Starting with Equation 19.10, show that the total pressureP in a container filled with a

Markings to indicate length are placed on a steel tape in a room that is at a temperature of 228C. Measurements are then made with the same tape on a day when the temperature is 278C. Assume the objects you are measuring have a smaller coefficient of linear expansion than steel. Are the measurements (a) too long, (b) too short, or (c) accurate?

When a certain gas under a pressure of 5.00 3 106 Pa at 25.08C is allowed to expand to 3.00 times its original volume, its final pressure is 1.07 3 106 Pa. What is its final temperature? (a) 450 K (b) 233 K (c) 212 K (d) 191 K (e) 115 K

If the volume of an ideal gas is doubled while its temperature is quadrupled, does the pressure (a) remain the same, (b) decrease by a factor of 2, (c) decrease by a factor of 4, (d) increase by a factor of 2, or (e) increase by a factor of 4

The pendulum of a certain pendulum clock is made of brass. When the temperature increases, what happens to the period of the clock? (a) It increases. (b) It decreases. (c) It remains the same.

A temperature of 1628F is equivalent to what temperature in kelvins? (a) 373 K (b) 288 K (c) 345 K (d) 201 K (e) 308 K

A cylinder with a piston holds 0.50 m3 of oxygen at an absolute pressure of 4.0 atm. The piston is pulled outward, increasing the volume of the gas until the pressure drops to 1.0 atm. If the temperature stays constant, what new volume does the gas occupy? (a) 1.0 m3 (b) 1.5 m3 (c) 2.0 m3 (d) 0.12 m3 (e) 2.5 m3

What would happen if the glass of a thermometer expanded more on warming than did the liquid in the tube? (a) The thermometer would break. (b) It could be used only for temperatures below room temperature. (c) You would have to hold it with the bulb on top. (d) The scale on the thermometer is reversed so that higher temperature values would be found closer to the bulb. (e) The numbers would not be evenly spaced.

A cylinder with a piston contains a sample of a thin gas. The kind of gas and the sample size can be changed. The cylinder can be placed in different constanttemperature baths, and the piston can be held in different positions. Rank the following cases according to the pressure of the gas from the highest to the lowest, displaying any cases of equality. (a) A 0.002-mol sample of oxygen is held at 300 K in a 100-cm3 container. (b) A 0.002-mol sample of oxygen is held at 600 K in a 200-cm3 container. (c) A 0.002-mol sample of oxygen is held at 600 K in a 300-cm3 container. (d) A 0.004-mol sample of helium is held at 300 K in a 200-cm3 container. (e) A 0.004-mol sample of helium is held at 250 K in a 200-cm3 container.

Two cylinders A and B at the same temperature contain the same quantity of the same kind of gas. Cylinder A has three times the volume of cylinder B. What can you conclude about the pressures the gases exert? (a) We can conclude nothing about the pressures (b) The pressure in A is three times the pressure in B. (c) The pressures must be equal. (d) The pressure in A must be one-third the pressure in B.

A rubber balloon is filled with 1 L of air at 1 atm and 300 K and is then put into a cryogenic refrigerator at 100 K. The rubber remains flexible as it cools. (i) What happens to the volume of the balloon? (a) It decreases to 13 L. (b) It decreases to 1/!3 L. (c) It is constant. (d) It increases to !3 L. (e) It increases to 3 L. (ii) What happens to the pressure of the air in the balloon? (a) It decreases to 13 atm. (b) It decreases to 1/!3 atm. (c) It is constant. (d) It increases to !3 atm. (e) It increases to 3 atm.

The average coefficient of linear expansion of copper is 17 3 1026 (8C)21. The Statue of Liberty is 93 m tall on a summer morning when the temperature is 258C. Assume the copper plates covering the statue are mounted edge to edge without expansion joints and do not buckle or bind on the framework supporting them as the day grows hot. What is the order of magnitude of the statues increase in height? (a) 0.1 mm (b) 1 mm (c) 1 cm (d) 10 cm (e) 1 m

Suppose you empty a tray of ice cubes into a bowl partly full of water and cover the bowl. After one-half hour, the contents of the bowl come to thermal equilibrium, with more liquid water and less ice than you started with. Which of the following is true? (a) The temperature of the liquid water is higher than the temperature of the remaining ice. (b) The temperature of the liquid water is the same as that of the ice. (c) The temperature of the liquid water is less than that of the ice. (d) The comparative temperatures of the liquid water and ice depend on the amounts present.

