Thermal Expansion

Thermal Expansion: 

MFS Thermal Expansion Created by: Marlon Flores Sacedon Physics section, DMPS June 2010

Thermal Expansion: 

Thermal Expansion Two Kinds of Thermal Expansion Linear Expansion Volume Expansion Most materials expand when their temperature increase. The decks of bridges need special joints and supports to allow for expansion. A completely filled and tightly capped bottle of water cracks when it is heated You can loosen a metal jar lid by running hot water over it. . These are examples of Thermal Expansion MFS

Linear Expansion: 

Linear Expansion L o T 2 T 2 >T 1 T 1 Volume Expansion MFS

Linear Expansion: 

MFS Linear Expansion L o T 2 T 2 >T 1 T 1 Volume Expansion V o @ T 1 @ T 2 T 2 >T 1 Where: = coef. of linear expansion (1/K) L = change in length (m) L & L o = final & initial length (m) T 2 &T 1 = final & initial temperature ( o C) Where: = coef. of volume expansion (1/K) V = change in volume (m 3 ) V & V o = final & initial volume (m 3 ) T 2 &T 1 = final & initial temperature (o C) L

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Material [K -1 or (C o ) -1 ] Aluminum 2.4x10 -5 Brass 2.0x10 -5 Copper 1.7x10 -5 Glass 0.4-0.9x10 -5 Invar 0.09x10 -5 Quartz 0.04x10 -5 Steel 1.2x10 -5 Coefficients of Linear Expansion Material [K -1 or (C o ) -1 ] Aluminum 7.2x10 -5 Brass 6.0x10 -5 Copper 5.110 -5 Glass 1.2-2.7x10 -5 Invar 0.27x10 -5 Quartz 0.12x10 -5 Steel 3.6x10 -5 Ethanol 75x10 -5 Carbon Disulfide 115x10 -5 Glycerine 49x10 -5 Mercury 18x10 -5 Coefficients of Volume Expansion Relationship between coefficient of volume expansion & coefficient of linear expansion MFS

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MFS Example 1: A surveyor uses a steel measuring tape that is exactly 50.000 m long at a temperature of 20 o C. What is its length on a hot summer day when the temperature is 35 o C? Answer: 50.009 m 50 m Temperature of tape @ 20 o C Transforming From the formula: Solution L o L = ? Temperature of tape @ 35 o C From table, the coefficient of linear expansion = 50.0009 m Ans

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Example 2: The surveyor uses the measuring tape (in Example 1) to measure a distance when the temperature is 35 o C; the value that she reads off the tape is 35.794m. What is the actual distance? Answer: 35.800 m Solution L o = 50 m Temperature of tape @ 20 o C L = 50.009 m Temperature of tape @ 35 o C Let: x = actual distance @ T=35 o C Ans

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Example 3: A glass flask with volume 200 cm 3 is filled to the brim with mercury at 20 o C. How much mercury overflows when the temperature of the system is raised to 100 o C? The coefficient of linear expansion of the glass is 0.40x10 -5 K -1 . Answer: 2.7 cm3 Solution Glass flask filled w/ mercury @ T 1 =20 o C @ T 2 =100 o C Mercury column Glass flask Mercury, expanded volume( V Hg ) Mercury overflows ( V over) Glass flask, expanded volume ( V glass ) From table:

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Seat Work1 : The Humber Bridge in England has the world’s longest single span, 1410m in length. Calculate the change in length of steel deck of the span when the temperature increases from -5.0 o C to 18.0 o C. Answer: 0.39 m Seat Work2 : A metal rod is 40.125 cm long at 20.0 o C and 40.148 cm long at 45.0 o C. Calculate the average coefficient of linear expansion of the rod for this temperature range. Answer: 2.3x10 -5 (C o ) -1 Seat Work3 : A glass flask whose volume is 1000.00 cm 3 at 0.0 o C is completely filled with mercury at this temperature. When flask and mercury are warmed to 55.0 o C, 8.95 cm 3 of mercury overflow. If the coefficient of volume expansion of mercury is 18.0x10 -5 K -1, compute the coefficient of volume expansion of the glass. Answer: 1.7x10 -5 (C o ) -1

