University Physics with Modern Physics

Wolfgang Bauer, Gary D. Westfall

Chapter 17

Temperature - all with Video Answers

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Chapter Questions

Problem 1

Two mercury-expansion thermometers have identical reservoirs and cylindrical tubes made of the same glass but of different diameters. Which of the two thermometers can be graduated to a better resolution?
a) The thermometer with the smaller diameter tube will have better resolution.
b) The thermometer with the larger diameter tube will have better resolution.
c) The diameter of the tube is irrelevant; it is only the volume expansion coefficient of mercury that matters.
d) Not enough information is given to tell.

Ankur S

Ankur S

Numerade Educator

Problem 2

For a class demonstration, your physics instructor uniformly heats a bimetallic strip that is held in a horizontal orientation. As a result, the bimetallic strip bends upward. This tells you that the coefficient of linear thermal expansion for metal T, on the top is _____ that of metal B, on the bottom.
a) smaller than
b) larger than
c) equal to

Shahab Ullah

Numerade Educator

Problem 3

Two solid objects, $A$ and $B$, are in contact. In which case will thermal energy transfer from $\mathrm{A}$ to $\mathrm{B} ?$
a) $\mathrm{A}$ is at $20{ }^{\circ} \mathrm{C},$ and $\mathrm{B}$ is at $27{ }^{\circ} \mathrm{C}$
b) $A$ is at $15^{\circ} \mathrm{C},$ and $\mathrm{B}$ is at $15^{\circ} \mathrm{C}$.
c) $\mathrm{A}$ is at $0{ }^{\circ} \mathrm{C},$ and $\mathrm{B}$ is at $-10{ }^{\circ} \mathrm{C}$.

Shahab Ullah

Numerade Educator

Problem 4

Which of the following bimetallic strips will exhibit the greatest sensitivity to temperature changes? That is, which one will bend the most as temperature increases?
a) copper and steel
b) steel and aluminum
c) copper and aluminum
d) aluminum and brass
e) copper and brass

Shahab Ullah

Numerade Educator

Problem 5

The background temperature of the universe is
a) $6000 \mathrm{~K}$.
b) $288 \mathrm{~K}$.
c) $3 \mathrm{~K}$.
d) $2.73 \mathrm{~K}$.
e) $0 \mathrm{~K}$.

Shahab Ullah

Numerade Educator

Problem 6

Which air temperature feels coldest?
a) $-40^{\circ} \mathrm{C}$
c) $233 \mathrm{~K}$
b) $-40^{\circ} \mathrm{F}$
d) All three are equal.

Shahab Ullah

Numerade Educator

Problem 7

At what temperature do the Celsius and Fahrenheit temperature scales have the same numeric value?
a) -40 degrees
b) 0 degrees
c) 40 degrees
d) 100 degrees

Shahab Ullah

Numerade Educator

Problem 8

The city of Yellowknife in the Northwest Territories of Canada is on the shore of Great Slave Lake. The average high in July is $21^{\circ} \mathrm{C}$ and the average low in January is $-31{ }^{\circ} \mathrm{C}$. Great Slave Lake has a volume of $2090 \mathrm{~km}^{3}$ and is the deepest lake in North America, with a depth of $614 \mathrm{~m}$. What is the temperature of the water at the bottom of Great Slave Lake in January?
a) $-31^{\circ} \mathrm{C}$
b) $-10^{\circ} \mathrm{C}$
c) $0^{\circ} \mathrm{C}$
d) $4^{\circ} \mathrm{C}$
e) $32^{\circ} \mathrm{C}$

Matthew Baker

Numerade Educator

Problem 9

Which object has the higher temperature after being left outside for an entire winter night: a metal door knob or a rug?
a) The metal door knob has the higher temperature.
b) The rug has the higher temperature.
c) Both have the same temperature.
d) It depends on the outside temperature.

Matthew Baker

Numerade Educator

Problem 10

A common way of opening a tight lid on a glass jar is to place it under warm water. The thermal expansion of the metal lid is greater than that of the glass jar; thus, the space between the two expands and it is easier to open the jar. Will this work for a metal lid on a container of the same kind of metal?