A hole is drilled in a metal plate. When the metal is raised to a higher temperature, what happens to the diameter of the hole? (a) It decreases. (b) It increases. (c) It remains the same. (d) The answer depends on the initial temperature of the metal. (e) None of those answers is correct.

On a very cold day in upstate New York, the temperature is 2258C, which is equivalent to what Fahrenheit temperature? (a) 2468F (b) 2778F (c) 188F (d) 2258F (e) 2138F

A square hole 8.00 cm along each side is cut in a sheet of copper. (a) Calculate the change in the area of this hole resulting when the temperature of the sheet is increased by 50.0 K. (b) Does this change represent an increase or a decrease in the area enclosed by the hole?

The average coefficient of volume expansion for carbon tetrachloride is 5.81 3 104 (C)1. If a 50.0-gal steel container is filled completely with carbon tetrachloride when the temperature is 10.0C, how much will spill over when the temperature rises to 30.0C?

At 20.08C, an aluminum ring has an inner diameter of 5.000 0 cm and a brass rod has a diameter of 5.050 0 cm. (a) If only the ring is warmed, what temperature must it reach so that it will just slip over the rod? (b) What If? If both the ring and the rod are warmed together, what temperature must they both reach so that the ring barely slips over the rod? (c) Would this latter process work? Explain. Hint: Consult Table 20.2 in the next chapter.

Why is the following situation impossible? A thin brass ring has an inner diameter 10.00 cm at 20.08C. A solid aluminum cylinder has diameter 10.02 cm at 20.08C. Assume the average coefficients of linear expansion of the two metals are constant. Both metals are cooled together to a temperature at which the ring can be slipped over the end of the cylinder.

A volumetric flask made of Pyrex is calibrated at 20.08C. It is filled to the 100-mL mark with 35.08C acetone. After the flask is filled, the acetone cools and the flask warms so that the combination of acetone and flask reaches a uniform temperature of 32.08C. The combination is then cooled back to 20.08C. (a) What is the volume of the acetone when it cools to 20.08C? (b) At the temperature of 32.08C, does the level of acetone lie above or below the 100-mL mark on the flask? Explain.

Review. On a day that the temperature is 20.08C, a concrete walk is poured in such a way that the ends of the walk are unable to move. Take Youngs modulus for concrete to be 7.00 3 109 N/m2 and the compressive strength to be 2.00 3 109 N/m2. (a) What is the stress in the cement on a hot day of 50.08C? (b) Does the concrete fracture?

A hollow aluminum cylinder 20.0 cm deep has an internal capacity of 2.000 L at 20.08C. It is completely filled with turpentine at 20.08C. The turpentine and the aluminum cylinder are then slowly warmed together to 80.08C. (a) How much turpentine overflows? (b) What is the volume of turpentine remaining in the cylinder at 80.08C? (c) If the combination with this amount of turpentine is then cooled back to 20.08C, how far below the cylinders rim does the turpentines surface recede?

Review. The Golden Gate Bridge in San Francisco has a main span of length 1.28 km, one of the longest in the world. Imagine that a steel wire with this length and a cross-sectional area of 4.00 3 1026 m2 is laid in a straight line on the bridge deck with its ends attached to the towers of the bridge. On a summer day the temperature of the wire is 35.08C. (a) When winter arrives, the towers stay the same distance apart and the bridge deck keeps the same shape as its expansion joints open. When the temperature drops to 210.08C, what is the tension in the wire? Take Youngs modulus for steel to be 20.0 3 1010 N/m2. (b) Permanent deformation occurs if the stress in the steel exceeds its elastic limit of 3.00 3 108 N/m2. At what temperature would the wire reach its elastic limit? (c) What If? Explain how your answers to parts (a) and (b) would change if the Golden Gate Bridge were twice as long.

A sample of lead has a mass of 20.0 kg and a density of 11.3 3 103 kg/m3 at 08C. (a) What is the density of lead at 90.08C? (b) What is the mass of the sample of lead at 90.08C?

A sample of a solid substance has a mass m and a density r0 at a temperature T0. (a) Find the density of the substance if its temperature is increased by an amount DT in terms of the coefficient of volume expansion b. (b) What is the mass of the sample if the temperature is raised by an amount DT ?

