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# College Physics 2017

## Educators

### Problem 1

For each of the following temperatures, find the equivalent temperature on the indicated scale: (a) $-273.15^{\circ} \mathrm{C}$ on the Fahrenheit scale, (b) $98.6^{\circ} \mathrm{F}$ on the Celsius scale, and (c) $1.00 \times 10^{2} \mathrm{K}$ on the Fahrenheit scale.

Salamat A.

### Problem 2

The pressure in a constant-volume gas thermometer is 0.700 atm at $1.00 \times 10^{2 \circ} \mathrm{C}$ and 0.512 atm at $0^{\circ} \mathrm{C}$ (a) What is the temperature when the pressure is 0.0400 $\mathrm{atm}$ ? (b) What is the pressure at $450^{\circ} \mathrm{C}$ ?

Salamat A.

### Problem 3

The boiling point of liquid hydrogen is 20.3 $\mathrm{K}$ at atmospheric pressure. What is this temperature on (a) the Celsius scale and (b) the Fahrenheit scale?

Salamat A.

### Problem 4

Death Valley holds the record for the highest recorded temperature in the United States. On July $10,1913$ , at a place called Furnace Creek Ranch, the temperature rose to $134^{\circ} \mathrm{F}$ . The lowest U.S. temperature ever recorded occurred at Prospect Creek Camp in Alaska on January $23,1971$ , when the temperature plummeted to $-79.8^{\circ} \mathrm{F}$ . (a) Convert these temperatures to the Celsius scale. (b) Convert the Celsius temperatures to Kelvin.

Salamat A.

### Problem 5

On January $22,1943,$ in Spearfish, South Dakota, the temperature rose from $-4.00^{\circ} \mathrm{F}$ to $45.0^{\circ} \mathrm{F}$ over the course of two minutes (the current world record for the fastest recorded temperature change). By how much did the temperature change on the Kelvin scale?

Salamat A.

### Problem 6

In a student experiment, a constant-volume gas thermometer is calibrated in dry ice $\left(-78.5^{\circ} \mathrm{C}\right)$ and in boiling ethyl alcohol $\left(78.0^{\circ} \mathrm{C}\right) .$ The separate pressures are 0.900 atm and 1.635 atm. (a) What value of absolute zero in degrees Celsius does the calibration yield? What pressures would be found at $(\mathrm{b})$ the freezing and $(\mathrm{c})$ boiling points of water? Hint: Use the linear relationship $P=A+B T$ , where $A$ and $B$ are constants.

Salamat A.

### Problem 7

A person's body temperature is $101.6^{\circ} \mathrm{F}$ , indicating a fever of $3.0^{\circ} \mathrm{F}$ above the normal average body temperature of $98.6^{\circ} \mathrm{F}$ . How many degrees above normal is this body temperature on the Celsius scale?

Salamat A.

### Problem 8

The temperature difference between the inside and the outside of a home on a cold winter day is $57.0^{\circ} \mathrm{F}$ . Express this difference on (a) the Celsius scale and (b) the Kelvin scale.

Salamat A.

### Problem 9

A nurse measures the temperature of a patient to be 41.5 $\mathrm{C}$ . (a) What is this temperature on the Fahrenheit scale? (b) Do you think the patient is seriously ill? Explain.

Salamat A.

### Problem 10

Temperature differences on the Rankine scale are identical to differences on the Fahrenheit scale, but absolute zero is given as $0^{\circ} \mathrm{R}$ . (a) Find a relationship converting the temperatures $T_{F}$ of the Fahrenheit scale to the corresponding temperatures $T_{R}$ of the Rankine scale. (b) Find a second relationship converting temperatures $T_{R}$ of the Rankine scale to the temperatures $T_{K}$ of the Kelvin scale.

Salamat A.

### Problem 11

The New River Gorge bridge in West Virginia is a $518-\mathrm{long}$ . steel arch. How much will its length change between temperature extremes of $-20.0^{\circ} \mathrm{C}$ and $35.0^{\circ} \mathrm{C}$ ?

Salamat A.

