Chapter 6: Fluid Mechanics
Answers to Chapter 6 Review Questions
1. Liquid and gas, which both flow.
2. The volume decreases; the mass
remains the same; the density
increases.
3. Mass density is mass/volume; weight
density is weight/volume.
4. Force is push or pull, or an
interaction; pressure is push or pull
per amount of surface area.
5. Both directly proportional; more
depth, more pressure. More density,
more pressure.
6. Twice as deep produces twice the
water pressure. Greater pressure at
the same depth in salt water because it
is denser.
7. The same.
8. Initial flow is perpendicular to the
surface of the container.
9. Because pressure up against bottom is
greater than pressure down against top.
10. Same.
11. An immersed body is buoyed up by a
force equal to the weight of the fluid it
displaces.
12. Both imply water displacement and
submerged means completely
immersed with complete volume
displacement.
13. Same.
14. 1 kilogram. 10 N (or more precisely,
9.8 N).
15. 1/2 L. Weight of 1/2 L of water, 5 N
(or more precisely, 4.9 N).
16. Depends only on weight of water
displaced. Volume of object, because
amount of buoyant force depends on
that volume of water displaced.
17. Floating. Then buoyant force and
weight of object have the same
magnitude.
18. Yes, for the volume of the submerged
object is the same volume of fluid
displaced. The weight of this
displaced fluid is the buoyant force.
19. When floating, buoyant force depends
on both, because weight of object =
weight of displaced fluid. In either
case the net force on the object is zero,
meaning weight = buoyant force.
20. 100 tons; 100 tons.
21. Distance between molecules is greater
in a gas.
22. By twice.
23. Increases by twice.
24. “Pressure × volume” for a quantity of
gas at one time is equal to any
“different pressure × different
volume” at any other time.
25. Weight of air in the atmosphere.
26. About 1.2 kilograms.
27. About 1 kilogram with a weight of
9.8 N.
28. Same.
29. Same.
30. Must be 13.6 times taller to have the
same weight as a column of water of
the same cross section—because
density of mercury is 13.6 times
greater.
31. Pushed, because atm pressure is doing
the pushing. When you suck on the
straw, all you do is reduce the
pressure within the straw, allowing
the push of the atm to raise the fluid
level in the straw. See Figures 6.26
and 6.27.
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32. Atm pressure will push water only as
high as 10.3 m.
33. 1 N. If buoyant force decreases,
balloon moves downward. If it
increases, the balloon moves upward.
34. Bernoulli’s Principle briefly: Where
the speed of a fluid increases, internal
pressure in the fluid decreases.
35. Steamlines are smooth paths of a
fluid. Pressure is less where
streamlines are closer together.
36. Bernoulli’s principle applies only to
internal pressures in a fluid itself, not
on the pressures that a fluid may exert
when its momentum changes.
37. They all experience reduced
atmospheric pressure on their top
surfaces. The greater pressure beneath
produces a net upward force—lift.
Solutions to Chapter 6 Exercises
1. Stand on a bathroom scale and you read your weight. When you lift one foot off so you’re standing on
one foot, does the reading change? Does a scale read force or pressure?
The scale measures force, not pressure, and is calibrated to read your weight. That’s
why your weight on the scale is the same whether you stand on one foot or both.
2. The photo shows physics teacher Marshall Ellenstein walking barefoot on broken glass bottles in his
class. What physics concept is Marshall demonstrating, and why is he careful that the broken pieces
are small and numerous? (The Band-Aids on his feet are for humor!)
The concept of pressure is being demonstrated. He is careful that the pieces are small
and numerous so that his weight is applied over a large area of contact. Then the sharp
glass provides insufficient pressure to cut the feet.
3. In a deep dive, a whale is appreciably compressed by the pressure of the surrounding water. What
happens to the whale’s density?
Like the loaf of bread in Figure 6.1, its volume is decreased. Its mass stays the same so
the density increases. A whale is denser when it swims deeper in the ocean.
4. The density of a rock doesn’t change when it is submerged in water. Does your density change
when you are submerged in water? Defend your answer.
This is similar to Exercise 3. Whereas a rock is not compressible, you are. When you’re
submerged, water pressure squeezes in on you and reduces your volume. This increases
your density. (Be careful when swimming—at shallow depths you may still be
less dense than water and be buoyed to the surface without effort, but at greater depths
you may be pressed to a density greater than water and you’ll have to swim to the
surface.)
5. Why are persons confined to bed less likely to develop bedsores on their bodies if they use a
waterbed rather than an ordinary mattress?
A person lying on a waterbed experiences less bodyweight pressure because more of
the body is in contact with the supporting surface. The greater area reduces the support
pressure.
6. If water faucets upstairs and downstairs are turned fully on, will more water per second flow out
the downstairs faucet? Or both the same?
More water will flow from a downstairs open faucet because of the greater pressure.
Since pressure depends on depth, the downstairs faucet is effectively “deeper” than the
upstairs faucet. The pressure downstairs is greater by an amount = weight density ×
depth, where the depth is the vertical distance between faucets.
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7. Which do you suppose exerts more pressure on the ground—an elephant or a woman standing on
spike heels? (Which will be more likely to make dents in a linoleum floor?) Can you approximate a
rough calculation for each?
A woman with spike heels exerts considerably more pressure on the ground than an
elephant! Example: A 500-N woman with 1-cm2 spike heels puts half her weight on
each foot, distributed (let’s say) half on her heel and half on her sole. So the pressure
exerted by each heel will be (125 N/1 cm2) = 125 N/cm2. A 20,000-N elephant with
1000 cm2 feet exerting 1/4 its weight on each foot produces (5000N/1000 cm2) = 5N/cm2;
about 25 times less pressure. (So a woman with spike heels will make greater dents in a
new linoleum floor than an elephant will.)
8. Suppose you wish to lay a level foundation for a home on hilly and bushy terrain. How can you use
a garden hose filled with water to determine equal elevations for distant points?
The use of a water-filled garden hose as an elevation indicator is a practical example of
water seeking its own level. The water surface at one end of the hose will be at the
same elevation above sea level as the water surface at the other end of the hose.
9. When you are bathing on a stony beach, why do the stones hurt your feet less when you get in deep
water?
In deep water, you are buoyed up by the water displaced and as a result, you don’t exert
as much pressure against the stones on the bottom. When you are up to your neck in
water, you hardly feel the bottom at all.
10. If liquid pressure were the same at all depths, would there be a buoyant force on an object
submerged in the liquid? Explain.
Buoyant force is the result of differences in pressure; if there are no pressure
differences, there is no buoyant force. This can be illustrated by the following example:
A Ping-Pong ball pushed beneath the surface of water will normally float back to the
surface when released. If the container of water is in free fall, however, a submerged
Ping-Pong ball will fall with the container and make no attempt to reach the surface. In
this case there is no buoyant force acting on the ball because there are no pressure
differences—the local effects of gravity are absent.
11. The Himalayan Mountains are slightly less dense than the mantle material upon which they “float.”
Do you suppose that, like floating icebergs, they are deeper than they are high?
As per the Link to Geology box in the chapter, mountain ranges are very similar to
icebergs: Both float in a denser medium, and extend farther down into that medium
than they extend above it.
12. How much force is needed to push a nearly weightless but rigid 1-L carton beneath a surface of water?
The force needed will be the weight of 1 L of water, which is 9.8 N. If the weight of the
carton is not negligible, then the force needed would be 9.8 N minus the carton’s
weight, for then the carton would be “helping” to push itself down.
13. Why is it inaccurate to say that heavy objects sink and that light objects float? Give exaggerated
examples to support your answer.
Heavy objects may or may not sink, depending on their densities (a heavy log floats
while a small rock sinks, or a boat floats while a paper clip sinks, for example). People
who say that heavy objects sink really mean that dense objects sink. Be careful to
distinguish between how heavy an object is and how dense it is.
14. Compared to an empty ship, would a ship loaded with a cargo of Styrofoam sink deeper into water
or rise in water? Defend your answer.
When a ship is empty its weight is least and it displaces the least water and floats
highest. Carrying a load of anything increases its weight and makes it float lower. It
will float as low when carrying a few tons of Styrofoam as it will carrying the same
number of tons of iron ore. So the ship floats lower in the water when loaded with
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Styrofoam than when empty. If the Styrofoam were outside the ship, below water line,
then the ship would float higher as a person would with a life preserver.
15. A barge filled with scrap iron is in a canal lock. If the iron is thrown overboard, does the water level
at the side of the lock rise, fall, or remain unchanged? Explain.
