PHYS 305 - INTRODUCTION TO MODERN PHYSICS Thermal expansion Question Bank Set 5 | PHYS 305

PHYS 305 - INTRODUCTION TO

MODERN PHYSICS - Thermal

expansion

Question Bank - Set 5

Liberty University

Question 1

Question

A steel rod is initially 2 meters long at 20

C. If the coefficient of linear expansion

for steel is 12×10−6per degree Celsius, what will be the length of the rod when

the temperature increases to 50

Solution

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

is 20

C and it increases to 50

C, the change in temperature is ∆T= 50C−20C=

30C.

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

(∆L). The formula for linear expansion is ∆L=L·α·∆T, where: - Lis the

initial length of the rod, - αis the coefficient of linear expansion, - ∆Tis the

change in temperature.

Substitute the values into the formula:

∆L= 2 m ·12 ×10−6per

C·30C

Step 3: Calculate the change in length.

∆L= 2 ×12 ×10−6×30 = 0.00072 m = 0.72 mm

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

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

Lfinal = 2 m + 0.00072 m = 2.00072 m

Therefore, when the temperature of the steel rod increases to 50

C, its length

will be 2.00072 meters.

Question 2

Question

A brass rod of length 2 m and diameter 1 cm is heated from 20

C to 120

Calculate the increase in length of the rod due to thermal expansion. (Coefficient

of linear expansion of brass = 1.9×10−5per degree Celsius).

Solution

Step 1: Calculate the initial volume of the brass rod. The initial volume Viof

the brass rod can be calculated using the formula for the volume of a cylinder:

Vi=πr2h

where ris the initial radius and his the initial length of the rod. Given that

the diameter is 1 cm, the initial radius rcan be calculated as:

r=1 cm

2= 0.5 cm = 0.005 m

Therefore, the initial volume Viis:

Vi=π×(0.005)2×2≈1.57 ×10−4m3

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

∆Lof the rod can be calculated using the formula:

∆L=α·L·∆T

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

is the change in temperature. Given that the coefficient of linear expansion

of brass is 1.9×10−5per degree Celsius, the initial length Lis 2 m, and the

change in temperature is 120

C - 20

C = 100

C. Substitute these values into the

formula:

∆L= 1.9×10−5×2×100 = 0.0038 m

Therefore, the increase in length of the brass rod due to thermal expansion

is 0.0038 meters.

Question 3

Question

A steel rod has a length of 2 meters at 20

C. If the coefficient of linear expansion

for steel is 1.2×10−5K−1, by how much does the length of the rod increase

when the temperature is raised to 100

Solution

Let’s denote the original length of the rod at 20

C as L0= 2 m, the final tem-

perature as Tf= 100C, the initial temperature as Ti= 20C, and the coefficient

of linear expansion for steel as α= 1.2×10−5K−1.

Step 1: Calculate the change in temperature.

∆T=Tf−Ti= 100C −20C = 80C

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

sion:

∆L=L0·α·∆T

∆L= 2 m ·1.2×10−5K−1·80C = 0.00192 m

Therefore, the length of the rod increases by 0.00192 meters when the tem-

perature is raised to 100

Question 4

Question

A steel rod of length 2 m is heated from 20

C to 120

C. If the coefficient of

linear expansion of steel is 12 ×10−6

C−1, calculate the change in length of the

rod.

Solution

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

Final temperature, Tf= 120

Initial temperature, Ti= 20

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

C−1

Recall the formula for linear expansion:

∆L=L0α∆T

Step 1: Calculate the change in temperature

∆T=Tf−Ti= 120 −20 = 100

Step 2: Calculate the change in length of the rod

∆L= 2 ×12 ×10−6×100 = 0.0024 m

Therefore, the change in length of the steel rod is 0.0024 m.

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 is the final length of the rod? Given that the linear

expansion coefficient of steel is 1.2×10−5per degree Celsius.

Solution

Step 1: Calculate the change in temperature. Since the rod is heated from 20

to 100

C, the change in temperature (∆T) is:

∆T= 100C−20C= 80C

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

be calculated using the formula:

∆L=αL0∆T

where: ∆L= change in length, α= linear expansion coefficient of steel, L0=

initial length of the rod, ∆T= change in temperature.

Substitute the values into the formula:

∆L= (1.2×10−5per

C) ×(2 m) ×(80

∆L= 0.000024 m

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

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

Lfinal =L0+ ∆L

Lfinal = 2 m + 0.000024 m

Lfinal ≈2.000024 m

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

C is approx-

imately 2.000024 meters.

Question 6

Question

A solid metal rod has a length of 1.5 m at 0

C. If the coefficient of linear

expansion of the metal is 2 ×10−5/C, find the change in length of the rod when

the temperature is raised to 100

Solution

Step 1: First, we need to calculate the change in temperature. Given: Initial

length of the rod, L0= 1.5 m Coefficient of linear expansion, α= 2 ×10−5/C

Final temperature, Tf= 100

The change in temperature, ∆T=Tf−Ti= 100C−0C= 100C

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

linear expansion

∆L=α·L0·∆T

Substitute the values:

∆L= 2 ×10−5/C ×1.5 m ×100C

Step 3: Calculate the change in length

∆L= 2 ×10−5×1.5×100 = 3 ×10−3m

Therefore, the change in length of the rod when the temperature is raised

to 100

C is 0.003 m.

Question 7

Question

A steel rod with a length of 2 meters has a coefficient of linear expansion of

1.2×10−5per degree Celsius. If the temperature increases by 50 degrees Celsius,

what will be the change in length of the rod?

Solution

Step 1: We can use the formula for linear expansion to find the change in length

of the rod. Step 2: The formula for linear expansion is given by ∆L=α·L·∆T,

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

is the original length of the rod, and - ∆Tis the change in temperature. Step 3:

Substituting the given values into the formula, we have: ∆L= (1.2×10−5)·2·50.

Step 4: Solving the expression, we get: ∆L= 1.2×10−5·2·50 = 0.0012 meters.

Step 5: Therefore, the change in length of the steel rod will be 0.0012 meters.

Question 8

Question

A steel rod of length 2 m at 20

C has a hole drilled through its diameter. At

what temperature will the rod be at if the hole closes up? Assume the coefficient

of linear expansion for steel is 1.2×10−5

C−1.

Solution

Step 1: Identify the given information. Let L0= 2 m be the original length

of the steel rod at 20

C and Lbe the length of the rod at the unknown final

temperature, T.

Step 2: Determine the change in length of the steel rod due to temperature

change. The change in length, ∆L, can be calculated using the formula:

∆L=αL0∆T,

where α= 1.2×10−5

C−1is the coefficient of linear expansion for steel and

∆T=T−20 is the change in temperature.

Step 3: Find the condition for the hole to close up. For the hole to close up,

the change in length of the rod must be equal to the length of the hole. Thus:

∆L=L0.

Step 4: Substitute and solve for the final temperature, T. Substitute the

expressions for ∆Land L0into the equation:

αL0∆T=L0.

Solving for ∆Tgives:

∆T=1

α.

Finally, substitute the known value of αinto the equation and solve for Tto

find the final temperature at which the hole closes up.

Question 9

Question

A copper rod has a length of 2.0 m at 20

C. If the temperature is increased to

120

C, what is the new length of the rod? (Coefficient of linear expansion for

copper is 1.7×10−5/

Solution

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

T1= 20CFinal temperature T2= 120C

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

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

∆L=α·L·∆T

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

length, ∆T= change in temperature.

Substitute the values:

∆L= (1.7×10−5)×2.0×100

Step 3: Now, calculate the new length of the rod:

Lfinal =Linitial + ∆L

Substitute the values:

Lfinal = 2.0+∆L

Step 4: Calculate the final length. Thus,

Lfinal = 2.0 + (1.7×10−5×2.0×100)

Final length is:

Lfinal = 2.0+0.0034 = 2.0034 m

Question 10

Question

A steel rod of length 2 m and a brass rod of length 3 m are both heated from

an initial temperature of 20

C to a final temperature of 100

C. Calculate the

difference in the increase in length between the two rods. The coefficient of

linear expansion for steel is 12 ×10−6/◦Cand for brass is 18 ×10−6/◦C.

Solution

Step 1: Calculate the increase in length of the steel rod. Given the coefficient

of linear expansion for steel is 12 ×10−6/◦C, the change in length of the steel

rod can be calculated using the formula:

∆Lsteel =L0·αsteel ·∆T

where L0is the initial length of the steel rod, αsteel is the coefficient of linear

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

into the formula:

∆Lsteel = 2 ·12 ×10−6·(100 −20)

∆Lsteel = 2 ·12 ×10−6·80

∆Lsteel = 1.92 ×10−3m

Step 2: Calculate the increase in length of the brass rod. Given the coefficient

of linear expansion for brass is 18 ×10−6/◦C, the change in length of the brass

rod can be calculated using the same formula:

∆Lbrass =L0·αbrass ·∆T

Substitute the values into the formula:

∆Lbrass = 3 ·18 ×10−6·(100 −20)

∆Lbrass = 3 ·18 ×10−6·80

∆Lbrass = 4.32 ×10−3m

Step 3: Calculate the difference in the increase in length between the two

rods. The difference in the increase in length between the two rods is:

∆Lbrass −∆Lsteel = 4.32 ×10−3−1.92 ×10−3

∆Lbrass −∆Lsteel = 2.4×10−3m

Therefore, the difference in the increase in length between the steel and brass

rods is 2.4×10−3meters.

Question 11

Question

A steel beam of length 10 m is heated from 20

C to 120

C. Calculate the change

in length of the beam if the coefficient of linear expansion of steel is 12 ×10−6

per degree Celsius.

Solution

Step 1: Determine the initial length of the beam Given: Initial length of the

beam, L0= 10 m

Step 2: Calculate the change in temperature The change in temperature,

∆T=Tf−TiGiven: Initial temperature, Ti= 20

C Final temperature, Tf=

120

C Therefore, ∆T= 120 −20 = 100

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

α·L0·∆TGiven: Coefficient of linear expansion, α= 12 ×10−6per

C Initial

length, L0= 10 m Change in temperature, ∆T= 100

C Substitute the values

into the formula: ∆L= 12 ×10−6·10 ·100

Step 4: Calculate the change in length ∆L= 12 ×10−6·10 ·100 = 0.012 m

Therefore, the change in length of the steel beam is 0.012 meters when heated

from 20

C to 120

Question 12

Question

A steel bridge is constructed to have a length of 250 meters at a temperature of

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

C−1, what

will be the length of the bridge when the temperature rises to 40

Solution

Let’s denote the initial length of the bridge as L0= 250 m, the coefficient of

linear expansion as α= 1.2×10−5

C−1, and the change in temperature as

∆T= 40 −20 = 20

Step 1: Calculate the change in length of the bridge. The change in length

of the bridge can be calculated using the formula:

∆L=L0·α·∆T

∆L= 250 ×1.2×10−5×20

∆L= 0.06 m

Step 2: Determine the final length of the bridge. The final length of the

bridge can be found by adding the change in length to the initial length.

Lfinal =L0+ ∆L

Lfinal = 250 + 0.06

Lfinal = 250.06 m

Therefore, when the temperature rises to 40

C, the length of the steel bridge

will be 250.06 meters.

Question 13

Question

A steel pipe of length 10 m is heated from 20

C to 120

C. If the coefficient of

linear expansion of steel is 12 ×10−6K−1, find the change in length of the pipe.

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= 120C−20C= 100C

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

pipe can be determined using the formula:

∆L=L·α·∆T

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

and ∆Tis the change in temperature.

Step 3: Substitute the known values into the formula. Given that L= 10 m

and α= 12 ×10−6K−1, we can calculate the change in length:

∆L= 10 m ·12 ×10−6K−1·100C

Step 4: Solve for the change in length.

∆L= 10 m ·12 ×10−6K−1·100 = 1.2 mm

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

C to

120

C is 1.2 mm.

