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					                                  Mr. Kovall’s Neighborhood
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                                LINEAR and VOLUME EXPANSION
                                           SAMPLE SOLUTIONS

 1. A      steel cable 50 m (= 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?

          L =Lo(T) = (12 x 10-6/ oC)(50 m)[40 oC - (-15o C)]

            L = 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 2,600o C?

           L =Lo(T)

           L = 0.6192 cm;

             Ln = Lo + L = 40 cm + 0.6192 cm = 40.6192 cm

 3. An ordinary glass (= 3 x 10-6/ oC) bottle is filled to the brim with Hg (= 9 x 10-6/oC) at
     10 oC. If the volume of the Hg is 100 cc at 10o C, how much Hg will overflow (or more
     could be added) if the temperature of the glass and mercury is changed to 90o C?
  
       Vglass =Vo(T) ;= 3() = 3(3 x 10o C)

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

                     = 7.2 x 10-2 cc

             VHg =Vo(T)

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

                  = 1.44 cc

            Volume(overflow) =  VHg - Vglass

                            = 1.44 cc - 7.2 x 10-2 cc = 1.368 cc

                                              PROBLEMS

1. A 200 m steel (= 12 x 10-6/ oC) cable is used to support a suspension bridge. If the air
   temperature of the environment changes from -30o F to 95o F, how much will the cable
   expand?

2. A 20 m concrete (= 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 -30o F to 120o F?

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

4. A steel (= 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 (= 2.07 x 10-4/ oC) if its temperature is raised from 2o C to
   20o C?

6. A 25 m steel (= 12 x 10-6/ oC) tape is correctly calibrated at 0o C. The tape measure is
   then used in an environment where the temperature is 25o C to measure the distance
   between two marks on a concrete (= 15 x 10-6/ oC) floor. The tape measure indicates
   that the distance between the two marks is 25 m at 25o C.
      (A) What is the actual distance between the two marks?
      (B) What would the tape measure the distance between these two marks at 0o C?

7. An iron (= 11 x 10-6/ oC) pipe is 100 m long at -10o C. How much room must be left for
   expansion if its temperatures vary from -10o C to 105o C?

8. What must be the length of a steel (= 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 (= 19 x 10-6/ oC) is 1.5 cm when
   the temperature is 20o C. What will be the hole’s diameter at 5o C?

10. The space between 12 m steel is 0.0075 m at - 18 oC. If the space completely closes up
    at 32o C, what is the coefficient of linear expansion for these rails?

11. A copper (= 17 x 10-6/ oC) has a diameter of 3.98 cm at 20o C. To what temperature
    must this ring be heated to exactly fit a shaft 4 cm in diameter?

12. A steel (= 12 x 10-6/ oC) is 1,500 m long at 30o C. What is its length at 10o C?
13. Some railroad rails are 30 m long and made of steel (= 11 x 10-6/ oC). How much
    change in length will they undergo if they are exposed to seasonal temperature ranges
    from -25o C to 35o 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 (= 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 20o C. With the steam in the
    pipe, its temperature reaches 120o C. What is the maximum change in length of the
    pipe?

16. A 69 m wire is made of invar (= 9 x 10-6/ oC). How much will its length increase if its
    temperature is raised from -5o F to 40o F?

17. If 25 cc of a gas is collected under pressure at a temperature of 23o 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 10o C, what will it be at 40o C, if the volume
    remains constant?

20. A certain glass (= 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 -10o C to 110o C?

21. The steel (= 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 -10o C to
    105o C?

22. An aluminum (= 23 x 10-26/ oC) measuring rule measures a piece of copper
    (= 17 x 10-6/ oC) at 5o C. What is the true length of the copper at 105o C if the tape
    measure was calibrated at 5o C?

23. Compute the increase in length of 500 m of copper (= 17 x 10-6/ oC) when its
    temperature changes from 32o F to 12o 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 15o C, a bare wheel has a diameter of 30 cm and the inside diameter of a steel rim
    (= 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 (= 12 x 10-6/ oC) ball has a diameter of 6 cm and is 0.01 mm smaller than the
    hole in a brass (= 19 x 10-6/ oC) plate when both are at 30o 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 (= 10 x 10-6/ oC) block
    5 inches x 10 inches by 6 inches when the temperature changes from 15o C to 47o C?