The water level will fall. This is because the iron will displace a greater amount of
water while being supported than when submerged. A floating object displaces its
weight of water, which is more than its own volume, while a submerged object
displaces only its volume. (This may be illustrated in the kitchen sink with a dish
floating in a dishpan full of water. Silverware in the dish takes the place of the scrap
iron. Note the level of water at the side of the dishpan, and then throw the silverware
overboard. The floating dish will float higher and the water level at the side of the
dishpan will fall. Will the volume of the silverware displace enough water to bring the
level to its starting point? No, not as long as it is denser than water.)
16. Would the water level in a canal lock go up or down if a battleship in the lock sank?
For the same reason as in the previous exercise, the water level will fall. (Try this one in
your kitchen sink also. Note the water level at the side of the dishpan when a bowl
floats in it. Tip the bowl so it fills and submerges, and you’ll see the water level at the
side of the dishpan fall.)
17. A balloon is weighted so that it is barely able to float in water. If it is pushed beneath the surface,
will it come back to the surface, stay at the depth to which it is pushed, or sink? Explain. (Hint: Does
the balloon’s density change?)
The balloon will sink to the bottom because its density increases with depth. The balloon
is compressible, so the increase in water pressure beneath the surface compresses
it and reduces its volume, thereby increasing its density. Density is further increased as
it sinks to regions of greater pressure and compression. This sinking is understood also
from a buoyant force point of view. As its volume is reduced by increasing pressure as
it descends, the amount of water it displaces becomes less. The result is a decrease in
the buoyant force that initially was sufficient to barely keep it afloat.
18. A ship sailing from the ocean into a fresh-water harbor sinks slightly deeper into the water. Does
the buoyant force on it change? If so, does it increase or decrease?
The buoyant force does not change. The buoyant force on a floating object is always
equal to that object’s weight, no matter what the fluid.
19. Suppose you are given the choice between two life preservers that are identical in size, the first a
light one filled with Styrofoam and the second a very heavy one filled with lead pellets. If you
submerge these life preservers in the water, upon which will the buoyant force be greater? Upon
which will the buoyant force be ineffective? Why are your answers different?
Since both preservers are the same size, they will displace the same amount of water
when submerged and be buoyed up with equal forces. Effectiveness is another story.
The amount of buoyant force exerted on the heavy lead-filled preserver is much less
than its weight. If you wear it, you’ll sink. The same amount of buoyant force exerted
on the lighter Styrofoam preserver is greater than its weight and it will keep you afloat.
The amount of the force and the effectiveness of the force are two different things.
20. The relative densities of water, ice, and alcohol are 1.0, 0.9, and 0.8 respectively. Do ice cubes float
higher or lower in a mixed alcoholic drink? What can you say about a cocktail in which the ice cubes
lie submerged at the bottom of the glass?
Ice cubes will float lower in a mixed drink because the mixture of alcohol and water is
less dense than water. In a less dense liquid a greater volume of liquid must be
displaced to equal the weight of the floating ice. In pure alcohol, the volume of alcohol
equal to that of the ice cubes weighs less than the ice cubes, and buoyancy is less than
weight and ice cubes will sink. Submerged ice cubes in a cocktail indicate that it is
predominantly alcohol.
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21. When an ice cube in a glass of water melts, does the water level in the glass rise, fall, or remain
unchanged? Does your answer change if the ice cube contains many air bubbles? How about if the
ice cube contains many grains of heavy sand?
When the ice cube melts the water level at the side of the glass is unchanged
(neglecting temperature effects). To see this, suppose the ice cube to be a 5 gram cube;
then while floating it will displace 5 grams of water. But when melted it becomes the
same 5 grams of water. Hence the water level is unchanged. The same occurs when the
ice cube with the air bubbles melts. Whether the ice cube is hollow or solid, it will
displace as much water floating as it will melted. If the ice cube contains grains of
heavy sand, however, upon melting, the water level at the edge of the glass will drop.
This is similar to the case of the scrap iron of Exercise 15.
22. A half-filled bucket of water is on a spring scale. Will the reading of the scale increase or remain the
same if a fish is placed in the bucket? (Will your answer be different if the bucket is initially filled to
the brim?)
If water doesn’t overflow, the reading on the scale will increase by the ordinary weight
of the fish. However, if the bucket is brim filled so a volume of water equal to the
volume of the fish overflows, then the reading will not change. We assume here that
the fish and water have the same density.
23. We say that the shape of a liquid is that of its container. But with no container and no gravity, what
is the natural shape of a blob of water? Why?
Because of surface tension, which tends to minimize the surface of a blob of water, its
shape without gravity and other distorting forces will be a sphere—the shape with the
least surface area for a given volume.
24. If you release a Ping-Pong ball beneath the surface of water, it will rise to the surface. Would it do
the same if submerged in a big blob of water floating weightless in an orbiting spacecraft?
A Ping-Pong ball in water in a zero-g environment would experience no buoyant force.
This is because buoyancy depends on a pressure difference on different sides of a submerged
body. In this weightless state, no pressure difference would exist because no
water pressure exists.
25. It is said that a gas fills all the space available to it. Why then doesn’t the atmosphere go off into
space?
Some of the molecules in the Earth’s atmosphere do go off into outer space—those like
helium with speeds greater than escape speed. But the average speeds of most
molecules in the atmosphere are well below escape speed, so the atmosphere is held to
Earth by Earth gravity.
26. Count the tires on a large tractor trailer that is unloading food at your local supermarket, and you
may be surprised to count 18 tires. Why so many tires? (Hint: See Activity 5.)
The weight of a truck is distributed over the part of the tires that make contact with the
road. Weight/surface area = pressure, so the greater the surface area, or equivalently,
the greater the number of tires, the greater the weight of the truck can be for a given
pressure. What pressure? The pressure exerted by the tires on the road, which is
determined by (but is somewhat greater than) the air pressure in its tires. Can you see
how this relates to Activity 5?
27. How does the density of air in a deep mine compare with the air density at the Earth’s surface?
The density of air in a deep mine is greater than at the surface. The air filling up the
mine adds weight and pressure at the bottom of the mine, and according to Boyle’s law,
greater pressure in a gas means greater density.
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28. Two teams of eight horses each were unable to pull the Magdeburg hemispheres apart (Figure 6.21).
Why? Suppose two teams of nine horses each could pull them apart. Then would one team of nine
horses succeed if the other team were replaced with a strong tree? Defend your answer.
To begin with, the two teams of horses used in the Magdeburg hemispheres
demonstration were for showmanship and effect, for a single team and a strong tree
would have provided the same force on the hemispheres. So if two teams of nine
horses each could pull the hemispheres apart, a single team of nine horses could also, if
a tree or some other strong object were used to hold the other end of the rope.
29. Before boarding an airplane, you buy a bag of chips, or any item packaged in an airtight foil
package, and while in flight you notice that it is puffed up. Explain why this occurs.
If the item is sealed in an air-tight package at sea level, then the pressure in the package
is about 1 atmosphere. Cabin pressure is reduced somewhat for high altitude flying, so
the pressure in the package is greater than the surrounding pressure and the package
therefore puffs outwards.
30. Why do you suppose that airplane windows are smaller than bus windows?
Airplane windows are small because the pressure difference between the inside and
outside surfaces result in large net forces that are directly proportional to the window’s
surface area. (Larger windows would have to be proportionately thicker to withstand
the greater net force—windows on underwater research vessels are similarly small.)
31. A half cup or so of water is poured into a 5-L can, which is placed on a source of heat until most of
the water has boiled away. Then the top of the can is screwed on tightly and the can is removed
from the source of heat and allowed to cool. What happens to the can and why?
The can collapses under the weight of the atmosphere. When water was boiling in the
can, much of the air inside was driven out and replaced by steam. Then, with the cap
tightly fastened, the steam inside cooled and condensed back to the liquid state,
creating a partial vacuum in the can which could not withstand the crushing force of
the atmosphere outside.
32. We can understand how pressure in water depends on depth by considering a stack of bricks. The
pressure below the bottom brick is determined by the weight of the entire stack. Halfway up the
stack, the pressure is half because the weight of the bricks above is half. To explain atmospheric
pressure, we should consider compressible bricks, like foam rubber. Why is this so?
Unlike water, air is easily compressed. In fact, its density is proportional to its pressure.
So, near the surface, where the pressure is greater, the air’s density is greater, and at
high altitude, where the pressure is less, the air’s density is less.