An underground gasoline tank can hold 1.00 3 103 gallons of gasoline at 52.0F. Suppose the tank is being filled on a day when the outdoor temperature (and the temperature of the gasoline in a tanker truck) is 95.0F. When the underground tank registers that it is full, how many gallons have been transferred from the truck, according to a non-temperature-compensated gauge on the truck? Assume the temperature of the gasoline quickly cools from 95.0F to 52.0F upon entering the tank.

A rigid tank contains 1.50 moles of an ideal gas. Determine the number of moles of gas that must be withdrawn from the tank to lower the pressure of the gas from 25.0 atm to 5.00 atm. Assume the volume of the tank and the temperature of the gas remain constant during this operation.

Gas is confined in a tank at a pressure of 11.0 atm and a temperature of 25.08C. If two-thirds of the gas is withdrawn and the temperature is raised to 75.08C, what is the pressure of the gas remaining in the tank?

Your father and your younger brother are confronted with the same puzzle. Your fathers garden sprayer and your brothers water cannon both have tanks with a capacity of 5.00 L (Fig. P19.28). Your father puts a negligible amount of concentrated fertilizer into his tank. They both pour in 4.00 L of water and seal up their tanks, so the tanks also contain air at atmospheric pressure. Next, each uses a hand-operated pump to inject more air until the absolute pressure in the tank reaches 2.40 atm. Now each uses his device to spray out waternot airuntil the stream becomes feeble, which it does when the pressure in the tank reaches 1.20 atm. To accomplish spraying out all the water, each finds he must pump up the tank three times. Here is the puzzle: most of the water sprays out after the second pumping. The first and the third pumpingup processes seem just as difficult as the second but result in a much smaller amount of water coming out. Account for this phenomenon.

Gas is contained in an 8.00-L vessel at a temperature of 20.08C and a pressure of 9.00 atm. (a) Determine the number of moles of gas in the vessel. (b) How many molecules are in the vessel?

A container in the shape of a cube 10.0 cm on each edge contains air (with equivalent molar mass 28.9 g/mol) at atmospheric pressure and temperature 300 K. Find (a) the mass of the gas, (b) the gravitational force exerted on it, and (c) the force it exerts on each face of the cube. (d) Why does such a small sample exert such a great force?

An auditorium has dimensions 10.0 m 3 20.0 m 3 30.0 m. How many molecules of air fill the auditorium at 20.08C and a pressure of 101 kPa (1.00 atm)?

The pressure gauge on a tank registers the gauge pressure, which is the difference between the interior pressure and exterior pressure. When the tank is full of oxygen (O2), it contains 12.0 kg of the gas at a gauge pressure of 40.0 atm. Determine the mass of oxygen that has been withdrawn from the tank when the pressure reading is 25.0 atm. Assume the temperature of the tank remains constant.

(a) Find the number of moles in one cubic meter of an ideal gas at 20.08C and atmospheric pressure. (b) For air, Avogadros number of molecules has mass 28.9 g. Calculate the mass of one cubic meter of air. (c) State how this result compares with the tabulated density of air at 20.08C.

Use the definition of Avogadros number to find the mass of a helium atom.

A popular brand of cola contains 6.50 g of carbon dioxide dissolved in 1.00 L of soft drink. If the evaporating carbon dioxide is trapped in a cylinder at 1.00 atm and 20.0C, what volume does the gas occupy?

In state-of-the-art vacuum systems, pressures as low as 1.00 3 1029 Pa are being attained. Calculate the number of molecules in a 1.00-m3 vessel at this pressure and a temperature of 27.08C.

An automobile tire is inflated with air originally at 10.08C and normal atmospheric pressure. During the process, the air is compressed to 28.0% of its original volume and the temperature is increased to 40.08C. (a) What is the tire pressure? (b) After the car is driven at high speed, the tires air temperature rises to 85.08C and the tires interior volume increases by 2.00%. What is the new tire pressure (absolute)?

Review. To measure how far below the ocean surface a bird dives to catch a fish, a scientist uses a method originated by Lord Kelvin. He dusts the interiors of plastic tubes with powdered sugar and then seals one end of each tube. He captures the bird at nighttime in its nest and attaches a tube to its back. He then catches the same bird the next night and removes the tube. In one trial, using a tube 6.50 cm long, water washes away the sugar over a distance of 2.70 cm from the open end of the tube. Find the greatest depth to which the bird dived, assuming the air in the tube stayed at constant temperature.