Question 14

Question

A steel rod has a length of 2 meters at 20

C. If the coefficient of linear expansion

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

is increased to 100

Solution

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

of linear expansion as α= 1.2×10−5/C, the original temperature as T0= 20C,

and the final temperature as Tf= 100C.

Step 1: Calculate the change in temperature. Since the temperature changes

from 20Cto 100C, the change in temperature is:

∆T=Tf−T0= 100C−20C= 80C

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

a material with a linear expansion coefficient αdue to a temperature change

∆Tcan be calculated using the formula:

∆L=L0α∆T

Step 3: Calculate the change in length. Substitute the given values into

the formula:

∆L= 2 ×1.2×10−5/C ×80C= 0.00192 meters

Step 4: Determine the final length. The final length Lfof the steel rod can

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

Lf=L0+ ∆L= 2 meters + 0.00192 meters = 2.00192 meters

Therefore, when the temperature is increased to 100

C, the length of the

steel rod will be 2.00192 meters.

Question 15

Question

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

1.2×10−5◦C−1, determine how much its length will increase when heated to

120◦C.

Solution

Step 1: Let’s first calculate the change in temperature (∆T). Given that the

initial temperature is 20◦C and the final temperature is 120◦C, we have:

∆T= 120◦C−20◦C = 100◦C

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

in length (∆L) of the steel rod. The formula for linear expansion is given by:

∆L=Lα∆T

where: ∆L= change in length, L= initial length of the rod, α= coefficient of

linear expansion, and ∆T= change in temperature.

Substitute the given values:

∆L= 2 ×1.2×10−5×100

Step 3: Now, we can calculate the change in length (∆L).

∆L= 2 ×1.2×10−5×100 = 0.0024 m

Therefore, the length of the steel rod will increase by 0.0024 meters when

heated to 120◦C.

Question 16

Question

A brass rod and an aluminum rod are both initially 1 meter in length at 0◦C.

If the temperature increases by 100◦C, by how much will the total length of the

two rods increase? The linear expansion coefficients for brass and aluminum are

19 ×10−6/◦Cand 23 ×10−6/◦Crespectively.

Solution

Step 1: Calculate the increase in length of the brass rod. The increase in length

of a material is given by the formula:

∆L=αL∆T

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

the change in temperature. For brass:

∆Lbrass = (19 ×10−6)×1×100 = 0.0019 meters

Step 2: Calculate the increase in length of the aluminum rod. For aluminum:

∆Laluminum = (23 ×10−6)×1×100 = 0.0023 meters

Step 3: Calculate the total increase in length of the two rods. The total

increase in length is the sum of the increases in length of the brass and aluminum

rods:

∆Ltotal = ∆Lbrass + ∆Laluminum = 0.0019 + 0.0023 = 0.0042 meters

Therefore, the total length of the two rods will increase by 0.0042 meters

when the temperature increases by 100◦C.

Question 17

Question

A steel rod with an initial length of 2 meters is heated from 20

C to 80

C. If the

linear expansion coefficient of steel is 1.2×10−5per degree Celsius, calculate

the final length of the rod.

Solution

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

C Final temperature, Tf= 80

The change in temperature, ∆T=Tf−Ti= 80 −20 = 60

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

given by:

∆L=αL∆T

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

change in temperature.

Given: α= 1.2×10−5L= 2 m ∆T= 60

Substitute the values into the formula:

∆L= (1.2×10−5)×2×60

∆L= 1.44 ×10−4×2×60

∆L= 1.44 ×10−4×120

∆L= 0.01728 m

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

by:

Lf=L+ ∆L

Substitute the values into the formula:

Lf= 2 + 0.01728

Lf= 2.01728 m

Therefore, the final length of the rod after heating is 2.01728 meters.

Question 18

Question

A steel rod has a length of 2.0 m at 20◦C. If the rod is heated to 120◦C, how

much longer will the rod become? The coefficient of linear expansion for steel

is 1.2×10−5K−1.

Solution

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

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

Tf−Ti= 120◦C - 20◦C = 100K

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

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

∆L=L0α∆T

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

for steel, ∆Tis the change in temperature.

Given: Initial length, L0= 2.0 m Coefficient of linear expansion, α= 1.2×

10−5K−1Change in temperature, ∆T= 100K

Substitute the values into the formula:

∆L= (2.0 m)(1.2×10−5K−1)(100 K)

Step 3: Calculate the change in length.

∆L= 2.4×10−4m=0.24 mm

Therefore, the steel rod will become 0.24 mm longer when heated from 20◦C

to 120◦C.

Question 19

Question

A steel rod has a length of 2.0 m at 20

C. If the rod is heated to 75

C, what

will be its new length? (Assume the linear expansion coefficient of steel is

12 ×10−6/C)

Solution

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

Tiis 20

C and the final temperature Tfis 75

C, the change in temperature is:

∆T=Tf−Ti= 75C−20C= 55C

Step 2: Calculate the linear expansion of the steel rod. The linear expansion

of the steel rod can be calculated using the formula:

∆L=L0·α·∆T

where ∆Lis the change in length, L0is the original length, αis the linear

expansion coefficient, and ∆Tis the change in temperature. Substituting the

values:

∆L= 2.0 m ·12 ×10−6/C ·55C= 0.00132 m = 1.32 mm

Step 3: Calculate the new length of the steel rod. The new length Lfof

the steel rod can be calculated by adding the change in length to the original

length:

Lf=L0+∆L= 2.0 m+0.00132 m = 2.00132 m = 2.001 m (to 3 decimal places)

Therefore, the new length of the steel rod when heated to 75

C is 2.001 m.

Question 20

Question

A copper rod is 2.5 m long at 20

C. If the rod is heated to 80

C, what will be

its new length? The coefficient of linear expansion for copper is 1.7×10−5K−1.

Solution

Step 1: First, calculate the change in temperature:

∆T=Tf−Ti= 80C−20C= 60C

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

(∆L) of the rod:

∆L=L0α∆T

where L0is the original length, αis the coefficient of linear expansion, and ∆T

is the change in temperature.

Step 3: Substitute the given values into the formula:

∆L= 2.5 m ×(1.7×10−5K−1)×60C

Step 4: Calculate the change in length:

∆L= 2.5×1.7×10−5×60

Step 5: Simplify the expression:

∆L= 2.55 ×10−4m

Step 6: Finally, find the new length by adding the change in length to the

original length:

Lf=L0+ ∆L= 2.5 m + 2.55 ×10−4m

Step 7: Calculate the new length:

Lf= 2.5+2.55 ×10−4

Lf= 2.500255 m

Therefore, the new length of the copper rod when heated to 80

C is approx-

imately 2.500255 meters.

Question 21

Question

A brass rod with a length of 2.5 m and a diameter of 1.0 cm is heated from 20◦C

to 70◦C. Calculate the change in length of the rod due to thermal expansion.

(Coefficient of linear expansion for brass is 19 ×10−6K−1)

Solution

Step 1: Calculate the initial volume of the rod. The initial volume of the rod

can be calculated using the formula for the volume of a cylinder:

V=πr2h

where ris the radius of the rod, his the initial length of the rod, and π≈

3.14159. Given r= 0.01 m and h= 2.5 m, we have:

V=π(0.01)2×2.5

Step 2: Calculate the final volume of the rod. The final volume of the rod can

be calculated using the same formula, but with the new length after expansion.

The new length can be calculated using the formula for linear expansion:

∆L=αL0∆T

where αis the coefficient of linear expansion, L0is the initial length, and ∆T

is the change in temperature. With α= 19 ×10−6K−1,L0= 2.5 m, and

∆T= 70◦C−20◦C = 50◦C, we can find the new length.

Step 3: Calculate the change in volume. The change in volume can be

calculated as the difference between the final volume and the initial volume:

∆V=Vfinal −Vinitial

Step 4: Calculate the change in length. Since the rod expands uniformly,

the change in length is related to the change in volume as follows:

∆V=πr2∆L

We can solve for ∆Lto find the change in length of the rod due to thermal

expansion.

Question 22

Question

A rod made of steel is initially 1.5 m long at 20◦C. If the coefficient of linear

expansion for steel is 12 ×10−6◦C−1, find the temperature at which the rod’s

length will increase to 1.51 m.

Solution

Let Lbe the original length of the steel rod, ∆Lbe the change in length, α

be the coefficient of linear expansion, T0be the original temperature, Tfbe the

final temperature, and Tbe the temperature at which the rod’s length increases

to 1.51 m.

Step 1: First, we find the change in length when the temperature changes

from T0to Tf:

∆L=Lα(Tf−T0)

Step 2: Substitute the given values into the formula:

∆L= 1.5×12 ×10−6(Tf−20)

Step 3: Since the final length of the rod is 1.51 m, we have:

L+ ∆L= 1.51

1.5+1.5×12 ×10−6(Tf−20) = 1.51

Step 4: Solve for Tf:

1.5+1.5×12 ×10−6Tf−1.5×12 ×10−6×20 = 1.51

1.5+1.8×10−5Tf−3.6×10−5= 1.51

1.8×10−5Tf= 1.51 −1.5+3.6×10−5

1.8×10−5Tf= 1.01 + 3.6×10−5

Tf=1.01 + 3.6×10−5

1.8×10−5

Tf≈62.22◦C

Therefore, the temperature at which the rod’s length will increase to 1.51 m

is approximately 62.22◦C.

Question 23

Question

A steel rod of length 2 m is heated from 20

C to 100

C. If the coefficient of

linear expansion for steel is 1.2×10−5

C−1, determine the change in length of

the rod.

Solution

Step 1: Calculate the initial length change due to heating from 20

C to 100

Given that the initial length of the steel rod is 2 m, we have:

∆L=L·α·∆T

∆L= 2 m ·1.2×10−5

C−1·(100 −20)

∆L= 2 ×1.2×10−3m

∆L= 2.4×10−3m

Step 2: Determine the change in length of the rod. Since the steel rod

expands in both directions, the total change in length is twice the initial change

in length:

Change in Length = 2 ×2.4×10−3m

Change in Length = 4.8×10−3m

Therefore, the change in length of the steel rod is 4.8×10−3m.

Question 24

Question

A brass rod of length 2.0 m is heated from 20

C to 120

C. If the coefficient of

linear expansion of brass is 1.9×10−5K−1, what is the change in length of the

rod?

Solution

Step 1: Calculate the initial length of the rod Given: Initial length of the brass

rod, L0= 2.0 m Coefficient of linear expansion of brass, α= 1.9×10−5K−1

Change in temperature, ∆T= 120C−20C= 100CUsing the formula for linear

expansion:

∆L=L0α∆T

Substitute the values:

∆L= 2.0×1.9×10−5×100

∆L= 2.0×1.9×10−3

∆L= 3.8×10−3m

Therefore, the change in length of the brass rod is 3.8 mm.

Question 25

Question

A steel rod is initially 2 meters long at 20◦C. What temperature increase is

required to increase the length of the rod by 1 cm if the coefficient of linear

expansion of steel is 12 ×10−6/◦C?

Solution

Let Lbe the original length of the steel rod, ∆Lbe the change in length, ∆T

be the change in temperature, and αbe the coefficient of linear expansion.

Step 1: Write out the formula for linear expansion: The change in length

∆Lof a material is given by the formula:

∆L=Lα∆T

Step 2: Use the given information: We know that L= 2m, ∆L= 1cm =

0.01m, and α= 12 ×10−6/◦C. We want to find ∆T.

Step 3: Plug the values into the formula:

∆L=Lα∆T

0.01 = (2)(12 ×10−6)(∆T)

Step 4: Solve for ∆T: Dividing both sides by (2)(12 ×10−6),

∆T=0.01

2×12 ×10−6

∆T=0.01

24 ×10−6

∆T=0.01

24 ×106

∆T=1

2400 ×106

∆T=1000

2400

∆T= 0.4167 ◦C

So, the temperature needs to be increased by 0.4167◦Cto increase the length

of the rod by 1 cm.

