Mr. Kovall’s Neighborhood

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LINEAR and VOLUME EXPANSION

SAMPLE SOLUTIONS

1. A steel cable 50 m (a = 12 x 10^-6/ ^oC) in length is exposed to a temperature range from

-15 ^oC. What is the increase in length of the cable?

D L = aL_o(DT) = (12 x 10^-6/ ^oC)(50 m)[40 ^oC - (-15^o C)]

DL = 3.3 x 10^-2 m

2. A tungsten filament in a lamp is 40 cm long. Assuming that the average value of alpha

over this temperature range is 6 x 10^-6/ ^oC, how long will the filament be at 2,600^o C?

DL = aL_o(DT)

DL = 0.6192 cm;

L_n = L_o + DL = 40 cm + 0.6192 cm = 40.6192 cm

3. An ordinary glass (a = 3 x 10^-6/ ^oC) bottle is filled to the brim with Hg (b = 9 x 10^-6/^oC) at

10 ^oC. If the volume of the Hg is 100 cc at 10^o C, how much Hg will overflow (or more

could be added) if the temperature of the glass and mercury is changed to 90^o C?

D V_glass = bV_o(DT) ; b = 3(a) = 3(3 x 10^o C)

= (9 x 10^-6/ ^oC)(100 cc)(90^o C - 10^o C)

= 7.2 x 10^-2 cc

D VHg = bV_o(DT)

= (18 x 10 ^-5/ ^oC)(100 cc)(90^o C - 10^o C)

= 1.44 cc

Volume(overflow) = D VHg - DVglass

= 1.44 cc - 7.2 x 10^-2 cc = 1.368 cc

PROBLEMS

1. A 200 m steel (a = 12 x 10^-6/ ^oC) cable is used to support a suspension bridge. If the air

temperature of the environment changes from -30^o F to 95^o F, how much will the cable

expand?

2. A 20 m concrete (a = 15 x 10^-6/ ^oC) walk runs from the house to the side walk. What total

distance must be left between the house and walk and the walk and sidewalk if seasonal

temperatures are expected to range from -30^o F to 120^o F?

3. A steel (a = 12 x 10^-6/ ^oC) rule is 1 meter long at 0^o C. At what temperature will it be 1 mm

longer?

4. A steel (a = 12 x 10^-6/ ^oC) cable encircles the Earth at the equator (R = 4 x 10³ mi). The

air temperature rises 20 Centigrade degrees:

(A) How much will the cable increase in length?

(B) What will be the change in radius of the circle formed by the cable?

5. How much does 1 liter of water (b = 2.07 x 10^{-4/ oC}) if its temperature is raised from 2^o C to

20^o C?

6. A 25 m steel (a = 12 x 10^-6/ ^oC) tape is correctly calibrated at 0^o C. The tape measure is

then used in an environment where the temperature is 25^o C to measure the distance

between two marks on a concrete (a = 15 x 10^-6/ ^oC) floor. The tape measure indicates

that the distance between the two marks is 25 m at 25^o C.

(A) What is the actual distance between the two marks?

(B) What would the tape measure the distance between these two marks at 0^o C?

7. An iron (a = 11 x 10^-6/ ^oC) pipe is 100 m long at -10^o C. How much room must be left for

expansion if its temperatures vary from -10^o C to 105^o C?

8. What must be the length of a steel (a = 12 x 10^-6/ ^oC) rod if its length increases by

1.02 x 10^-3 m as a result of a 20 Centigrade degree temperature change?

9. The diameter of a hole drilled through a piece of brass (a = 19 x 10^-6/ ^oC) is 1.5 cm when

the temperature is 20^o C. What will be the hole’s diameter at 5^o C?

10. The space between 12 m steel is 0.0075 m at - 18 ^oC. If the space completely closes up

at 32^o C, what is the coefficient of linear expansion for these rails?

11. A copper (a = 17 x 10^-6/ ^oC) has a diameter of 3.98 cm at 20^o C. To what temperature

must this ring be heated to exactly fit a shaft 4 cm in diameter?

12. A steel (a = 12 x 10^-6/ ^oC) is 1,500 m long at 30^o C. What is its length at 10^o C?

13. Some railroad rails are 30 m long and made of steel (a = 11 x 10^-6/ ^oC). How much

change in length will they undergo if they are exposed to seasonal temperature ranges

from -25^o C to 35^o C?