Review. The mass of a hot-air balloon and its cargo (not including the air inside) is 200 kg. The air outside is at 10.08C and 101 kPa. The volume of the balloon is 400 m3. To what temperature must the air in the balloon be warmed before the balloon will lift off? (Air density at 10.08C is 1.244 kg/m3.)

A room of volume V contains air having equivalent molar mass M (in g/mol). If the temperature of the room is raised from T1 to T2, what mass of air will leave the room? Assume that the air pressure in the room is maintained at P0.

Review. At 25.0 m below the surface of the sea, where the temperature is 5.008C, a diver exhales an air bubble having a volume of 1.00 cm3. If the surface temperature of the sea is 20.08C, what is the volume of the bubble just before it breaks the surface?

Estimate the mass of the air in your bedroom. State the quantities you take as data and the value you measure or estimate for each.

A cook puts 9.00 g of water in a 2.00-L pressure cooker that is then warmed to 5008C. What is the pressure inside the container?

The pressure gauge on a cylinder of gas registers the gauge pressure, which is the difference between the interior pressure and the exterior pressure P0. Lets call the gauge pressure Pg. When the cylinder is full, the mass of the gas in it is mi at a gauge pressure of Pgi. Assuming the temperature of the cylinder remains constant, show that the mass of the gas remaining in the cylinder when the pressure reading is Pg f is given by mf 5 mi a Pg f 1 P0 Pg i 1 P0 b

Long-term space missions require reclamation of the oxygen in the carbon dioxide exhaled by the crew. In one method of reclamation, 1.00 mol of carbon dioxide produces 1.00 mol of oxygen and 1.00 mol of methane as a byproduct. The methane is stored in a tank under pressure and is available to control the attitude of the spacecraft by controlled venting. A single astronaut exhales 1.09 kg of carbon dioxide each day. If the methane generated in the respiration recycling of three astronauts during one week of flight is stored in an originally empty 150-L tank at 245.0C, what is the final pressure in the tank?

A steel beam being used in the construction of a skyscraper has a length of 35.000 m when delivered on a cold day at a temperature of 15.0008F. What is the length of the beam when it is being installed later on a warm day when the temperature is 90.0008F?

A spherical steel ball bearing has a diameter of 2.540 cm at 25.008C. (a) What is its diameter when its temperature is raised to 100.08C? (b) What temperature change is required to increase its volume by 1.000%?

A bicycle tire is inflated to a gauge pressure of 2.50 atm when the temperature is 15.08C. While a man rides the bicycle, the temperature of the tire rises to 45.08C. Assuming the volume of the tire does not change, find the gauge pressure in the tire at the higher temperature.

In a chemical processing plant, a reaction chamber of fixed volume V0 is connected to a reservoir chamber of fixed volume 4V0 by a passage containing a thermally insulating porous plug. The plug permits the chambers to be at different temperatures. The plug allows gas to pass from either chamber to the other, ensuring that the pressure is the same in both. At one point in the processing, both chambers contain gas at a pressure of 1.00 atm and a temperature of 27.0C. Intake and exhaust valves to the pair of chambers are closed. The reservoir is maintained at 27.0C while the reaction chamber is heated to 400C. What is the pressure in both chambers after that is done?

Why is the following situation impossible? An apparatus is designed so that steam initially at T 5 1508C, P 5 1.00 atm, and V 5 0.500 m3 in a pistoncylinder apparatus undergoes a process in which (1) the volume remains constant and the pressure drops to 0.870 atm, followed by (2) an expansion in which the pressure remains constant and the volume increases to 1.00 m3, followed by (3) a return to the initial conditions. It is important that the pressure of the gas never fall below 0.850 atm so that the piston will support a delicate and very expensive part of the apparatus. Without such support, the delicate apparatus can be severely damaged and rendered useless. When the design is turned into a working prototype, it operates perfectly.

A mercury thermometer is constructed as shown in Figure P19.51. The Pyrex glass capillary tube has a diameter of 0.004 00 cm, and the bulb has a diameter of 0.250 cm. Find the change in height of the mercury column that occurs with a temperature change of 30.08C.