Question 26

Question

A steel rod with a length of 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, what is the change in length of

the rod?

Solution

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

Step 2: Determine the change in length of the rod using the coefficient of linear

expansion.

Step 1: Calculate the initial length change due to the change in tempera-

ture.

The initial length change due to the change in temperature can be calculated

using the formula:

∆L=L0·α·∆T

where: ∆L= change in length, L0= 2.0 m (initial length), α= 1.2×10−5/

(coefficient of linear expansion for steel), ∆T= 120◦C−20◦C = 100◦C.

Substitute the given values and calculate:

∆L= 2.0 m ×1.2×10−5

C−1×100

∆L= 0.0024 m

Step 2: Determine the change in length of the rod using the coefficient of

linear expansion.

The final change in length of the rod is equal to the initial length change

due to the change in temperature:

∆L= 0.0024 m

Therefore, the change in length of the steel rod is 0.0024 m.

Question 27

Question

A brass rod of length 2 m at 0◦C is clamped at its two ends. By how much

does its length change when the temperature of the rod is raised to 100◦C?

(Coefficient of linear expansion of brass = 1.9×10−5/◦C)

Solution

Let L0be the original length of the brass rod at 0◦C, ∆Tbe the change in

temperature, αbe the coefficient of linear expansion of brass, and ∆Lbe the

change in length.

Step 1: Calculate the change in temperature. Given ∆T= 100◦C - 0◦C =

100◦C

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

sion:

∆L=α·L0·∆T

∆L= (1.9×10−5/◦C) ·(2 m) ·(100◦C)

∆L= 0.000038 m

Therefore, the length of the brass rod changes by 0.000038 m when the

temperature is raised to 100◦C.

Question 28

Question

A steel rod of length 2.0 m undergoes a temperature increase of 100

C. If the

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

C−1, what is the change in

length of the rod?

Solution

Step 1: We can use the formula for linear expansion ∆L=α·L·∆T, where:

∆L= change in length, α= coefficient of linear expansion, L= original length,

and ∆T= change in temperature.

Step 2: Substituting the given values into the formula, we have: ∆L=

(12 ×10−6

C−1)·(2.0 m) ·(100

Step 3: Calculating the change in length, we get: ∆L= 12×10−6·2.0·100 =

0.0024 m

Step 4: Therefore, the change in length of the steel rod is 0.0024 meters.

Question 29

Question

A solid iron rod has a length of 2.0 m at 20◦C. If the rod is heated to 120◦C,

what will be its new length? The coefficient of linear expansion for iron is

1.2×10−5per degree Celsius.

Solution

Let’s denote the initial length of the iron rod as Liat 20◦C and the final length

after heating as Lfat 120◦C. The change in length, ∆L, can be calculated using

the formula for linear expansion:

∆L=Li·α·∆T

where: Li= 2.0 m (initial length), α= 1.2×10−5K−1(coefficient of linear

expansion for iron), and ∆T= 120◦C−20◦C = 100 K (change in temperature).

Step 1: Calculate the change in length ∆L.

∆L= 2.0 m ×1.2×10−5K−1×100 K

∆L= 0.0024 m = 2.4 mm

Step 2: Calculate the final length Lf.

Lf=Li+ ∆L

Lf= 2.0 m + 0.0024 m

Lf= 2.0024 m

The new length of the iron rod when heated to 120◦C is 2.0024 m, or 2.0024×

103mm.

Question 30

Question

A steel rod is 1 meter long 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

100◦C.

Solution

Given: Initial length of the steel rod, L0= 1 m

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

C−1

Change in temperature, ∆T= 100◦C−20◦C = 80◦C

From the formula for linear expansion:

∆L=L0α∆T

Step 1: Calculate the change in length of the steel rod.

∆L= 1 ×12 ×10−6×80 = 0.000096 m

So, the change in length of the rod when heated to 100◦C is 0.000096 meters.

Question 31

Question

A steel rod has a length of 2.00 m at 20

C. If the temperature of the rod is

increased to 120

C, what is the new length of the rod? The linear expansion

coefficient of steel is 12 ×10−6

C−1.

Solution

Step 1: Calculate the change in length of the steel rod. The change in length of

the rod can be calculated using the formula:

∆L=α·L·∆T

where ∆L= change in length, α= 12 ×10−6

C−1(linear expansion coefficient

of steel), L= 2.00 m (initial length of the rod), ∆T= 120 −20 = 100

C (change

in temperature).

Substitute the values into the formula:

∆L= 12 ×10−6·2.00 ·100

Step 2: Calculate the new length of the steel rod. The new length of the rod

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

Lnew =L+ ∆L

Substitute the values and calculate:

Lnew = 2.00 + 12 ×10−6·2.00 ·100

Question 32

Question

A brass container is filled with 0.5 L of water at 20

C. If the container is then

heated to 70

C, calculate the change in volume of the container. Assume the

linear expansion coefficient of brass is 19 ×10−6K−1and the coefficient of

volume expansion for water is 2.1×10−4K−1.

Solution

Step 1: Calculate the initial volume of the water: Given that the initial volume

of water is 0.5 L, we have Vi= 0.5 L.

Step 2: Calculate the change in temperature: The change in temperature is

∆T= 70C−20C= 50C.

Step 3: Calculate the change in volume of the water: We will calculate the

change in volume of water first. The coefficient of volume expansion for water,

βwater = 2.1×10−4K−1.

The change in volume of water is given by:

∆Vwater =Viβwater∆T

∆Vwater = 0.5 L ×2.1×10−4K−1×50 K

∆Vwater = 0.000525 L

Step 4: Calculate the change in volume of the brass container: Now, we will

calculate the change in volume of the brass container. The linear expansion

coefficient of brass, αbrass = 19 ×10−6K−1.

The change in volume of the brass container is given by:

∆Vbrass =Viαbrass∆T

∆Vbrass = 0.5 L ×19 ×10−6K−1×50 K

∆Vbrass = 0.000475 L

Step 5: Calculate the total change in volume of the container: The total

change in volume is the sum of the changes in volume of the water and the

brass container:

∆Vtotal = ∆Vwater + ∆Vbrass

∆Vtotal = 0.000525 L + 0.000475 L

∆Vtotal = 0.001 L

Therefore, the change in volume of the brass container when heated to 70

is 0.001 L.

Question 33

Question

A steel rod has a length of 2 meters at a temperature of 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 ×10−6per degree Celsius.

Solution

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

expansion to find the new length of the steel rod.

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

is given by:

∆T=Tf−Ti= 120C−20C= 100C

Step 2: Use the formula for linear expansion to find the new length of the

steel rod. The change in length of the steel rod can be calculated using the

formula:

∆L=α·L·∆T

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

and ∆Tis the change in temperature.

Substitute the values into the formula:

∆L= (12 ×10−6)·2·100 = 0.0024 meters

The new length of the steel rod is:

Lf=Li+ ∆L= 2 + 0.0024 = 2.0024 meters

Therefore, when the steel rod is heated to 120

C, its new length will be

2.0024 meters.

Question 34

Question

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

of linear expansion for steel is 1.2×10−5◦C−1, calculate the change in length

of the rod when the temperature is increased to 100◦C.

Solution

Let L1be the initial length of the rod and L2be the final length of the rod.

Given: Initial length, L1= 2 m Change in temperature, ∆T= 100◦C−20◦C =

80◦C Coefficient of linear expansion, α= 1.2×10−5◦C−1

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

sion:

L2=L1(1 + α∆T)

Step 2: Substitute the given values into the formula:

L2= 2(1 + 1.2×10−5×80)

L2= 2(1 + 0.00096)

L2= 2.00192

Step 3: Calculate the change in length:

∆L=L2−L1

∆L= 2.00192 −2

∆L= 0.00192 m

Therefore, the change in length of the steel rod when the temperature is

increased to 100◦C is 0.00192 meters.

Question 35

Question

A copper rod initially measures 2 meters in length at a temperature of 20◦C.

If the coefficient of linear expansion for copper is 17 ×10−6◦C−1, what is the

length of the rod when the temperature increases to 120◦C?

Solution

Step 1: First, we need to calculate the change in temperature. Given that the

initial temperature is 20◦C and the final temperature is 120◦C, the change in

temperature is:

∆T= 120◦C−20◦C = 100◦C

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

∆L=α·L·∆T

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

and ∆Tis the change in temperature.

Substitute the values:

∆L= 17 ×10−6◦C−1·2 m ·100◦C=0.0034 m

Step 3: Finally, we can find the length of the rod at 120◦C by adding the

change in length to the initial length:

Lfinal =Linitial + ∆L= 2 m + 0.0034 m = 2.0034 m

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

120◦C is 2.0034 meters.

Question 2

Question

A brass rod of length 2 m and diameter 1 cm is heated from 20

C to 120

Calculate the increase in length of the rod due to thermal expansion. (Coefficient

of linear expansion of brass = 1.9×10−5per degree Celsius).

Solution

Step 1: Calculate the initial volume of the brass rod. The initial volume Viof

the brass rod can be calculated using the formula for the volume of a cylinder:

Vi=πr2h

where ris the initial radius and his the initial length of the rod. Given that

the diameter is 1 cm, the initial radius rcan be calculated as:

r=1 cm

2= 0.5 cm = 0.005 m

Therefore, the initial volume Viis:

Vi=π×(0.005)2×2≈1.57 ×10−4m3

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

∆Lof the rod can be calculated using the formula:

∆L=α·L·∆T

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

is the change in temperature. Given that the coefficient of linear expansion

of brass is 1.9×10−5per degree Celsius, the initial length Lis 2 m, and the

change in temperature is 120

C - 20

C = 100

C. Substitute these values into the

formula:

∆L= 1.9×10−5×2×100 = 0.0038 m

Therefore, the increase in length of the brass rod due to thermal expansion

is 0.0038 meters.

Question 3

Question

A steel rod has a length of 2 meters at 20

C. If the coefficient of linear expansion

for steel is 1.2×10−5K−1, by how much does the length of the rod increase

when the temperature is raised to 100

Solution

Let’s denote the original length of the rod at 20

C as L0= 2 m, the final tem-

perature as Tf= 100C, the initial temperature as Ti= 20C, and the coefficient

of linear expansion for steel as α= 1.2×10−5K−1.

Step 1: Calculate the change in temperature.

∆T=Tf−Ti= 100C −20C = 80C

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

sion:

∆L=L0·α·∆T

∆L= 2 m ·1.2×10−5K−1·80C = 0.00192 m

Therefore, the length of the rod increases by 0.00192 meters when the tem-

perature is raised to 100

Question 4

Question

A steel rod of length 2 m is heated from 20

C to 120

C. If the coefficient of

linear expansion of steel is 12 ×10−6

C−1, calculate the change in length of the

rod.

Solution

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

Final temperature, Tf= 120

Initial temperature, Ti= 20

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

C−1

Recall the formula for linear expansion:

∆L=L0α∆T

Step 1: Calculate the change in temperature

∆T=Tf−Ti= 120 −20 = 100

Step 2: Calculate the change in length of the rod

∆L= 2 ×12 ×10−6×100 = 0.0024 m

Therefore, the change in length of the steel rod is 0.0024 m.

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 is the final length of the rod? Given that the linear

expansion coefficient of steel is 1.2×10−5per degree Celsius.

Solution

Step 1: Calculate the change in temperature. Since the rod is heated from 20

to 100

C, the change in temperature (∆T) is:

∆T= 100C−20C= 80C

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

be calculated using the formula:

∆L=αL0∆T

where: ∆L= change in length, α= linear expansion coefficient of steel, L0=

initial length of the rod, ∆T= change in temperature.

Substitute the values into the formula:

∆L= (1.2×10−5per

C) ×(2 m) ×(80

∆L= 0.000024 m

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

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

Lfinal =L0+ ∆L

Lfinal = 2 m + 0.000024 m

Lfinal ≈2.000024 m

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

C is approx-

imately 2.000024 meters.

