 m = 0.192 mm

Step 4: Determine the final length of the rod. The final length will be the

sum of the initial length and the change in length:

Final length = 2 m + 0.000192 m = 2.000192 m

Therefore, the length of the steel rod when the temperature increases to

100

C will be 2.000192 meters.

Question 8

Question

A steel rod has an original length of 1.5 m at 20

C. If the rod expands to 1.505

m when heated to 90

C, calculate the coefficient of linear expansion of steel.

Solution

Step 1: Identify the known values and the formula for linear expansion: Given:

Initial length, L0= 1.5 m

Final length, Lf= 1.505 m

Initial temperature, T0= 20

Final temperature, Tf= 90

The formula for linear expansion is:

Lf=L0(1 + α·∆T)

where: Lf= final length

L0= initial length

α= coefficient of linear expansion

∆T= change in temperature

Step 2: Calculate the change in temperature:

∆T=Tf−T0= 90 −20 = 70

Step 3: Substitute the known values into the formula for linear expansion:

1.505 = 1.5(1 + α·70)

Step 4: Solve for the coefficient of linear expansion, α:

1.505 = 1.5 + 105α

0.005 = 105α

α=0.005

105

α≈4.76 ×10−5

C−1

Therefore, the coefficient of linear expansion of steel is approximately 4.76 ×

10−5

C−1.

Question 9

Question

A steel rod has a length of 1.0 m at 20

C. If the rod is heated to 120

C, what will

be its new length? The coefficient of linear expansion for steel is 12 ×10−6K−1.

Solution

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

20CFinal temperature, Tf= 120CChange in temperature, ∆T=Tf−Ti=

120C−20C= 100C

Step 2: Use the formula for linear expansion: The change in length, ∆L, of

the steel rod can be calculated using the formula:

∆L=L0α∆T

where: L0= 1.0 m (initial length) α= 12 ×10−6K−1(coefficient of linear

expansion for steel) ∆T= 100C

Step 3: Calculate the change in length.

∆L= 1.0 m ×12 ×10−6K−1×100C

∆L= 0.0012 m = 1.2 mm

Step 4: Calculate the new length of the steel rod. The new length, Lf, can

be found by adding the change in length to the initial length:

Lf=L0+ ∆L= 1.0 m + 0.0012 m = 1.0012 m

Therefore, when the steel rod is heated to 120

C, its new length will be

1.0012 meters.

Question 10

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, find the change in length of the rod

when it is heated to 150◦C.

Solution

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

is 20◦C and the final temperature is 150◦C, the change in temperature is:

∆T= 150◦C−20◦C = 130◦C

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

the rod is given by:

∆L=α·L·∆T

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

the change in temperature.

Plugging in the values, we get:

∆L= (1.2×10−5)·2·130

Step 3: Calculate the change in length.

∆L= 1.56 ×10−3meters

Therefore, the change in length of the rod when heated to 150◦C is 1.56

millimeters.

Question 11

Question

A steel rod with an initial length of 2 meters undergoes a temperature change

of 100 degrees Celsius. If the coefficient of linear expansion for steel is 12×10−6

per degree Celsius, what is the final length of the rod?

Solution

Step 1: Calculate the change in length of the rod using the formula for linear

expansion:

∆L=α·L·∆T

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

length of the rod ∆T= change in temperature

Given: α= 12 ×10−6L= 2 m ∆T= 100 degrees Celsius

Substitute the given values into the formula:

∆L= 12 ×10−6·2·100

∆L= 0.0024 m

Step 2: Calculate the final length of the rod using the formula:

Lfinal =Linitial + ∆L

Lfinal = 2 + 0.0024

Lfinal = 2.0024 m

Therefore, the final length of the steel rod after a temperature change of 100

degrees Celsius is 2.0024 meters.

Question 12

Question

A 10 m long aluminum rod is heated from 20◦C to 120◦C. If the coefficient of

linear expansion of aluminum is 23 ×10−6C−1, what is the change in length of

the rod?

Solution

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

(Ti) is 20◦C and the final temperature (Tf) is 120◦C, the change in temperature

(∆T) is:

∆T=Tf−Ti= 120◦C−20◦C = 100◦C

Step 2: Use the formula for linear expansion. The change in length (∆L) of

the aluminum rod can be calculated using the formula for linear expansion:

∆L=α·L·∆T

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

and ∆Tis the change in temperature.

Step 3: Plug in the values and solve for ∆L. Substitute the known values

into the formula:

∆L= (23 ×10−6C−1)·(10 m) ·(100◦C)

Step 4: Calculate the change in length.

∆L= 23 ×10−5m·10 m ·100 = 0.023 m = 2.3 cm

Thus, the change in length of the aluminum rod is 2.3 cm.

Question 13

Question

A steel rod has an initial length of 2 meters at 20

C. If the coefficient of linear

expansion for steel is 12 ×10−6per degree Celsius, determine the length of the

rod when its temperature reaches 100

Solution

Step 1: Calculate the change in length of the steel rod due to the temperature

increase. Step 2: Determine the final length of the rod after expansion.

Step 1: Given: Initial length of the steel rod, L0= 2 meters, Change in

temperature, ∆T= 100 −20 = 80

C, Coefficient of linear expansion for steel,

α= 12 ×10−6per

The change in length of the steel rod, ∆L, can be calculated using the

formula:

∆L=α·L0·∆T

Substitute the given values to find ∆L:

∆L= 12 ×10−6·2·80

∆L= 1.92 ×10−3meters

Step 2: The final length of the steel rod after expansion is given by:

Lf=L0+ ∆L

Substitute the values for L0and ∆Lto find Lf:

Lf= 2 + 1.92 ×10−3

Lf= 2.00192 meters

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

C is

2.00192 meters.

Question 14

Question

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

of linear expansion of the metal is 2×10−5◦C−1, find the temperature at which

the rod’s length will be 2.01 meters.

Solution

Step 1: Let Lbe the final length of the metal rod, L0be the initial length of

the metal rod, Tbe the final temperature, T0be the initial temperature, and α

be the coefficient of linear expansion. Step 2: We can use the formula for linear

expansion: ∆L

L0=α∆T. Step 3: We can rearrange the formula to solve for ∆T:

∆T=∆L

αL0. Step 4: Given that L0= 2 m, ∆L= 2.01 m −2 m = 0.01 m, and

α= 2 ×10−5◦C−1, we substitute these values into the formula to find ∆T.

Step 5: ∆T=0.01 m

2×10−5m−1×2 m . Step 6: ∆T=0.01

4×10−5. Step 7: ∆T= 250 K.

Step 8: Therefore, the final temperature Tis T0+ ∆T= 20◦C + 250 K = 270 K

or 270◦C.

Question 15

Question

A brass rod is initially 1.0 m long at 20

C. If the rod is heated to 120

C, what

will be its final length? Given that the linear expansion coefficient of brass is

2.0×10−5K−1.

Solution

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

20◦C Final temperature, T2= 120◦C

The change in temperature, ∆T=T2−T1= 120◦C−20◦C = 100◦C.

Step 2: Calculate the final length of the brass rod. The linear expansion of

a material is given by the formula:

∆L=L0α∆T

where ∆L= change in length, L0= initial length, α= linear expansion coeffi-

cient, ∆T= change in temperature.

Substitute the given values into the formula:

∆L= (1.0 m) ×(2.0×10−5K−1)×(100◦C)

∆L= 2.0×10−4m

Step 3: Calculate the final length of the brass rod. The final length, Lf, is

the sum of the initial length and the change in length:

Lf=L0+ ∆L

Substitute the values:

Lf= 1.0 m + 2.0×10−4m

Lf= 1.0002 m

Therefore, the final length of the brass rod when heated to 120

C is 1.0002

meters.