Shahab Ullah

Numerade Educator

Problem 11

Would it be possible to have a temperature scale defined in such a way that the hotter an object or system got, the lower (less positive or more negative) its temperature was?

Matthew Baker

Numerade Educator

Problem 12

The solar corona has a temperature of about $1 \cdot 10^{6} \mathrm{~K}$. However, a spaceship flying in the corona will not be burned up. Why is this?

Shahab Ullah

Numerade Educator

Problem 13

Explain why it might be difficult to weld aluminum to steel or to weld any two unlike metals together.

Matthew Baker

Numerade Educator

Problem 14

Two solid objects are made of different materials. Their volumes and volume expansion coefficients are $V_{1}$ and $V_{2}$ and $\beta_{1}$ and $\beta_{2}$, respectively. It is observed that during a temperature change of $\Delta T$, the volume of each object changes by the same amount. If $V_{1}=2 V_{2}$ what is the ratio of the volume expansion coefficients?

Narayan Hari

Numerade Educator

Problem 15

Some textbooks use the unit $\mathrm{K}^{-1}$ rather than ${ }^{\circ} \mathrm{C}^{-1}$ for values of the linear expansion coefficient; see Table 17.2 How will the numerical values of the coefficient differ if expressed in $\mathrm{K}^{-1}$ ?

Narayan Hari

Numerade Educator

Problem 16

You are outside on a hot day, with the air temperature at $T_{0}$. Your sports drink is at a temperature $T_{\mathrm{d}}$ in a sealed plastic bottle. There are a few remaining ice cubes in the sports drink, which are at a temperature $T_{\mathrm{i}}$, but they are melting fast.
a) Write an inequality expressing the relationship among the three temperatures.
b) Give reasonable values for the three temperatures in degrees Celsius.

Ankur S

Numerade Educator

Problem 17

The Rankine temperature scale is an absolute temperature scale that uses Fahrenheit degrees; that is, temperatures are measured in Fahrenheit degrees, starting at absolute zero. Find the relationships between temperature values on the Rankine scale and those on the Fahrenheit, Kelvin, and Celsius scales.

Matthew Baker

Numerade Educator

Problem 18

The Zeroth Law of Thermodynamics forms the basis for the definition of temperature with regard to thermal energy. But the concept of temperature is used in other areas of physics. In a system with energy levels, such as electrons in an atom or protons in a magnetic field, the population of a level with energy $E$ is proportional to the factor $e^{-E / k_{\mathrm{B}} T},$ where $T$ is the absolute temperature of the system and $k_{\mathrm{B}}=1.381 \cdot 10^{-23} \mathrm{~J} / \mathrm{K}$ is Boltzmann's constant. In a two-level system with the levels" energies differing by $\Delta E$, the ratio of the populations of the higher-energy and lower-energy levels is $p_{\text {high }} / p_{\text {low }}=e^{-\Delta E / k_{\mathrm{B}} T}$. Such a system can have an infinite or even negative absolute temperature. Explain the meaning of such temperatures.

Morgan Cheatham

Morgan Cheatham

Numerade Educator

Problem 19

Suppose a bimetallic strip is constructed of two strips of metals with linear expansion coefficients $\alpha_{1}$ and $\alpha_{2}$, where $\alpha_{1}>\alpha_{2}$
a) If the temperature of the bimetallic strip is reduced by $\Delta T$, what way will the strip bend (toward the side made of metal 1 or the side made of metal 2)? Briefly explain.
b) If the temperature is increased by $\Delta T$, which way will the strip bend?

Shahab Ullah

Numerade Educator

Problem 20

For food storage, what is the advantage of placing a metal lid on a glass jar? (Hint: Why does running the metal lid under hot water for a minute help you open such a jar?)

Shahab Ullah

Numerade Educator

Problem 21

A solid cylinder and a cylindrical shell, of identical radius and length and made of the same material, experience the same temperature increase $\Delta T .$ Which of the two will expand to a larger outer radius?

