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

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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

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Linear Expansion LoLo T2T2 T 2 >T 1 T1T1 Volume Expansion MFS

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Linear Expansion LoLo T2T2 T 2 >T 1 T1T1 Volume Expansion V T 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 ] Aluminum2.4x10 -5 Brass2.0x10 -5 Copper1.7x10 -5 Glass x10 -5 Invar0.09x10 -5 Quartz0.04x10 -5 Steel1.2x10 -5 Coefficients of Linear Expansion Material [K -1 or (C o ) -1 ] Aluminum7.2x10 -5 Brass6.0x10 -5 Copper Glass x10 -5 Invar0.27x10 -5 Quartz0.12x10 -5 Steel3.6x10 -5 Ethanol75x10 -5 Carbon Disulfide115x10 -5 Glycerine49x10 -5 Mercury18x10 -5 Coefficients of Volume Expansion Relationship between coefficient of volume expansion & coefficient of linear expansion MFS

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Example 1: A surveyor uses a steel measuring tape that is exactly 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: m 50 m Temperature of 20 o C Transforming From the formula: Solution LoLo L = ? Temperature of 35 o C From table, the coefficient of linear expansion = 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 m. What is the actual distance? Answer: m Solution L o = 50 m Temperature of 20 o C L = m Temperature of 35 o C Let: x = actual T=35 o C Ans

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MFS 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/ T 1 =20 o 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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MFS Seat Work1: The Humber Bridge in England has the worlds 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 cm long at 20.0 o C and 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 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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MFS 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 o C to 5.00 o C? Answer: -2.9x ) 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 doesnt change.) Answer: 2.3x10 -5 (C o ) -1 3) A metal rod that is 30.0 cm long expands by 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 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 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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MFS F Thermal Stress so (thermal stress) F A Tensile Stress Tensile Strain LoLo Where: F = Tensile force, (N) A = cross-section area, (m 2) Y = Youngs Modulus, (Pa or N/m 2 ) = coef. of linear expansion, (K -1 ) = change in tempersture, (K)

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Approximate Youngs Modulus SubstanceYoungs Modulus, Y (Pa) Aluminum7.0x10 10 Brass9.0x10 10 Copper11x10 10 Crown glass6.0x10 10 Iron21x10 10 Lead1.6x10 10 Nickel21x10 10 Steel20x10 10

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MFS 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 6 and -1.7x10 4 T 1 =17.2 o C F = 0 Area A=20cm 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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MFS Assignment: 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. Youngs modulus for the wire is 20.x10 11 Pa. Answer: a) 3.2x10 -5 (C o ) -1 ; b)2.5x10 9 Pa o C o C o C o C F>0 Ans

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MFS 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

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