Question 16

Question

A steel rod of length 2.00 m at 20.0◦C is heated until its temperature reaches

200.0◦C. If the linear expansion coefficient of steel is 12.0×10−6/◦C, what is

the final length of the rod?

Solution

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

20.0◦C and final temperature Tf= 200.0◦C, we have

∆T=Tf−Ti= 200.0◦C−20.0◦C = 180.0◦C

Step 2: Calculate the expansion in length. The linear expansion equation is

given by:

∆L=α·L·∆T

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

the change in temperature.

Substitute the given values into the formula:

∆L= (12.0×10−6/◦C) ·(2.00 m) ·(180.0◦C)

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

by:

Lf=L+ ∆L

Substitute the original length L= 2.00 m and the expansion ∆Lcalculated

in step 2:

Lf= 2.00 m + (12.0×10−6/◦C) ·(2.00 m) ·(180.0◦C)

After calculating, the final length of the steel rod when heated to 200.0◦C

is found to be the final length of the rod.

Question 17

Question

A steel rod has a length of 2.0 m at 20

C. If the coefficient of linear expansion

for steel is 1.2×10−5/C, what is the length of the rod at 100

Solution

Step 1: Identify the given values and the coefficient of linear expansion. The

initial length of the steel rod is L0= 2.0 m, the initial temperature is T0= 20C,

the final temperature is Tf= 100C, and the coefficient of linear expansion for

steel is α= 1.2×10−5/C.

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

The change in length of the rod can be calculated using the formula:

∆L=L0·α·∆T

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

linear expansion, and ∆Tis the change in temperature. Substitute the values

into the formula:

∆L= 2.0 m ·(1.2×10−5/C)·(100C−20C)

Step 3: Calculate the change in length.

∆L= 2.0 m ·1.2×10−5/C ·80C

∆L= 0.00192 m = 1.92 mm

Step 4: Find the final length of the rod. The final length of the rod can be

found by adding the change in length to the initial length:

Lf=L0+ ∆L

Lf= 2.0 m + 0.00192 m

Lf= 2.00192 m = 2.002 m (rounded to 3 decimal places)

Therefore, the length of the steel rod at 100

C is 2.002 meters.

Question 18

Question

A brass rod of length 1.5 m at 20

C is heated until its temperature reaches

120

C. If the linear expansion coefficient of brass is 2.0×10−5per

C, what is

the final length of the rod?

Solution

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

formula to find the final length of the rod.

Step 1: The change in temperature is given by:

∆T= (120C)−(20C) = 100C

Step 2: The linear expansion formula is given by:

∆L=α·L·∆T

where: ∆L= Change in length, α= Linear expansion coefficient, L= Original

length, and ∆T= Change in temperature.

Substitute the values into the formula:

∆L= (2.0×10−5/C)·(1.5m)·(100C)=0.0003m= 0.3mm

Therefore, the final length of the brass rod is:

1.5m+ 0.0003m= 1.5003m

Question 19

Question

A brass rod of length 2.0 m and an aluminium rod of length 1.0 m are rigidly

attached end-to-end. The rods are initially at a temperature of 20

C. If the

temperature is increased to 120

C, find the increase in the length of the rods

and the final separation between their ends assuming no external force is applied.

Given: - Brass: αBrass = 19 ×10−6

C−1- Aluminium: αAluminium = 23 ×

10−6

C−1

Solution

Let LBrass and LAluminium be the lengths of the brass and aluminum rods re-

spectively at temperature T, and LBrass0 and LAluminium0 be their lengths at

the initial temperature of 20

Step 1: Find the increase in length of the brass rod. At temperature T, the

increase in length of the brass rod is given by:

∆LBrass =LBrass −LBrass0 =LBrass0αBrass (T−20)

Substitute the values:

∆LBrass = 2.0 m ×19 ×10−6

C−1×(120 −20)

C=0.036 m

Step 2: Find the increase in length of the aluminium rod. Similarly, the

increase in length of the aluminum rod is given by:

∆LAluminium =LAluminium −LAluminium0 =LAluminium0αAluminium (T−20)

Substitute the values:

∆LAluminium = 1.0 m ×23 ×10−6

C−1×(120 −20)

C=0.023 m

Step 3: Find the final separation between their ends. Since the rods are

rigidly attached end-to-end, the final separation between their ends is:

∆LBrass −∆LAluminium = 0.036 m −0.023 m = 0.013 m

Therefore, the increase in the length of the rods is 0.036 m for brass and

0.023 m for aluminum, with a final separation of 0.013 m.

Question 20

Question

A steel rod has a length of 2 meters at 20

C. If the coefficient of linear expansion

of steel is 12 ×10−6

C−1, find the change in length of the rod when it is heated

to 200

Solution

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

T1= 20CFinal temperature, T2= 200CChange in temperature, ∆T=T2−

T1= 200C−20C= 180C

Step 2: Next, use the formula for linear expansion to find the change in

length: The formula for linear expansion is:

∆L=L0α∆T

where: ∆L= change in length L0= initial length α= coefficient of linear

expansion ∆T= change in temperature

Step 3: Substitute the given values into the formula:

∆L= 2 m ×12 ×10−6

C−1×180C

Step 4: Calculate the change in length:

∆L= 2 ×12 ×10−6×180

∆L= 4.32 ×10−3m

∆L= 4.32 mm

Therefore, the change in length of the steel rod when heated to 200

C is 4.32

mm.

Question 21

Question

A steel rod has an initial length of 2.0 m at 20

C. If the rod is heated to 120

what will be its final length? Assume the linear expansion coefficient for steel

is 1.2×10−5per degree Celsius.

Solution

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

formula to find the final length.

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

T1= 20

C Final temperature, T2= 120

C Change in temperature, ∆T=

T2−T1= 120 −20 = 100

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

expansion formula is given by:

∆L=α·L·∆T

where: ∆L= change in length α= linear expansion coefficient = 1.2×10−5L

= initial length = 2.0 m ∆T= change in temperature = 100

Substitute the values into the formula to find the change in length:

∆L= (1.2×10−5)·2.0·100

∆L= 0.000024 m

Now, the final length Lfcan be found by adding the change in length to the

initial length:

Lf= 2.0+0.000024

Lf= 2.000024 m

Therefore, the final length of the steel rod when heated to 120

C will be

2.000024 meters.

Question 22

Question

A copper rod, initially at 20◦C, has a length of 2 meters. If the rod is heated

to 120◦C, determine the final length of the rod given that the linear coefficient

of thermal expansion for copper is 16.8×10−6◦C−1.

Solution

Step 1: We can use the equation for linear thermal expansion:

∆L=α·L·∆T

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

expansion, - Lis the original length, and - ∆Tis the change in temperature.

Step 2: First, calculate the change in temperature:

∆T= 120◦C−20◦C = 100◦C

Step 3: Now, substitute the known values into the formula for thermal ex-

pansion to find the change in length:

∆L= (16.8×10−6◦C−1)·(2 m) ·(100 ◦C)

Step 4: Calculating the change in length:

∆L= 16.8×10−6×2×100 m = 0.00336 m = 3.36 mm

Step 5: Finally, determine the final length of the rod by adding the change

in length to the original length:

Final length = 2 m + 0.00336 m = 2.00336 m

Therefore, the final length of the copper rod when heated to 120◦C is 2.00336

meters.