33. The “pump” in a vacuum cleaner is merely a high-speed fan. Would a vacuum cleaner pick up dust
from a rug on the moon? Explain.
A vacuum cleaner wouldn’t work on the moon. A vacuum cleaner operates on Earth because
the atmospheric pressure pushes dust into the machine’s region of reduced
pressure. On the moon there is no atmospheric pressure to push the dust anywhere.
34. If you could somehow replace the mercury in a mercury barometer with a denser liquid, would the
height of the liquid column be greater or less than with mercury? Why?
The height would be less. The weight of the column balances the weight of an equalarea
column of air. The denser liquid would need less height to have the same weight
as the mercury column.
35. Would it be slightly more difficult to draw soda through a straw at sea level or on top of a very high
mountain? Explain.
Drinking through a straw is slightly more difficult atop a mountain. This is because the
reduced atmospheric pressure is less effective in pushing soda up into the straw.
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36. Your friend says that the buoyant force of the atmosphere on an elephant is significantly greater
than the buoyant force of the atmosphere on a small helium-filled balloon. What do you say?
You agree with your friend, for the elephant displaces far more air than a small heliumfilled
balloon, or small anything. The effects of the buoyant forces, however, is a
different story. The large buoyant force on the elephant is insignificant relative to its
enormous weight. The tiny buoyant force acting on the balloon of tiny weight,
however, is significant.
37. Why is it so difficult to breathe when snorkeling at a depth of 1 m, and practically impossible at a
2-m depth? Why can’t a diver simply breathe through a hose that extends to the surface?
One’s lungs, like an inflated balloon, are compressed when submerged in water, and
the air within is compressed. Air will not of itself flow from a region of low pressure
into a region of higher pressure. The diaphragm in one’s body reduces lung pressure to
permit breathing, but this limit is strained when nearly 1 m below the water surface.
The limit is exceeded at more than a 1-m depth.
38. When you replace helium in a balloon with less-dense hydrogen, does the buoyant force on the
balloon change if the balloon remains the same size? Explain.
The buoyant force does not change, because the volume of the balloon does not change.
The buoyant force is the weight of air displaced, and doesn’t depend on what is doing
the displacing.
39. A steel tank filled with helium gas doesn’t rise in air, but a balloon containing the same helium
easily does? Why?
An object rises in air only when buoyant force exceeds its weight. A steel tank of anything
weighing more than the air it displaces, so won’t rise. A helium-filled balloon
weighs less than the air it displaces and rises.
40. Two identical balloons of the same volume are pumped up with air to more than atmospheric
pressure and suspended on the ends of a stick that is horizontally balanced. One of the balloons is
then punctured. Is there a change in the stick’s balance? If so, which way does it tip?
The end supporting the punctured balloon tips upwards as it is lightened by the
amount of air that escapes. There is also a loss of buoyant force on the punctured
balloon, but that loss of upward force is less than the loss of downward force, since the
density of air in the balloon before puncturing was greater than the density of
surrounding air.
41. Imagine a huge space colony that consists of a rotating air-filled cylinder. How would the density of
air at “ground level” compare to the air densities “above”?
The rotating habitat is a centrifuge, and denser air is “thrown to” the outer wall. Just as
on Earth, the maximum air density is at “ground level,” and becomes less with
increasing altitude (distance toward the center). Air density in the rotating habitat is
least at the zero-g region, the hub.
42. Would a helium-filled balloon “rise” in the atmosphere of a rotating space habitat? Defend your
answer.
The helium-filled balloon will be buoyed from regions of greater pressure to regions of
lesser pressure, and will “rise” in a rotating air-filled habitat.
43. The force of the atmosphere at sea level against the outside of a 10-m2 store window is about a
million N. Why does this not shatter the window? Why might the window shatter in a strong wind
blowing past the window?
The force of the atmosphere is on both sides of the window; the net force is zero, so
windows don’t normally break under the weight of the atmosphere. In a strong wind,
however, pressure will be reduced on the windward side (Bernoulli’s Principle) and
the forces no longer cancel to zero. Many windows are blown outward in strong winds.
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44. In a department store, an airstream from a hose connected to the exhaust of a vacuum cleaner blows
upward at an angle and supports a beach ball in midair. Does the air blow mostly under or over the
ball to provide support?
More air blows over the top of the beach ball than under it, causing the pressure above
the ball to be less than the pressure under it (Bernoulli’s principle). The pressure
difference provides a support force to counteract the gravitational force.
45. When a steadily flowing gas flows from a larger-diameter pipe to a smaller-diameter pipe, what
happens to (a) its speed, (b) its pressure, and (c) the spacing between its streamlines?
(a) Speed increases (so that the same quantity of gas can move through the pipe in the
same time). (b) Pressure decreases (Bernoulli’s principle). (c) The spacing between the
streamlines decreases, because the same number of streamlines fit in a smaller area.
46. How is an airplane able to fly upside down?
An airplane flies upside down by tilting its fuselage so that there is an angle of attack
of the wing with oncoming air. (It does the same when flying right side up, but then,
because the wings are designed for right-side-up flight, the tilt of the fuselage may not
need to be as great.)
47. When a jet plane is cruising at high altitude, the flight attendants have more of a “hill” to climb as
they walk forward along the aisle than when the plane is cruising at a lower altitude. Why does the
pilot have to fly with a greater “angle of attack” at high altitude than at low?
The air density and pressure are less at higher altitude, so the wings (and, with them,
the whole airplane) are tilted to a greater angle to produce the needed pressure
difference between the upper and lower surfaces of the wing. In terms of force and air
deflection, the greater angle of attack is needed to deflect a greater volume of lowerdensity
air downward to give the same upward force.
48. What physics principle underlies these three observations? When passing an oncoming truck on the
highway, your car tends to sway toward the truck. The canvas roof of a convertible automobile
bulges upward when the car is traveling at high speeds. The windows of older trains sometimes
break when a high-speed train passes by on the next track.
Bernoulli’s Principle. For the moving car the pressure will be less on the side of the car
where the air is moving fastest—the side nearest the truck, resulting in the car’s being
pushed by the atmosphere towards the truck. Inside the convertible, atmospheric
pressure is greater than outside, and the canvas roof top is pushed upwards towards the
region of lesser pressure. Similarly for the train windows, where the interior air is at
rest relative to the window and the air outside is in motion. Air pressure against the
inner surface of the window is greater than the atmospheric pressure outside. When the
difference in pressures is significant enough, the window is blown out.
49. A steady wind blows over the waves of an ocean. Why does the wind increase the humps and
troughs of the waves?
The troughs are partially shielded from the wind, so the air moves faster over the crests
than in the troughs. Pressure is therefore lower at the top of the crests than down below
in the troughs. The greater pressure in the troughs pushes the water into even higher
crests.
50. Wharves are made with pilings that permit the free passage of water. Why would a solid-walled
wharf be disadvantageous to ships attempting to pull alongside?
A solid-walled wharf is disadvantageous to ships pulling alongside because water currents
are constrained and speed up between the ship and the wharf. This results in a
reduced water pressure, and the normal pressure on the other side of the ship then
forces the ship against the wharf. The pilings avoid this mishap by allowing the freer
passage of water between the wharf and the ship.
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Solutions to Chapter 6 Problems
1. How much pressure do you experience when you balance a 5-kg ball on the tip of your finger, say
of area 1 cm2?
A 5-kg ball weighs 49 N, so the pressure is 49 N/cm2 = 490 kPa.
2. A 6 kg piece of metal displaces 1 liter of water when submerged. What is its density?
Density = m/V = 6 kg/1 liter = 6 kg/liter. (Since there are 1000 liters in 1 cubic meter,
density may be expressed in units kg/m3. Density = 6 kg/1 liter × 1000 liter/m3 =
6000 kg/m3, six times the density of water.)
3. The depth of water behind the Hoover Dam in Nevada is 220 m. What is the water pressure at the
base of this dam? (Neglect the pressure due to the atmosphere.)
Pressure = weight density × depth = 9800 N/m3 × 220 m = 2,160,000 N/m2 = 2160 kPa.
4. A rectangular barge, 5 m long and 2 m wide, floats in fresh water. (a) Find how much deeper it
floats when its load is a 400-kg horse. (b) If the barge can only be pushed 15 cm deeper into the
water before water overflows to sink it, how many 400-kg horses can it carry?
(a) The volume of the extra water displaced will weigh as much as the 400-kg horse.
And the volume of extra water displaced will also equal the area of the barge times the
extra depth. That is, V = Ah, where A is the horizontal area of the barge;
Then h =
VA
.