### Problem 12

A grandfather clock is controlled by a swinging brass pendulum that is 1.3 $\mathrm{m}$ long at a temperature of $20.0^{\circ} \mathrm{C}$ . (a) What is the length of the pendulum rod when the temperature drops to $0.0^{\circ} \mathrm{C}$ ? (b) If a pendulum's period is given by $T=2 \pi \sqrt{L / g}$ , where $L$ is its length, does the change in length of the rod cause the clock to run fast or slow?

Salamat A.

### Problem 13

A pair of eyeglass frames are made of epoxy plastic (coefficient of linear expansion $=1.30 \times 10^{-4}\left(^{\circ} \mathrm{C}\right)^{-1} )$ . At room temperature $\left(20.0^{\circ} \mathrm{C}\right)$ , the frames have circular lens holes 2.20 $\mathrm{cm}$ in radius. To what temperature must the frames be heated if lenses 2.21 $\mathrm{cm}$ in radius are to be inserted into them?

Salamat A.

### Problem 14

A spherical steel ball bearing has a diameter of 2.540 $\mathrm{cm}$ at $25.00^{\circ} \mathrm{C}$ (a) What is its diameter when its temperature is raised to $100.0^{\circ} \mathrm{C}$ ? (b) What temperature change is required to increase its volume by 1.000$\%$ ?

Salamat A.

### Problem 15

A brass ring of diameter 10.00 $\mathrm{cm}$ at $20.0^{\circ} \mathrm{C}$ is heated and slipped over an aluminum rod of diameter 10.01 $\mathrm{cm}$ at $20.0^{\circ} \mathrm{C}$ . Assuming the average coefficients of linear expansion are constant, (a) to what temperature must the combination be cooled to separate the two metals? Is that temperature attainable? (b) What if the aluminum rod were 10.02 $\mathrm{cm}$ in diameter?

Salamat A.

### Problem 16

A wire is 25.0 $\mathrm{m}$ long at $2.00^{\circ} \mathrm{C}$ and is 1.19 $\mathrm{cm}$ longer at $30.0^{\circ} \mathrm{C}$ . Find the wire's coefficient of linear expansion.

Salamat A.

### Problem 17

The density of lead is $1.13 \times 10^{4} \mathrm{kg} / \mathrm{m}^{3}$ at $20.0^{\circ} \mathrm{C} .$ Find its density at $105^{\circ} \mathrm{C} .$

Salamat A.

### Problem 18

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 $4.00 \times 10^{-6} \mathrm{m}^{2}$ is laid on the bridge deck with its ends attached to the towers of the bridge, on a summer day when the temperature of the wire is $35.0^{\circ} \mathrm{C}$ . (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 $-10.0^{\circ} \mathrm{C},$ what is the tension in the wire? Take Young's modulus for steel to be $20.0 \times 10^{10} \mathrm{N} / \mathrm{m}^{2}$ . (b) Permanent deformation occurs if the stress in the steel exceeds its elastic limit of $3.00 \times 10^{8} \mathrm{N} / \mathrm{m}^{2}$ . At what temperature would the wire reach its elastic limitt? (c) Explain how your answers to (a) and (b) would change if the Golden Gate Bridge were twice as long.

Salamat A.

### Problem 19

An underground gasoline tank can hold $1.00 \times 10^{9}$ gallons of gasoline at $52.0^{\circ} \mathrm{F}$ . If the tank is being filled on a day when the outdoor temperature (and the gasoline in a tanker truck) is $95.0^{\circ} \mathrm{F}$ , how many gallons from the truck can be poured into the tank? Assume the temperature of the gasoline quickly cools from $95.0^{\circ} \mathrm{F}$ to $52.0^{\circ} \mathrm{F}$ upon entering the tank.

Salamat A.

### Problem 20

Show that the coefficient of volume expansion, $\beta,$ is related to the coefficient of linear expansion, $\alpha,$ through the expression $\beta=3 \alpha$ .

Salamat A.

### Problem 21

A hollow aluminum cylinder 20.0 $\mathrm{cm}$ deep has an internal capacity of 2.000 $\mathrm{L}$ at $20.0^{\circ} \mathrm{C} .$ It is completely filled with turpentine at $20.0^{\circ} \mathrm{C}$ . The turpentine and the aluminum cylinder are then slowly warmed together to $80.0^{\circ} \mathrm{C}$ . (a) How much turpentine overflows? (b) What is the volume of the turpentine remaining in the cylinder at $80.0^{\circ} \mathrm{C}$ ? (c) If the combination with this amount of turpentine is then cooled back to $20.0^{\circ} \mathrm{C},$ how far below the cylinder's rim does the turpentine's surface recede?