A liquid with a coefficient of volume expansion b just fills a spherical shell of volume V (Fig. P19.51). The shell and the open capillary of area A projecting from the top of the sphere are made of a material with an average coefficient of linear expansion a. The liquid is free to expand into the capillary. Assuming the temperature increases by DT, find the distance Dh the liquid rises in the capillary.

Review. An aluminum pipe is open at both ends and used as a flute. The pipe is cooled to 5.008C, at which its length is 0.655 m. As soon as you start to play it, the pipe fills with air at 20.08C. After that, by how much does its fundamental frequency change as the metal rises in temperature to 20.08C?

Two metal bars are made of invar and a third bar is made of aluminum. At 08C, each of the three bars is drilled with two holes 40.0 cm apart. Pins are put through the holes to assemble the bars into an equilateral triangle as in Figure P19.54. (a) First ignore the expansion of the invar. Find the angle between the invar bars as a function of Celsius temperature. (b) Is your answer accurate for negative as well as positive temperatures? (c) Is it accurate for 08C? (d) Solve the problem again, including the expansion of the invar. Aluminum melts at 6608C and invar at 1 4278C. Assume the tabulated expansion coefficients are constant. What are (e) the greatest and (f) the smallest attainable angles between the invar bars?

A student measures the length of a brass rod with a steel tape at 20.08C. The reading is 95.00 cm. What will the tape indicate for the length of the rod when the rod and the tape are at (a) 215.08C and (b) 55.08C?

The density of gasoline is 730 kg/m3 at 08C. Its average coefficient of volume expansion is 9.60 3 1024 (8C)21. Assume 1.00 gal of gasoline occupies 0.003 80 m3.How many extra kilograms of gasoline would you receive if you bought 10.0 gal of gasoline at 08C rather than at 20.08C from a pump that is not temperature compensated?

A liquid has a density r. (a) Show that the fractional change in density for a change in temperature DT is Dr/r 5 2b DT. (b) What does the negative sign signify? (c) Fresh water has a maximum density of 1.000 0 g/cm3 at 4.08C. At 10.08C, its density is 0.999 7 g/cm3. What is b for water over this temperature interval? (d) At 08C, the density of water is 0.999 9 g/cm3. What is the value for b over the temperature range 08C to 4.008C?

(a) Take the definition of the coefficient of volume expansion to be b 5 1 V dV dT ` P 5constant 5 1 V 'V 'T Use the equation of state for an ideal gas to show that the coefficient of volume expansion for an ideal gas at constant pressure is given by b 5 1/T, where T is the absolute temperature. (b) What value does this expression predict for b at 08C? State how this result compares with the experimental values for (c) helium and (d) air in Table 19.1. Note: These values are much larger than the coefficients of volume expansion for most liquids and solids.

Review. A clock with a brass pendulum has a period of 1.000 s at 20.08C. If the temperature increases to 30.08C, (a) by how much does the period change and (b) how much time does the clock gain or lose in one week?

A bimetallic strip of length L is made of two ribbons of different metals bonded together. (a) First assume the strip is originally straight. As the strip is warmed, the metal with the greater average coefficient of expansion expands more than the other, forcing the strip into an arc with the outer radius having a greater circumference (Fig. P19.60). Derive an expression for the angle of bending u as a function of the initial length of the strips, their average coefficients of linear expansion, the change in temperature, and the separation of the centers of the strips (Dr 5 r2 2 r1). (b) Show that the angle of bending decreases to zero when DT decreases to zero and also when the two average coefficients of expansion become equal. (c) What If? What happens if the strip is cooled?

The rectangular plate shown in Figure P19.61 has an area Ai equal to ,w. If the temperature increases by DT, each dimension increases according to Equation 19.4, where a is the average coefficient of linear expansion. (a) Show that the increase in area is DA 5 2aAi DT. (b) What approximation does this expression assume?

The measurement of the average coefficient of volume expansion b for a liquid is complicated because the container also changes size with temperature. Figure P19.62 shows a simple means for measuring b despite the expansion of the container. With this apparatus, one arm of a U-tube is maintained at 08C in a waterice bath, and the other arm is maintained at a different temperature TC in a constant-temperature bath. The connecting tube is horizontal. A difference in the length or diameter of the tube between the two arms of the U-tube has no effect on the pressure balance at the bottom of the tube because the pressure depends only on the depth of the liquid. Derive an expression for b for the liquid in terms of h0, ht, and TC.