Matthew Baker

Numerade Educator

Problem 22

Express each of the following temperatures in degrees Celsius and in kelvins.
a) $-19^{\circ} \mathrm{F}$
b) $98.6^{\circ} \mathrm{F}$
c) $52^{\circ} \mathrm{F}$

Shahab Ullah

Numerade Educator

Problem 23

One thermometer is calibrated in degrees Celsius, and another in degrees Fahrenheit. At what temperature is the reading on the thermometer calibrated in degrees Celsius three times the reading on the other thermometer?

Shahab Ullah

Numerade Educator

Problem 24

During the summer of 2007 , temperatures as high as $47^{\circ} \mathrm{C}$ were recorded in Southern Europe. The highest temperature ever recorded in the United States was $134^{\circ} \mathrm{F}$ (at Death Valley, California, in 1913 ). What is the difference between these two temperatures, in degrees Celsius?

Shahab Ullah

Numerade Educator

Problem 25

The lowest air temperature recorded on Earth is $-129^{\circ} \mathrm{F}$ in Antarctica. Convert this temperature to the Celsius scale.

Shahab Ullah

Numerade Educator

Problem 26

What air temperature will feel twice as warm as $0^{\circ} \mathrm{F} ?$

Shahab Ullah

Numerade Educator

Problem 27

A piece of dry ice (solid carbon dioxide) sitting in a classroom has a temperature of approximately $-79^{\circ} \mathrm{C}$
a) What is this temperature in kelvins?
b) What is this temperature in degrees Fahrenheit?

Shahab Ullah

Numerade Educator

Problem 28

In $1742,$ the Swedish astronomer Anders Celsius proposed a temperature scale on which water boils at 0.00 degrees and freezes at $100 .$ degrees. In 1745 , after the death of Celsius, Carolus Linnaeus (another Swedish scientist) reversed those standards, yielding the scale that is most commonly used today. Find room temperature $\left(77.0^{\circ} \mathrm{F}\right)$ on Celsius's original temperature scale.

Narayan Hari

Numerade Educator

Problem 29

At what temperature do the Kelvin and Fahrenheit scales have the same numeric value?

Shahab Ullah

Numerade Educator

Problem 30

How does the density of copper that is just above its melting temperature of $1356 \mathrm{~K}$ compare to that of copper at room temperature?

Narayan Hari

Numerade Educator

Problem 31

The density of steel is $7800.0 \mathrm{~kg} / \mathrm{m}^{3}$ at $20.0^{\circ} \mathrm{C}$. Find the density at $100.0^{\circ} \mathrm{C}$.

Narayan Hari

Numerade Educator

Problem 32

Two cubes with sides of length $100.0 \mathrm{~mm}$ fit in a space that is $201.0 \mathrm{~mm}$ wide, as shown in the figure. One cube is made of aluminum, and the other is made of brass. What temperature increase is required for the cubes to completely fill the gap?

Ankur S

Numerade Educator

Problem 33

A brass piston ring is to be fitted onto a piston by first heating up the ring and then slipping it over the piston. The piston ring has an inner diameter of $10.00 \mathrm{~cm}$ and an outer diameter of $10.20 \mathrm{~cm} .$ The piston has an outer diameter of $10.10 \mathrm{~cm},$ and a groove for the piston ring has an outer diameter of $10.00 \mathrm{~cm} .$ To what temperature must the piston ring be heated so that it will slip onto the piston?

Matthew Baker

Numerade Educator

Problem 34

An aluminum sphere of radius $10.0 \mathrm{~cm}$ is heated from $100.0^{\circ} \mathrm{F}$ to $200.0^{\circ} \mathrm{F}$. Find (a) the volume change and (b) the radius change.