14. Compute the coefficient of linear expansion of a 60 cm metal rod which expands

1.02 mm when the temperature rises 100 ^o C?

15. An iron (a = 12 x 10^-6/ ^oC) pipe extends 300 m through a tunnel from a boiler house to a

school building. With no steam on, the pipe’s temperature is 20^o C. With the steam in the

pipe, its temperature reaches 120^o C. What is the maximum change in length of the

pipe?

16. A 69 m wire is made of invar (a = 9 x 10^-6/ ^oC). How much will its length increase if its

temperature is raised from -5^o F to 40^o F?

17. If 25 cc of a gas is collected under pressure at a temperature of 23^o C and the pressure

remains constant, what will be the resultant volume of the gas when the temperature is

0 ^oC?

18. 500 cc of hydrogen is collected when the barometer reads 730 mm pressure. If the

temperature remains constant, what will be the resultant volume of the gas when the

barometer reads 750 mm?

19. If the air pressure in a tire is 500 g/cm³ at 10^o C, what will it be at 40^o C, if the volume

remains constant?

20. A certain glass (a = 3 x 10^-6/ ^oC) thermometer bulb holds 5.44 g of mercury

(density = 13.6 g/cc). The thermometer stem is 25 cm long. If the coefficient of volume

expansion of mercury is 18 x 10^-5/ ^oC, what must be the inside diameter of the stem bore

in order that the tube may be graduated from -10^o C to 110^o C?

21. The steel (a = 11 x 10^-1/ ^oC) cables of a certain bridge are 3 km long. What will be their

seasonal change in length if they are exposed to temperature changes of -10^o C to

105^o C?

22. An aluminum (a = 23 x 10^-26/ ^oC) measuring rule measures a piece of copper

(a = 17 x 10^-6/ ^oC) at 5^o C. What is the true length of the copper at 105^o C if the tape

measure was calibrated at 5^o C?

23. Compute the increase in length of 500 m of copper (a = 17 x 10^-6/ ^oC) when its

temperature changes from 32^o F to 12^o F?

24. A 3 m rod is found to have expanded 0.036 mm for a rise in temperature of 100

Fahrenheit degrees. What is the coefficient of linear expansion of this metal rod?

25. At 15^o C, a bare wheel has a diameter of 30 cm and the inside diameter of a steel rim

(a = 11 x 10^-6/ ^oC) is 29.97 cm. To what temperature must the rim be taken so as to just

slip over the wheel?

26. An iron (a = 12 x 10^-6/ ^oC) ball has a diameter of 6 cm and is 0.01 mm smaller than the

hole in a brass (a = 19 x 10^-6/ ^oC) plate when both are at 30^o C. At what temperature, the

same for the brass plate and the iron ball, will the ball just stick inside the hole?

27. Determine the change in volume of a cast iron (a = 10 x 10^-6/ ^oC) block

5 inches x 10 inches by 6 inches when the temperature changes from 15^o C to 47^o C?

28. A glass (a = 9 x 10^-6/ ^oC) vessel is filled with 1 liter of turpentine (b = 97 x 10^-5/ ^oC) at

50^o C. What volume of liquid will overflow, if any, when the temperature is raised

36 Centigrade degrees?

29. The steel (a = 11 x 10^-6/ ^oC) tank on a gasoline truck is loaded with 5,000 gallons of

gasoline (b = 0.9 x 10^-3/ ^oC) at 15^o C. After traveling several hours in the hot sun where

the temperature of the gasoline reaches 40^o C, the driver stops and unloads 5,000

gallons.

(A) How many gallons of gasoline remain in the truck? (B) To whom does this gasoline

belong?

30. Copper expands by 0.0014% for every Celsius degree rise in temperature. How much does

a 1,000 ft copper electrical transmission wire expand or contract between winter (-10^o C) and

summer (30^o C)?

31. The iron (a = 12 x 10^-6/ ^oC) rim of a wagon wheel has a 1.5 m internal diameter when the

temperature is 120^o C. What is the rim’s internal diameter when the temperature is

20^o C?

32. A gold (a = 14.3 x 10^-5/ ^oC) strip is made into a ring whose inside diameter is 20 mm at

20^o C. What is the inside diameter of the ring when the temperature is 70^o C.

33. A steel (a = 12 x 10^-6/ ^oC) tape measure is standardized at 100 ft when it is at 75^o F.

What would it measure the length of a piece of wood at 100^o F?