A copper rod and a steel rod are different in length by 5.00 cm at 08C. The rods are warmed and cooled together. (a) Is it possible that the length difference remains constant at all temperatures? Explain. (b) If so, describe the lengths at 08C as precisely as you can. Can you tell which rod is longer? Can you tell the lengths of the rods?

A vertical cylinder of crosssectional area A is fitted with a tight-fitting, frictionless piston of mass m (Fig. P19.64). The piston is not restricted in its motion in any way and is supported by the gas at pressure P below it. Atmospheric pressure is P0. We wish to find the height h in Figure P19.64. (a) What analysis model is appropriate to describe the piston? (b) Write an appropriate force equation for the piston from this analysis model in terms of P, P0, m, A, and g. (c) Suppose n moles of an ideal gas are in the cylinder at a temperature of T. Substitute for P in your answer to part (b) to find the height h of the piston above the bottom of the cylinder.

Review. Consider an object with any one of the shapes displayed in Table 10.2. What is the percentage increase in the moment of inertia of the object when it is warmed from 08C to 1008C if it is composed of (a) copper or (b) aluminum? Assume the average linear expansion coefficients shown in Table 19.1 do not vary between 08C and 1008C. (c) Why are the answers for parts (a) and (b) the same for all the shapes?

(a) Show that the density of an ideal gas occupying a volume V is given by r 5 PM/RT, where M is the molar mass. (b) Determine the density of oxygen gas at atmospheric pressure and 20.08C.

Two concrete spans of a 250-m-long bridge are placed end to end so that no room is allowed for expansion (Fig. P19.67a). If a temperature increase of 20.08C occurs, what is the height y to which the spans rise when they buckle (Fig. P19.67b)?

Two concrete spans that form a bridge of length L are placed end to end so that no room is allowed for expansion (Fig. P19.67a). If a temperature increase of DT occurs, what is the height y to which the spans rise when they buckle (Fig. P19.67b)?

Review. (a) Derive an expression for the buoyant force on a spherical balloon, submerged in water, as a function of the depth h below the surface, the volume Vi of the balloon at the surface, the pressure P0 at the surface, and the density rw of the water. Assume the water temperature does not change with depth. (b) Does the buoyant force increase or decrease as the balloon is submerged? (c) At what depth is the buoyant force onehalf the surface value?

Review. Following a collision in outer space, a copper disk at 8508C is rotating about its axis with an angular speed of 25.0 rad/s. As the disk radiates infrared light, its temperature falls to 20.08C. No external torque acts on the disk. (a) Does the angular speed change as the disk cools? Explain how it changes or why it does not. (b) What is its angular speed at the lower temperature?

Starting with Equation 19.10, show that the total pressure P in a container filled with a mixture of several ideal gases is P 5 P1 1 P2 1 P3 1 . . . , where P1, P2, . . . are the pressures that each gas would exert if it alone filled the container. (These individual pressures are called the partial pressures of the respective gases.) This result is known as Daltons law of partial pressures.

Review. A steel wire and a copper wire, each of diameter 2.000 mm, are joined end to end. At 40.08C, each has an unstretched length of 2.000 m. The wires are connected between two fixed supports 4.000 m apart on a tabletop. The steel wire extends from x 5 22.000 m to x 5 0, the copper wire extends from x 5 0 to x 5 2.000 m, and the tension is negligible. The temperature is then lowered to 20.08C. Assume the average coefficient of linear expansion of steel is 11.0 3 1026 (8C)21 and that of copper is 17.0 3 1026 (8C)21. Take Youngs modulus for steel to be 20.0 3 1010 N/m2 and that for copper to be 11.0 3 1010 N/m2. At this lower temperature, find (a) the tension in the wire and (b) the x coordinate of the junction between the wires.

Review. A steel guitar string with a diameter of 1.00 mm is stretched between supports 80.0 cm apart. The temperature is 0.08C. (a) Find the mass per unit length of this string. (Use the value 7.86 3 103 kg/m3 for the density.) (b) The fundamental frequency of transverse oscillations of the string is 200 Hz. What is the tension in the string? Next, the temperature is raised to 30.08C. Find the resulting values of (c) the tension and (d) the fundamental frequency. Assume both the Youngs modulus of 20.0 3 1010 N/m2 and the average coefficient of expansion a 5 11.0 3 1026 (8C)21 have constant values between 0.08C and 30.08C.