Now A = 5m × 2m = 10 m2; to find the volume V of barge pushed into the water by the
horse’s weight, which equals the volume of water displaced, we know that
density =
mV
. Or from this, V = m
density = 400kg
1000kg/m3 = 0.4 m3.
So h =
VA
= 0.4 m3
10 m2 = 0.04 m, which is 4 cm deeper.
(b) If each horse will push the barge 4 cm deeper, the question becomes: How many
4-cm increments will make 15 cm? 15/4 = 3.75, so 3 horses can be carried without
sinking. 4 horses will sink the barge.
5. A merchant in Katmandu sells you a solid gold 1-kg statue for a very reasonable price. When you
get home, you wonder whether or not you got a bargain, so you lower the statue into a container of
water and measure the volume of displaced water. What volume will verify that it’s pure gold?
From Table 6.1 the density of gold is 19.3 g/cm3. Your gold has a mass of 1000 grams, so
1000 g
V = 19.3 g/cm3. Solving for V, V = 1000 g
19.3 g/cm3 = 51.8 cm3.
6. When a 2.0-kg object is suspended in water, it “masses” 1.5 kg. What is the density of the object?
Density = mass
volume = 2.0 kg
volume of (2.0 - 1.5) kg of water = 2.0 kg
0.5 l = 4 kg/liter.
And since 1 liter = 103 cm = 10-3 m, density = 4,000 kg/m3.
(Or this can be reasoned as follows: The buoyant force on the object is the force needed
to support 0.5 kg, so 0.5 kg of water is displaced. Since density is mass/volume, volume
is mass/density, and displaced volume = (0.5 kg)/(1000 kg/m3) = 5 × 10–4 m3. The object’s
volume is the same as the volume it displaces, so the object’s density is mass/volume =
(2 kg)/(5 × 10−4 m3) = 4000 kg/m3, four times the density of water.)
7. An ice cube measures 10 cm on a side, and floats in water. One cm extends above water level. If you
shaved off the 1-cm part, how many cm of the remaining ice would extend above water level?
10% of ice extends above water. So 10% of the 9-cm thick ice would float above the water
line; 0.9 cm. So the ice pops up. Interestingly, when mountains erode they become
lighter and similarly pop up! Hence it takes a long time for mountains to wear away.
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8. A vacationer floats lazily in the ocean with 90 percent of his body below the surface. The density of
the ocean water is 1,025 kg/m3. What is the vacationer’s average density?
The displaced water, with a volume 90 percent of the vacationer’s volume, weighs the
same as the vacationer (to provide a buoyant force equal to his weight). Therefore his
density is 90 percent of the water’s density. Vacationer’s density = (0.90)(1,025 kg/m3) =
923 kg/m3.
9. Air in a cylinder is compressed to one-tenth its original volume with no change in temperature.
What happens to its pressure?
According to Boyle’s law, the product of pressure and volume is constant (at constant
temperature), so one-tenth the volume means ten times the pressure.
10. In the previous problem, if a valve is opened to let out enough air to bring the pressure back down
to its original value, what percentage of the molecules escape?
To decrease the pressure ten-fold, back to its original value, in a fixed volume, 90% of
the molecules must escape, leaving one-tenth of the original number.
11. Estimate the buoyant force that air exerts on you. (To do this, you can estimate your volume by
knowing your weight and by assuming that your weight density is a bit less than that of water.)
To find the buoyant force that the air exerts on you, find your volume and multiply by
the weight density of air (From Table 6.1 we see that the mass of 1 m3 of air is about
1.25 kg. Multiply this by 9.8 N/kg and you get 12.25 N/m3). You can estimate your
volume by your weight and by assuming your density is approximately equal to that of
water (a little less if you can float). The weight density of water is 104N/m3, which we’ll
assume is your density. By ratio and proportion:
104N
m3 = (your weight in newtons)
(your volume in meters3)
If your weight is a heavy 1000 N, for example (about 220 lb), your volume is 0.1 m3. So
buoyant force = 12.25 N/m3 × 0.1 m3 = about 1.2 N, the weight of a big apple). (A useful
conversion factor is 4.45 N = 1 pound.) Another way to do this is to say that the ratio of
the buoyant force to your weight is the same as the ratio of air density to water density
(which is your density). This ratio is 1.25/1000 = 0.00125. Multiply this ratio by your
weight to get the buoyant force.
12. Nitrogen and oxygen in their liquid states have densities only 0.8 and 0.9 that of water. Atmospheric
pressure is due primarily to the weight of nitrogen and oxygen gas in the air. If the atmosphere
liquefied, would its depth be greater or less than 10.3 m?
If the atmosphere were composed of pure water vapor, the atmosphere would condense
to a depth of 10.3 m. Since the atmosphere is composed of gases that have less density
in the liquid state, their liquid depths would be more than 10.3 m, about 12 m. (A nice
reminder of how thin and fragile our atmosphere really is.)
13. A mountain-climber friend with a mass of 80 kg ponders the idea of attaching a helium-filled
balloon to himself to effectively reduce his weight by 25% when he climbs. He wonders what the
approximate size of such a balloon would be. Hearing of your physics skills, he asks you. What
answer can you come up with, showing your calculations?
To effectively lift (0.25)(80 kg) = 20 kg the mass of displaced air would be 20 kg.
Density of air is about 1.2 kg/m3. From density = mass/volume, the volume of 20 kg of
air, also the volume of the balloon (neglecting the weight of the hydrogen) would be
vol = mass/density = (20 kg)/(1.2 kg/m3) = 16.6 m3, slightly more than 3 m in diameter
for a spherical balloon.
14. On a perfect fall day, you are hovering at low altitude in a hot-air balloon, accelerated neither
upward nor downward. The total weight of the balloon, including its load and the hot air in it, is
Conceptual Physical Science—Third Edition 259
(a) The weight of the displaced air must be the same as the weight supported, since the total force
(gravity plus buoyancy) is zero. The displaced air weighs 20,000 N. (b) Since weight = mg, the mass
of the displaced air is m = W/g = (20,000 N)/(10 m/s2) = 2,000 kg. Since density is mass/volume, the
volume of the displaced air is vol = mass/density = (2,000 kg)/(1.2 kg/m3) = 1,700 m3 (same answer
to two figures if g = 9.8 m/s2 is used).
15. How much lift is exerted on the wings of an airplane that have a total surface area of 100 m2 when
the difference in air pressure below and above the wings is 4% of atmospheric pressure?
Lift will equal the difference in force below and above the wing surface. The difference
in force will equal the difference in air pressure × wing area.
Lift = 0.04 PA = (0.04)(105 N/m2)(100 m2) = 4 × 105 N. (That’s about 44 tons.)
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Chapter 7: Thermal Energy and Thermodynamics
Answers to Chapter 7 Review Questions
1. Because the hammer’s blow causes
atoms in the metal to jostle faster.
2. The thermal energy in a substance is
the total energy of all its atoms and
molecules, consisting both of the
potential and kinetic energy of the
particles in a substance
3. 0°C, 32°F. 100°C, 212°F.
4. Average.
5. Both the thermometer and whatever it
measures reach a common
temperature (thermal equilibrium).
So, a thermometer measures its own
temperature.
6. By 1/273.
7. Zero.
8. 273 K, 373 K.
9. None.
10. Energy travels, so when a cold surface
is touched, energy goes from your
hand to the surface.
11. Temperature is a measure of the
average translational kinetic energy
per molecule, while heat is the
thermal energy transferred from one
thing to another due to a temperature
difference.
12. Heat is energy in transit, while
thermal energy is the grand total of all
energies in a substance. Heat flows,
thermal energy is contained.
13. Temperature difference; heat flows
from high to low temperatures.
14. Cold isn’t an entity in itself, but is the
lack of thermal energy.
15. By burning food and measuring the
amount of energy released.
16. A Calorie is 1000 calories.
17. Both are units of energy; 4.18 J = 1 cal.
18. Whenever heat flows into or out of a
system, the energy gain or loss equals
the amount of heat transferred.
19. The first law is the conservation of
energy applied to thermal systems.
20. Increases. Increases.
21. Heat never spontaneously flows from
a cold substance to a hot substance.
22. It defines the direction—from hot to
cold.
23. No system can reach absolute zero.
24. In every transformation some energy
is diluted and is less useful for doing
work.
25. Entropy.
26. Silver (which has a lower specific heat).
27. Low.
28. Very high in comparison.
29. North Atlantic water cools, and
releases energy to the air, which blows
over Europe.