Salamat A.

### Problem 22

A construction worker uses a steel tape to measure the length of an aluminum support column. If the measured length is 18.700 $\mathrm{m}$ when the temperature is $21.2^{\circ} \mathrm{C},$ what is the measured length when the temperature rises to $29.4^{\circ} \mathrm{C} ?$ Note: Don't neglect the expansion of the tape.

Salamat A.

### Problem 23

The band in Figure $P 10.23$ is stainless steel (coefficient of linear expansion $=$ $17.3 \times 10^{-6}\left(^{\circ} \mathrm{C}\right)^{-1} ;$ Young's modulus $=18 \times$ $10^{10} \mathrm{N} / \mathrm{m}^{2} ) .$ It is essentially circular with an initial mean radius of $5.0 \mathrm{mm},$ a height of $4.0 \mathrm{mm},$ and a thickness of 0.50 $\mathrm{mm}$ . If the band just fits snugly over the tooth when heated to a temperature of $80.0^{\circ} \mathrm{C},$ what is the tension in the band when it cools to a temperature of $37^{\circ} \mathrm{C}$ ?

Salamat A.

### Problem 24

The Trans-Alaskan pipeline is 1 300 km long, reaching from Prudhoe Bay to the port of Valdez, and is subject to temperatures ranging from $-73^{\circ} \mathrm{C}$ to $+35^{\circ} \mathrm{C}$ (a) How much does the steel pipeline expand due to the difference in temperature? (b) How can one compensate for this expansion?

Salamat A.

### Problem 25

The average coefficient of volume expansion for carbon tetrachloride is $5.81 \times 10^{-4}\left(^{\circ} \mathrm{C}\right)^{-1}$ . If a 50.0 -gal steel container is filled completely with carbon tetrachloride when the temperature is $10.0^{\circ} \mathrm{C},$ how much will spill over when the temperature rises to $30.0^{\circ} \mathrm{C}$ ?

Salamat A.

### Problem 26

The density of gasoline is $7.30 \times 10^{2} \mathrm{kg} / \mathrm{m}^{3}$ at $0^{\circ} \mathrm{C}$ . Its average coefficient of volume expansion is $9.60 \times 10^{-4}\left(^{\circ} \mathrm{C}\right)^{-1}$ , and note that $1.00 \mathrm{gal}=0.00380 \mathrm{m}^{3} .$ (a) Calculate the mass of 10.0 gal of gas at $0^{\circ} \mathrm{C}$ . (b) If 1.000 $\mathrm{m}^{3}$ of gasoline at $0^{\circ} \mathrm{C}$ is warmed by $20.0^{\circ} \mathrm{C},$ calculate its new volume. (c) Using the answer to part (b), calculate the density of gasoline at $20.0^{\circ} \mathrm{C}$ . (d) Calculate the mass of 10.0 $\mathrm{gal}$ of gas at $20.0^{\circ} \mathrm{C}$ . (e) How many extra kilograms of gasoline would you get if you bought 10.0 gal of gasoline at $0^{\circ} \mathrm{C}$ rather than at $20.0^{\circ} \mathrm{C}$ from a pump that is not temperature compensated?

Salamat A.

### Problem 27

Figure $P 10.27$ shows a circular steel casting with a gap. If the casting is heated, (a) does the width of the gap increase or decrease? (b) The gap width is 1.600 cm when the temperature is $30.0^{\circ} \mathrm{C}$ . Determine the gap width when the temperature is $190^{\circ} \mathrm{C}$ .

Salamat A.

### Problem 28

The concrete sections of a certain superhighway are designed to have a length of 25.0 $\mathrm{m}$ . The sections are poured and cured at $10.0^{\circ} \mathrm{C}$ . What minimum spacing should the engineer leave between the sections to eliminate buckling if the concrete is to reach a temperature of $50.0^{\circ} \mathrm{C} ?$

Salamat A.