Question 23

Question

A steel rod of length 1.5 m is heated from 20

C to 120

C. If the coefficient of

linear expansion for steel is 11 ×10−6

C−1, by how much does the length of the

rod increase?

Solution

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

C Final temperature, Tf= 120

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

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

The formula for linear expansion is given by:

∆L=L·α·∆T

where: ∆L= Change in length L= Initial length α= Coefficient of linear

expansion ∆T= Change in temperature

Given that: Initial length, L= 1.5 m Coefficient of linear expansion, α=

11 ×10−6

C−1Change in temperature, ∆T= 100

Substitute these values into the formula:

∆L= 1.5·11 ×10−6·100

∆L= 1.65 ×10−4m

Therefore, the length of the steel rod increases by 1.65 ×10−4meters.

Question 24

Question

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

is 1.2×10−5/◦C, find the length of the rod at 100◦C.

Solution

Step 1: Let’s denote the original length of the steel rod as L0and the final

temperature as Tf. We also know the coefficient of linear expansion, α, is

1.2×10−5/◦C.

Step 2: We can use the formula for linear expansion:

∆L=αL0∆T

where ∆Lis the change in length of the rod, L0is the original length, αis

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

Step 3: We are given the original length L0= 2 meters, the coefficient of

linear expansion α= 1.2×10−5/◦C, and the change in temperature ∆T=

100◦C.

Step 4: Substituting the values into the formula, we get:

∆L= (1.2×10−5/◦·2m) ·100◦= 0.0024m

Step 5: The final length of the steel rod at 100◦C is obtained by adding the

change in length to the original length:

Lf=L0+ ∆L= 2m + 0.0024m = 2.0024m

Step 6: Therefore, the length of the steel rod at 100◦C is 2.0024 meters.

Question 25

Question

A steel rod of length 2.5 m at 20◦C is heated to 120◦C. If the coefficient of linear

expansion of steel is 1.2×10−5/◦C, find the change in length of the rod.

Solution

Step 1: Calculate the initial length change due to the temperature increase.

Given the coefficient of linear expansion α= 1.2×10−5/◦C, the initial length

of the steel rod L0= 2.5 m, and the temperature change ∆T= 120◦C - 20◦C

= 100◦C, we can use the formula for linear expansion:

∆L=αL0∆T

∆L= (1.2×10−5/◦C)(2.5 m)(100◦C)

∆L= 0.003 m

Therefore, the initial change in length due to temperature increase is 0.003

Step 2: Calculate the final length of the rod. The final length of the rod Lf

is given by:

Lf=L0+ ∆L

Lf= 2.5 m + 0.003 m

Lf= 2.503 m

Step 3: Calculate the change in length of the rod. The change in length is

given by:

∆L=Lf−L0

∆L= 2.503 m −2.5 m

∆L= 0.003 m

Therefore, the change in length of the steel rod when heated to 120◦C is

0.003 m.

Question 26

Question

A steel rod has an original length of 2 meters and a coefficient of linear expansion

of 1.2×10−5K−1. If the rod is heated from an initial temperature of 20

C to a

final temperature of 120

C, what is the final length of the rod?

Solution

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

20◦C Final temperature, Tf= 120◦C

The change in temperature, ∆T, is given by:

∆T=Tf−Ti= 120◦C−20◦C = 100◦C

Step 2: Calculate the change in length The change in length, ∆L, is given

by:

∆L=α·L·∆T

where: α= 1.2×10−5K−1(coefficient of linear expansion of steel) L= 2 m

(original length) ∆T= 100◦C

Therefore,

∆L= (1.2×10−5K−1)×(2 m) ×(100◦C)

∆L= 0.0024 m

Step 3: Calculate the final length The final length, Lf, is given by:

Lf=L+ ∆L= 2 m + 0.0024 m

Lf= 2.0024 m

Therefore, the final length of the rod after heating from 20

C to 120

C is

2.0024 meters.

Question 27

Question

A steel rod of length 2.0 m is heated from 20◦C to 120◦C. If the coefficient of

linear expansion for steel is 1.2×10−5/

C, calculate the change in length of the

rod.

Solution

Step 1: Calculate the initial length of the rod. Given: Original length of rod,

L0= 2.0 m

Step 2: Calculate the change in temperature. Given: Initial temperature,

T1= 20◦C Final temperature, T2= 120◦C

Change in temperature, ∆T=T2−T1= 120◦C - 20◦C = 100◦C

Step 3: Use the formula for linear expansion. The change in length, ∆L, is

given by:

∆L=L0α∆T

where: L0= initial length of the rod α= coefficient of linear expansion ∆T

= change in temperature

Step 4: Substitute the given values into the formula and calculate.

∆L= 2.0 m ×(1.2×10−5

C−1)×100

∆L= 2.0×1.2×10−3m

∆L= 2.4×10−3m

Therefore, the change in length of the steel rod is 2.4×10−3m.

Question 28

Question

A 2 m long steel rod is heated from 20

C to 120

C. If the linear coefficient of

thermal expansion for steel is 1.2×10−5per degree Celsius, what is the change

in length of the rod?

Solution

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

is 20

C and the final temperature is 120

C, we have:

∆T= 120C−20C= 100C

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

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

∆L=Liα∆T

where: Li= initial length of the rod α= linear coefficient of thermal expansion

∆T= change in temperature

Substitute the values into the formula:

∆L= 2 m ×1.2×10−5per

C×100

Step 3: Calculate the change in length.

∆L= 2 ×1.2×10−3m=2.4×10−3m

Therefore, the change in length of the steel rod when heated from 20

C to

120

C is 2.4×10−3m.

Question 29

Question

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

steel is 1.2×10−5/◦C, what will be the length of the beam when the temperature

rises to 100◦C?

Solution

Step 1: Let’s first calculate the change in length of the steel beam due to the

increase in temperature. Given: Initial length of the steel beam, L0= 20 m

Coefficient of linear expansion, α= 1.2×10−5/◦C Change in temperature,

∆T= 100◦−20◦= 80◦C

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

∆L=α·L0·∆T

Step 2: Now we can substitute the given values into the formula to find the

change in length.

∆L= (1.2×10−5/◦C) ·20 m ·80◦C

Step 3: Calculate the change in length.

∆L= 0.024 m

Step 4: Finally, we can find the final length of the steel beam at 100◦C by

adding the change in length to the initial length.

Final length = L0+ ∆L= 20 m + 0.024 m = 20.024 m

Therefore, the length of the steel beam when the temperature rises to 100◦C

will be 20.024 meters.

Question 30

Question

A steel rod has a length of 1 meter at 20◦C. If the coefficient of linear expansion

of steel is 1.2×10−5◦C−1, what will be the length of the rod at 200◦C?

Solution

Step 1: Calculate the change in temperature. Let ∆Tbe the change in temper-

ature. Given: initial temperature T1= 20◦C and final temperature T2= 200◦C.

Therefore, ∆T=T2−T1= 200◦C−20◦C = 180◦C.

Step 2: Calculate the change in length. The change in length ∆Lof the

steel rod is given by the formula:

∆L=α·L·∆T

where: α= coefficient of linear expansion = 1.2×10−5◦C−1L= initial length

of the steel rod = 1 meter ∆T= change in temperature = 180◦C

Substitute the values into the formula:

∆L= (1.2×10−5)·1·180

Step 3: Calculate the change in length.