30. Its two different kinds of metal have
different rates of expansion. One
expands or contracts more or less than
the other with changes in
temperature.
31. Liquids.
32. Contraction (until it reaches 4°C).
33. Ice crystals are open structured. So
when water freezes, the crystals
occupy more space. More volume
means lower density.
34. Less dense, for the slush has greater
volume. When temperature goes up, it
melts, decreasing the volume of water.
35. 4°C.
Conceptual Physical Science—Third Edition 261
36. As water goes through 4°C on the way
to freezing, it sinks to the bottom—not
cold enough to freeze. Any 0°C water
floats at the top, where it freezes.
Solutions to Chapter 7 Exercises
1. Why wouldn’t you expect all the molecules in a gas to have the same speed?
Gas molecules move haphazardly at random speeds. They continually run into one
another, sometimes giving kinetic energy to neighbors, sometimes receiving kinetic
energy. In this continual interaction, it would be statistically impossible for any large
number of molecules to have the same speed. Temperature has to do with average speeds.
2. In your room there are things such as tables, chairs, other people, and so forth. Which of these
things has a temperature (1) lower than, (2) greater than, and (3) equal to the temperature of the air?
Inanimate things such as tables, chairs, furniture, and so on, have the same temperature
as the surrounding air (assuming they are in thermal equilibrium with the air—i.e., no
sudden gush of different-temperature air or such). People and other mammals,
however, generate their own heat and have body temperatures that are normally higher
than air temperature.
3. Why can’t you establish whether you are running a high temperature by touching your own
forehead?
You cannot establish by your own touch whether or not you are running a fever
because there would be no temperature difference between your hand and forehead. If
your forehead is a couple of degrees higher in temperature than normal, your hand is
also a couple of degrees higher.
4. Which is greater, an increase in temperature of 1C° or one of 1F°?
Since Celsius degrees are larger than Fahrenheit degrees, an increase of 1C° is larger.
It’s 9/5 as large.
5. Which has the greater amount of internal energy, an iceberg or a cup of hot coffee? Explain.
The hot coffee has a higher temperature, but not a greater internal energy. Although the
iceberg has less internal energy per mass, its enormously greater mass gives it a greater
total energy than that in the small cup of coffee. (For a smaller volume of ice, the fewer
number of more energetic molecules in the hot cup of coffee may constitute a greater
total amount of internal energy—but not compared to an iceberg.)
6. On which temperature scale does the average kinetic energy of molecules double when the
temperature doubles?
The Kelvin temperature scale.
7. The temperature of the sun’s interior is about 107 degrees. Does it matter whether this is degrees
Celsius or kelvins? Defend your answer.
No, for a difference of 273 in 10,000,000 is insignificant.
8. Use the laws of thermodynamics to defend the statement that 100 percent of the electrical energy
that goes into lighting a lamp is converted to thermal energy.
Only a small percentage of the electric energy that goes into lighting a lamp becomes
light. The rest is thermal energy. But even the light is absorbed by the surroundings,
and also ends up as thermal energy. So by the first law, all the electrical energy is
ultimately converted to thermal energy. By the second law, organized electrical energy
degenerates to the more disorganized form, thermal energy.
9. When air is rapidly compressed, why does its temperature increase?
Work is done in compressing the air, which in accord with the first law of
thermodynamics, increases its thermal energy. This is evident by its increased
temperature.
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10. Which of the laws of thermodynamics has exceptions?
Only the second law is a probabilistic statement and has exceptions.
11. If you vigorously shake a can of liquid back and forth for more than a minute, will there be a
noticeable temperature increase? (Try it and see.)
You do work on the liquid when you vigorously shake it, which increases its thermal
energy. The temperature change should be noticeable.
12. What happens to the gas pressure within a sealed gallon can when it is heated? Cooled? Why?
Gas pressure increases in the can when heated, and decreases when cooled. The
pressure that a gas exerts depends on the average kinetic energy of its molecules,
therefore, on its temperature.
13. After driving a car for some distance, why does the air pressure in the tires increase?
The tires heat up, which heats the air within. The molecules in the heated air move
faster, which increases air pressure in the tires.
14. If you drop a hot rock into a pail of water, the temperature of the rock and the water will change
until both are equal. The rock will cool and the water will warm. Does this hold true if the hot rock
is dropped into the Atlantic Ocean? Explain.
The hot rock will cool and the cool water will warm, regardless of the relative amounts
of each. The amount of temperature change, however, does depend in great part on the
relative masses of the materials. For a hot rock dropped into the Atlantic Ocean, the
change in temperature would be too small to measure. Keep increasing the mass of the
rock or keep decreasing the mass of the ocean and the change will be evident.
15. In the old days, on a cold winter night it was common to bring a hot object to bed with you.
Which would be better to keep you warm through the cold night—a 10-kilogram iron brick or a
10-kilogram jug of hot water at the same temperature? Explain.
The brick will cool off too fast and you’ll be cold in the middle of the night. Bring a jug
of hot water with its higher specific heat to bed and you’ll make it through the night.
16. Desert sand is very hot in the day and very cool at night. What does this tell you about its specific heat?
Sand has a low specific heat, as evidenced by its relatively large temperature changes
for small changes in internal energy. A substance with a high specific heat, on the other
hand, must absorb or give off large amounts of internal energy for comparable
temperature changes.
17. Adding the same amount of heat to two different objects does not necessarily produce the same
increase in temperature. Why not?
Different substances have different thermal properties due to differences in the way
energy is stored internally in the substances. When the same amount of heat produces
different changes in temperatures in two substances of the same mass, we say they
have different specific heat capacities. Each substance has its own characteristic specific
heat capacity. Temperature measures the average kinetic energy of random motion, but
not other kinds of energy.
18. Why will a watermelon stay cool for a longer time than sandwiches when both are removed from a
cooler on a hot day?
Water has a high specific heat capacity, which is to say, it normally takes a long time to
heat up, or cool down. The water in the watermelon resists changes in temperature, so
once cooled it will stay cool longer than sandwiches or other nonwatery substances
under the same conditions. Be glad water has a high specific heat capacity the next time
you’re enjoying cool watermelon on a hot day!
19. Bermuda is about as far north of the Equator as North Carolina, but unlike North Carolina it has a
tropical climate year round. Why is this so?
The climate of Bermuda, like that of all islands, is moderated by the high specific heat
of water. What moderates the climates are the large amounts of energy given off and
Conceptual Physical Science—Third Edition 263
absorbed by water for small changes in temperature. When the air is cooler than the
water, the water warms the air; when the air is warmer than the water, the water cools
the air.
20. Iceland, so named to discourage conquest by expanding empires, is not at all ice-covered like
Greenland and parts of Siberia, even though it is nearly on the Arctic Circle. The average winter
temperature of Iceland is considerably higher than regions at the same latitude in eastern Greenland
and central Siberia. Why is this so?
The climate of Iceland, like that of Bermuda in the previous exercise, is moderated by
the surrounding water.
21. Why does the presence of large bodies of water tend to moderate the climate of nearby land—make
it warmer in cold weather, and cooler in hot weather?
In winter months when the water is warmer than the air, the air is warmed by the water
to produce a seacoast climate warmer than inland. In summer months when the air is
warmer than the water, the air is cooled by the water to produce a seacoast climate
cooler than inland. This is why seacoast communities and especially islands do not
experience the high and low temperature extremes that characterize inland locations.
22. If the winds at the latitude of San Francisco and Washington, D.C., were from the east rather than
from the west, why might San Francisco be able to grow only cherry trees and Washington, D.C.,
only palm trees?
As the ocean off the coast of San Francisco cools in the winter, the heat it loses warms
the atmosphere it comes in contact with. This warmed air blows over the California
coastline to produce a relatively warm climate. If the winds were easterly instead of
westerly, the climate of San Francisco would be chilled by winter winds from dry and
cold Nevada. The climate would be reversed also in Washington D.C., because air
warmed by the cooling of the Atlantic Ocean would blow over Washington D.C. and
produce a warmer climate in winter there.
23. Cite an exception to the claim that all substances expand when heated.
Water is an exception. Below 4 degrees Celsius, it expands when cooled.
24. Would a bimetallic strip function if the two different metals happened to have the same rates of
expansion? Is it important that they expand at different rates? Explain.
No, the different expansions are what bends the strip or coil. Without the different
expansions a bimetallic strip would not bend when heated.