### Problem 29

A sample of pure copper has a mass of 12.5 g. Calculate the number of (a) moles in the sample and (b) copper atoms in the sample.

Salamat A.

### Problem 30

Formaldehyde has the chemical formula $\mathrm{CH}_{2} \mathrm{O}$ . Calculate the number of (a) moles, and (b) $\mathrm{CH}_{2} \mathrm{O}$ molecules in 275 $\mathrm{g}$ of formaldehyde.

Salamat A.

### Problem 31

One mole of oxygen gas is at a pressure of 6.00 atm and a temperature of $27.0^{\circ} \mathrm{C}$ (a) If the gas is heated at constant volume until the pressure triples, what is the final temperature? (b) If the gas is heated so that both the pressure and volume are doubled, what is the final temperature?

Salamat A.

### Problem 32

A container holds 0.500 $\mathrm{m}^{\mathrm{s}}$ of oxygen at an absolute pressure of 4.00 $\mathrm{atm}$ . A valve is opened, allowing the gas to drive a piston, increasing the volume of the gas until the pressure drops to 1.00 $\mathrm{atm}$ . If the temperature remains constant, what new volume does the gas occupy?

Salamat A.

### Problem 33

(a) An ideal gas occupies a volume of 1.0 $\mathrm{cm}^{3}$ at $20 .^{\circ} \mathrm{C}$ and atmospheric pressure. Determine the number of molecules of gas in the container. (b) If the pressure of the $1.0-\mathrm{cm}^{3}$ volume is reduced to $1.0 \times 10^{-11} \mathrm{Pa}$ (an extremely good vacuum) while the temperature remains constant, how many moles of gas remain in the container?

Salamat A.

### Problem 34

An automobile tire is inflated with air originally at $10.0^{\circ} \mathrm{C}$ and normal atmospheric pressure. During the process, the air is compressed to 28.0$\%$ of its original volume and the temperature is increased to $40.0^{\circ} \mathrm{C}$ . (a) What is the tire pressure in pascals? (b) After the car is driven at high speed, the tire's air temperature rises to $85.0^{\circ} \mathrm{C}$ and the tire's interior volume increases by 2.00$\%$ . What is the new tire pressure (absolute) in pascals?

Salamat A.

### Problem 35

Gas is confined in a tank at a pressure of 11.0 atm and a temperature of $25.0^{\circ} \mathrm{C}$ . If two-thirds of the gas is withdrawn and the temperature is raised to $75.0^{\circ} \mathrm{C},$ what is the new pressure of the gas remaining in the tank?

Salamat A.

### Problem 36

Gas is contained in an 8.00-L vessel at a temperature of $20.0^{\circ} \mathrm{C}$ and a pressure of 9.00 $\mathrm{atm}$ . (a) Determine the number of moles of gas in the vessel. (b) How many molecules are in the vessel?

Salamat A.

### Problem 37

A weather balloon is designed to expand to a maximum radius of 20 $\mathrm{m}$ at its working altitude, where the air pressure is 0.030 $\mathrm{atm}$ and the temperature is 200 $\mathrm{K}$ . If the balloon is filled at atmospheric pressure and $300 \mathrm{K},$ what is its radius at liftoff?

Salamat A.

### Problem 38

The density of helium gas at $0^{\circ} \mathrm{C}$ is $\rho_{0}=0.179 \mathrm{kg} / \mathrm{m}^{3}$ . The temperature is then raised to $T=100^{\circ} \mathrm{C},$ but the pressure is kept constant. Assuming the helium is an ideal gas, calculate the new density $\rho_{f}$ of the gas.

Salamat A.

### Problem 39

An air bubble has a volume of 1.50 $\mathrm{cm}^{3}$ when it is released by a submarine $1.00 \times 10^{2} \mathrm{m}$ below the surface of a lake. What is the volume of the bubble when it reaches the surface? Assume the temperature and the number of air molecules in the bubble remain constant during its ascent.

Salamat A.

### Problem 40

During inhalation, a person’s diaphragm and intercostal muscles contract, expanding the chest cavity and lowering the internal air pressure below ambient so that air flows in through the mouth and nose to the lungs. Suppose a person’s lungs hold 1 250 mL of air at a pressure of 1.00 atm. If the person expands the chest cavity by 525 mL while keeping the nose and mouth closed so that no air is inhaled, what will be the air pressure in the lungs in atm? Assume the air temperature remains constant.