∆L= 2.16 ×10−3meters

Step 4: Calculate the final length. The final length L2of the steel rod at

200◦C is given by:

L2=L+ ∆L

Substitute the values into the formula:

L2= 1 + 2.16 ×10−3

Step 5: Calculate the final length.

L2= 1.00216 meters

Therefore, the length of the steel rod at 200◦C is 1.00216 meters.

Question 31

Question

A cylindrical rod made of steel with an initial length of 2 meters undergoes a

temperature change of 100

C. If the coefficient of linear expansion of steel is

11 ×10−6/

C, what is the final length of the rod?

Solution

Step 1: Identify the given values. The initial length of the rod, Li, is 2 meters;

the temperature change, ∆T, is 100

C; and the coefficient of linear expansion,

α, is 11 ×10−6/

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

rod, ∆L, can be calculated using the formula:

∆L=α·Li·∆T

Step 3: Substitute the given values into the formula.

∆L= (11 ×10−6/

C) ·(2 m) ·(100

Step 4: Calculate the change in length.

∆L= 0.0022 meters

Step 5: Determine the final length of the rod. The final length, Lf, can be

found by adding the change in length to the initial length:

Lf=Li+ ∆L

Step 6: Substitute the values to find the final length.

Lf= 2 m + 0.0022 m

Step 7: Calculate the final length of the rod.

Lf= 2.0022 meters

Therefore, the final length of the steel rod after a temperature change of

100

C is 2.0022 meters.

Question 32

Question

A steel rod is initially 2 meters long at 20

C. If the coefficient of linear expansion

for steel is 1.2×10−5/

C, find the temperature at which the length of the rod

will increase by 1 cm.

Solution

Step 1: Let’s start by determining the change in length of the steel rod. The

formula for linear expansion is given by:

∆L=L0α∆T

where: - ∆Lis the change in length, - L0is the initial length, - αis the coefficient

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

Step 2: We are given that the initial length L0= 2 m, the coefficient of linear

expansion α= 1.2×10−5/

C, and we need to find the change in temperature

∆Twhen the length increases by 1 cm (or 0.01 m).

Step 3: We can rewrite the formula as:

0.01 = 2 ×1.2×10−5×∆T

Step 4: Solve for ∆T:

∆T=0.01

2×1.2×10−5= 4166.67

Therefore, the temperature at which the length of the rod will increase by 1

cm is 4166.67

Question 33

Question

A copper rod has a length of 2.00 m at 20

C. If the rod is heated to 90

C, what

is the final length of the rod? (Coefficient of linear expansion for copper =

1.70 ×10−5per

Solution

Step 1: Calculate the change in temperature. At 20

C, the initial temperature,

and at 90

C, the final temperature, the change in temperature is given by:

∆T= 90C−20C= 70C

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

length. The change in length (∆L) of the rod can be calculated using the

formula:

∆L=αL∆T

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

the change in temperature. Substitute the values: α= 1.70 ×10−5,L= 2.00

m, and ∆T= 70

∆L= (1.70 ×10−5)×2.00 ×70 = 0.00238 m

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

rod can be calculated using:

Lf=Li+ ∆L

where Liis the initial length. Substitute Li= 2.00 m and ∆L= 0.00238 m.

Lf= 2.00 + 0.00238 = 2.00238 m

Therefore, the final length of the copper rod when heated to 90

C is 2.00238

Question 34

Question

A steel bridge is 200 meters long at a temperature of 20◦C. If the bridge expands

by 5 cm when the temperature rises to 40◦C, what is the coefficient of linear

expansion of steel?

Solution

Step 1: Let’s define the given values. Let L0= 200 m be the initial length of

the steel bridge at 20◦C, Lf=L0+ ∆L= 200 m + 5 cm = 200.05 m be the

final length of the bridge at 40◦C, ∆T= 40◦C - 20◦C = 20◦C be the change in

temperature, and αbe the coefficient of linear expansion of steel.

Step 2: Use the formula for linear expansion:

∆L=L0·α·∆T

Step 3: Plug in the known values and solve for α:

5 cm = 200 m ·α·20◦C

α=5 cm

200 m ·20 ◦C= 0.000125/◦C

Therefore, the coefficient of linear expansion of steel is 0.000125/◦C.

Question 2

Question

A steel rod is initially 2 meters long at 20

C. If the rod is heated to 120

C, what

is the final length of the rod if the linear coefficient of thermal expansion for

steel is 1.2×10−5K−1?

Solution

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

thermal expansion to find the final length of the rod.

Step 1: The change in temperature is given by:

∆T=Tf−Ti= 120◦C−20◦C = 100◦C = 100 K

Step 2: The change in length of the rod can be calculated using the formula

for linear thermal expansion:

∆L=α·L·∆T

where ∆L= change in length, α= linear coefficient of thermal expansion (1.2×

10−5K−1), L= original length of the rod (2 m), ∆T= change in temperature

(100 K).

Substitute the values into the formula:

∆L= (1.2×10−5K−1)·(2 m) ·(100 K)

∆L= 0.00024 m = 0.24 mm

Therefore, the final length of the steel rod at 120

C is:

Lf=Li+ ∆L= 2 m + 0.24 mm = 2.00024 m

Thus, the final length of the rod is 2.00024 meters.

Question 3

Question

A steel rod of length 2.0 m and diameter 1.0 cm is heated from 20

C to 120

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

C−1, calculate the

change in length of the rod.

Solution

Step 1: Calculate the initial length of the rod.

Given: Initial length, L0= 2.0 m

Initial temperature, T0= 20

Coefficient of linear expansion, α= 12 ×10−6

C−1

Using the formula for linear expansion:

∆L=α·L0·∆T

Substitute the given values:

∆L= 12 ×10−6·2.0·(120 −20)

∆L= 12 ×10−6·2.0·100

∆L= 12 ×10−6·200

∆L= 2.4×10−3m

∆L= 2.4 mm

The change in length of the steel rod is 2.4 mm.

Question 4

Question

A 2 m long steel rod is heated from 20

C to 120

C. Given that the linear coef-

ficient of thermal expansion for steel is 1.2×10−5per degree Celsius, calculate

the final length of the rod.

Solution

Step 1: Calculate the change in temperature ∆T. Given: Initial temperature

Ti= 20CFinal temperature Tf= 120C

The change in temperature is given by:

∆T=Tf−Ti= 120C−20C= 100C

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

∆Lof the rod is given by:

∆L=αL∆T

where: α= 1.2×10−5per degree Celsius (linear coefficient of thermal expansion

for steel) L= 2 m (initial length of the rod) ∆T= 100

Step 3: Calculate the change in length ∆L.

∆L= (1.2×10−5)(2)(100) = 0.0024 m

Step 4: Calculate the final length of the rod. The final length Lfis given

by:

Lf=L+ ∆L

Lf= 2 + 0.0024 = 2.0024 m

Therefore, the final length of the steel rod after being heated to 120

C is

2.0024 meters.

Question 5

Question

A steel rod is initially 2 meters long at a temperature of 20

C. If the rod is heated

to 100

C, what will be its new length? The coefficient of linear expansion for

steel is 1.2×10−5

C−1.

Solution

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

20Cand final temperature, Tf= 100C, we can find the change in temperature

using the equation:

∆T=Tf−Ti= 100C−20C= 80C

Step 2: Calculate 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. Substituting the values, we get:

∆L= (1.2×10−5

C−1)·2 m ·80C

∆L= 0.000024 m = 2.4×10−5m

Step 3: Calculate the final length. The final length can be found by adding

the change in length to the original length:

Lf=Li+ ∆L= 2 m + 2.4×10−5m=2.000024 m

Therefore, the final length of the steel rod when heated to 100

C is 2.000024

meters.