Question 6

Question

A solid metal rod has a length of 1.5 m at 0

C. If the coefficient of linear

expansion of the metal is 2 ×10−5/C, find the change in length of the rod when

the temperature is raised to 100

Solution

Step 1: First, we need to calculate the change in temperature. Given: Initial

length of the rod, L0= 1.5 m Coefficient of linear expansion, α= 2 ×10−5/C

Final temperature, Tf= 100

The change in temperature, ∆T=Tf−Ti= 100C−0C= 100C

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

linear expansion

∆L=α·L0·∆T

Substitute the values:

∆L= 2 ×10−5/C ×1.5 m ×100C

Step 3: Calculate the change in length

∆L= 2 ×10−5×1.5×100 = 3 ×10−3m

Therefore, the change in length of the rod when the temperature is raised

to 100

C is 0.003 m.

Question 7

Question

A steel rod with a length of 2 meters has a coefficient of linear expansion of

1.2×10−5per degree Celsius. If the temperature increases by 50 degrees Celsius,

what will be the change in length of the rod?

Solution

Step 1: We can use the formula for linear expansion to find the change in length

of the rod. Step 2: The formula for linear expansion is given by ∆L=α·L·∆T,

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

is the original length of the rod, and - ∆Tis the change in temperature. Step 3:

Substituting the given values into the formula, we have: ∆L= (1.2×10−5)·2·50.

Step 4: Solving the expression, we get: ∆L= 1.2×10−5·2·50 = 0.0012 meters.

Step 5: Therefore, the change in length of the steel rod will be 0.0012 meters.

Question 8

Question

A steel rod of length 2 m at 20

C has a hole drilled through its diameter. At

what temperature will the rod be at if the hole closes up? Assume the coefficient

of linear expansion for steel is 1.2×10−5

C−1.

Solution

Step 1: Identify the given information. Let L0= 2 m be the original length

of the steel rod at 20

C and Lbe the length of the rod at the unknown final

temperature, T.

Step 2: Determine the change in length of the steel rod due to temperature

change. The change in length, ∆L, can be calculated using the formula:

∆L=αL0∆T,

where α= 1.2×10−5

C−1is the coefficient of linear expansion for steel and

∆T=T−20 is the change in temperature.

Step 3: Find the condition for the hole to close up. For the hole to close up,

the change in length of the rod must be equal to the length of the hole. Thus:

∆L=L0.

Step 4: Substitute and solve for the final temperature, T. Substitute the

expressions for ∆Land L0into the equation:

αL0∆T=L0.

Solving for ∆Tgives:

∆T=1

α.

Finally, substitute the known value of αinto the equation and solve for Tto

find the final temperature at which the hole closes up.

Question 9

Question

A copper rod has a length of 2.0 m at 20

C. If the temperature is increased to

120

C, what is the new length of the rod? (Coefficient of linear expansion for

copper is 1.7×10−5/

Solution

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

T1= 20CFinal temperature T2= 120C

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

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

∆L=α·L·∆T

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

length, ∆T= change in temperature.

Substitute the values:

∆L= (1.7×10−5)×2.0×100

Step 3: Now, calculate the new length of the rod:

Lfinal =Linitial + ∆L

Substitute the values:

Lfinal = 2.0+∆L

Step 4: Calculate the final length. Thus,

Lfinal = 2.0 + (1.7×10−5×2.0×100)

Final length is:

Lfinal = 2.0+0.0034 = 2.0034 m

Question 10

Question

A steel rod of length 2 m and a brass rod of length 3 m are both heated from

an initial temperature of 20

C to a final temperature of 100

C. Calculate the

difference in the increase in length between the two rods. The coefficient of

linear expansion for steel is 12 ×10−6/◦Cand for brass is 18 ×10−6/◦C.

Solution

Step 1: Calculate the increase in length of the steel rod. Given the coefficient

of linear expansion for steel is 12 ×10−6/◦C, the change in length of the steel

rod can be calculated using the formula:

∆Lsteel =L0·αsteel ·∆T

where L0is the initial length of the steel rod, αsteel is the coefficient of linear

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

into the formula:

∆Lsteel = 2 ·12 ×10−6·(100 −20)

∆Lsteel = 2 ·12 ×10−6·80

∆Lsteel = 1.92 ×10−3m

Step 2: Calculate the increase in length of the brass rod. Given the coefficient

of linear expansion for brass is 18 ×10−6/◦C, the change in length of the brass

rod can be calculated using the same formula:

∆Lbrass =L0·αbrass ·∆T

Substitute the values into the formula:

∆Lbrass = 3 ·18 ×10−6·(100 −20)

∆Lbrass = 3 ·18 ×10−6·80

∆Lbrass = 4.32 ×10−3m

Step 3: Calculate the difference in the increase in length between the two

rods. The difference in the increase in length between the two rods is:

∆Lbrass −∆Lsteel = 4.32 ×10−3−1.92 ×10−3

∆Lbrass −∆Lsteel = 2.4×10−3m

Therefore, the difference in the increase in length between the steel and brass

rods is 2.4×10−3meters.

Question 11

Question

A steel beam of length 10 m is heated from 20

C to 120

C. Calculate the change

in length of the beam if the coefficient of linear expansion of steel is 12 ×10−6

per degree Celsius.

Solution

Step 1: Determine the initial length of the beam Given: Initial length of the

beam, L0= 10 m

Step 2: Calculate the change in temperature The change in temperature,

∆T=Tf−TiGiven: Initial temperature, Ti= 20

C Final temperature, Tf=

120

C Therefore, ∆T= 120 −20 = 100

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

α·L0·∆TGiven: Coefficient of linear expansion, α= 12 ×10−6per

C Initial

length, L0= 10 m Change in temperature, ∆T= 100

C Substitute the values

into the formula: ∆L= 12 ×10−6·10 ·100

Step 4: Calculate the change in length ∆L= 12 ×10−6·10 ·100 = 0.012 m

Therefore, the change in length of the steel beam is 0.012 meters when heated

from 20

C to 120

Question 12

Question

A steel bridge is constructed to have a length of 250 meters at a temperature of

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

C−1, what

will be the length of the bridge when the temperature rises to 40

Solution

Let’s denote the initial length of the bridge as L0= 250 m, the coefficient of

linear expansion as α= 1.2×10−5

C−1, and the change in temperature as

∆T= 40 −20 = 20

Step 1: Calculate the change in length of the bridge. The change in length

of the bridge can be calculated using the formula:

∆L=L0·α·∆T

∆L= 250 ×1.2×10−5×20

∆L= 0.06 m

Step 2: Determine the final length of the bridge. The final length of the

bridge can be found by adding the change in length to the initial length.

Lfinal =L0+ ∆L

Lfinal = 250 + 0.06

Lfinal = 250.06 m

Therefore, when the temperature rises to 40

C, the length of the steel bridge

will be 250.06 meters.

Question 13

Question

A steel pipe of length 10 m is heated from 20

C to 120

C. If the coefficient of

linear expansion of steel is 12 ×10−6K−1, find the change in length of the pipe.

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= 120C−20C= 100C

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

pipe can be determined using the formula:

∆L=L·α·∆T

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

and ∆Tis the change in temperature.

Step 3: Substitute the known values into the formula. Given that L= 10 m

and α= 12 ×10−6K−1, we can calculate the change in length:

∆L= 10 m ·12 ×10−6K−1·100C

Step 4: Solve for the change in length.

∆L= 10 m ·12 ×10−6K−1·100 = 1.2 mm

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

C to

120

C is 1.2 mm.

Question 14

Question

A steel rod has a length of 2 meters at 20

C. If the coefficient of linear expansion

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

is increased to 100

Solution

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

of linear expansion as α= 1.2×10−5/C, the original temperature as T0= 20C,

and the final temperature as Tf= 100C.

Step 1: Calculate the change in temperature. Since the temperature changes

from 20Cto 100C, the change in temperature is:

∆T=Tf−T0= 100C−20C= 80C

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

a material with a linear expansion coefficient αdue to a temperature change

∆Tcan be calculated using the formula:

∆L=L0α∆T

Step 3: Calculate the change in length. Substitute the given values into

the formula:

∆L= 2 ×1.2×10−5/C ×80C= 0.00192 meters

Step 4: Determine the final length. The final length Lfof the steel rod can

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

Lf=L0+ ∆L= 2 meters + 0.00192 meters = 2.00192 meters

Therefore, when the temperature is increased to 100

C, the length of the

steel rod will be 2.00192 meters.

Question 15

Question

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

1.2×10−5◦C−1, determine how much its length will increase when heated to

120◦C.

Solution

Step 1: Let’s first calculate the change in temperature (∆T). Given that the

initial temperature is 20◦C and the final temperature is 120◦C, we have:

∆T= 120◦C−20◦C = 100◦C

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

in length (∆L) of the steel rod. The formula for linear expansion is given by:

∆L=Lα∆T

where: ∆L= change in length, L= initial length of the rod, α= coefficient of

linear expansion, and ∆T= change in temperature.

Substitute the given values:

∆L= 2 ×1.2×10−5×100

Step 3: Now, we can calculate the change in length (∆L).

∆L= 2 ×1.2×10−5×100 = 0.0024 m

Therefore, the length of the steel rod will increase by 0.0024 meters when

heated to 120◦C.

Question 16

Question

A brass rod and an aluminum rod are both initially 1 meter in length at 0◦C.

If the temperature increases by 100◦C, by how much will the total length of the

two rods increase? The linear expansion coefficients for brass and aluminum are

19 ×10−6/◦Cand 23 ×10−6/◦Crespectively.

Solution

Step 1: Calculate the increase in length of the brass rod. The increase in length

of a material is given by the formula:

∆L=αL∆T

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

the change in temperature. For brass:

∆Lbrass = (19 ×10−6)×1×100 = 0.0019 meters

Step 2: Calculate the increase in length of the aluminum rod. For aluminum:

∆Laluminum = (23 ×10−6)×1×100 = 0.0023 meters

Step 3: Calculate the total increase in length of the two rods. The total

increase in length is the sum of the increases in length of the brass and aluminum

rods:

∆Ltotal = ∆Lbrass + ∆Laluminum = 0.0019 + 0.0023 = 0.0042 meters

Therefore, the total length of the two rods will increase by 0.0042 meters

when the temperature increases by 100◦C.

Question 17

Question

A steel rod with an initial length of 2 meters is heated from 20

C to 80

C. If the

linear expansion coefficient of steel is 1.2×10−5per degree Celsius, calculate

the final length of the rod.

Solution

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

C Final temperature, Tf= 80

The change in temperature, ∆T=Tf−Ti= 80 −20 = 60

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

given by:

∆L=αL∆T

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

change in temperature.

Given: α= 1.2×10−5L= 2 m ∆T= 60

Substitute the values into the formula:

∆L= (1.2×10−5)×2×60

∆L= 1.44 ×10−4×2×60

∆L= 1.44 ×10−4×120

∆L= 0.01728 m

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

by:

Lf=L+ ∆L

Substitute the values into the formula:

Lf= 2 + 0.01728

Lf= 2.01728 m

Therefore, the final length of the rod after heating is 2.01728 meters.

Question 18

Question

A steel rod has a length of 2.0 m at 20◦C. If the rod is heated to 120◦C, how

much longer will the rod become? The coefficient of linear expansion for steel

is 1.2×10−5K−1.

Solution

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

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

Tf−Ti= 120◦C - 20◦C = 100K

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

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

∆L=L0α∆T

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

for steel, ∆Tis the change in temperature.

Given: Initial length, L0= 2.0 m Coefficient of linear expansion, α= 1.2×

10−5K−1Change in temperature, ∆T= 100K

Substitute the values into the formula:

∆L= (2.0 m)(1.2×10−5K−1)(100 K)

Step 3: Calculate the change in length.