Salamat A.

### Problem 41

What is the average kinetic energy of a molecule of oxygen at a temperature of 300. K?

Salamat A.

### Problem 42

Calculate the root-mean-square (rms) speed of methane $\left(\mathrm{CH}_{4}\right)$ gas molecules at a temperature of 325 K .

Salamat A.

### Problem 43

Three moles of an argon gas are at a temperature of 275 K. Calculate (a) the kinetic energy per molecule, (b) the root-mean-square (rms) speed of an atom in the gas, and (c) the internal energy of the gas.

Salamat A.

### Problem 44

A sealed cubical container 20.0 $\mathrm{cm}$ on a side contains a gas with three times Avogadro's number of neon atoms at a temperature of $20.0^{\circ} \mathrm{C}$ . (a) Find the internal energy of the gas. (b) Find the total translational kinetic energy of the gas. (c) Calculate the average kinetic energy per atom. (d) Use Equation 10.13 to calculate the gas pressure. (e) Calculate the gas pressure using the ideal gas law (Eq. 10.8$)$

Salamat A.

### Problem 45

Use Avogadro’s number to find the mass of a helium atom.

Salamat A.

### Problem 46

Two gases in a mixture pass through a filter at rates proportional to the gases’ rms speeds. (a) Find the ratio of speeds for the two isotopes of chlorine, $^{35} \mathrm{Cl}$ and $^{37} \mathrm{Cl},$ as they pass through the air. (b) Which isotope moves faster?

Salamat A.

### Problem 47

At what temperature would the rms speed of helium atoms equal (a) the escape speed from Earth, $1.12 \times 10^{4} \mathrm{m} / \mathrm{s}$ and (b) the escape speed from the Moon, $2.37 \times 10^{3} \mathrm{m} / \mathrm{s}$ ? (See Topic 7 for a discussion of escape speed. ) Note: The mass of a helium atom is $6.64 \times 10^{-27} \mathrm{kg}$ .

Salamat A.

### Problem 48

A 7.00 -L vessel contains 3.50 moles of ideal gas at a pressure of $1.60 \times 10^{6} \mathrm{~Pa}$. Find (a) the temperature of the gas and (b) the average kinetic energy of a gas molecule in the vessel. (c) What additional information would you need if you were asked to find the average speed of a gas molecule?

Salamat A.

### Problem 49

Superman leaps in front of Lois Lane to save her from a volley of bullets. In a l-minute interval, an automatic weapon fires 150 bullets, each of mass $8.0 \mathrm{g},$ at $4.00 \times 10^{2} \mathrm{m} / \mathrm{s}$ . The bullets strike his mighty chest, which has an area of 0.75 $\mathrm{m}^{2} .$ Find the average force exerted on Superman's chest if the bullets bounce back after an elastic, head-on collision.

Salamat A.

### Problem 50

In a period of $1.0 \mathrm{s}, 5.0 \times 10^{23}$ nitrogen molecules strike a wall of area 8.0 $\mathrm{cm}^{2}$ . If the molecules move at $3.00 \times 10^{2} \mathrm{m} / \mathrm{s}$ and strike the wall head-on in a perfectly elastic collision, find the pressure exerted on the wall. (The mass of one $\mathrm{N}_{2}$ molecule is $4.68 \times 10^{-26} \mathrm{kg} . )$

Salamat A.

### Problem 51

Inside the wall of a house, an L-shaped section of hot-water pipe consists of three parts: a straight horizontal piece $h=28.0 \mathrm{cm}$ long, an elbow, and a straight, vertical piece $\ell=134 \mathrm{cm}$ long (Fig. $\mathrm{P} 10.51 )$ . A stud and a second- story floorboard hold the ends of this section of copper pipe stationary. Find section of copper pipe stationary. Find the magnitude and direction of the displacement of the pipe elbow when the water flow is turned on, raising the temperature of the pipe from $18.0^{\circ} \mathrm{C}$ to $46.5^{\circ} \mathrm{C}$

Salamat A.