Matthew Baker

Numerade Educator

Problem 35

Steel rails for a train track are laid in a region subject to extremes of temperature. The distance from one juncture to the next is $5.2000 \mathrm{~m},$ and the cross-sectional area of the rails is $60 . \mathrm{cm}^{2}$. If the rails touch each other without buckling at the maximum temperature, $50 .{ }^{\circ} \mathrm{C}$, how much space will there be between the rails at $-10 .{ }^{\circ} \mathrm{C} ?$

Ankur S

Numerade Educator

Problem 36

Even though steel has a relatively low linear expansion coefficient $\left(\alpha_{\text {steel }}=13 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1}\right),$ the expansion of steel railroad tracks can potentially create significant problems on very hot summer days. To accommodate for the thermal expansion, a gap is left between consecutive sections of the track. If each section is $25.0 \mathrm{~m}$ long at $20.0{ }^{\circ} \mathrm{C}$ and the gap between sections is $10.0 \mathrm{~mm}$ wide, what is the highest temperature the tracks can take before the expansion creates compressive forces between sections?

Matthew Baker

Numerade Educator

Problem 37

A medical device used for handling tissue samples has two metal screws, one $20.0 \mathrm{~cm}$ long and made from brass $\left(\alpha_{\mathrm{b}}=18.9 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1}\right)$ and the other $30.0 \mathrm{~cm}$ long and made from aluminum $\left(\alpha_{\mathrm{a}}=23.0 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1}\right)$. A gap of $1.00 \mathrm{~mm}$ exists between the ends of the screws at $22.0^{\circ} \mathrm{C}$. At what temperature will the two screws touch?

Averell Hause

Carnegie Mellon University

Problem 38

You are designing a precision mercury thermometer based on the thermal expansion of mercury $\left(\beta=1.8 \cdot 10^{-4}{ }^{\circ} \mathrm{C}^{-1}\right)$ which causes the mercury to expand up a thin capillary as the temperature increases. The equation for the change in volume of the mercury as a function of temperature is $\Delta V=\beta V_{0} \Delta T$ where $V_{0}$ is the initial volume of the mercury and $\Delta V$ is the change in volume due to a change in temperature, $\Delta T .$ In response to a temperature change of $1.0^{\circ} \mathrm{C}$, the column of mercury in your precision thermometer should move a distance $D=1.0 \mathrm{~cm}$ up a cylindrical capillary of radius $r=0.10 \mathrm{~mm} .$ Determine the initial volume of mercury that allows this change. Then find the radius of a spherical bulb that contains this volume of mercury.

Averell Hause

Carnegie Mellon University

Problem 39

On a hot summer day, a cubical swimming pool is filled to within $1.0 \mathrm{~cm}$ of the top with water at $21{ }^{\circ} \mathrm{C} .$ When the water warms to $37^{\circ} \mathrm{C}$, the pool overflows. What is the depth of the pool?

Averell Hause

Carnegie Mellon University

Problem 40

A steel rod of length $1.0 \mathrm{~m}$ is welded to the end of an aluminum rod of length $2.0 \mathrm{~m}$ (lengths measured at $22^{\circ} \mathrm{C}$ ). The combination rod is then heated to $200 .{ }^{\circ} \mathrm{C} .$ What is the
change in length of the combination rod at $200 .{ }^{\circ} \mathrm{C} ?$

Ankur S

Numerade Educator

Problem 41

$\cdot 17.41$ A clock based on a simple pendulum is situated outdoors in Anchorage, Alaska. The pendulum consists of a mass of 1.00 kg that is hanging from a thin brass rod that is $2.000 \mathrm{~m}$ long. The clock is calibrated perfectly during a summer day with an average temperature of $25.0^{\circ} \mathrm{C}$. During the winter, when the average temperature over one 24 -h period is $-20.0^{\circ} \mathrm{C}$, find the time elapsed for that period according to the simple pendulum clock.