∆L= 2.4×10−4m=0.24 mm

Therefore, the steel rod will become 0.24 mm longer when heated from 20◦C

to 120◦C.

Question 19

Question

A steel rod has a length of 2.0 m at 20

C. If the rod is heated to 75

C, what

will be its new length? (Assume the linear expansion coefficient of steel is

12 ×10−6/C)

Solution

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

Tiis 20

C and the final temperature Tfis 75

C, the change in temperature is:

∆T=Tf−Ti= 75C−20C= 55C

Step 2: Calculate the linear expansion of the steel rod. The linear expansion

of the steel rod can be calculated using the formula:

∆L=L0·α·∆T

where ∆Lis the change in length, L0is the original length, αis the linear

expansion coefficient, and ∆Tis the change in temperature. Substituting the

values:

∆L= 2.0 m ·12 ×10−6/C ·55C= 0.00132 m = 1.32 mm

Step 3: Calculate the new length of the steel rod. The new length Lfof

the steel rod can be calculated by adding the change in length to the original

length:

Lf=L0+∆L= 2.0 m+0.00132 m = 2.00132 m = 2.001 m (to 3 decimal places)

Therefore, the new length of the steel rod when heated to 75

C is 2.001 m.

Question 20

Question

A copper rod is 2.5 m long at 20

C. If the rod is heated to 80

C, what will be

its new length? The coefficient of linear expansion for copper is 1.7×10−5K−1.

Solution

Step 1: First, calculate the change in temperature:

∆T=Tf−Ti= 80C−20C= 60C

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

(∆L) of the rod:

∆L=L0α∆T

where L0is the original length, αis the coefficient of linear expansion, and ∆T

is the change in temperature.

Step 3: Substitute the given values into the formula:

∆L= 2.5 m ×(1.7×10−5K−1)×60C

Step 4: Calculate the change in length:

∆L= 2.5×1.7×10−5×60

Step 5: Simplify the expression:

∆L= 2.55 ×10−4m

Step 6: Finally, find the new length by adding the change in length to the

original length:

Lf=L0+ ∆L= 2.5 m + 2.55 ×10−4m

Step 7: Calculate the new length:

Lf= 2.5+2.55 ×10−4

Lf= 2.500255 m

Therefore, the new length of the copper rod when heated to 80

C is approx-

imately 2.500255 meters.

Question 21

Question

A brass rod with a length of 2.5 m and a diameter of 1.0 cm is heated from 20◦C

to 70◦C. Calculate the change in length of the rod due to thermal expansion.

(Coefficient of linear expansion for brass is 19 ×10−6K−1)

Solution

Step 1: Calculate the initial volume of the rod. The initial volume of the rod

can be calculated using the formula for the volume of a cylinder:

V=πr2h

where ris the radius of the rod, his the initial length of the rod, and π≈

3.14159. Given r= 0.01 m and h= 2.5 m, we have:

V=π(0.01)2×2.5

Step 2: Calculate the final volume of the rod. The final volume of the rod can

be calculated using the same formula, but with the new length after expansion.

The new length can be calculated using the formula for linear expansion:

∆L=αL0∆T

where αis the coefficient of linear expansion, L0is the initial length, and ∆T

is the change in temperature. With α= 19 ×10−6K−1,L0= 2.5 m, and

∆T= 70◦C−20◦C = 50◦C, we can find the new length.

Step 3: Calculate the change in volume. The change in volume can be

calculated as the difference between the final volume and the initial volume:

∆V=Vfinal −Vinitial

Step 4: Calculate the change in length. Since the rod expands uniformly,

the change in length is related to the change in volume as follows:

∆V=πr2∆L

We can solve for ∆Lto find the change in length of the rod due to thermal

expansion.

Question 22

Question

A rod made of steel is initially 1.5 m long at 20◦C. If the coefficient of linear

expansion for steel is 12 ×10−6◦C−1, find the temperature at which the rod’s

length will increase to 1.51 m.

Solution

Let Lbe the original length of the steel rod, ∆Lbe the change in length, α

be the coefficient of linear expansion, T0be the original temperature, Tfbe the

final temperature, and Tbe the temperature at which the rod’s length increases

to 1.51 m.

Step 1: First, we find the change in length when the temperature changes

from T0to Tf:

∆L=Lα(Tf−T0)

Step 2: Substitute the given values into the formula:

∆L= 1.5×12 ×10−6(Tf−20)

Step 3: Since the final length of the rod is 1.51 m, we have:

L+ ∆L= 1.51

1.5+1.5×12 ×10−6(Tf−20) = 1.51

Step 4: Solve for Tf:

1.5+1.5×12 ×10−6Tf−1.5×12 ×10−6×20 = 1.51

1.5+1.8×10−5Tf−3.6×10−5= 1.51

1.8×10−5Tf= 1.51 −1.5+3.6×10−5

1.8×10−5Tf= 1.01 + 3.6×10−5

Tf=1.01 + 3.6×10−5

1.8×10−5

Tf≈62.22◦C

Therefore, the temperature at which the rod’s length will increase to 1.51 m

is approximately 62.22◦C.

Question 23

Question

A steel rod of length 2 m is heated from 20

C to 100

C. If the coefficient of

linear expansion for steel is 1.2×10−5

C−1, determine the change in length of

the rod.

Solution

Step 1: Calculate the initial length change due to heating from 20

C to 100

Given that the initial length of the steel rod is 2 m, we have:

∆L=L·α·∆T

∆L= 2 m ·1.2×10−5

C−1·(100 −20)

∆L= 2 ×1.2×10−3m

∆L= 2.4×10−3m

Step 2: Determine the change in length of the rod. Since the steel rod

expands in both directions, the total change in length is twice the initial change

in length:

Change in Length = 2 ×2.4×10−3m

Change in Length = 4.8×10−3m

Therefore, the change in length of the steel rod is 4.8×10−3m.

Question 24

Question

A brass rod of length 2.0 m is heated from 20

C to 120

C. If the coefficient of

linear expansion of brass is 1.9×10−5K−1, what is the change in length of the

rod?

Solution

Step 1: Calculate the initial length of the rod Given: Initial length of the brass

rod, L0= 2.0 m Coefficient of linear expansion of brass, α= 1.9×10−5K−1

Change in temperature, ∆T= 120C−20C= 100CUsing the formula for linear

expansion:

∆L=L0α∆T

Substitute the values:

∆L= 2.0×1.9×10−5×100

∆L= 2.0×1.9×10−3

∆L= 3.8×10−3m

Therefore, the change in length of the brass rod is 3.8 mm.

Question 25

Question

A steel rod is initially 2 meters long at 20◦C. What temperature increase is

required to increase the length of the rod by 1 cm if the coefficient of linear

expansion of steel is 12 ×10−6/◦C?

Solution

Let Lbe the original length of the steel rod, ∆Lbe the change in length, ∆T

be the change in temperature, and αbe the coefficient of linear expansion.

Step 1: Write out the formula for linear expansion: The change in length

∆Lof a material is given by the formula:

∆L=Lα∆T

Step 2: Use the given information: We know that L= 2m, ∆L= 1cm =

0.01m, and α= 12 ×10−6/◦C. We want to find ∆T.

Step 3: Plug the values into the formula:

∆L=Lα∆T

0.01 = (2)(12 ×10−6)(∆T)

Step 4: Solve for ∆T: Dividing both sides by (2)(12 ×10−6),

∆T=0.01

2×12 ×10−6

∆T=0.01

24 ×10−6

∆T=0.01

24 ×106

∆T=1

2400 ×106

∆T=1000

2400

∆T= 0.4167 ◦C

So, the temperature needs to be increased by 0.4167◦Cto increase the length

of the rod by 1 cm.

Question 26

Question

A steel rod with a length of 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, what is the change in length of

the rod?

Solution

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

Step 2: Determine the change in length of the rod using the coefficient of linear

expansion.

Step 1: Calculate the initial length change due to the change in tempera-

ture.

The initial length change due to the change in temperature can be calculated

using the formula:

∆L=L0·α·∆T

where: ∆L= change in length, L0= 2.0 m (initial length), α= 1.2×10−5/

(coefficient of linear expansion for steel), ∆T= 120◦C−20◦C = 100◦C.

Substitute the given values and calculate:

∆L= 2.0 m ×1.2×10−5

C−1×100

∆L= 0.0024 m

Step 2: Determine the change in length of the rod using the coefficient of

linear expansion.

The final change in length of the rod is equal to the initial length change

due to the change in temperature:

∆L= 0.0024 m

Therefore, the change in length of the steel rod is 0.0024 m.

Question 27

Question

A brass rod of length 2 m at 0◦C is clamped at its two ends. By how much

does its length change when the temperature of the rod is raised to 100◦C?

(Coefficient of linear expansion of brass = 1.9×10−5/◦C)

Solution

Let L0be the original length of the brass rod at 0◦C, ∆Tbe the change in

temperature, αbe the coefficient of linear expansion of brass, and ∆Lbe the

change in length.

Step 1: Calculate the change in temperature. Given ∆T= 100◦C - 0◦C =

100◦C

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

sion:

∆L=α·L0·∆T

∆L= (1.9×10−5/◦C) ·(2 m) ·(100◦C)

∆L= 0.000038 m

Therefore, the length of the brass rod changes by 0.000038 m when the

temperature is raised to 100◦C.

Question 28

Question

A steel rod of length 2.0 m undergoes a temperature increase of 100

C. If the

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

C−1, what is the change in

length of the rod?

Solution

Step 1: We can use the formula for linear expansion ∆L=α·L·∆T, where:

∆L= change in length, α= coefficient of linear expansion, L= original length,

and ∆T= change in temperature.

Step 2: Substituting the given values into the formula, we have: ∆L=

(12 ×10−6

C−1)·(2.0 m) ·(100

Step 3: Calculating the change in length, we get: ∆L= 12×10−6·2.0·100 =

0.0024 m

Step 4: Therefore, the change in length of the steel rod is 0.0024 meters.

Question 29

Question

A solid iron rod has a length of 2.0 m at 20◦C. If the rod is heated to 120◦C,

what will be its new length? The coefficient of linear expansion for iron is

1.2×10−5per degree Celsius.

Solution

Let’s denote the initial length of the iron rod as Liat 20◦C and the final length

after heating as Lfat 120◦C. The change in length, ∆L, can be calculated using

the formula for linear expansion:

∆L=Li·α·∆T

where: Li= 2.0 m (initial length), α= 1.2×10−5K−1(coefficient of linear

expansion for iron), and ∆T= 120◦C−20◦C = 100 K (change in temperature).

Step 1: Calculate the change in length ∆L.

∆L= 2.0 m ×1.2×10−5K−1×100 K

∆L= 0.0024 m = 2.4 mm

Step 2: Calculate the final length Lf.

Lf=Li+ ∆L

Lf= 2.0 m + 0.0024 m

Lf= 2.0024 m

The new length of the iron rod when heated to 120◦C is 2.0024 m, or 2.0024×

103mm.

Question 30

Question

A steel rod is 1 meter long 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

100◦C.

Solution

Given: Initial length of the steel rod, L0= 1 m

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

C−1

Change in temperature, ∆T= 100◦C−20◦C = 80◦C

From the formula for linear expansion:

∆L=L0α∆T

Step 1: Calculate the change in length of the steel rod.

∆L= 1 ×12 ×10−6×80 = 0.000096 m

So, the change in length of the rod when heated to 100◦C is 0.000096 meters.

Question 31

Question

A steel rod has a length of 2.00 m at 20

C. If the temperature of the rod is

increased to 120

C, what is the new length of the rod? The linear expansion

coefficient of steel is 12 ×10−6

C−1.

Solution

Step 1: Calculate the change in length of the steel rod. The change in length of

the rod can be calculated using the formula:

∆L=α·L·∆T

where ∆L= change in length, α= 12 ×10−6

C−1(linear expansion coefficient

of steel), L= 2.00 m (initial length of the rod), ∆T= 120 −20 = 100

C (change

in temperature).