### Problem 52

The active element of a certain laser is made of a glass rod 30.0 cm long and 1.50 cm in diameter. Assume the average coefficient of linear expansion of the glass is $9.00 \times 10^{-6}$ $\left(^{\circ} \mathrm{C}\right)^{-1}$ . If the temperature of the rod increases by $65.0^{\circ} \mathrm{C}$ , what is the increase in $(\mathrm{a})$ its length, (b) its diameter, and (c) its volume?

Salamat A.

### Problem 53

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

Salamat A.

### Problem 54

Consider an object with any one of the shapes displayed in Table 8.1. What is the percentage increase in the moment of inertia of the object when it is warmed from $0^{\circ} \mathrm{C}$ to 100$\cdot^{\circ} \mathrm{C}$ if it is composed of (a) copper or (b) aluminum? Assume the average linear expansion coefficients shown in Table 10.1 $\mathrm{do}$ not vary between $0^{\circ} \mathrm{C}$ and $100 .^{\circ} \mathrm{C}$ . (c) Why are the answers for parts (a) and (b) the same for all the shapes?

Salamat A.

### Problem 55

A steel beam being used in the construction of a skyscraper has a length of 35.000 $\mathrm{m}$ when delivered on a cold day at a temperature of $15.000^{\circ} \mathrm{F}$ . What is the length of the beam when it is being installed later on a warm day when the temperature is $90.000^{\circ} \mathrm{F} ?$

Salamat A.

### Problem 56

A 1.5 -long glass tube that is closed at one end is weighted and lowered to the bottom of a freshwater lake. When the tube is recovered, an indicator mark shows that water rose to within 0.40 $\mathrm{m}$ of the closed end. Determine the depth of the lake. Assume constant temperature.

Salamat A.

### Problem 57

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, with 1.00 mol of methane as a by-product. 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 recycling of three astronauts’ respiration during one week of flight is stored in an originally empty 150-L tank at $-45.0^{\circ} \mathrm{C},$ what is the final pressure in the tank?

Salamat A.

### Problem 58

A vertical cylinder of cross-sectional area A is fitted with a tight-fitting, frictionless piston of mass $m(\text { Fig. } \mathrm{P} 10.58),$ (a) If $n$ moles of an ideal gas are in the cylinder at a temperature of $T,$ use Newton's second law for equilibrium to show that the height $h$ at which the piston is in equilibrium under its own weight is given by
$$h=\frac{n R T}{m g+P_{0} A}$$
where $P_{0}$ is atmospheric pressure.
(b) Is the pressure inside the cylinder less than, equal to, or greater than atmospheric pressure? (c) If the gas in the cylinder is warmed, how would the answer for $h$ be affected?

Salamat A.

### Problem 59

A flask made of Pyrex is calibrated at $20.0^{\circ} \mathrm{C}$ . It is filled to the 100 -mL mark on the flask with $35.0^{\circ} \mathrm{C}$ acetone. (a) What is the volume of the acetone when both it and the flask cool to $20.0^{\circ} \mathrm{C} ?$ (b) Would the temporary increase in the Pyrex flask's volume make an appreciable difference in the answer? Why or why not?

Salamat A.

### Problem 60

A 20.0 -L. tank of carbon dioxide gas $\left(\mathrm{CO}_{2}\right)$ is at a pressure of $9.50 \times 10^{5}$ Pa and temperature of $19.0^{\circ} \mathrm{C}$ . (a) Calculate the temperature of the gas in Kelvin. (b) Use the ideal gas law to calculate the number of moles of gas in the tank. (c) Use the periodic table to compute the molecular weight of carbon dioxide, expressing it in grams per mole. (d) Obtain the number of grams of carbon dioxide in the tank. (e) A fire breaks out, raising the ambient temperature by 224.0 $\mathrm{K}$ while 82.0 $\mathrm{g}$ of gas leak out of the tank. Calculate the new temperature and the number of moles of gas remaining in the tank. (f) Using a technique analogous to that in Example $10.6 \mathrm{b},$ find a symbolic expression for the final pressure, neglecting the change in volume of the tank. (g) Calculate the final pressure in the tank as a result of the fire and leakage.

Salamat A.