Averell Hause

Carnegie Mellon University

Problem 42

In a thermometer manufacturing plant, a type of mercury thermometer is built at room temperature $\left(20^{\circ} \mathrm{C}\right)$ to measure temperatures in the $20^{\circ} \mathrm{C}$ to $70^{\circ} \mathrm{C}$ range, with $\mathrm{a}$ $1-\mathrm{cm}^{3}$ spherical reservoir at the bottom and a $0.5-\mathrm{mm}$ inner diameter expansion tube. The wall thickness of the reservoir and tube is negligible, and the $20^{\circ} \mathrm{C}$ mark is at the junction between the spherical reservoir and the tube. The tubes and reservoirs are made of fused silica, a transparent glass form of $\mathrm{SiO}_{2}$ that has a very low linear expansion coefficient $(\alpha=$ $\left.0.4 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1}\right) .$ By mistake, the material used for one batch of thermometers was quartz, a transparent crystalline form of $\mathrm{SiO}_{2}$ with a much higher linear expansion coefficient $\left(\alpha=12.3 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1}\right) .$ Will the manufacturer have to scrap the batch, or will the thermometers work fine, within the expected uncertainty of $5 \%$ in reading the temperature? The volume expansion coefficient of mercury is $\beta=181 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1}$.

Averell Hause

Carnegie Mellon University

Problem 43

The ends of the two rods shown in the figure are separated by $5.0 \mathrm{~mm}$ at $25^{\circ} \mathrm{C}$. The left-hand rod is brass and $1.0 \mathrm{~m}$ long; the right-hand rod is steel and $1 \mathrm{~m}$ long. Assuming that the outside ends of both rods rest firmly against rigid supports, at what temperature will the ends of the rods that face each other just touch?

Ankur S

Numerade Educator

Problem 44

The figure shows a temperature compensated pendulum in which lead and steel rods are arranged so that the pendulum's length is unaffected by changes in the temperature. Determine the length $L$ of the two lead bars.

Narayan Hari

Numerade Educator

Problem 45

Consider a bimetallic strip consisting of a 0.50 -mmthick brass upper strip welded to a 0.50 -mm-thick steel lower strip. When the temperature of the bimetallic strip is increased by $20 . \mathrm{K}$, the unattached tip deflects by $3.0 \mathrm{~mm}$ from its original straight position, as shown in the figure. What is the length of the strip at its original position?

Keshav Singh

Numerade Educator

Problem 46

Thermal expansion seems like a small effect, but it can engender tremendous, often damaging, forces. For example, steel has a linear expansion coefficient of $\alpha=1.2 \cdot 10^{-5}{ }^{\circ} \mathrm{C}^{-1}$ and a bulk modulus of $B=160$ GPa. Calculate the pressure engendered in steel by a $1.0^{\circ} \mathrm{C}$ temperature increase.

Ankur S

Numerade Educator

Problem 47

At room temperature, an iron horseshoe, when dunked into a cylindrical tank of water (radius of $10.0 \mathrm{~cm})$ causes the water level to rise $0.25 \mathrm{~cm}$ above the level without the horseshoe in the tank. When heated in the blacksmith's stove from room temperature to a temperature of $7.00 \cdot 10^{2} \mathrm{~K}$ worked into its final shape, and then dunked back into the water, how much does the water level rise above the "no horseshoe" level (ignore any water that evaporates as the horseshoe enters the water)? Note: The linear expansion coefficient for iron is roughly that of steel: $11 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1}$.

Ankur S

Numerade Educator

Problem 48

A clock has an aluminum pendulum with a period of $1.000 \mathrm{~s}$ at $20.0{ }^{\circ} \mathrm{C}$. Suppose the clock is moved to a location where the average temperature is $30.0^{\circ} \mathrm{C} .$ Determine
(a) the new period of the clock's pendulum and (b) how much time the clock will lose or gain in 1 week.