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Assignment 1 ) A Pendulum shaft of a clock is made of brass. What is the fractional change in length of the shaft when it is cooled from 19.50 o C to 5.00 o C? Answer: -2.9x10 -4 2 ) An underground tank with a capacity of 1700L (1.70m 3 ) is filled with ethanol that has an initial temperature of 19.0 o C. After the ethanol has cooled off to the temperature of the tank and ground, which is 10.0 o C, how much air space will there be above the ethanol in the tank? (Assume that the volume of the tank doesn’t change.) Answer: 2.3x10 -5 (C o ) -1 3 ) A metal rod that is 30.0 cm long expands by 0.0650 cm when its temperature is raised from 0 o C to 100 o C. A rod of a different metal and of the same length expands by 0.0350 cm for the same rise in temperature. A third rod, also 30.0 cm long is made up of pieces of each of the above metals placed end-to-end and expands 0.0580 cm between 0 o C and 100 o C. Find the length of each portion of the composite bar. Answer: 23.0cm, 7.0cm

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F Thermal Stress so (thermal stress) F A Tensile Stress Tensile Strain L o Where: F = Tensile force, (N) A = cross-section area, (m 2) Y = Young’s Modulus, (Pa or N/m 2 ) = coef. of linear expansion, (K -1 ) = change in tempersture, (K)

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Approximate Young’s Modulus Substance Young’s Modulus, Y (Pa) Aluminum 7.0x10 10 Brass 9.0x10 10 Copper 11x10 10 Crown glass 6.0x10 10 Iron 21x10 10 Lead 1.6x10 10 Nickel 21x10 10 Steel 20x10 10

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L o =10cm L o =10cm Example 1 : An aluminum cylinder 10 cm long, with a cross-section area of 20 cm 3 , is to be used as a spacer between two steel walls. At 17.2 o C it just slips in between the walls. When it warms to 22.3 o C, calculate the stress in the cylinder and the total force it exerts on each wall, assuming that the walls are perfectly rigid and a constant distance apart. Answer: -8.6x10 4 and -1.7x10 4 N @ T 1 =17.2 o C F = 0 Area A=20cm 3 @ T 1 =22.3 o C F > 0 Stress is - 8.6x10 6 Pa Negative sign indicates compression F = -1.7x10 4 N (compression)

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Seat Work1 : a) A wire that is 1.50 m long at 20 o C is found to increase in length by 1.9 cm when warmed to 420 o C. Compute its average coefficient of linear expansion for this temperature range. b) The wire is stretched just taut (zero tension) at 420 o C. Find the stress in the wire if it is cooled to 20 o C without being allowed to contact. Young’s modulus for the wire is 20.x10 11 Pa. Answer: a) 3.2x10 -5 (C o ) -1 ; b)2.5x10 9 Pa 1.50 m @ 20 o C 0.019 m @ 420 o C 1.519 m @ 420 o C 1.519 m @ 20 o C F>0 F>0 Ans

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Assignment 1 ) A brass rod is 185 cm long and 1.60 cm in diameter. What force must be applied to each end of the rod to prevent it from contracting when it is cooled from 120 o C to 10 o C? Answer: 4.0x10 4 N 2 ) Steel train rails are laid in 12.0 m-long segments placed end-to-end. The rails are laid on a winter day when their temperature is -2.0 o C. a) How much space must be left between adjacent rails if they are to just touch on a summer day when their temperature is 33.0 o C? b) If the rails are originally laid in contact, what is the stress in them on a summer day when their temperature is 33.0 o C? Answer: a) 5.0x10 -3 m; b) 8.4x10 7 Pa