25. Steel plates are commonly attached to each other with rivets, which are slipped into holes in the
plates and rounded over with hammers. The hotness of the rivets makes them easier to round over,
but their hotness has another important advantage in providing a tight fit. What is it?
When the rivets cool they contract. This tightens the plates being attached.
26. A method for breaking boulders used to be putting them in a hot fire, then dousing them with cold
water. Why would this fracture the boulders?
When doused, the outer part of the boulders cooled while the insides were still hot.
This caused a difference in contraction, which fractured the boulders.
27. An old remedy for a pair of nested drinking glasses that stick together is to run water at different
temperatures into the inner glass and over the surface of the outer glass. Which water should be hot,
and which cold?
Cool the inner glass and heat the outer glass. If it’s done the other way around, the
glasses will stick even tighter (if not break).
28. Would you or the gas company gain by having gas warmed before it passed through your
gas meter?
Gas is sold by volume. The gas meter that tallies your gas bill operates by measuring
the number of volume units (such as cubic feet) that pass through it. Warm gas is
expanded gas and occupies more space, and if it passes through your meter, it will be
registered as more gas than if it were cooled and more compact. The gas company gains
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if gas is warm when it goes through your meter because the same amount of warmer
gas has a greater volume.
29. A metal ball is just able to pass through a metal ring. When the ball is heated, however, it will not
pass through the ring. What would happen if the ring, rather than the ball, were heated? Does the
size of the hole increase, stay the same, or decrease?
Every part of a metal ring expands when it is heated—not only the thickness, but the
outer and inner circumference as well. Hence the ball that normally passes through the
hole when the temperatures are equal will more easily pass through the expanded hole
when the ring is heated. (Interestingly enough, the hole will expand as much as a disk
of the same metal undergoing the same increase in temperature. Blacksmiths mounted
metal rims in wooden wagon wheels by first heating the rims. Upon cooling, the
contraction resulted in a snug fit.)
30. After a machinist very quickly slips a hot, snugly fitting iron ring over a very cold brass cylinder,
there is no way that the two can be separated intact. Can you explain why this is so?
Brass expands and contracts more than iron for the same changes in temperature. Since
they are both good conductors and are in contact with each other, one cannot be heated
or cooled without also heating or cooling the other. If the iron ring is heated, it
expands—but the brass expands even more. Cooling the two will not result in
separation either, for even at the lowest temperatures the shrinkage of brass over iron
would not produce separation.
31. Suppose you cut a small gap in a metal ring. If you heat the ring, will the gap become wider or
narrower?
The gap in the ring will become wider when the ring is heated. Try this: draw a couple
of lines on a ring where you pretend a gap to be. When you heat the ring, the lines will
be farther apart—the same amount as if a real gap were there. Every part of the ring
expands proportionally when heated uniformly—thickness, length, gap and all.
32. One of the reasons the first light bulbs were expensive was that the electrical lead wires into the
bulb were made of platinum, which expands at about the same rate as glass when heated. Why is it
important that the metal leads and the glass have the same coefficient of expansion?
In the construction of a light bulb, it is important that the metal leads and the glass
have the same rate of heat expansion. If the metal leads expand more than glass, the
glass may crack. If the metal expands less than glass upon being heated, air will leak in
through the resulting gaps.
33. Suppose that water is used in a thermometer instead of mercury. If the temperature is at 4°C and
then changes, why can’t the thermometer indicate whether the temperature is rising or falling?
Water has the greatest density at 4°C; therefore, either cooling or heating at this temperature
will result in an expansion of the water. A small rise in water level would be
ambiguous and make a water thermometer impractical in this temperature region.
34. How does the combined volume of the billions and billions of hexagonal open spaces in the
structures of ice crystals in a piece of ice compare to the portion of ice that floats above the water
line?
The combined volume of all the billions of “open rooms” in the hexagonal ice crystals
of a piece of ice is equal to the volume of the part of the ice that extends above water
when ice floats. When the ice melts, the open spaces are filled in by the amount of ice
that extends above the water level. This is why the water level doesn’t rise when ice in
a glass of ice water melts—the melting ice “caves in” and nicely fills the open spaces.
35. State whether water at the following temperatures will expand or contract when warmed a little:
0°C; 4°C; 6°C.
At 0°C it will contract when warmed a little; at 4°C it will expand, and at 6°C it will
expand.
Conceptual Physical Science—Third Edition 265
36. Why is it important to protect water pipes so they don’t freeze?
It is important to keep water in pipes from freezing because when the temperature
drops below freezing, the water expands as it freezes whereas the pipe (if metal) will
fracture if water in them freezes.
37. If cooling occurred at the bottom of a pond instead of at the surface, would a lake freeze from the
bottom up? Explain.
If cooling occurred at the bottom of a pond instead of at the surface, ice would still
form at the surface, but it would take much longer for ponds to freeze. This is because
all the water in the pond would have to be reduced to a temperature of 0°C rather than
4°C before the first ice would form. Ice that forms at the bottom where the cooling
process is occurring would be less dense and would float to the surface (except for ice
that may form on material anchored to the bottom of the pond).
38. Make up a multiple-choice question that distinguishes between heat and temperature.
Open.
Solutions to Chapter 7 Problems
Quantity of heat, Q, is equal to the specific heat capacity of the substance c multiplied by its mass m and the
temperature change ΔT; That is, Q = cmΔT.
1. Will burns a 0.6-g peanut beneath 50 g of water, which increases in temperature from 22°C to 50°C.
(a) Assuming 40% efficiency, what is the food value in calories of the peanut? (b) What is the food
value in calories per gram?
(a) The amount of heat absorbed by the water is Q = cmΔT =
(1.0 cal/g C°)(50.0 g)(50°C – 22°C) = 1400 cal. At 40% efficiency only 0.4 the energy from
the peanut raises the water temperature, so the calorie content of the peanut is 1400/0.4
= 3500 cal. (b) The food value of a peanut is 3500 cal/0.6 g = 5.8 kilocalories per gram.
2. Pounding a nail into wood makes the nail warmer. Consider a 5 gram steel nail 6 cm long, and a
hammer that exerts an average force of 500 N on it when it is being driven into a piece of wood. The
nail becomes hotter. Calculate the increase in the nail’s temperature. (Assume the specific heat
capacity of steel is 450 Joules/kg°C.)
Work the hammer does on the nail is given by F × d, and the temperature change of the
nail can be found from using Q = cm ΔT. First, we get everything into more convenient
units for calculating: 5 grams = 0.005 kg; 6 cm = 0.06 m. Then F × d = 500 N × 0.06 m = 30 J,
and 30 J = (0.005 kg)(450 J/kg°C)(ΔT) which we can solve to get ΔT = 30/(0.005 × 450) =
13.3°C. (You will notice a similar effect when you remove a nail from a piece of wood.
The nail that you pull out is noticeably warm.)
3. If you wish to warm 100 kg of water by 20°C for your bath, how much heat is required? (Give your
answer in calories and joules.)
Each kilogram requires 1 kilocalorie for each degree change, so 100 kg needs
100 kilocalories for each degree change. Twenty degrees means twenty times this,
which is 2,000 kcal.
By formula, Q = mcΔT = (100,000 g)(1 cal/g°C)(20°C) = 2000 kcal. We can convert this to
joules knowing that 4.18 J = 1 cal. In joules this quantity of heat is 8360 kJ.
4. The specific heat capacity of copper is 0.092 calories per gram per degree Celsius. How much heat is
required to raise the temperature of a 10-gm piece of copper from 0°C to 100°C? How does this
compare with the heat needed to raise the temperature of the same mass of water through the same
temperature difference?
Raising the temperature of 10 gm of copper by one degree takes 10 × 0.092 =
0.92 calories, and raising it through 100 degrees takes 100 times as much, or 92 calories.
By formula, Q = mcΔT = (10 g)(0.092 cal/g°C)(100°C) = 92 cal.
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Heating 10 grams of water through the same temperature difference takes
1,000 calories, more than ten times more than for the copper—another reminder that
water has a large specific heat capacity.
5. What will be the final temperature of 100 g of 20°C water when 100 g of 40° iron nails are
submerged in it? (The specific heat of iron is 0.12 cal/g C°. Here you should equate the heat gained
by the water to the heat lost by the nails.)
Heat gained by water= heat lost by nails
(cm ΔT)water = (cm ΔT)nails
(1)(100) (T − 20) = (0.12)(100)(40 − T), giving T = 22.1°C.