Question 6

Question

A brass cylinder with a radius of 2 cm and a height of 10 cm is heated from

C to 120

C. If the coefficient of linear expansion of brass is 1.9×10−5per

degree Celsius, by what percentage does the volume of the cylinder increase?

Solution

Step 1: Calculate the increase in height of the cylinder due to thermal expansion.

Step 2: Calculate the increase in radius of the cylinder due to thermal expansion.

Step 3: Use the formula for the volume of a cylinder to find the percentage

increase in volume.

Step 1: Given: Initial height of the cylinder, hi= 10 cm Initial temperature,

Ti= 20

C Final temperature, Tf= 120

C Coefficient of linear expansion, α=

1.9×10−5per

The increase in height, ∆h, can be calculated as:

∆h=hi·α·∆T

∆h= 10 ×1.9×10−5×(120 −20)

∆h= 0.018 cm

Step 2: The increase in radius, ∆r, can be calculated in a similar way:

∆r=ri·α·∆T

∆r= 2 ×1.9×10−5×(120 −20)

∆r= 0.038 cm

Step 3: The percentage increase in volume, %∆V, can be calculated using

the formula for the volume of a cylinder:

Vi=πr2

ihi

Vf=π(ri+ ∆r)2(hi+ ∆h)

%∆V=Vf−Vi

Vi×100

%∆V=π(2 + 0.038)2(10 + 0.018) −π×22×10

π×22×10 ×100

%∆V= 2.33%

Therefore, the volume of the cylinder increases by approximately 2.33

Question 7

Question

A steel rod is initially 2 meters long at a temperature of 20 degrees Celsius. If

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

C, what will be the

length of the rod when the temperature increases to 100 degrees Celsius?

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= 100C−20C= 80C

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

∆L= 12 ×10−6/

C×2 m ×80C

Step 3: Calculate the change in length:

∆L= 0.000192 m = 0.192 mm

Step 4: Determine the final length of the rod. The final length will be the

sum of the initial length and the change in length:

Final length = 2 m + 0.000192 m = 2.000192 m

Therefore, the length of the steel rod when the temperature increases to

100

C will be 2.000192 meters.

Question 8

Question

A steel rod has an original length of 1.5 m at 20

C. If the rod expands to 1.505

m when heated to 90

C, calculate the coefficient of linear expansion of steel.

Solution

Step 1: Identify the known values and the formula for linear expansion: Given:

Initial length, L0= 1.5 m

Final length, Lf= 1.505 m

Initial temperature, T0= 20

Final temperature, Tf= 90

The formula for linear expansion is:

Lf=L0(1 + α·∆T)

where: Lf= final length

L0= initial length

α= coefficient of linear expansion

∆T= change in temperature

Step 2: Calculate the change in temperature:

∆T=Tf−T0= 90 −20 = 70

Step 3: Substitute the known values into the formula for linear expansion:

1.505 = 1.5(1 + α·70)

Step 4: Solve for the coefficient of linear expansion, α:

1.505 = 1.5 + 105α

0.005 = 105α

α=0.005

105

α≈4.76 ×10−5

C−1

Therefore, the coefficient of linear expansion of steel is approximately 4.76 ×

10−5

C−1.

Question 9

Question

A steel rod has a length of 1.0 m at 20

C. If the rod is heated to 120

C, what will

be its new length? The coefficient of linear expansion for steel is 12 ×10−6K−1.

Solution

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

20CFinal temperature, Tf= 120CChange in temperature, ∆T=Tf−Ti=

120C−20C= 100C

Step 2: Use the formula for linear expansion: The change in length, ∆L, of

the steel rod can be calculated using the formula:

∆L=L0α∆T

where: L0= 1.0 m (initial length) α= 12 ×10−6K−1(coefficient of linear

expansion for steel) ∆T= 100C

Step 3: Calculate the change in length.

∆L= 1.0 m ×12 ×10−6K−1×100C

∆L= 0.0012 m = 1.2 mm

Step 4: Calculate the new length of the steel rod. The new length, Lf, can

be found by adding the change in length to the initial length:

Lf=L0+ ∆L= 1.0 m + 0.0012 m = 1.0012 m

Therefore, when the steel rod is heated to 120

C, its new length will be

1.0012 meters.

Question 10

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, find the change in length of the rod

when it is heated to 150◦C.

Solution

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

is 20◦C and the final temperature is 150◦C, the change in temperature is:

∆T= 150◦C−20◦C = 130◦C

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

the rod is given by:

∆L=α·L·∆T

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

the change in temperature.

Plugging in the values, we get:

∆L= (1.2×10−5)·2·130

Step 3: Calculate the change in length.

∆L= 1.56 ×10−3meters

Therefore, the change in length of the rod when heated to 150◦C is 1.56

millimeters.

Question 11

Question

A steel rod with an initial length of 2 meters undergoes a temperature change

of 100 degrees Celsius. If the coefficient of linear expansion for steel is 12×10−6

per degree Celsius, what is the final length of the rod?

Solution

Step 1: Calculate the change in length of the rod using the formula for linear

expansion:

∆L=α·L·∆T

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

length of the rod ∆T= change in temperature

Given: α= 12 ×10−6L= 2 m ∆T= 100 degrees Celsius

Substitute the given values into the formula:

∆L= 12 ×10−6·2·100

∆L= 0.0024 m

Step 2: Calculate the final length of the rod using the formula:

Lfinal =Linitial + ∆L

Lfinal = 2 + 0.0024

Lfinal = 2.0024 m

Therefore, the final length of the steel rod after a temperature change of 100

degrees Celsius is 2.0024 meters.

Question 12

Question

A 10 m long aluminum rod is heated from 20◦C to 120◦C. If the coefficient of

linear expansion of aluminum is 23 ×10−6C−1, what is the change in length of

the rod?

Solution

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

(Ti) is 20◦C and the final temperature (Tf) is 120◦C, the change in temperature

(∆T) is:

∆T=Tf−Ti= 120◦C−20◦C = 100◦C

Step 2: Use the formula for linear expansion. The change in length (∆L) of

the aluminum rod can be calculated using the formula for linear expansion:

∆L=α·L·∆T

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

and ∆Tis the change in temperature.

Step 3: Plug in the values and solve for ∆L. Substitute the known values

into the formula:

∆L= (23 ×10−6C−1)·(10 m) ·(100◦C)

Step 4: Calculate the change in length.

∆L= 23 ×10−5m·10 m ·100 = 0.023 m = 2.3 cm

Thus, the change in length of the aluminum rod is 2.3 cm.

Question 13

Question

A steel rod has an initial length of 2 meters at 20

C. If the coefficient of linear

expansion for steel is 12 ×10−6per degree Celsius, determine the length of the

rod when its temperature reaches 100

Solution

Step 1: Calculate the change in length of the steel rod due to the temperature

increase. Step 2: Determine the final length of the rod after expansion.

Step 1: Given: Initial length of the steel rod, L0= 2 meters, Change in

temperature, ∆T= 100 −20 = 80

C, Coefficient of linear expansion for steel,

α= 12 ×10−6per

The change in length of the steel rod, ∆L, can be calculated using the

formula:

∆L=α·L0·∆T

Substitute the given values to find ∆L:

∆L= 12 ×10−6·2·80

∆L= 1.92 ×10−3meters

Step 2: The final length of the steel rod after expansion is given by:

Lf=L0+ ∆L

Substitute the values for L0and ∆Lto find Lf:

Lf= 2 + 1.92 ×10−3

Lf= 2.00192 meters

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

C is

2.00192 meters.