Substitute the values into the formula:

∆L= 12 ×10−6·2.00 ·100

Step 2: Calculate the new length of the steel rod. The new length of the rod

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

Lnew =L+ ∆L

Substitute the values and calculate:

Lnew = 2.00 + 12 ×10−6·2.00 ·100

Question 32

Question

A brass container is filled with 0.5 L of water at 20

C. If the container is then

heated to 70

C, calculate the change in volume of the container. Assume the

linear expansion coefficient of brass is 19 ×10−6K−1and the coefficient of

volume expansion for water is 2.1×10−4K−1.

Solution

Step 1: Calculate the initial volume of the water: Given that the initial volume

of water is 0.5 L, we have Vi= 0.5 L.

Step 2: Calculate the change in temperature: The change in temperature is

∆T= 70C−20C= 50C.

Step 3: Calculate the change in volume of the water: We will calculate the

change in volume of water first. The coefficient of volume expansion for water,

βwater = 2.1×10−4K−1.

The change in volume of water is given by:

∆Vwater =Viβwater∆T

∆Vwater = 0.5 L ×2.1×10−4K−1×50 K

∆Vwater = 0.000525 L

Step 4: Calculate the change in volume of the brass container: Now, we will

calculate the change in volume of the brass container. The linear expansion

coefficient of brass, αbrass = 19 ×10−6K−1.

The change in volume of the brass container is given by:

∆Vbrass =Viαbrass∆T

∆Vbrass = 0.5 L ×19 ×10−6K−1×50 K

∆Vbrass = 0.000475 L

Step 5: Calculate the total change in volume of the container: The total

change in volume is the sum of the changes in volume of the water and the

brass container:

∆Vtotal = ∆Vwater + ∆Vbrass

∆Vtotal = 0.000525 L + 0.000475 L

∆Vtotal = 0.001 L

Therefore, the change in volume of the brass container when heated to 70

is 0.001 L.

Question 33

Question

A steel rod has a length of 2 meters at a temperature of 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 ×10−6per degree Celsius.

Solution

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

expansion to find the new length of the steel rod.

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

is given by:

∆T=Tf−Ti= 120C−20C= 100C

Step 2: Use the formula for linear expansion to find the new length of the

steel rod. The change in length of the steel rod can be calculated using the

formula:

∆L=α·L·∆T

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

and ∆Tis the change in temperature.

Substitute the values into the formula:

∆L= (12 ×10−6)·2·100 = 0.0024 meters

The new length of the steel rod is:

Lf=Li+ ∆L= 2 + 0.0024 = 2.0024 meters

Therefore, when the steel rod is heated to 120

C, its new length will be

2.0024 meters.

Question 34

Question

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

of linear expansion for steel is 1.2×10−5◦C−1, calculate the change in length

of the rod when the temperature is increased to 100◦C.

Solution

Let L1be the initial length of the rod and L2be the final length of the rod.

Given: Initial length, L1= 2 m Change in temperature, ∆T= 100◦C−20◦C =

80◦C Coefficient of linear expansion, α= 1.2×10−5◦C−1

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

sion:

L2=L1(1 + α∆T)

Step 2: Substitute the given values into the formula:

L2= 2(1 + 1.2×10−5×80)

L2= 2(1 + 0.00096)

L2= 2.00192

Step 3: Calculate the change in length:

∆L=L2−L1

∆L= 2.00192 −2

∆L= 0.00192 m

Therefore, the change in length of the steel rod when the temperature is

increased to 100◦C is 0.00192 meters.

Question 35

Question

A copper rod initially measures 2 meters in length at a temperature of 20◦C.

If the coefficient of linear expansion for copper is 17 ×10−6◦C−1, what is the

length of the rod when the temperature increases to 120◦C?

Solution

Step 1: First, we need to calculate the change in temperature. Given that the

initial temperature is 20◦C and the final temperature is 120◦C, the change in

temperature is:

∆T= 120◦C−20◦C = 100◦C

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

∆L=α·L·∆T

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

and ∆Tis the change in temperature.

Substitute the values:

∆L= 17 ×10−6◦C−1·2 m ·100◦C=0.0034 m

Step 3: Finally, we can find the length of the rod at 120◦C by adding the

change in length to the initial length:

Lfinal =Linitial + ∆L= 2 m + 0.0034 m = 2.0034 m

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

120◦C is 2.0034 meters.

Question 2

Question

A brass rod of length 2 m and diameter 1 cm is heated from 20

C to 120

Calculate the increase in length of the rod due to thermal expansion. (Coefficient

of linear expansion of brass = 1.9×10−5per degree Celsius).

Solution

Step 1: Calculate the initial volume of the brass rod. The initial volume Viof

the brass rod can be calculated using the formula for the volume of a cylinder:

Vi=πr2h

where ris the initial radius and his the initial length of the rod. Given that

the diameter is 1 cm, the initial radius rcan be calculated as:

r=1 cm

2= 0.5 cm = 0.005 m

Therefore, the initial volume Viis:

Vi=π×(0.005)2×2≈1.57 ×10−4m3

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

∆Lof the rod can be calculated using the formula:

∆L=α·L·∆T

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

is the change in temperature. Given that the coefficient of linear expansion

of brass is 1.9×10−5per degree Celsius, the initial length Lis 2 m, and the

change in temperature is 120

C - 20

C = 100

C. Substitute these values into the

formula:

∆L= 1.9×10−5×2×100 = 0.0038 m

Therefore, the increase in length of the brass rod due to thermal expansion

is 0.0038 meters.

Question 3

Question

A steel rod has a length of 2 meters at 20

C. If the coefficient of linear expansion

for steel is 1.2×10−5K−1, by how much does the length of the rod increase

when the temperature is raised to 100

Solution

Let’s denote the original length of the rod at 20

C as L0= 2 m, the final tem-

perature as Tf= 100C, the initial temperature as Ti= 20C, and the coefficient

of linear expansion for steel as α= 1.2×10−5K−1.

Step 1: Calculate the change in temperature.

∆T=Tf−Ti= 100C −20C = 80C

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

sion:

∆L=L0·α·∆T

∆L= 2 m ·1.2×10−5K−1·80C = 0.00192 m

Therefore, the length of the rod increases by 0.00192 meters when the tem-

perature is raised to 100

Question 4

Question

A steel rod of length 2 m is heated from 20

C to 120

C. If the coefficient of

linear expansion of steel is 12 ×10−6

C−1, calculate the change in length of the

rod.

Solution

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

Final temperature, Tf= 120

Initial temperature, Ti= 20

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

C−1

Recall the formula for linear expansion:

∆L=L0α∆T

Step 1: Calculate the change in temperature

∆T=Tf−Ti= 120 −20 = 100

Step 2: Calculate the change in length of the rod

∆L= 2 ×12 ×10−6×100 = 0.0024 m

Therefore, the change in length of the steel rod is 0.0024 m.

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 is the final length of the rod? Given that the linear

expansion coefficient of steel is 1.2×10−5per degree Celsius.

Solution

Step 1: Calculate the change in temperature. Since the rod is heated from 20

to 100

C, the change in temperature (∆T) is:

∆T= 100C−20C= 80C

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

be calculated using the formula:

∆L=αL0∆T

where: ∆L= change in length, α= linear expansion coefficient of steel, L0=

initial length of the rod, ∆T= change in temperature.

Substitute the values into the formula:

∆L= (1.2×10−5per

C) ×(2 m) ×(80

∆L= 0.000024 m

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

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

Lfinal =L0+ ∆L

Lfinal = 2 m + 0.000024 m

Lfinal ≈2.000024 m

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

C is approx-

imately 2.000024 meters.

Question 6

Question

A solid metal rod has a length of 1.5 m at 0

C. If the coefficient of linear

expansion of the metal is 2 ×10−5/C, find the change in length of the rod when

the temperature is raised to 100

Solution

Step 1: First, we need to calculate the change in temperature. Given: Initial

length of the rod, L0= 1.5 m Coefficient of linear expansion, α= 2 ×10−5/C

Final temperature, Tf= 100

The change in temperature, ∆T=Tf−Ti= 100C−0C= 100C

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

linear expansion

∆L=α·L0·∆T

Substitute the values:

∆L= 2 ×10−5/C ×1.5 m ×100C

Step 3: Calculate the change in length

∆L= 2 ×10−5×1.5×100 = 3 ×10−3m

Therefore, the change in length of the rod when the temperature is raised

to 100

C is 0.003 m.

Question 7

Question

A steel rod with a length of 2 meters has a coefficient of linear expansion of

1.2×10−5per degree Celsius. If the temperature increases by 50 degrees Celsius,

what will be the change in length of the rod?

Solution

Step 1: We can use the formula for linear expansion to find the change in length

of the rod. Step 2: The formula for linear expansion is given by ∆L=α·L·∆T,

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

is the original length of the rod, and - ∆Tis the change in temperature. Step 3:

Substituting the given values into the formula, we have: ∆L= (1.2×10−5)·2·50.

Step 4: Solving the expression, we get: ∆L= 1.2×10−5·2·50 = 0.0012 meters.

Step 5: Therefore, the change in length of the steel rod will be 0.0012 meters.

Question 8

Question

A steel rod of length 2 m at 20

C has a hole drilled through its diameter. At

what temperature will the rod be at if the hole closes up? Assume the coefficient

of linear expansion for steel is 1.2×10−5

C−1.

Solution

Step 1: Identify the given information. Let L0= 2 m be the original length

of the steel rod at 20

C and Lbe the length of the rod at the unknown final

temperature, T.

Step 2: Determine the change in length of the steel rod due to temperature

change. The change in length, ∆L, can be calculated using the formula:

∆L=αL0∆T,

where α= 1.2×10−5

C−1is the coefficient of linear expansion for steel and

∆T=T−20 is the change in temperature.

Step 3: Find the condition for the hole to close up. For the hole to close up,

the change in length of the rod must be equal to the length of the hole. Thus:

∆L=L0.

Step 4: Substitute and solve for the final temperature, T. Substitute the

expressions for ∆Land L0into the equation:

αL0∆T=L0.

Solving for ∆Tgives:

∆T=1

α.

Finally, substitute the known value of αinto the equation and solve for Tto

find the final temperature at which the hole closes up.

Question 9

Question

A copper rod has a length of 2.0 m at 20

C. If the temperature is increased to

120

C, what is the new length of the rod? (Coefficient of linear expansion for

copper is 1.7×10−5/

Solution

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

T1= 20CFinal temperature T2= 120C

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

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

∆L=α·L·∆T

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

length, ∆T= change in temperature.

Substitute the values:

∆L= (1.7×10−5)×2.0×100

Step 3: Now, calculate the new length of the rod:

Lfinal =Linitial + ∆L

Substitute the values:

Lfinal = 2.0+∆L

Step 4: Calculate the final length. Thus,

Lfinal = 2.0 + (1.7×10−5×2.0×100)

Final length is:

Lfinal = 2.0+0.0034 = 2.0034 m

Question 10

Question

A steel rod of length 2 m and a brass rod of length 3 m are both heated from

an initial temperature of 20

C to a final temperature of 100

C. Calculate the

difference in the increase in length between the two rods. The coefficient of

linear expansion for steel is 12 ×10−6/◦Cand for brass is 18 ×10−6/◦C.

Solution

Step 1: Calculate the increase in length of the steel rod. Given the coefficient

of linear expansion for steel is 12 ×10−6/◦C, the change in length of the steel

rod can be calculated using the formula:

∆Lsteel =L0·αsteel ·∆T

where L0is the initial length of the steel rod, αsteel is the coefficient of linear

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

into the formula:

∆Lsteel = 2 ·12 ×10−6·(100 −20)

∆Lsteel = 2 ·12 ×10−6·80

∆Lsteel = 1.92 ×10−3m

Step 2: Calculate the increase in length of the brass rod. Given the coefficient

of linear expansion for brass is 18 ×10−6/◦C, the change in length of the brass

rod can be calculated using the same formula:

∆Lbrass =L0·αbrass ·∆T

Substitute the values into the formula:

∆Lbrass = 3 ·18 ×10−6·(100 −20)

∆Lbrass = 3 ·18 ×10−6·80

∆Lbrass = 4.32 ×10−3m

Step 3: Calculate the difference in the increase in length between the two

rods. The difference in the increase in length between the two rods is:

∆Lbrass −∆Lsteel = 4.32 ×10−3−1.92 ×10−3

∆Lbrass −∆Lsteel = 2.4×10−3m

Therefore, the difference in the increase in length between the steel and brass

rods is 2.4×10−3meters.