28. A glass (= 9 x 10-6/ oC) vessel is filled with 1 liter of turpentine (= 97 x 10-5/ oC) at
    50o C. What volume of liquid will overflow, if any, when the temperature is raised
    36 Centigrade degrees?

29. The steel (= 11 x 10-6/ oC) tank on a gasoline truck is loaded with 5,000 gallons of
    gasoline (= 0.9 x 10-3/ oC) at 15o C. After traveling several hours in the hot sun where
    the temperature of the gasoline reaches 40o 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 (-10o C) and
    summer (30o C)?

31. The iron (= 12 x 10-6/ oC) rim of a wagon wheel has a 1.5 m internal diameter when the
    temperature is 120o C. What is the rim’s internal diameter when the temperature is
    20o C?

32. A gold (= 14.3 x 10-5/ oC) strip is made into a ring whose inside diameter is 20 mm at
   20o C. What is the inside diameter of the ring when the temperature is 70o C.

33. A steel (= 12 x 10-6/ oC) tape measure is standardized at 100 ft when it is at 75o F.
    What would it measure the length of a piece of wood at 100o F?

34. A spool is loaded with 1,000 m of copper(= 16.8 x 10-6/ oC) wire at 30o C. How much
    wire will there be on the spool at
      (A) 0o C?
      (B) 70o C?

35. A 1 liter soda glass (= 3.3 x 10-6/ oC) is filled to the brim with mercury
    (= 1.82 x 10-4/ oC) at 15o C. How many cubic centimeters of mercury will overflow (or
    additional could be added) when the temperature is 45o C?
36. A gold (= 14.3 x 10-6/ oC) disk at 20o 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 50o C?

37. A brass (= 19.3 x 10-6/ oC) storage tank is filled to the top with 500 gallons of alcohol
    (= 11.2 x 10-4/ oC) when the temperature is -22o F. What condition exists at 95o F?

38. The Eiffel Tower in Paris is 984 ft high at 0o C and is made of iron (= 11 x 10-6/ oC).
    What will be its actual height at 40o C?

39. A 1,000 gallon iron (= 12 x 10-6/ oC) storage tank is filled to the top with alcohol
    (= 11.2 x 10-4/ oC) when the temperature is -10o C. What condition will exist when the
    temperature is:
       (A) -40o C?
        (B) 50o C?

40. Your class ring (= 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 25o C. While you are outside engaged in a snowball
    fight, the temperature of the ring reaches -20o 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 110o 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 68o 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 (= 23.8 x 10-6/ oC tape measure is used to measure a piece of
    copper (= 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 (= 10.5 x 10-6/ oC) of your sports car with high octane gasoline
    ( = 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 (= 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 (= 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 (= 23 x 10-6/ oC) measure is correctly calibrated at 75 oF. The tape is
    then used to measure a piece of copper (= 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 100o F?

52. A brass (= 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 (= 1.3 x 10-6/ oC) tape measure is accurately calibrated at 15 oF. It is then used to
    measure a piece of glass (= 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 (= 10.5 x 10-6/ oC) holds exactly 750 gallons of water (= 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 (= 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 (= 22 x 10-6/ oC) measuring rod, which was calibrated at 5 oC, measures a
    certain distance on a piece of steel (= 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 (= 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 (= 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 (= 1 x 10-5/ oC) is filled to the top with 50 cc of mercury (= 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 (= 17 x 10-6/ oC) is measured as 65 ft at 110 oC by a steel (= 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 (= 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 (= 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 (= 1 x 10-5/ oC) is completely filled with 50 cc of mercury
    (= 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 (= 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 373o:
      (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 562o Kelvin is equivalent to:
      (A) 289 oF                  (C) 825 oF
              o
       (B) 289 C                  (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
                o
       (B) 100 C                     (D) 373 oK.
20. A temperature of 15 oF is equivalent to:
      (A) -3.2 oC                (C) -9.4 oC
               o
      (B) -7.9 C                 (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

				
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