A cylinder is closed by a piston connected to a spring of constant 2.00 3 103 N/m (see Fig. P19.74). With the spring relaxed, the cylinder is filled with 5.00 L of gas at a pressure of 1.00 atm and a temperature of 20.08C. (a) If the piston has a cross-sectional area of 0.010 0 m2 and negligible mass, how high will it rise when the temperature is raised to 2508C? (b) What is the pressure of the gas at 2508C?

Helium gas is sold in steel tanks that will rupture if subjected to tensile stress greater than its yield strength of 5 3 108 N/m2. If the helium is used to inflate a balloon, could the balloon lift the spherical tank the helium came in? Justify your answer. Suggestion: You may consider a spherical steel shell of radius r and thickness t having the density of iron and on the verge of breaking apart into two hemispheres because it contains helium at high pressure.

A cylinder that has a 40.0-cm radius and is 50.0 cm deep is filled with air at 20.08C and 1.00 atm (Fig. P19.76a). A 20.0-kg piston is now lowered into the cylinder, compressing the air trapped inside as it takes equilibrium height hi (Fig. P19.76b). Finally, a 25.0-kg dog stands on the piston, further compressing the air, which remains at 208C (Fig. P19.76c). (a) How far down (Dh) does the piston move when the dog steps onto it? (b) To what temperature should the gas be warmed to raise the piston and dog back to hi?

The relationship L 5 Li 1 aLi DT is a valid approximation when a DT is small. If a DT is large, one must integrate the relationship dL 5 aL dT to determine the final length. (a) Assuming the coefficient of linear expansion of a material is constant as L varies, determine a general expression for the final length of a rod made of the material. Given a rod of length 1.00 m and a temperature change of 100.08C, determine the error caused by the approximation when (b) a 5 2.00 3 1025 (8C)21 (a typical value for a metal) and (c) when a 5 0.020 0 (8C)21 (an unrealistically large value for comparison). (d) Using the equation from part (a), solve Problem 21 again to find more accurate results.

Review. A house roof is a perfectly flat plane that makes an angle u with the horizontal. When its temperature changes, between Tc before dawn each day and Th in the middle of each afternoon, the roof expands and contracts uniformly with a coefficient of thermal expansion a1. Resting on the roof is a flat, rectangular metal plate with expansion coefficient a2, greater than a1. The length of the plate is L, measured along the slope of the roof. The component of the plates weight perpendicular to the roof is supported by a normal force uniformly distributed over the area of the plate. The coefficient of kinetic friction between the plate and the roof is mk. The plate is always at the same temperature as the roof, so we assume its temperature is continuously changing. Because of the difference in expansion coefficients, each bit of the plate is moving relative to the roof below it, except for points along a certain horizontal line running across the plate called the stationary line. If the temperature is rising, parts relative to the roof and feel a force of kinetic friction acting up the roof. Elements of area above the stationary line are sliding up the roof, and on them kinetic friction acts downward parallel to the roof. The stationary line occupies no area, so we assume no force of static friction acts on the plate while the temperature is changing. The plate as a whole is very nearly in equilibrium, so the net friction force on it must be equal to the component of its weight acting down the incline. (a) Prove that the stationary line is at a distance of L 2 a1 2 tan u mk b below the top edge of the plate. (b) Analyze the forces that act on the plate when the temperature is falling and prove that the stationary line is at that same distance above the bottom edge of the plate. (c) Show that the plate steps down the roof like an inchworm, moving each day by the distance L mk 1a2 2 a1 2 1Th 2 Tc 2 tan u (d) Evaluate the distance an aluminum plate moves each day if its length is 1.20 m, the temperature cycles between 4.008C and 36.08C, and if the roof has slope 18.5, coefficient of linear expansion 1.50 3 1025 (8C)21, and coefficient of friction 0.420 with the plate. (e) What If? What if the expansion coefficient of the plate is less than that of the roof? Will the plate creep up the roof?

A 1.00-km steel railroad rail is fastened securely at both ends when the temperature is 20.08C. As the temperature increases, the rail buckles, taking the shape of an arc of a vertical circle. Find the height h of the center of the rail when the temperature is 25.08C. (You will need to solve a transcendental equation.)