To solve the problems below you need to know about the average coefficient of linear expansion, α, which
differs for different materials. We define α to be the change in length per unit length—or the fractional change
in length—for a one degree Celsius temperature change. That is, α = ΔL/L per C°. For aluminum,
α = 24 × 10−6/C°, and for steel, α = 11 × 10−6/C°.
The change in length ΔL of a material is given by ΔL = Lα ΔT.
6. Suppose a bar 1 m long expands 0.5 cm when heated. By how much will a bar 100 m long of the
same material expand when similarly heated?
If a 1-m long bar expands 1/2 cm when heated, a bar of the same material that is
100 times as long will expand 100 times as much, 0.5 cm for each meter, or 50 cm.
(The heated bar will be 100.5 m long.)
7. Suppose the 1.3-km main span of steel for the Golden Gate Bridge had no expansion joints. How
much longer would it be for an increase in temperature of 15°C?
By formula: ΔL = LoαΔT = (1300 m)(11 × 10−6/°C)(15°C) = 0.21 m.
8. Consider a 40,000-km steel pipe that forms a ring to fit snugly all around the circumference of the
world. Suppose people along its length breathe on it so as to raise its temperature 1 Celsius degree.
The pipe gets longer. It also is no longer snug. How high does it stand above ground level? (To
simplify, consider only the expansion of its radial distance from the center of the Earth, and apply
the geometry formula that relates circumference C and radius r, C = 2πr. The result is surprising!)
If a snugly fitting steel pipe that girdled the world were heated by 1 Celsius degree, it
would stand about 70 meters off the ground! The most straight-forward way to see this
is to consider the radius of the 40,000 long kilometer pipe, which is the radius of the
Earth, 6370 kilometers. Steel will expand 11 parts in a million for each C° increase in
temperature; the radius as well as the circumference will expand by this fraction. So
11 millionths of 6370 kilometers = 70 meters. Is this not astounding?
Or by formula for the Earth’s radius, ΔL = LoαΔT = (6370 × 103m)(11 × 10-6/°C)(1°C) = 70 m.
Conceptual Physical Science—Third Edition 267
Chapter 8: Heat Transfer and Change of Phase
Answers to Chapter 8 Review Questions
1. Conduction, convection, radiation.
2. Loose electrons conduct energy by
collisions throughout a substance.
3. Electrons are free to roam in
conductors, which easily conduct
heat. Electrons are firmly attached in
insulators, which therefore don’t
conduct heat well.
4. Wood is a poor conductor, even when
red hot. So very little heat conducts
from the coal to your feet.
5. Because they are composed largely of
air spaces, which are good insulators.
6. No insulator completely prevents heat
flow. Instead, an insulator slows the
rate of heat penetration.
7. By movement of heated fluid—by
currents.
8. Heated fluid expands, becomes less
dense, and is buoyed upward like any
fluid that is less dense than its
surroundings.
9. The temperature decreases (as is
evidenced by blowing on your hand
with puckered lips).
10. Her hand is in a region of rapidly
expanding water vapor, which
quickly becomes relatively cool.
Cooling is enhanced by the mixture of
surrounding cool air drawn in.
11. In day, the shore is warmed more
than water, so winds blow from water
toward shore. At night the reverse
occurs; the shore cools more than
water and winds blow in the opposite
direction.
12. Energy of electromagnetic waves.
13. Frequency and absolute temperature
are directly proportional.
14. Terrestrial radiation is that emitted by
Planet Earth. It differs in amount
(lower) and frequency (lower) from
solar radiation.
15. Electromagnetic radiation in the infrared
part of the spectrum.
16. All objects are also absorbing energy
from the surroundings. Temperature
will decrease only if the object is a net
emitter—if it emits more than it
absorbs.
17. Whether it is colder or warmer than
the surroundings. If colder, it is a net
absorber. If warmer, it is a net emitter.
18. No. Absorption and reflection are
opposite processes.
19. Because multiple reflections inside the
eye absorb energy, which therefore is
not reflected back through the pupil.
20. If it’s a good conductor, energy
radiated away is returned by the
warm Earth, so it doesn’t get very
cold. But if it’s a poor conductor, less
energy from the Earth warms it and it
can become appreciably colder than
the air.
21. It does matter, for ΔT will be greater if
it’s in the freezer. So it will cool faster
in the freezer.
22. ΔT will be greater when the poker is
in the cold room, so its rate of cooling
is faster there.
23. Yes.
24. Solid, liquid, gas, and plasma.
25. Molecules have a wide variety of
speeds.
26. Evaporation is a change of phase from
liquid to gas at the surface of a liquid.
Faster molecules are the ones that
evaporate, leaving slower ones
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behind. Therefore the remaining
liquid cools.
27. The change of phase direct from solid
to gas.
28. Condensation is a change of phase
from gas to liquid. Molecules gain
speed when attracted to the liquid’s
surface and therefore warm the liquid.
29. Steam gives up considerable energy
when it changes phase, condensing to
a liquid and wetting the skin.
30. You are warmed by the condensation
of water vapor in the air upon your
skin.
31. Evaporation is a change of phase at
the surface of a liquid; boiling is
evaporation beneath the surface and
throughout the liquid.
32. At greater pressure molecules need to
move faster to form bubbles. The
speeds corresponding to 100°C are
insufficient.
33. Higher temperature.
34. Molecules vibrate violently and break
apart from one another.
35. Molecules slow down enough to
cohere.
36. Foreign molecules in the water
interfere with crystal formation of ice.
37. Liquid absorbs in turning to gas;
releases when turning to a solid.
38. A gas gives off energy when it turns
to a liquid. A solid absorbs energy in
turning to a liquid.
39. Heat of fusion refers to the energy
needed to change from a solid to a
liquid, (or vice versa), and heat of
vaporization refers to the energy
needed to change from a liquid to a
gas (or vice versa).
40. Because energy that ordinarily would
go into burning your finger goes
instead into changing the phase of
moisture.
Solutions to Chapter 8 Exercises
1. Wrap a fur coat around a thermometer. Will the temperature rise?
No, the coat is not a source of heat, but merely keeps the thermal energy of the wearer
from leaving rapidly.
2. If you hold one end of a metal nail against a piece of ice, the end in your hand soon becomes cold.
Does cold flow from the ice to your hand? Explain.
Energy “flows” from higher to lower temperature, from your hand to the ice. It is the
energy, heat, flowing from your hand that produces the sensation of coolness. There is
no flow from cold to hot; only from hot to cold.
3. What is the purpose of a layer or copper or aluminum on the bottom of stainless steel cookware?
Copper and aluminum are better conductors than stainless steel, and therefore more
quickly transfer heat to the cookware’s interior.
4. In terms of physics, why do restaurants serve baked potatoes wrapped in aluminum foil?
The main reason for serving potatoes wrapped in aluminum foil is to increase the time
that the potatoes remain hot after being removed from the oven. Heat transfer by
radiation is minimized as radiation from the potatoes is internally reflected, and heat
transfer by convection is minimized as circulating air cannot make contact with the
shielded potatoes. The foil also serves to retain moisture.
5. Many tongues have been injured by licking a piece of metal on a very cold day. Why would no
harm result if a piece of wood were licked on the same day?
In touching the tongue to very cold metal, enough heat can be quickly conducted away
from the tongue to bring the saliva to sub-zero temperature where it freezes, locking
Conceptual Physical Science—Third Edition 269
the tongue to the metal. In the case of relatively nonconducting wood, much less heat is
conducted from the tongue and freezing does not take place fast enough for sudden
sticking to occur.
6. Wood is a better insulator than glass. Yet fiberglass is commonly used as an insulator in wooden
buildings. Explain.
Air is an excellent insulator. The reason that fiberglass is a good insulator is principally
because of the vast amount of air spaces trapped in it.
7. Visit a snow-covered cemetery and note the snow does not slope upward against the gravestones,
but instead forms depressions as shown. Can you think of a reason for this?
Heat from the relatively warm ground is conducted by the gravestone to melt the snow
in contact with the gravestone. Likewise for trees or any materials that are better
conductors of heat than snow, and that extend into the ground.
8. You can bring water in a paper cup to a boil by placing it over a hot flame. Why doesn’t the paper
cup burn?
Much of the energy of the flame is readily conducted through the paper to the water.
The large amount of water relative to the paper absorbs the energy that would
otherwise raise the temperature of the paper. The upper limit of 212°F for the water is
well below the ignition temperature of the paper, 451°F (hence the title “451” of Ray
Bradbury’s science fiction novel about book burning).