Narayan Hari

Numerade Educator

Problem 49

Using techniques similar to those that were originally developed for miniaturized semiconductor electronics, scientists and engineers are creating Micro-Electro-Mechanical Systems (MEMS). An example is an electrothermal actuator that is driven by heating its different parts using an electrical current. The device is used to position $125-\mu \mathrm{m}$ -diameter optical fibers with submicron resolution and consists of thin and thick silicon arms connected in the shape of a U, as shown in the figure. The arms are not attached to the substrate under the device but are free to move, whereas the electrical contacts (marked $+$ and $-$ in the figure) are attached to the substrate and cannot move. The thin arm is $3.0 \cdot 10^{1} \mu \mathrm{m}$ wide, and the thick arm is $130 \mu \mathrm{m}$ wide. Both arms are $1800 \mu \mathrm{m}$ long. Electrical current flows through the arms, causing them to heat up. Although the same current flows through both arms, the thin arm has a greater electrical resistance than the thick arm and therefore dissipates more electrical power and gets substantially hotter. When current is made to flow through the beams, the thin beam reaches a temperature of $4.0 \cdot 10^{2}{ }^{\circ} \mathrm{C},$ and the thick beam reaches a temperature of $2.0 \cdot 10^{2}{ }^{\circ} \mathrm{C} .$ Assume that the temperature in each beam is constant along the entire length of that beam (strictly speaking, this is not the case) and that the two arms remain parallel and bend only in the plane of the paper at higher temperatures. How much and in which direction will the tip move? The linear expansion coefficient for silicon is $3.2 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1}$.

Averell Hause

Carnegie Mellon University

Problem 50

Another MEMS device, used for the same purpose as the one in Problem 17.49 , has a different design. This electrothermal actuator consists of a thin $V$ -shaped silicon beam, as shown in the figure. The beam is not attached to the substrate under the device but is free to move, whereas the electrical contacts (marked $+$ and $-$ in the figure) are attached to the substrate and cannot move. The beam spans a gap between the electrical contacts that is $1800 \mu \mathrm{m}$ wide, and the two halves of the beam slant up from horizontal by 0.10 rad. Electrical current flows through the beam, causing it to heat up. When current is made to flow across the beam, the beam reaches a temperature of $500 .{ }^{\circ} \mathrm{C}$. Assume that the temperature is constant along the entire length of the beam (strictly speaking, this is not the case). How much and in which direction will the tip move? The linear expansion coefficient for silicon is $3.2 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1}$.

Averell Hause

Carnegie Mellon University

Problem 51

The volume of $1.00 \mathrm{~kg}$ of liquid water over the temperature range from $0.00^{\circ} \mathrm{C}$ to $50.0^{\circ} \mathrm{C}$ fits reasonably well to the polynomial function $V=1.00016-\left(4.52 \cdot 10^{-5}\right) T+$ $\left(5.68 \cdot 10^{-6}\right) T^{2}$, where the volume is measured in cubic meters and $T$ is the temperature in degrees Celsius.
a) Use this information to calculate the volume expansion coefficient for liquid water as a function of temperature.
b) Evaluate your expression at $20.0^{\circ} \mathrm{C}$, and compare the value to that listed in Table $17.3 .$

Averell Hause

Carnegie Mellon University

Problem 52

a) Suppose a bimetallic strip is constructed of copper and steel strips of thickness $1.0 \mathrm{~mm}$ and length $25 \mathrm{~mm},$ and the temperature of the strip is reduced by $5.0 \mathrm{~K}$. Determine the radius of curvature of the cooled strip (the radius of curvature of the interface between the two strips).
b) If the strip is $25 \mathrm{~mm}$ long, how far is the maximum deviation of the strip from the straight orientation?

Keshav Singh

Numerade Educator

Problem 53

A copper cube of side length $40 . \mathrm{cm}$ is heated from $20 .{ }^{\circ} \mathrm{C}$ to $120{ }^{\circ} \mathrm{C}$. What is the change in the volume of the cube? The linear expansion coefficient of copper is $17 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1}$.

Narayan Hari

Numerade Educator

Problem 54

When a 50.0 -m-long metal pipe is heated from $10.0^{\circ} \mathrm{C}$ to $40.0^{\circ} \mathrm{C}$, it lengthens by $2.85 \mathrm{~cm}$.
a) Determine the linear expansion coefficient.
b) What type of metal is the pipe made of?

Prabhat Tyagi

Numerade Educator

Problem 55

On a cool morning, with the temperature at $15.0^{\circ} \mathrm{C}$, a painter fills a 5.00 -gal aluminum container to the brim with turpentine. When the temperature reaches $27.0^{\circ} \mathrm{C}$, how much fluid spills out of the container? The volume expansion coefficient for this brand of turpentine is $9.00 \cdot 10^{-4}{ }^{\circ} \mathrm{C}^{-1}$.