34. A spool is loaded with 1,000 m of copper(a = 16.8 x 10^-6/ ^oC) wire at 30^o C. How much

wire will there be on the spool at

(A) 0^o C?

(B) 70^o C?

35. A 1 liter soda glass (a = 3.3 x 10^-6/ ^oC) is filled to the brim with mercury

(b = 1.82 x 10-⁴/ ^oC) at 15^o C. How many cubic centimeters of mercury will overflow (or

additional could be added) when the temperature is 45^o C?

36. A gold (a = 14.3 x 10^-6/ ^oC) disk at 20^o C has a diameter of 10 cm and has a hole cut in

the center that is 5 cm in diameter. Calculate the diameter of the hole when the

temperature is 50^o C?

37. A brass (a = 19.3 x 10^-6/ ^oC) storage tank is filled to the top with 500 gallons of alcohol

(b = 11.2 x 10^-4/ ^oC) when the temperature is -22^o F. What condition exists at 95^o F?

38. The Eiffel Tower in Paris is 984 ft high at 0^o C and is made of iron (a = 11 x 10^-6/ ^oC).

What will be its actual height at 40^o C?

39. A 1,000 gallon iron (a = 12 x 10^-6/ ^oC) storage tank is filled to the top with alcohol

(b = 11.2 x 10^-4/ ^oC) when the temperature is -10^o C. What condition will exist when the

temperature is:

(A) -40^o C?

(B) 50^o C?

40. Your class ring (a = 15.6 x 10^-6/ ^oC) has a diameter of 18.01 mm when you put it on in

the house where the temperature is 25^o C. While you are outside engaged in a snowball

fight, the temperature of the ring reaches -20^o C. If your finger has a diameter of 18 mm,

will you have to worry about the ring slipping off your finger? (Remember the human

body is 90% water). Explain!!!

41. You are about to go swimming in the ocean where the air temperature is 110^o F. You are

wearing the same ring as in problem #40. You also forgot to take off the ring in your

air-conditioned room where the temperature is 68^o F before going out for the swim. Will

you have to worry about the ring slipping off your finger? Explain!!!

42. A 95 m aluminum (a = 23.8 x 10^-6/ ^oC tape measure is used to measure a piece of

copper (a = 16.8 x 10^-6/ ^oC) at 75 ^oC. The tape was standardized at 10 ^oC.

(A) What is the actual length of the copper at 75 ^oC?

(B) What would be the accurate length of the copper at 0 ^oC?

43. You fill the gas tank (a = 10.5 x 10^-6/ ^oC) of your sports car with high octane gasoline

(b = 9.55 x 10^-4/ ^oC) when the temperature is 80 ^oF. What condition exists that evening

when the temperature reaches 35 ^oF?

44. A gold (a = 14.3 x 10^-6/ ^oC) disk at 20 ^oC has a diameter of 10 cm and has a whole cut in

the center that is 4 cm in diameter. What is the change in surface area of the disk when

the temperature is changed to 85 ^oC?

45. What would be the change in volume of the disk in problem #44?

46. The brass (a = 19.3 x 10^-6/ ^oC) retainer ring for a special assembly is 2.345 mm in

diameter at 25 ^oC. To what temperature must this retainer ring be taken so that it will just

fit inside its specially ground slot that is 2.3449 mm in diameter?

47. Upon what factors does the coefficient of linear expansion depend?

48. What is the relationship between the coefficient of volume expansion and that of linear

expansion? (Show all work)

49. Why does the measurement of the expansion of a gas trapped beneath a liquid (such as

mercury) cause problems in determining the expansion of that gas as the temperature

rises?

50. What affect does a gas bubble in a thermometer have on the accurate measurement of

temperature? Explain!!!

51. A 50 m aluminum (a = 23 x 10^-6/ ^oC) measure is correctly calibrated at 75 ^oF. The tape is

then used to measure a piece of copper (a = 16.8 x 10-6/ ^oC) when they are both at

55 ^oF.

(A) What is the true length of the copper at 55 ^oF?

(B) Will the tape be long enough to measure that same piece of copper at 100^o F?

52. A brass (a = 19 x 10-⁶/ ^oC) has a diameter of 9.655 cm at 25 ^oC. To what temperature

must it be taken to exactly fit a slot that is 9.6564 cm in diameter?