### Problem 61

A liquid with a coefficient of volume expansion of $\beta$ just fills a spherical flask of volume $V_{0}$ at temperature $T_{i}$ (Fig. $\mathrm{P} 10.61 ) .$ The flask is made of a material that has a coefficient of linear expansion of $\alpha$ . The liquid is free to expand into a capillary of cross-sectional into a capillary of cross-sectional area $A$ at the top. (a) Show that if the temperature increases by $\Delta T,$ the liquid rises in the capillary by the amount $\Delta h=\left(V_{0} / A\right)(\beta-3 \alpha) \Delta T$ . (b) For a typical system, such as a mercury thermometer, why is it a good approximation to neglect the expansion of the flask?

Salamat A.

### Problem 62

Before beginning a long trip on a hot day, a driver inflates an automobile tire to a gauge pressure of 1.80 atm at 300. K. At the end of the trip, the gauge pressure has increased to 2.20 atm. (a) Assuming the volume has remained constant, what is the temperature of the air inside the tire? (b) What percent- age of the original mass of air in the tire should be released so the pressure returns to its original value? Assume the temperature remains at the value found in part (a) and the volume of the tire remains constant as air is released.

Salamat A.

### Problem 63

Two concrete spans of a 250-m-long bridge are placed end to end so that no room is allowed for expansion (Fig. P10.63a). If the temperature increases by $20.0^{\circ} \mathrm{C},$ what is the height $y$ to which the spans rise when they buckle Fig. P10.63b)?

Salamat A.

### Problem 64

An expandable cylinder has its top connected to a spring with force constant $2.00 \times 10^{3} \mathrm{N} / \mathrm{m}$ (Fig. $\mathrm{P} 10.64 )$ . The cylinder is filled with 5.00 $\mathrm{L}$ of gas with the spring relaxed at a pressure of 1.00 $\mathrm{atm}$ and a temperature of $20.0^{\circ} \mathrm{C}$ . (a) If the lid has a cross-sectional area of 0.0100 $\mathrm{m}^{2}$ and negligible mass, how high will the lid rise when the temperature is raised to $250^{\circ} \mathrm{C}^{2}$ (b) What is the pressure of the gas at $250^{\circ} \mathrm{C}$ ?

Salamat A.

### Problem 65

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. P10.65). Derive an expression for the angle of bending, $\theta,$ 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 $\left(\Delta r=r_{2}-r_{1}\right) .$ (b) Show that the angle of bending goes to zero when $\Delta T$ goes to zero and also when the two average coefficients of expansion become equal. (c) What happens if the strip is cooled?

Salamat A.

### Problem 66

A 250 -m-long bridge is improperly designed so that it cannot expand with temperature. It is made of concrete with $\alpha=12 \times$ $10^{-6}\left(^{\circ} \mathrm{C}\right)^{-1}$ .( a) Assuming the maximum change in temperature at the site is expected to be $20^{\circ} \mathrm{C}$ , find the change in length the span would undergo if it were free to expand. (b) Show that the stress on an object with Young's modulus $Y$ when raised by $\Delta T$ with its ends firmly fixed is given by $\alpha Y \Delta T$ (c) If the maximum stress the bridge can withstand without crumbling is $2.0 \times 10^{7} \mathrm{Pa}$ , will it crumble because of this temperature increase? Young's modulus for concrete is about $2.0 \times 10^{10} \mathrm{Pa}$ .

Salamat A.

### Problem 67

Following a collision in outer space, a copper disk at $850^{\circ} \mathrm{C}$ is rotating about its axis with an angular speed of 25.0 $\mathrm{rad} / \mathrm{s}$ . As the disk radiates infrared light, its temperature falls to $20.0^{\circ} \mathrm{C}$ . 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?

Salamat A.
Two small containers, each with a volume of $1.00 \times 10^{2} \mathrm{cm}^{3}$ , contain helium gas at $0^{\circ} \mathrm{C}$ and 1.00 atm pressure. The two containers are joined by a small open tube of negligible volume, allowing gas to flow from one container to the other. What common pressure will exist in the two containers if the temperature of one container is raised to $1.00 \times 10^{2}$ ' $\mathrm{C}$ while the other container is kept at $0^{\circ} \mathrm{C}$ ?