Narayan Hari

Numerade Educator

Problem 56

A building having a steel infrastructure is $6.00 \cdot 10^{2} \mathrm{~m}$ high on a day when the temperature is $0.00^{\circ} \mathrm{C} .$ How much taller is the building on a day when the temperature is $45.0^{\circ} \mathrm{C} ?$ The linear expansion coefficient of steel is $1.30 \cdot 10^{-5}{ }^{\circ} \mathrm{C}^{-1}$.

Narayan Hari

Numerade Educator

Problem 57

In order to create a tight fit between two metal parts, machinists sometimes make the interior part larger than the hole into which it will fit and then either cool the interior part or heat the exterior part until they fogether. Suppose an aluminum rod with diameter $D_{1}$ (at $\left.2.0 \cdot 10^{1}{ }^{\circ} \mathrm{C}\right)$ is to be fit into a hole in a brass plate that has a diameter $D_{2}=10.000 \mathrm{~mm}$ (at $\left.2.0 \cdot 10^{1}{ }^{\circ} \mathrm{C}\right) .$ The machinists can cool the rod to $77.0 \mathrm{~K}$ by immersing it in liquid nitrogen. What is the largest possible diameter that the rod can have at $2.0 \cdot 10^{1}{ }^{\circ} \mathrm{C}$ and just fit into the hole if the rod is cooled to $77.0 \mathrm{~K}$ and the brass plate is left at $2.0 \cdot 10^{1}{ }^{\circ} \mathrm{C} ?$ The linear expansion coefficients for aluminum and brass are $22 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1}$ and $19 \cdot 10^{-6}{ }^{\circ} \mathrm{C}^{-1}$, respectively.

Averell Hause

Carnegie Mellon University

Problem 58

A military vehicle is fueled with gasoline in the United States, preparatory to being shipped overseas. Its fuel tank has a capacity of $213 \mathrm{~L}$. When it is fueled, the temperature is $57^{\circ} \mathrm{F}$. At its destination, it may be required to operate in temperatures as high as $120^{\circ} \mathrm{F}$. What is the maximum volume of gasoline that should be put in its tank?

Narayan Hari

Numerade Educator

Problem 59

A mercury thermometer contains $8.0 \mathrm{~mL}$ of mercury. If the tube of the thermometer has a cross-sectional area of $1.0 \mathrm{~mm}^{2}$, what should the spacing between the ${ }^{\circ} \mathrm{C}$ marks be?

Ankur S

Numerade Educator

Problem 60

A 14 -gal container is filled with gasoline. Neglect the change in volume of the container and find how many gallons are lost if the temperature increases by $27^{\circ} \mathrm{F}$. The volume expansion coefficient of gasoline is $9.6 \cdot 10^{-4}{ }^{\circ} \mathrm{C}^{-1}$.

Narayan Hari

Numerade Educator

Problem 61

A highway of concrete slabs is to be built in the Libyan desert, where the highest air temperature recorded is $57.8^{\circ} \mathrm{C}$. The temperature is $20.0^{\circ} \mathrm{C}$ during construction of the highway. The slabs are measured to be $12.0 \mathrm{~m}$ long at this temperature. How wide should the expansion cracks between the slabs be (at $20.0^{\circ} \mathrm{C}$ ) in order to prevent buckling at the highest temperatures?

Narayan Hari

Numerade Educator

Problem 62

An aluminum vessel with a volume capacity of $500 . \mathrm{cm}^{3}$ is filled with water to the brim at $20 .{ }^{\circ} \mathrm{C} .$ The vessel and contents are heated to $50 .{ }^{\circ} \mathrm{C} .$ During the heating process, will the water spill over the top, will there be more room for water to be added, or will the water level remain the same? Calculate the volume of water that will spill over or that could be added if either is the case.

Averell Hause

Carnegie Mellon University

Problem 63

By how much does the temperature of a given mass of kerosene need to change in order for its volume to increase by $1.0 \%$ ?