53. A steel (a = 1.3 x 10^-6/ ^oC) tape measure is accurately calibrated at 15 ^oF. It is then used to

measure a piece of glass (a = 9 x 10^-6/ ^oC) at 85 ^oF. The tape measure indicates that the

glass is 95 m.

(A) Is this the actual measurement of the glass? [EXPLAIN]

(B) Will the tape measure be long enough to measure the same piece of glass when

they are both at 0 ^oF? [EXPLAIN]

54. A cast iron (a = 10.5 x 10^-6/ ^oC) holds exactly 750 gallons of water (b = 2.07 x 10^-4/ ^oC)

when it is filled to the top at 60 ^oF. What condition exists at 35 ^oF?

55. A silver (a = 18.8 x 10^-6/ ^oC) disk has a diameter 10 mm at 20 ^oC. A whole is cut in this

disk that has a diameter of 3 mm. What is the change in the surface area of the disk

when the temperature is changed to -40 ^oC?

56. What would be the change in volume of the disk in problem # 55?

57. An aluminum (a = 22 x 10^-6/ ^oC) measuring rod, which was calibrated at 5 ^oC, measures a

certain distance on a piece of steel (a = 12 x 10^-6/ ^oC) as 88.42 cm at 35 ^oC.

(A) Determine the error in measuring this distance.

(B) What would be the correct distance on the steel at 0 ^oC?

58. An iron (a = 12 x 10^-6/ ^oC) ball is 6 cm in diameter. This ball is 0.010 mm to large to pass

through a hole in a brass (a = 19 x 10^-6/ ^oC) plate at a temperature of 30 ^oC. At what

temperature, the same for the ball and plate, will the ball have exactly the same diameter

as the hole?

59. A glass flask (a = 1 x 10^-5/ ^oC) is filled to the top with 50 cc of mercury (b = 1.8 x 10^-4/ ^oC)

at 0 ^oC. What condition will exist when the temperature of the flask and mercury are both

raised to 100 ^oC?

60. A copper (a = 17 x 10^-6/ ^oC) is measured as 65 ft at 110 ^oC by a steel (a = 11 x 10^-6/ ^oC)

tape measure. The tape measure was calibrated at 10 ^oC. Is the tape measure long

enough to measure the piece of copper when they are both at 10 ^oC?

61. Determine the change in volume of a cast iron (a = 10 x 10^-6/ ^oC) block whose dimensions

are 5 inches by 10 inches by 6 inches when the temperature changes 32 Centigrade

degrees?

62. Steel (a = 12 x 10^-6/ ^oC) rails 55 ft long are to be used as railroad rails through a region

where the temperature varies from -5 ^oF to 115 ^oF. If the tracks are placed when the

temperature is 45 ^oF, how much space should be left between the ends of the rails?

63. If 535 cc of air is collected at 785 mm of mercury and 35 ^oC; what will be the new volume

of this gas when the temperature is 22 ^oC and a pressure of 875 mm of mercury?

64. A student collect 500 cc of air at 760 mm of mercury and a temperature of 25 ^oC. What

is the volume of this air at -25 ^oC and 800 mm of mercury?

65. A glass flask (a = 1 x 10^-5/ ^oC) is completely filled with 50 cc of mercury

(b = 1.8 x 10^-4/ ^oC) at 0 ^oC. How much mercury will overflow through a small hole in the

top as the temperature changes to 100 ^oC?

66. Calculate and prove the temperature reading at which a Centigrade and a Fahrenheit

thermometer readings are numerically identical.

67. A copper (a = 17 x 10^-6/ ^oC) rod is measured at 110 ^oC as being 65 ft in length. The tape

measures accurately at 10 ^oC. What would be the correct length of the copper rod at

10 ^oC as measured by the steel measuring?

SELF TEST

Part I: Multiple Choice

1. If the temperature of the gas in a container having a fixed volume is increased, the

number of impacts which the molecules make against the walls of the container:

(A) increase (C) remain unchanged

(B) decrease

2. The expansion of objects when heated is caused by:

(A) an increase in the size of the molecules;

(B) the additional space taken up by the heat;

(C) an increase in the average distance between the molecules;

(D) an increase in the attraction of the molecules for each other.

3. When using the general gas laws or equations, the temperature must always be

expressed in:

(A) Fahrenheit degrees (C) Celsius degrees

(B) Centigrade degrees (D) Kelvin degrees

4. The temperature at which molecules have the least energy is:

(A) 0 ^oC; (C) -257 ^oC;

(B) -40 ^oC; (D) -240 ^oC.