Question 11

Question

A steel beam of length 10 m is heated from 20

C to 120

C. Calculate the change

in length of the beam if the coefficient of linear expansion of steel is 12 ×10−6

per degree Celsius.

Solution

Step 1: Determine the initial length of the beam Given: Initial length of the

beam, L0= 10 m

Step 2: Calculate the change in temperature The change in temperature,

∆T=Tf−TiGiven: Initial temperature, Ti= 20

C Final temperature, Tf=

120

C Therefore, ∆T= 120 −20 = 100

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

α·L0·∆TGiven: Coefficient of linear expansion, α= 12 ×10−6per

C Initial

length, L0= 10 m Change in temperature, ∆T= 100

C Substitute the values

into the formula: ∆L= 12 ×10−6·10 ·100

Step 4: Calculate the change in length ∆L= 12 ×10−6·10 ·100 = 0.012 m

Therefore, the change in length of the steel beam is 0.012 meters when heated

from 20

C to 120

Question 12

Question

A steel bridge is constructed to have a length of 250 meters at a temperature of

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

C−1, what

will be the length of the bridge when the temperature rises to 40

Solution

Let’s denote the initial length of the bridge as L0= 250 m, the coefficient of

linear expansion as α= 1.2×10−5

C−1, and the change in temperature as

∆T= 40 −20 = 20

Step 1: Calculate the change in length of the bridge. The change in length

of the bridge can be calculated using the formula:

∆L=L0·α·∆T

∆L= 250 ×1.2×10−5×20

∆L= 0.06 m

Step 2: Determine the final length of the bridge. The final length of the

bridge can be found by adding the change in length to the initial length.

Lfinal =L0+ ∆L

Lfinal = 250 + 0.06

Lfinal = 250.06 m

Therefore, when the temperature rises to 40

C, the length of the steel bridge

will be 250.06 meters.

Question 13

Question

A steel pipe of length 10 m is heated from 20

C to 120

C. If the coefficient of

linear expansion of steel is 12 ×10−6K−1, find the change in length of the pipe.

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= 120C−20C= 100C

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

pipe can be determined using the formula:

∆L=L·α·∆T

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

and ∆Tis the change in temperature.

Step 3: Substitute the known values into the formula. Given that L= 10 m

and α= 12 ×10−6K−1, we can calculate the change in length:

∆L= 10 m ·12 ×10−6K−1·100C

Step 4: Solve for the change in length.

∆L= 10 m ·12 ×10−6K−1·100 = 1.2 mm

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

C to

120

C is 1.2 mm.

Question 14

Question

A steel rod has a length of 2 meters at 20

C. If the coefficient of linear expansion

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

is increased to 100

Solution

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

of linear expansion as α= 1.2×10−5/C, the original temperature as T0= 20C,

and the final temperature as Tf= 100C.

Step 1: Calculate the change in temperature. Since the temperature changes

from 20Cto 100C, the change in temperature is:

∆T=Tf−T0= 100C−20C= 80C

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

a material with a linear expansion coefficient αdue to a temperature change

∆Tcan be calculated using the formula:

∆L=L0α∆T

Step 3: Calculate the change in length. Substitute the given values into

the formula:

∆L= 2 ×1.2×10−5/C ×80C= 0.00192 meters

Step 4: Determine the final length. The final length Lfof the steel rod can

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

Lf=L0+ ∆L= 2 meters + 0.00192 meters = 2.00192 meters

Therefore, when the temperature is increased to 100

C, the length of the

steel rod will be 2.00192 meters.

Question 15

Question

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

1.2×10−5◦C−1, determine how much its length will increase when heated to

120◦C.

Solution

Step 1: Let’s first calculate the change in temperature (∆T). Given that the

initial temperature is 20◦C and the final temperature is 120◦C, we have:

∆T= 120◦C−20◦C = 100◦C

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

in length (∆L) of the steel rod. The formula for linear expansion is given by:

∆L=Lα∆T

where: ∆L= change in length, L= initial length of the rod, α= coefficient of

linear expansion, and ∆T= change in temperature.

Substitute the given values:

∆L= 2 ×1.2×10−5×100

Step 3: Now, we can calculate the change in length (∆L).

∆L= 2 ×1.2×10−5×100 = 0.0024 m

Therefore, the length of the steel rod will increase by 0.0024 meters when

heated to 120◦C.

Question 16

Question

A brass rod and an aluminum rod are both initially 1 meter in length at 0◦C.

If the temperature increases by 100◦C, by how much will the total length of the

two rods increase? The linear expansion coefficients for brass and aluminum are

19 ×10−6/◦Cand 23 ×10−6/◦Crespectively.

Solution

Step 1: Calculate the increase in length of the brass rod. The increase in length

of a material is given by the formula:

∆L=αL∆T

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

the change in temperature. For brass:

∆Lbrass = (19 ×10−6)×1×100 = 0.0019 meters

Step 2: Calculate the increase in length of the aluminum rod. For aluminum:

∆Laluminum = (23 ×10−6)×1×100 = 0.0023 meters

Step 3: Calculate the total increase in length of the two rods. The total

increase in length is the sum of the increases in length of the brass and aluminum

rods:

∆Ltotal = ∆Lbrass + ∆Laluminum = 0.0019 + 0.0023 = 0.0042 meters

Therefore, the total length of the two rods will increase by 0.0042 meters

when the temperature increases by 100◦C.

Question 17

Question

A steel rod with an initial length of 2 meters is heated from 20

C to 80

C. If the

linear expansion coefficient of steel is 1.2×10−5per degree Celsius, calculate

the final length of the rod.

Solution

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

C Final temperature, Tf= 80

The change in temperature, ∆T=Tf−Ti= 80 −20 = 60

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

given by:

∆L=αL∆T

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

change in temperature.

Given: α= 1.2×10−5L= 2 m ∆T= 60

Substitute the values into the formula:

∆L= (1.2×10−5)×2×60

∆L= 1.44 ×10−4×2×60

∆L= 1.44 ×10−4×120

∆L= 0.01728 m

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

by:

Lf=L+ ∆L

Substitute the values into the formula:

Lf= 2 + 0.01728

Lf= 2.01728 m

Therefore, the final length of the rod after heating is 2.01728 meters.

Question 18

Question

A steel rod has a length of 2.0 m at 20◦C. If the rod is heated to 120◦C, how

much longer will the rod become? The coefficient of linear expansion for steel

is 1.2×10−5K−1.

Solution

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

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

Tf−Ti= 120◦C - 20◦C = 100K

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

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

∆L=L0α∆T

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

for steel, ∆Tis the change in temperature.

Given: Initial length, L0= 2.0 m Coefficient of linear expansion, α= 1.2×

10−5K−1Change in temperature, ∆T= 100K

Substitute the values into the formula:

∆L= (2.0 m)(1.2×10−5K−1)(100 K)

Step 3: Calculate the change in length.

∆L= 2.4×10−4m=0.24 mm

Therefore, the steel rod will become 0.24 mm longer when heated from 20◦C

to 120◦C.

Question 19

Question

A steel rod has a length of 2.0 m at 20

C. If the rod is heated to 75

C, what

will be its new length? (Assume the linear expansion coefficient of steel is

12 ×10−6/C)

Solution

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

Tiis 20

C and the final temperature Tfis 75

C, the change in temperature is:

∆T=Tf−Ti= 75C−20C= 55C

Step 2: Calculate the linear expansion of the steel rod. The linear expansion

of the steel rod can be calculated using the formula:

∆L=L0·α·∆T

where ∆Lis the change in length, L0is the original length, αis the linear

expansion coefficient, and ∆Tis the change in temperature. Substituting the

values:

∆L= 2.0 m ·12 ×10−6/C ·55C= 0.00132 m = 1.32 mm

Step 3: Calculate the new length of the steel rod. The new length Lfof

the steel rod can be calculated by adding the change in length to the original

length:

Lf=L0+∆L= 2.0 m+0.00132 m = 2.00132 m = 2.001 m (to 3 decimal places)

Therefore, the new length of the steel rod when heated to 75

C is 2.001 m.

Question 20

Question

A copper rod is 2.5 m long at 20

C. If the rod is heated to 80

C, what will be

its new length? The coefficient of linear expansion for copper is 1.7×10−5K−1.

Solution

Step 1: First, calculate the change in temperature:

∆T=Tf−Ti= 80C−20C= 60C

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

(∆L) of the rod:

∆L=L0α∆T

where L0is the original length, αis the coefficient of linear expansion, and ∆T

is the change in temperature.

Step 3: Substitute the given values into the formula:

∆L= 2.5 m ×(1.7×10−5K−1)×60C

Step 4: Calculate the change in length:

∆L= 2.5×1.7×10−5×60

Step 5: Simplify the expression:

∆L= 2.55 ×10−4m

Step 6: Finally, find the new length by adding the change in length to the

original length:

Lf=L0+ ∆L= 2.5 m + 2.55 ×10−4m

Step 7: Calculate the new length:

Lf= 2.5+2.55 ×10−4

Lf= 2.500255 m

Therefore, the new length of the copper rod when heated to 80

C is approx-

imately 2.500255 meters.

Question 21

Question

A brass rod with a length of 2.5 m and a diameter of 1.0 cm is heated from 20◦C

to 70◦C. Calculate the change in length of the rod due to thermal expansion.

(Coefficient of linear expansion for brass is 19 ×10−6K−1)

Solution

Step 1: Calculate the initial volume of the rod. The initial volume of the rod

can be calculated using the formula for the volume of a cylinder:

V=πr2h

where ris the radius of the rod, his the initial length of the rod, and π≈

3.14159. Given r= 0.01 m and h= 2.5 m, we have:

V=π(0.01)2×2.5

Step 2: Calculate the final volume of the rod. The final volume of the rod can

be calculated using the same formula, but with the new length after expansion.

The new length can be calculated using the formula for linear expansion:

∆L=αL0∆T

where αis the coefficient of linear expansion, L0is the initial length, and ∆T

is the change in temperature. With α= 19 ×10−6K−1,L0= 2.5 m, and

∆T= 70◦C−20◦C = 50◦C, we can find the new length.

Step 3: Calculate the change in volume. The change in volume can be

calculated as the difference between the final volume and the initial volume:

∆V=Vfinal −Vinitial

Step 4: Calculate the change in length. Since the rod expands uniformly,

the change in length is related to the change in volume as follows:

∆V=πr2∆L

We can solve for ∆Lto find the change in length of the rod due to thermal

expansion.

Question 22

Question

A rod made of steel is initially 1.5 m long at 20◦C. If the coefficient of linear

expansion for steel is 12 ×10−6◦C−1, find the temperature at which the rod’s

length will increase to 1.51 m.

Solution

Let Lbe the original length of the steel rod, ∆Lbe the change in length, α

be the coefficient of linear expansion, T0be the original temperature, Tfbe the

final temperature, and Tbe the temperature at which the rod’s length increases

to 1.51 m.