Narayan Hari

Numerade Educator

Problem 64

A plastic-epoxy sheet has uniform holes of radius $1.99 \mathrm{~cm}$. The holes are intended to allow solid ball bear-
ings with an outer radius of $2.00 \mathrm{~cm}$ to just go through. Over what temperature rise must the plastic-epoxy sheet be heated so that the ball bearings will go through the holes? The linear expansion coefficient of plastic-epoxy is about $1.3 \cdot 10^{-4}{ }^{\circ} \mathrm{C}^{-1}$.

Averell Hause

Carnegie Mellon University

Problem 65

A uniform brass disk of radius $R$ and mass $M$ with a moment of inertia $I$ about its cylindrical axis of symmetry is at a temperature $T=20 .{ }^{\circ} \mathrm{C} .$ Determine the fractional change in its moment of inertia if it is heated to a temperature of $100 .{ }^{\circ} \mathrm{C}$.

Narayan Hari

Numerade Educator

Problem 66

A 25.01 -mm-diameter brass ball sits at room temperature on a 25.00 - mm-diameter hole made in an aluminum plate. The ball and plate are heated uniformly in a furnace, so both are at the same temperature at all times. At what temperature will the ball fall through the plate?

Narayan Hari

Numerade Educator

Problem 67

In a pickup basketball game, your friend cracked one of his teeth in a collision with another player while attempting to make a basket. To correct the problem, his dentist placed a steel band of initial internal diameter $4.4 \mathrm{~mm},$ and a crosssectional area of width $3.5 \mathrm{~mm},$ and thickness $0.45 \mathrm{~mm}$ on the tooth. Before placing the band on the tooth, he heated the band to $70 .{ }^{\circ} \mathrm{C}$. What will be the tension in the band once it cools down to the temperature in your friend's mouth $\left(37^{\circ} \mathrm{C}\right) ?$

Averell Hause

Carnegie Mellon University

Problem 68

Your physics teacher assigned a thermometerbuilding project. He gave you a glass tube with an inside diameter of $1.00 \mathrm{~mm}$ and a receptacle at one end. He also gave you $8.63 \mathrm{~cm}^{3}$ of mercury to pour into the tube, which filled the receptacle and some of the tube. You are to add marks indicating degrees Celsius on the glass tube. At what increments should the marks be put on? You know the volume expansion coefficient of mercury is $1.82 \cdot 10^{-4}{ }^{\circ} \mathrm{C}^{-1}$.

Ankur S

Numerade Educator

Problem 69

You are building a device for monitoring ultracold environments. Because the device will be used in environments where its temperature will change by $200 .{ }^{\circ} \mathrm{C}$ in $3.00 \mathrm{~s}$, it must have the ability to withstand thermal shock (rapid temperature changes). The volume of the device is $5.00 \cdot 10^{-5} \mathrm{~m}^{3}$, and if the volume changes by $1.00 \cdot 10^{-7} \mathrm{~m}^{3}$ in a time interval of $5.00 \mathrm{~s}$, the device will crack and be rendered useless. What is the maximum volume expansion coefficient that the material you use to build the device can have?

Narayan Hari

Numerade Educator

Problem 70

A steel rod of length $1.0000 \mathrm{~m}$ and cross-sectional area $5.00 \cdot 10^{-4} \mathrm{~m}^{2}$ is placed snugly against two immobile end points. The rod is initially placed when the temperature is $0^{\circ} \mathrm{C}$. Find the stress in the rod when the temperature rises to $40.0^{\circ} \mathrm{C}$.

Narayan Hari

Numerade Educator

Problem 71

A brass bugle can be thought of as a tube that is open on both ends (the actual physics is complicated by the interaction of the bugler's mouth and the mouthpiece and the bell at the end). The overall length if the bugle is stretched out is $183.0 \mathrm{~cm}$ (at $\left.20.0^{\circ} \mathrm{C}\right)$. A bugle is played on a hot summer day $\left(41.0^{\circ} \mathrm{C}\right)$. Find the fundamental frequency if
a) only the change in air temperature is considered,
b) only the change in the length of the bugle is considered, and
c) both effects in parts (a) and (b) are taken into account.

Averell Hause

Carnegie Mellon University