Question 14

Question

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

of linear expansion of the metal is 2×10−5◦C−1, find the temperature at which

the rod’s length will be 2.01 meters.

Solution

Step 1: Let Lbe the final length of the metal rod, L0be the initial length of

the metal rod, Tbe the final temperature, T0be the initial temperature, and α

be the coefficient of linear expansion. Step 2: We can use the formula for linear

expansion: ∆L

L0=α∆T. Step 3: We can rearrange the formula to solve for ∆T:

∆T=∆L

αL0. Step 4: Given that L0= 2 m, ∆L= 2.01 m −2 m = 0.01 m, and

α= 2 ×10−5◦C−1, we substitute these values into the formula to find ∆T.

Step 5: ∆T=0.01 m

2×10−5m−1×2 m . Step 6: ∆T=0.01

4×10−5. Step 7: ∆T= 250 K.

Step 8: Therefore, the final temperature Tis T0+ ∆T= 20◦C + 250 K = 270 K

or 270◦C.

Question 15

Question

A brass rod is initially 1.0 m long at 20

C. If the rod is heated to 120

C, what

will be its final length? Given that the linear expansion coefficient of brass is

2.0×10−5K−1.

Solution

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

20◦C Final temperature, T2= 120◦C

The change in temperature, ∆T=T2−T1= 120◦C−20◦C = 100◦C.

Step 2: Calculate the final length of the brass rod. The linear expansion of

a material is given by the formula:

∆L=L0α∆T

where ∆L= change in length, L0= initial length, α= linear expansion coeffi-

cient, ∆T= change in temperature.

Substitute the given values into the formula:

∆L= (1.0 m) ×(2.0×10−5K−1)×(100◦C)

∆L= 2.0×10−4m

Step 3: Calculate the final length of the brass rod. The final length, Lf, is

the sum of the initial length and the change in length:

Lf=L0+ ∆L

Substitute the values:

Lf= 1.0 m + 2.0×10−4m

Lf= 1.0002 m

Therefore, the final length of the brass rod when heated to 120

C is 1.0002

meters.

Question 16

Question

A steel rod of length 2.00 m at 20.0◦C is heated until its temperature reaches

200.0◦C. If the linear expansion coefficient of steel is 12.0×10−6/◦C, what is

the final length of the rod?

Solution

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

20.0◦C and final temperature Tf= 200.0◦C, we have

∆T=Tf−Ti= 200.0◦C−20.0◦C = 180.0◦C

Step 2: Calculate the expansion in length. The linear expansion equation is

given by:

∆L=α·L·∆T

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

the change in temperature.

Substitute the given values into the formula:

∆L= (12.0×10−6/◦C) ·(2.00 m) ·(180.0◦C)

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

by:

Lf=L+ ∆L

Substitute the original length L= 2.00 m and the expansion ∆Lcalculated

in step 2:

Lf= 2.00 m + (12.0×10−6/◦C) ·(2.00 m) ·(180.0◦C)

After calculating, the final length of the steel rod when heated to 200.0◦C

is found to be the final length of the rod.

Question 17

Question

A steel rod has a length of 2.0 m at 20

C. If the coefficient of linear expansion

for steel is 1.2×10−5/C, what is the length of the rod at 100

Solution

Step 1: Identify the given values and the coefficient of linear expansion. The

initial length of the steel rod is L0= 2.0 m, the initial temperature is T0= 20C,

the final temperature is Tf= 100C, and the coefficient of linear expansion for

steel is α= 1.2×10−5/C.

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

The change in length of the rod can be calculated using the formula:

∆L=L0·α·∆T

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

linear expansion, and ∆Tis the change in temperature. Substitute the values

into the formula:

∆L= 2.0 m ·(1.2×10−5/C)·(100C−20C)

Step 3: Calculate the change in length.

∆L= 2.0 m ·1.2×10−5/C ·80C

∆L= 0.00192 m = 1.92 mm

Step 4: Find the final length of the rod. The final length of the rod can be

found by adding the change in length to the initial length:

Lf=L0+ ∆L

Lf= 2.0 m + 0.00192 m

Lf= 2.00192 m = 2.002 m (rounded to 3 decimal places)

Therefore, the length of the steel rod at 100

C is 2.002 meters.

Question 18

Question

A brass rod of length 1.5 m at 20

C is heated until its temperature reaches

120

C. If the linear expansion coefficient of brass is 2.0×10−5per

C, what is

the final length of the rod?

Solution

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

formula to find the final length of the rod.

Step 1: The change in temperature is given by:

∆T= (120C)−(20C) = 100C

Step 2: The linear expansion formula is given by:

∆L=α·L·∆T

where: ∆L= Change in length, α= Linear expansion coefficient, L= Original

length, and ∆T= Change in temperature.

Substitute the values into the formula:

∆L= (2.0×10−5/C)·(1.5m)·(100C)=0.0003m= 0.3mm

Therefore, the final length of the brass rod is:

1.5m+ 0.0003m= 1.5003m

Question 19

Question

A brass rod of length 2.0 m and an aluminium rod of length 1.0 m are rigidly

attached end-to-end. The rods are initially at a temperature of 20

C. If the

temperature is increased to 120

C, find the increase in the length of the rods

and the final separation between their ends assuming no external force is applied.

Given: - Brass: αBrass = 19 ×10−6

C−1- Aluminium: αAluminium = 23 ×

10−6

C−1

Solution

Let LBrass and LAluminium be the lengths of the brass and aluminum rods re-

spectively at temperature T, and LBrass0 and LAluminium0 be their lengths at

the initial temperature of 20

Step 1: Find the increase in length of the brass rod. At temperature T, the

increase in length of the brass rod is given by:

∆LBrass =LBrass −LBrass0 =LBrass0αBrass (T−20)

Substitute the values:

∆LBrass = 2.0 m ×19 ×10−6

C−1×(120 −20)

C=0.036 m

Step 2: Find the increase in length of the aluminium rod. Similarly, the

increase in length of the aluminum rod is given by:

∆LAluminium =LAluminium −LAluminium0 =LAluminium0αAluminium (T−20)

Substitute the values:

∆LAluminium = 1.0 m ×23 ×10−6

C−1×(120 −20)

C=0.023 m

Step 3: Find the final separation between their ends. Since the rods are

rigidly attached end-to-end, the final separation between their ends is:

∆LBrass −∆LAluminium = 0.036 m −0.023 m = 0.013 m

Therefore, the increase in the length of the rods is 0.036 m for brass and

0.023 m for aluminum, with a final separation of 0.013 m.

Question 20

Question

A steel rod has a length of 2 meters at 20

C. If the coefficient of linear expansion

of steel is 12 ×10−6

C−1, find the change in length of the rod when it is heated

to 200

Solution

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

T1= 20CFinal temperature, T2= 200CChange in temperature, ∆T=T2−

T1= 200C−20C= 180C

Step 2: Next, use the formula for linear expansion to find the change in

length: The formula for linear expansion is:

∆L=L0α∆T

where: ∆L= change in length L0= initial length α= coefficient of linear

expansion ∆T= change in temperature

Step 3: Substitute the given values into the formula:

∆L= 2 m ×12 ×10−6

C−1×180C

Step 4: Calculate the change in length:

∆L= 2 ×12 ×10−6×180

∆L= 4.32 ×10−3m

∆L= 4.32 mm

Therefore, the change in length of the steel rod when heated to 200

C is 4.32

mm.

