CHAPTER 30 THERMAL EXPANSION
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1 CHAPTER 30 THERMAL EXPANSION EXERCISE 140, Page A length of lead piping is 50.0 m long at a temperature of 16 C. When hot water flows through it the temperature of the pipe rises to 80 C. Determine the length of the hot pipe if the coefficient of linear expansion of lead is = 50.0 m, temperature t 1 = 16 C, t 2 = 80 C and = K 1 Length of pipe at 80 C is given by: L 2 ] = 50.0[1 + ( (80-16] = 50.0[ ] i.e. an increase in length of m or mm = 50.0[ ] = m 2. A rod of metal is measured at 285 K and is m long. At 373 K the rod is m long. Determine the value of the coefficient of linear expansion for the metal. = m, L 2 = m, temperature t 1 = 285 K and temperature t 2 = 373 K ] i.e = 3.521[1 + ( ] = (3.521( (88 i.e = (3.521( (88 Hence, the coefficient of linear expansion, = (3.521(88 = i.e. coefficient of linear expansion, = K 1 345
2 3. A copper overhead transmission line has a length of 40.0 m between its supports at 20 C. Determine the increase in length at 50 C if the coefficient of linear expansion of copper is ] Hence, increase in length = (40.0 m( K 1 = (40.0( (30 = m or 20.4 mm (50 20 C 4. A brass measuring tape measures 2.10 m at a temperature of 15 C. Determine (a the increase in length when the temperature has increased to 40 C (b the percentage error in measurement at 40 C. Assume the coefficient of linear expansion of brass to be = 2.10 m, temperature t 1 = 15 C, t 2 = 40 C and = K 1 (a ] Hence, increase in length = (2.10 m( K 1 = (2.10( (25 = m or mm (40 15 C (b Percentage error in measurement at 40 C = increasein length % originallength 2.10 = 0.045% 346
3 5. A pendulum of a grandfather clock is 2.0 m long and made of steel. Determine the change in length of the pendulum if the temperature rises by 15 K. Assume the coefficient of linear expansion of steel to be ] Hence, increase in length = (2.0 m( K 1 (15 K = (2.0( (15 = m or 0.45 mm 6. A temperature control system is operated by the expansion of a zinc rod which is 200 mm long at 15 C. If the system is set so that the source of heat supply is cut off when the rod has expanded by 0.20 mm, determine the temperature to which the system is limited. Assume the coefficient of linear expansion of zinc to be = 200 mm = 0.20 m, L 2 = mm = mm = , temperature t 1 = 15 C ] Hence, increase in length i.e = (0.20( ( t = (0.20( ( t i.e. ( t 2-15 = = C 6 (0.20(31 10 i.e. the temperature to which the system is limited, t 2 = = C 347
4 7. A length of steel railway line is 30.0 m long when the temperature is 288 K. Determine the increase in length of the line when the temperature is raised to 303 K. Assume the coefficient of linear expansion of steel to be ] Hence, increase in length = (30.0 m( K 1 ( K = (30.0( (15 = m or 6.75 mm 8. A brass shaft is mm in diameter and has to be inserted in a hole of diameter 15.0 mm. Determine by how much the shaft must be cooled to make this possible, without using force. Take the coefficient of linear expansion of brass as = mm = m, L 2 = 15 mm = m ] i.e = [1 + ( K 1 (t 2 ] = ( ( ( t 2 i.e = ( ( ( t 2 Hence, (t 2 = ( = K i.e. the shaft must be cooled by 74 K 348
5 EXERCISE 141, Page A silver plate has an area of 800 mm 2 at 15 C. Determine the increase in the area of the plate when the temperature is raised to 100 C. Assume the coefficient of linear expansion of silver to be A 2 ] i.e. A 2 [1 + 2 ] since = 2, to a very close approximation i.e. A 2 + A 1 2 Hence, area increase = m K C = = m 2 or mm 2 2. At 283 K a thermometer contains 440 mm 3 of alcohol. Determine the temperature at which the volume is 480 mm 3 assuming that the coefficient of cubic expansion of the alcohol is ] i.e = [1 + ( (t 2-283] from which, 480 = ( (t and = 440( (t from which, (t = = K and temperature, t 2 = = K 349
6 3. A zinc sphere has a radius of 30.0 mm at a temperature of 20 C. If the temperature of the sphere is raised to 420 C, determine the increase in: (a the radius, (b the surface area, (c the volume of the sphere. Assume the coefficient of linear expansion for zinc to be (a Initial radius, L 1 = 30.0 mm, initial temperature, t 1 = = 293 K, final temperature, t 2 = = 693 K and = New radius at 693 K is given by: L 2 ] i.e. L 2 = 30.0[1 + ( ( ] = 30.0[ ] = mm Hence the increase in the radius is mm (b Initial surface area of sphere, A 1 = 4 r 2 = = 3600 mm 2 New surface area at 693 K is given by: A 2 ] i.e. A 2 [1 + 2 ] since = 2, to a very close approximation Thus A 2 = 3600 [1 + 2( (400] = 3600 [ ] = ( Hence increase in surface area = 3600 ( = mm 2 (c Initial volume of sphere, V 1 = 3 4 r 3 = mm 3 New volume at 693 K is given by: ] i.e. [1 + 3 ] since = 3, to a very close approximation 350
7 4 Thus = ( [1 + 3( (400] 3 = 3 4 ( [ ] = 3 4 ( ( ( Hence, the increase in volume = 3 4 ( ( = 4207 mm 3 4. A block of cast iron has dimensions of 50 mm by 30 mm by 10 mm at 15 C. Determine the increase in volume when the temperature of the block is raised to 75 C. Assume the coefficient of linear expansion of cast iron to be Initial volume of sphere, V 1 = = mm 3 New volume at 75 C is given by: ] i.e. [1 + 3 ] since = 3, to a very close approximation Thus = [1 + 3( (75-15] = 15000[ ] = ( Hence, the increase in volume = ( = 29.7 mm 3 5. Two litres of water, initially at 20 C, is heated to 40 C. Determine the volume of water at 40 C if the coefficient of volumetric expansion of water within this range is New volume at 40 C is given by: ] = 2[1 + ( (40 20] = 2[ ] = 2[1.006] = litres 351
8 6. Determine the increase in volume, in litres, of 3 m 3 of water when heated from 293 K to boiling point if the coefficient of cubic expansion is K 1 (1 litre 10 3 m 3. Initial volume of sphere, V 1 = = 3000 litres New volume at boiling point (i.e. 373 K is given by: ] Thus = 3000 [1 + ( ( ] = 3000[ ] = ( Hence, the increase in volume = 3000 ( = 50.4 litres 7. Determine the reduction in volume when the temperature of 0.5 litre of ethyl alcohol is reduced from 40 C to - 15 C. Take the coefficient of cubic expansion for ethyl alcohol as New volume at - 15 C is given by: ] Thus = 0.5 [1 + ( ( ] = 0.5 [1 + ( (- 55] = (0.5( (- 55 Hence, the reduction in volume = (0.5( (55= litres 352
9 EXERCISE 142, Page 313 Answers found from within the text of the chapter, pages 306 to 312. EXERCISE 143, Page (b 2. (c 3. (a 4. (d 5. (b 6. (c 7. (c 8. (a 9. (c 10. (b 353
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