Step 1: First, we find the change in length when the temperature changes

from T0to Tf:

∆L=Lα(Tf−T0)

Step 2: Substitute the given values into the formula:

∆L= 1.5×12 ×10−6(Tf−20)

Step 3: Since the final length of the rod is 1.51 m, we have:

L+ ∆L= 1.51

1.5+1.5×12 ×10−6(Tf−20) = 1.51

Step 4: Solve for Tf:

1.5+1.5×12 ×10−6Tf−1.5×12 ×10−6×20 = 1.51

1.5+1.8×10−5Tf−3.6×10−5= 1.51

1.8×10−5Tf= 1.51 −1.5+3.6×10−5

1.8×10−5Tf= 1.01 + 3.6×10−5

Tf=1.01 + 3.6×10−5

1.8×10−5

Tf≈62.22◦C

Therefore, the temperature at which the rod’s length will increase to 1.51 m

is approximately 62.22◦C.

Question 23

Question

A steel rod of length 2 m is heated from 20

C to 100

C. If the coefficient of

linear expansion for steel is 1.2×10−5

C−1, determine the change in length of

the rod.

Solution

Step 1: Calculate the initial length change due to heating from 20

C to 100

Given that the initial length of the steel rod is 2 m, we have:

∆L=L·α·∆T

∆L= 2 m ·1.2×10−5

C−1·(100 −20)

∆L= 2 ×1.2×10−3m

∆L= 2.4×10−3m

Step 2: Determine the change in length of the rod. Since the steel rod

expands in both directions, the total change in length is twice the initial change

in length:

Change in Length = 2 ×2.4×10−3m

Change in Length = 4.8×10−3m

Therefore, the change in length of the steel rod is 4.8×10−3m.

Question 24

Question

A brass rod of length 2.0 m is heated from 20

C to 120

C. If the coefficient of

linear expansion of brass is 1.9×10−5K−1, what is the change in length of the

rod?

Solution

Step 1: Calculate the initial length of the rod Given: Initial length of the brass

rod, L0= 2.0 m Coefficient of linear expansion of brass, α= 1.9×10−5K−1

Change in temperature, ∆T= 120C−20C= 100CUsing the formula for linear

expansion:

∆L=L0α∆T

Substitute the values:

∆L= 2.0×1.9×10−5×100

∆L= 2.0×1.9×10−3

∆L= 3.8×10−3m

Therefore, the change in length of the brass rod is 3.8 mm.

Question 25

Question

A steel rod is initially 2 meters long at 20◦C. What temperature increase is

required to increase the length of the rod by 1 cm if the coefficient of linear

expansion of steel is 12 ×10−6/◦C?

Solution

Let Lbe the original length of the steel rod, ∆Lbe the change in length, ∆T

be the change in temperature, and αbe the coefficient of linear expansion.

Step 1: Write out the formula for linear expansion: The change in length

∆Lof a material is given by the formula:

∆L=Lα∆T

Step 2: Use the given information: We know that L= 2m, ∆L= 1cm =

0.01m, and α= 12 ×10−6/◦C. We want to find ∆T.

Step 3: Plug the values into the formula:

∆L=Lα∆T

0.01 = (2)(12 ×10−6)(∆T)

Step 4: Solve for ∆T: Dividing both sides by (2)(12 ×10−6),

∆T=0.01

2×12 ×10−6

∆T=0.01

24 ×10−6

∆T=0.01

24 ×106

∆T=1

2400 ×106

∆T=1000

2400

∆T= 0.4167 ◦C

So, the temperature needs to be increased by 0.4167◦Cto increase the length

of the rod by 1 cm.

Question 26

Question

A steel rod with a length of 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, what is the change in length of

the rod?

Solution

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

Step 2: Determine the change in length of the rod using the coefficient of linear

expansion.

Step 1: Calculate the initial length change due to the change in tempera-

ture.

The initial length change due to the change in temperature can be calculated

using the formula:

∆L=L0·α·∆T

where: ∆L= change in length, L0= 2.0 m (initial length), α= 1.2×10−5/

(coefficient of linear expansion for steel), ∆T= 120◦C−20◦C = 100◦C.

Substitute the given values and calculate:

∆L= 2.0 m ×1.2×10−5

C−1×100

∆L= 0.0024 m

Step 2: Determine the change in length of the rod using the coefficient of

linear expansion.

The final change in length of the rod is equal to the initial length change

due to the change in temperature:

∆L= 0.0024 m

Therefore, the change in length of the steel rod is 0.0024 m.

Question 27

Question

A brass rod of length 2 m at 0◦C is clamped at its two ends. By how much

does its length change when the temperature of the rod is raised to 100◦C?

(Coefficient of linear expansion of brass = 1.9×10−5/◦C)

Solution

Let L0be the original length of the brass rod at 0◦C, ∆Tbe the change in

temperature, αbe the coefficient of linear expansion of brass, and ∆Lbe the

change in length.

Step 1: Calculate the change in temperature. Given ∆T= 100◦C - 0◦C =

100◦C

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

sion:

∆L=α·L0·∆T

∆L= (1.9×10−5/◦C) ·(2 m) ·(100◦C)

∆L= 0.000038 m

Therefore, the length of the brass rod changes by 0.000038 m when the

temperature is raised to 100◦C.

Question 28

Question

A steel rod of length 2.0 m undergoes a temperature increase of 100

C. If the

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

C−1, what is the change in

length of the rod?

Solution

Step 1: We can use the formula for linear expansion ∆L=α·L·∆T, where:

∆L= change in length, α= coefficient of linear expansion, L= original length,

and ∆T= change in temperature.

Step 2: Substituting the given values into the formula, we have: ∆L=

(12 ×10−6

C−1)·(2.0 m) ·(100

Step 3: Calculating the change in length, we get: ∆L= 12×10−6·2.0·100 =

0.0024 m

Step 4: Therefore, the change in length of the steel rod is 0.0024 meters.

Question 29

Question

A solid iron rod has a length of 2.0 m at 20◦C. If the rod is heated to 120◦C,

what will be its new length? The coefficient of linear expansion for iron is

1.2×10−5per degree Celsius.

Solution

Let’s denote the initial length of the iron rod as Liat 20◦C and the final length

after heating as Lfat 120◦C. The change in length, ∆L, can be calculated using

the formula for linear expansion:

∆L=Li·α·∆T

where: Li= 2.0 m (initial length), α= 1.2×10−5K−1(coefficient of linear

expansion for iron), and ∆T= 120◦C−20◦C = 100 K (change in temperature).

Step 1: Calculate the change in length ∆L.

∆L= 2.0 m ×1.2×10−5K−1×100 K

∆L= 0.0024 m = 2.4 mm

Step 2: Calculate the final length Lf.

Lf=Li+ ∆L

Lf= 2.0 m + 0.0024 m

Lf= 2.0024 m

The new length of the iron rod when heated to 120◦C is 2.0024 m, or 2.0024×

103mm.

Question 30

Question

A steel rod is 1 meter long 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

100◦C.

Solution

Given: Initial length of the steel rod, L0= 1 m

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

C−1

Change in temperature, ∆T= 100◦C−20◦C = 80◦C

From the formula for linear expansion:

∆L=L0α∆T

Step 1: Calculate the change in length of the steel rod.

∆L= 1 ×12 ×10−6×80 = 0.000096 m

So, the change in length of the rod when heated to 100◦C is 0.000096 meters.

Question 31

Question

A steel rod has a length of 2.00 m at 20

C. If the temperature of the rod is

increased to 120

C, what is the new length of the rod? The linear expansion

coefficient of steel is 12 ×10−6

C−1.

Solution

Step 1: Calculate the change in length of the steel rod. The change in length of

the rod can be calculated using the formula:

∆L=α·L·∆T

where ∆L= change in length, α= 12 ×10−6

C−1(linear expansion coefficient

of steel), L= 2.00 m (initial length of the rod), ∆T= 120 −20 = 100

C (change

in temperature).

Substitute the values into the formula:

∆L= 12 ×10−6·2.00 ·100

Step 2: Calculate the new length of the steel rod. The new length of the rod

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

Lnew =L+ ∆L

Substitute the values and calculate:

Lnew = 2.00 + 12 ×10−6·2.00 ·100

Question 32

Question

A brass container is filled with 0.5 L of water at 20

C. If the container is then

heated to 70

C, calculate the change in volume of the container. Assume the

linear expansion coefficient of brass is 19 ×10−6K−1and the coefficient of

volume expansion for water is 2.1×10−4K−1.

Solution

Step 1: Calculate the initial volume of the water: Given that the initial volume

of water is 0.5 L, we have Vi= 0.5 L.

Step 2: Calculate the change in temperature: The change in temperature is

∆T= 70C−20C= 50C.

Step 3: Calculate the change in volume of the water: We will calculate the

change in volume of water first. The coefficient of volume expansion for water,

βwater = 2.1×10−4K−1.

The change in volume of water is given by:

∆Vwater =Viβwater∆T

∆Vwater = 0.5 L ×2.1×10−4K−1×50 K

∆Vwater = 0.000525 L

Step 4: Calculate the change in volume of the brass container: Now, we will

calculate the change in volume of the brass container. The linear expansion

coefficient of brass, αbrass = 19 ×10−6K−1.

The change in volume of the brass container is given by:

∆Vbrass =Viαbrass∆T

∆Vbrass = 0.5 L ×19 ×10−6K−1×50 K

∆Vbrass = 0.000475 L

Step 5: Calculate the total change in volume of the container: The total

change in volume is the sum of the changes in volume of the water and the

brass container:

∆Vtotal = ∆Vwater + ∆Vbrass

∆Vtotal = 0.000525 L + 0.000475 L

∆Vtotal = 0.001 L

Therefore, the change in volume of the brass container when heated to 70

is 0.001 L.

Question 33

Question

A steel rod has a length of 2 meters at a temperature of 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 ×10−6per degree Celsius.

Solution

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

expansion to find the new length of the steel rod.

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

is given by:

∆T=Tf−Ti= 120C−20C= 100C

Step 2: Use the formula for linear expansion to find the new length of the

steel rod. The change in length of the steel rod can be calculated using the

formula:

∆L=α·L·∆T

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

and ∆Tis the change in temperature.

Substitute the values into the formula:

∆L= (12 ×10−6)·2·100 = 0.0024 meters

The new length of the steel rod is:

Lf=Li+ ∆L= 2 + 0.0024 = 2.0024 meters

Therefore, when the steel rod is heated to 120

C, its new length will be

2.0024 meters.

Question 34

Question

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

of linear expansion for steel is 1.2×10−5◦C−1, calculate the change in length

of the rod when the temperature is increased to 100◦C.

Solution

Let L1be the initial length of the rod and L2be the final length of the rod.

Given: Initial length, L1= 2 m Change in temperature, ∆T= 100◦C−20◦C =

80◦C Coefficient of linear expansion, α= 1.2×10−5◦C−1

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

sion:

L2=L1(1 + α∆T)

Step 2: Substitute the given values into the formula:

L2= 2(1 + 1.2×10−5×80)

L2= 2(1 + 0.00096)

L2= 2.00192

Step 3: Calculate the change in length:

∆L=L2−L1

∆L= 2.00192 −2

∆L= 0.00192 m

Therefore, the change in length of the steel rod when the temperature is

increased to 100◦C is 0.00192 meters.

Question 35

Question

A copper rod initially measures 2 meters in length at a temperature of 20◦C.

If the coefficient of linear expansion for copper is 17 ×10−6◦C−1, what is the

length of the rod when the temperature increases to 120◦C?

Solution

Step 1: First, we need to calculate the change in temperature. Given that the

initial temperature is 20◦C and the final temperature is 120◦C, the change in

temperature is:

∆T= 120◦C−20◦C = 100◦C

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

∆L=α·L·∆T

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

and ∆Tis the change in temperature.

Substitute the values:

∆L= 17 ×10−6◦C−1·2 m ·100◦C=0.0034 m

Step 3: Finally, we can find the length of the rod at 120◦C by adding the

change in length to the initial length:

Lfinal =Linitial + ∆L= 2 m + 0.0034 m = 2.0034 m

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

120◦C is 2.0034 meters.

PHYS 305 - INTRODUCTION TO MODERN PHYSICS Thermal expansion Question Bank Set 5

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