Question 21

Question

A steel rod has an initial length of 2.0 m at 20

C. If the rod is heated to 120

what will be its final length? Assume the linear expansion coefficient for steel

is 1.2×10−5per degree Celsius.

Solution

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

formula to find the final length.

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

T1= 20

C Final temperature, T2= 120

C Change in temperature, ∆T=

T2−T1= 120 −20 = 100

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

expansion formula is given by:

∆L=α·L·∆T

where: ∆L= change in length α= linear expansion coefficient = 1.2×10−5L

= initial length = 2.0 m ∆T= change in temperature = 100

Substitute the values into the formula to find the change in length:

∆L= (1.2×10−5)·2.0·100

∆L= 0.000024 m

Now, the final length Lfcan be found by adding the change in length to the

initial length:

Lf= 2.0+0.000024

Lf= 2.000024 m

Therefore, the final length of the steel rod when heated to 120

C will be

2.000024 meters.

Question 22

Question

A copper rod, initially at 20◦C, has a length of 2 meters. If the rod is heated

to 120◦C, determine the final length of the rod given that the linear coefficient

of thermal expansion for copper is 16.8×10−6◦C−1.

Solution

Step 1: We can use the equation for linear thermal expansion:

∆L=α·L·∆T

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

expansion, - Lis the original length, and - ∆Tis the change in temperature.

Step 2: First, calculate the change in temperature:

∆T= 120◦C−20◦C = 100◦C

Step 3: Now, substitute the known values into the formula for thermal ex-

pansion to find the change in length:

∆L= (16.8×10−6◦C−1)·(2 m) ·(100 ◦C)

Step 4: Calculating the change in length:

∆L= 16.8×10−6×2×100 m = 0.00336 m = 3.36 mm

Step 5: Finally, determine the final length of the rod by adding the change

in length to the original length:

Final length = 2 m + 0.00336 m = 2.00336 m

Therefore, the final length of the copper rod when heated to 120◦C is 2.00336

meters.

Question 23

Question

A steel rod of length 1.5 m is heated from 20

C to 120

C. If the coefficient of

linear expansion for steel is 11 ×10−6

C−1, by how much does the length of the

rod increase?

Solution

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

C Final temperature, Tf= 120

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

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

The formula for linear expansion is given by:

∆L=L·α·∆T

where: ∆L= Change in length L= Initial length α= Coefficient of linear

expansion ∆T= Change in temperature

Given that: Initial length, L= 1.5 m Coefficient of linear expansion, α=

11 ×10−6

C−1Change in temperature, ∆T= 100

Substitute these values into the formula:

∆L= 1.5·11 ×10−6·100

∆L= 1.65 ×10−4m

Therefore, the length of the steel rod increases by 1.65 ×10−4meters.

Question 24

Question

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

is 1.2×10−5/◦C, find the length of the rod at 100◦C.

Solution

Step 1: Let’s denote the original length of the steel rod as L0and the final

temperature as Tf. We also know the coefficient of linear expansion, α, is

1.2×10−5/◦C.

Step 2: We can use the formula for linear expansion:

∆L=αL0∆T

where ∆Lis the change in length of the rod, L0is the original length, αis

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

Step 3: We are given the original length L0= 2 meters, the coefficient of

linear expansion α= 1.2×10−5/◦C, and the change in temperature ∆T=

100◦C.

Step 4: Substituting the values into the formula, we get:

∆L= (1.2×10−5/◦·2m) ·100◦= 0.0024m

Step 5: The final length of the steel rod at 100◦C is obtained by adding the

change in length to the original length:

Lf=L0+ ∆L= 2m + 0.0024m = 2.0024m

Step 6: Therefore, the length of the steel rod at 100◦C is 2.0024 meters.

Question 25

Question

A steel rod of length 2.5 m at 20◦C is heated to 120◦C. If the coefficient of linear

expansion of steel is 1.2×10−5/◦C, find the change in length of the rod.

Solution

Step 1: Calculate the initial length change due to the temperature increase.

Given the coefficient of linear expansion α= 1.2×10−5/◦C, the initial length

of the steel rod L0= 2.5 m, and the temperature change ∆T= 120◦C - 20◦C

= 100◦C, we can use the formula for linear expansion:

∆L=αL0∆T

∆L= (1.2×10−5/◦C)(2.5 m)(100◦C)

∆L= 0.003 m

Therefore, the initial change in length due to temperature increase is 0.003

Step 2: Calculate the final length of the rod. The final length of the rod Lf

is given by:

Lf=L0+ ∆L

Lf= 2.5 m + 0.003 m

Lf= 2.503 m

Step 3: Calculate the change in length of the rod. The change in length is

given by:

∆L=Lf−L0

∆L= 2.503 m −2.5 m

∆L= 0.003 m

Therefore, the change in length of the steel rod when heated to 120◦C is

0.003 m.

Question 26

Question

A steel rod has an original length of 2 meters and a coefficient of linear expansion

of 1.2×10−5K−1. If the rod is heated from an initial temperature of 20

C to a

final temperature of 120

C, what is the final length of the rod?

Solution

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

20◦C Final temperature, Tf= 120◦C

The change in temperature, ∆T, is given by:

∆T=Tf−Ti= 120◦C−20◦C = 100◦C

Step 2: Calculate the change in length The change in length, ∆L, is given

by:

∆L=α·L·∆T

where: α= 1.2×10−5K−1(coefficient of linear expansion of steel) L= 2 m

(original length) ∆T= 100◦C

Therefore,

∆L= (1.2×10−5K−1)×(2 m) ×(100◦C)

∆L= 0.0024 m

Step 3: Calculate the final length The final length, Lf, is given by:

Lf=L+ ∆L= 2 m + 0.0024 m

Lf= 2.0024 m

Therefore, the final length of the rod after heating from 20

C to 120

C is

2.0024 meters.

Question 27

Question

A steel rod of length 2.0 m is heated from 20◦C to 120◦C. If the coefficient of

linear expansion for steel is 1.2×10−5/

C, calculate the change in length of the

rod.

Solution

Step 1: Calculate the initial length of the rod. Given: Original length of rod,

L0= 2.0 m

Step 2: Calculate the change in temperature. Given: Initial temperature,

T1= 20◦C Final temperature, T2= 120◦C

Change in temperature, ∆T=T2−T1= 120◦C - 20◦C = 100◦C

Step 3: Use the formula for linear expansion. The change in length, ∆L, is

given by:

∆L=L0α∆T

where: L0= initial length of the rod α= coefficient of linear expansion ∆T

= change in temperature

Step 4: Substitute the given values into the formula and calculate.

∆L= 2.0 m ×(1.2×10−5

C−1)×100

∆L= 2.0×1.2×10−3m

∆L= 2.4×10−3m

Therefore, the change in length of the steel rod is 2.4×10−3m.

Question 28

Question

A 2 m long steel rod is heated from 20

C to 120

C. If the linear coefficient of

thermal expansion for steel is 1.2×10−5per degree Celsius, what is the change

in length of the rod?

Solution

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

is 20

C and the final temperature is 120

C, we have:

∆T= 120C−20C= 100C

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

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

∆L=Liα∆T

where: Li= initial length of the rod α= linear coefficient of thermal expansion

∆T= change in temperature

Substitute the values into the formula:

∆L= 2 m ×1.2×10−5per

C×100

Step 3: Calculate the change in length.

∆L= 2 ×1.2×10−3m=2.4×10−3m

Therefore, the change in length of the steel rod when heated from 20

C to

120

C is 2.4×10−3m.