5. If the coefficient of expansion of glass were greater than that of mercury, an increase in

temperature would, in a glass-mercury thermometer, would cause the mercury to:

(A) rise (C) remain stationary

(B) fall

6. If 1 liter of water at its maximum density cools, its:

(A) volume will increase (C) volume will not change

(B) mass will be changed (D) volume will decrease

7. The heating of a body:

(A) increases the number of molecules;

(B) increases the attraction of the molecules for each other;

(C) decreases the average distance between the molecules;

(D) increases the average kinetic energy of the molecules.

8. The coefficient of expansion of a brass rod is 1.8 x 10^-5/ ^oC. This coefficient of expansion

is also:

(A) 3.22 x 10^-5/ ^oF (C) 3.2 x 10^-4/ ^oF

(B) 1.0 x 10^-5/ ^oF (D) 2.2 x 10^-4/ ^oF

9. Absolute zero is defined as the temperature at which

(A) no form of life can exist

(B) molecular motion is at a maximum

(C) the maximum amount of thermal energy has been liberated

(D) all substances solidify

10. The boiling point of pure water is 373^o:

(A) Centigrade (C) Fahrenheit

(B) Kelvin (D) Celsius

11. If the pressure is constant, the volume of a gas is directly proportional to the

temperature, which must be expressed in degrees:

(A) Celsius (C) Fahrenheit

(B) Centigrade (D) Kelvin

12. A reading of 562^o Kelvin is equivalent to:

(A) 289 ^oF (C) 825 ^oF

(B) 289 ^oC (D) 835 ^oC

13. A large copper plate has a hole drilled in the center. When the temperature of the plate

is cooled, the hole become:

(A) smaller; (C) remains constant;

(B) distorted; (D) larger.

14. The coefficient of volume expansion is least in:

(A) solids; (C) gases.

(B) liquids;

15. To obtain a greater mass of gasoline per unit volume, the fuel should be purchased when

the:

(A) temperature of the fuel is low (C) temperature of the fuel is high

(B) the tank is empty (D) tank is half-full

16. To design an increased sensitive glass-alcohol thermometer:

(A) the diameter of the glass bore should be larger

(B) the alcohol should be colored

(C) the diameter of the glass bore should be smaller

(D) the glass should be colored

17. A container filled with helium at 50 ^oK. To double its volume at constant pressure, the

helium must be taken to:

(A) -173 ^oC (C) -273 ^oC

(B) 273 ^oK (D) 150 ^oK

18. Most of the Earth’s heat comes from:

(A) the Earth’s interior (C) the burning of fuels

(B) friction (D) the sun

19. All of the following belong to the group EXCEPT:

(A) 32 ^oF (C) 212 ^oF

(B) 100 ^oC (D) 373 ^oK.

20. A temperature of 15 ^oF is equivalent to:

(A) -3.2 ^oC (C) -9.4 ^oC

(B) -7.9 ^oC (D) 5.2 ^oC

21. When a brass ball is heated in a flame, it:

(A) remains unchanged (C) contracts

(B) expands D) melts

22. Studies of the expansion of solids show that different solids:

(A) expand at different rates when heated

(B) expand at different rates when cooled

(C) expand at the same rate when heated

(D) expand at the same rate when cooled

23. Thermostats contain a:

(A) bar of silver (C) bar of iron

(B) compound bar (D) bar of steel

24. For small temperature changes, as the temperature of a solid rises, its coefficient of

expansion:

(A) decreases (C) remains unchanged

(B) increases

25. As the original length of a set of otherwise identical solids increase, the change in length

caused by a given temperature change:

(A) decreases (C) remains unchanged

(B) increases

26. As the temperature change experienced by a solid increases, the resulting change in

length:

(A) decreases (C) remains unchanged

(B) increases

27. The property of a substance which is measured by a thermometer might be correctly

described as:

(A) hotness (D) potential energy

(B) heat (E) temperature

(C) expansion

28. The change of state from a solid to a gas is called:

(A) fusion (C) sublimation

(B) vaporization (D) solidification

29. As ice melts, its temperature:

(A) rises (C) remains unchanged

(B) falls (D) may rise or fall depending upon the environment

30. Calories:

(A) are a measure of temperature (C) are units for measuring quantities of heat

(B) can be weighted on a balance (D) are equal to a definite quantity of water