Question 29

Question

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

steel is 1.2×10−5/◦C, what will be the length of the beam when the temperature

rises to 100◦C?

Solution

Step 1: Let’s first calculate the change in length of the steel beam due to the

increase in temperature. Given: Initial length of the steel beam, L0= 20 m

Coefficient of linear expansion, α= 1.2×10−5/◦C Change in temperature,

∆T= 100◦−20◦= 80◦C

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

∆L=α·L0·∆T

Step 2: Now we can substitute the given values into the formula to find the

change in length.

∆L= (1.2×10−5/◦C) ·20 m ·80◦C

Step 3: Calculate the change in length.

∆L= 0.024 m

Step 4: Finally, we can find the final length of the steel beam at 100◦C by

adding the change in length to the initial length.

Final length = L0+ ∆L= 20 m + 0.024 m = 20.024 m

Therefore, the length of the steel beam when the temperature rises to 100◦C

will be 20.024 meters.

Question 30

Question

A steel rod has a length of 1 meter at 20◦C. If the coefficient of linear expansion

of steel is 1.2×10−5◦C−1, what will be the length of the rod at 200◦C?

Solution

Step 1: Calculate the change in temperature. Let ∆Tbe the change in temper-

ature. Given: initial temperature T1= 20◦C and final temperature T2= 200◦C.

Therefore, ∆T=T2−T1= 200◦C−20◦C = 180◦C.

Step 2: Calculate the change in length. The change in length ∆Lof the

steel rod is given by the formula:

∆L=α·L·∆T

where: α= coefficient of linear expansion = 1.2×10−5◦C−1L= initial length

of the steel rod = 1 meter ∆T= change in temperature = 180◦C

Substitute the values into the formula:

∆L= (1.2×10−5)·1·180

Step 3: Calculate the change in length.

∆L= 2.16 ×10−3meters

Step 4: Calculate the final length. The final length L2of the steel rod at

200◦C is given by:

L2=L+ ∆L

Substitute the values into the formula:

L2= 1 + 2.16 ×10−3

Step 5: Calculate the final length.

L2= 1.00216 meters

Therefore, the length of the steel rod at 200◦C is 1.00216 meters.

Question 31

Question

A cylindrical rod made of steel with an initial length of 2 meters undergoes a

temperature change of 100

C. If the coefficient of linear expansion of steel is

11 ×10−6/

C, what is the final length of the rod?

Solution

Step 1: Identify the given values. The initial length of the rod, Li, is 2 meters;

the temperature change, ∆T, is 100

C; and the coefficient of linear expansion,

α, is 11 ×10−6/

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

rod, ∆L, can be calculated using the formula:

∆L=α·Li·∆T

Step 3: Substitute the given values into the formula.

∆L= (11 ×10−6/

C) ·(2 m) ·(100

Step 4: Calculate the change in length.

∆L= 0.0022 meters

Step 5: Determine the final length of the rod. The final length, Lf, can be

found by adding the change in length to the initial length:

Lf=Li+ ∆L

Step 6: Substitute the values to find the final length.

Lf= 2 m + 0.0022 m

Step 7: Calculate the final length of the rod.

Lf= 2.0022 meters

Therefore, the final length of the steel rod after a temperature change of

100

C is 2.0022 meters.

Question 32

Question

A steel rod is initially 2 meters long at 20

C. If the coefficient of linear expansion

for steel is 1.2×10−5/

C, find the temperature at which the length of the rod

will increase by 1 cm.

Solution

Step 1: Let’s start by determining the change in length of the steel rod. The

formula for linear expansion is given by:

∆L=L0α∆T

where: - ∆Lis the change in length, - L0is the initial length, - αis the coefficient

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

Step 2: We are given that the initial length L0= 2 m, the coefficient of linear

expansion α= 1.2×10−5/

C, and we need to find the change in temperature

∆Twhen the length increases by 1 cm (or 0.01 m).

Step 3: We can rewrite the formula as:

0.01 = 2 ×1.2×10−5×∆T

Step 4: Solve for ∆T:

∆T=0.01

2×1.2×10−5= 4166.67

Therefore, the temperature at which the length of the rod will increase by 1

cm is 4166.67

Question 33

Question

A copper rod has a length of 2.00 m at 20

C. If the rod is heated to 90

C, what

is the final length of the rod? (Coefficient of linear expansion for copper =

1.70 ×10−5per

Solution

Step 1: Calculate the change in temperature. At 20

C, the initial temperature,

and at 90

C, the final temperature, the change in temperature is given by:

∆T= 90C−20C= 70C

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

length. The change in length (∆L) of the rod can be calculated using the

formula:

∆L=αL∆T

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

the change in temperature. Substitute the values: α= 1.70 ×10−5,L= 2.00

m, and ∆T= 70

∆L= (1.70 ×10−5)×2.00 ×70 = 0.00238 m

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

rod can be calculated using:

Lf=Li+ ∆L

where Liis the initial length. Substitute Li= 2.00 m and ∆L= 0.00238 m.

Lf= 2.00 + 0.00238 = 2.00238 m

Therefore, the final length of the copper rod when heated to 90

C is 2.00238

Question 34

Question

A steel bridge is 200 meters long at a temperature of 20◦C. If the bridge expands

by 5 cm when the temperature rises to 40◦C, what is the coefficient of linear

expansion of steel?

Solution

Step 1: Let’s define the given values. Let L0= 200 m be the initial length of

the steel bridge at 20◦C, Lf=L0+ ∆L= 200 m + 5 cm = 200.05 m be the

final length of the bridge at 40◦C, ∆T= 40◦C - 20◦C = 20◦C be the change in

temperature, and αbe the coefficient of linear expansion of steel.

Step 2: Use the formula for linear expansion:

∆L=L0·α·∆T

Step 3: Plug in the known values and solve for α:

5 cm = 200 m ·α·20◦C

α=5 cm

200 m ·20 ◦C= 0.000125/◦C

Therefore, the coefficient of linear expansion of steel is 0.000125/◦C.

Question 35

Question

A brass rod of length 2.0 m and a steel rod of length 1.0 m are touching each

other at a temperature of 20

C. If the temperature is increased to 120

C, find

the final gap between the rods. The linear expansion coefficients for brass and

steel are 19 ×10−6

C−1and 11 ×10−6

C−1, respectively.

Solution

Step 1: Calculate the increase in length for the brass rod and the steel rod. The

change in length (∆L) for a material is given by the formula:

∆L=L0α∆T

where: - L0is the original length, - αis the linear expansion coefficient, and -

∆Tis the change in temperature.

For the brass rod: Given: L0= 2.0 m, αbrass = 19 ×10−6

C−1, ∆T=

120 −20 = 100

∆Lbrass = 2.0×19 ×10−6×100 = 0.0038 m = 3.8 mm

For the steel rod: Given: L0= 1.0 m, αsteel = 11 ×10−6

C−1, ∆T=

120 −20 = 100

∆Lsteel = 1.0×11 ×10−6×100 = 0.0011 m = 1.1 mm

Step 2: Calculate the final gap between the rods. At the higher temperature,

the total gap between the rods will be the sum of the change in lengths for the

brass and steel rods.

Total gap = ∆Lbrass + ∆Lsteel = 0.0038 + 0.0011 = 0.0049 m = 4.9 mm

Therefore, the final gap between the two rods when the temperature is in-

creased to 120

C is 4.9 mm.

PHYS 305 - INTRODUCTION TO MODERN PHYSICS Thermal expansion Question Bank Set 3

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