9. Why is it that you can safely hold your bare hand in a hot pizza oven for a few seconds, but if you
momentarily touch the metal inside you’ll burn yourself?
Air is a poor conductor, whatever the temperature. So holding your hand in hot air for a
short time is not harmful because very little heat is conducted by the air to your hand.
But if you touch the hot conducting surface of the oven, heat readily conducts to
you—ouch!
10. Wood has a very low conductivity. Does it still have a low conductivity if it is very hot—that is, in
the stage of smoldering red hot coals? Could you safely walk across a bed of red-hot wooden coals
with bare feet? Although the coals are hot, does much heat conduct from them to your feet if you
step quickly? Could you do the same on red-hot iron coals? Explain. (Caution: Coals can stick to
your feet, so OUCH—don’t try it!)
The conductivity of wood is relatively low whatever the temperature—even in the stage
of red-hot coals. You can safely walk barefoot across red-hot wooden coals if you step
quickly (like removing the wooden-handled pan with bare hands quickly from the hot
oven in the previous exercise) because very little heat is conducted to your feet. Because
of the poor conductivity of the coals, energy from within the coals does not readily
replace the energy that transfers to your feet. This is evident in the diminished redness
of the coal after your foot has left it. Stepping on red-hot iron coals, however, is a
different story. Because of the excellent conductivity of iron, very damaging amounts of
heat would transfer to your feet. More than simply ouch!
11. A friend says that in a mixture of gas in thermal equilibrium that the molecules have the same
average kinetic energy. Do you agree or disagree? Explain.
Agree, for your friend is correct.
12. Why would you not expect the molecules of air in your room to all have the same average speed?
Air molecules in your room have the same average kinetic energy, but not the same
average speed. Air is made up of molecules of different mass—some nitrogen, some
oxygen, and a small percentage of other gases. So even though they have the same
average kinetic energy, they won’t have the same average speed. The lighter molecules
will have average speeds greater than the heavier molecules.
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13. In a still room, smoke from a candle will sometimes rise only so far, not reaching the ceiling. Explain why.
The smoke, like hot air, is less dense than the surroundings and is buoyed upward. It
cools with contact with the surrounding air and becomes more dense. When its density
matches that of the surrounding air, its buoyancy and weight balance and rising ceases.
14. What does the high specific heat of water have to do with convection currents in the air at the
seashore?
Because of the high specific heat of water, sunshine warms water much less than it
warms land. As a result, air is warmed over the land and rises. Cooler air from above
the cool water takes its place and convection currents are formed. If land and water
were heated equally by the sun, such convection currents (and the winds they produce)
wouldn’t be.
15. How do the average kinetic energies per molecule compare in a mixture of hydrogen and oxygen
gases at the same temperature?
If they have the same temperature, then by definition, they have the same kinetic
energies per molecule.
16. In a mixture of hydrogen and oxygen gases at the same temperature, which molecules move faster?
Why?
Hydrogen molecules will be the faster moving when mixed with oxygen molecules.
They will have the same temperature, which means they will have the same average
kinetic energy. Recall that KE = 1/2 mv2. Since the mass of hydrogen is considerably
less than oxygen, the velocity must correspondingly be greater.
17. One container is filled with argon gas and the other with krypton gas. If both gases have the same
temperature, in which container are the atoms moving faster? Why?
As in the explanation of the previous exercise, the molecules of gas with the lesser mass
will have the higher average speeds. A look at the periodic table will show that argon
(A = 18) has less massive atoms than krypton (A = 36). The faster atoms are those of
argon. This is the case whether or not the gases are in separate containers.
18. Which atoms have the greatest average speed in a mixture, U-238 or U-235? How would this affect
diffusion through a porous membrane of otherwise identical gases made from these isotopes?
Again, molecules of gas with less mass have higher average speeds. So molecules
containing heavier U-238 are slower on the average. This favors the diffusion of the
faster gas containing U-235 through a porous membrane (which is how U-235 was
separated from U-238 by scientists in the 1940s!).
19. If we warm a volume of air, it expands. Does it then follow that if we expand a volume of air, it
warms? Explain.
When we warm a volume of air, we add energy to it. When we expand a volume of air,
we normally take energy out of it (because the expanding air does work on its
surroundings). So the conditions are quite different and the results will be different.
Expanding a volume of air actually lowers its temperature.
20. A snow-making machine used for ski areas blows a mixture of compressed air and water through a
nozzle. The temperature of the mixture may initially be well above the freezing temperature of
water, yet crystals of snow are formed as the mixture is ejected from the nozzle. Explain how this
happens.
The mixture expands when it is ejected from the nozzle, and therefore cools. At the
freezing temperature of 0°C, ice forms.
21. Turn an incandescent lamp on and off quickly while you are standing near it. You feel its heat but
find when you touch the bulb that it is not hot. Explain why you felt heat from it.
The heat you received was from radiation.
Conceptual Physical Science—Third Edition 271
22. A number of bodies at different temperatures placed in a closed room share radiant energy and
ultimately come to the same temperature. Would this thermal equilibrium be possible if good
absorbers were poor emitters and poor absorbers were good emitters? Explain.
If good absorbers were not also good emitters, then thermal equilibrium would not be
possible. If a good absorber only absorbed, then its temperature would climb above
that of poorer absorbers in the vicinity. And if poor absorbers were good emitters, their
temperatures would fall below that of better absorbers.
23. From the rules that a good absorber of radiation is a good radiator and a good reflector is a poor
absorber, state a rule relating the reflecting and radiating properties of a surface.
A good reflector is a poor radiator of heat, and a poor reflector is a good radiator of heat.
24. The heat of volcanoes and natural hot springs comes from trace amounts of radioactive minerals in
common rock in the Earth’s interior. Why isn’t the same kind of rock at the Earth’s surface warm to
the touch?
The energy given off by rock at the Earth’s surface transfers to the surroundings
practically as fast as it is generated. Hence there isn’t the buildup of energy that occurs
in the Earth’s interior.
25. Suppose at a restaurant you are served coffee before you are ready to drink it. In order that it be
hottest when you are ready for it, would you be wiser to add cream to it right away or when you are
ready to drink it?
Put the cream in right away for at least three reasons. Since black coffee radiates more
heat than white coffee, make it whiter right away so it won’t radiate and cool so
quickly while you are waiting. Also, by Newton’s law of cooling, the higher the
temperature of the coffee above the surroundings, the greater will be the rate of
cooling—so again add cream right away and lower the temperature to that of a reduced
cooling rate, rather than allowing it to cool fast and then bring the temperature down
still further by adding the cream later. Also—by adding the cream, you increase the
total amount of liquid, which for the same surface area, cools more slowly.
26. Even though metal is a good conductor, frost can be seen on parked cars in the early morning even
when the air temperature is above freezing. Can you explain this?
Heat radiates into the clear night air and the temperature of the car goes down.
Normally, heat is conducted to the car by the relatively warmer ground, but the rubber
tires prevent the conduction of heat from the ground. So heat radiated away is not
easily replaced and the car cools to temperatures below that of the surroundings. In this
way frost can form on a below-freezing car in the above-freezing environment.
27. When there is morning frost on the ground in an open park, why is it likely that none is on the
ground beneath park benches?
Under open skies, the ground radiates upward but the sky radiates almost nothing
back down. Under the benches, downward radiation of the benches decreases the net
radiation from the ground, resulting in warmer ground and, likely, no frost.
28. Outer space is not “nothingness.” It is full of radiation with a temperature of about 3 K. Since this
radiation is shining on the Earth at night, why does the Earth get cold at night?
Outer space does illuminate the Earth with its weak radiation at night, but the warmer
Earth gives out much more radiation than it receives.
29. Is it important to convert temperatures to the Kelvin scale when we use Newton’s law of cooling?
Why or why not?
Kelvins and Celsius degrees are the same size, and although ratios of these two scales
will produce very different results, differences in Kelvins and differences in Celsius
degrees will be the same. Since Newton’s law of cooling involves temperature
differences, either scale may be used.
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30. If you wish to save fuel and you’re going to leave your warm house for a half hour or so on a very
cold day, should you turn your thermostat down a few degrees, turn it off altogether, or let it
remain at the room temperature you desire?
Turn your heater off altogether and save fuel. When it is cold outside, your house is
constantly losing heat. How much is lost depends on the insulation and the difference
in inside and outside temperature (Newton’s law of cooling). Keeping ΔT high
consumes more fuel. To consume less fuel, keep ΔT low and turn your heater off
altogether. Will more fuel be required to reheat the house when you return than would
have been required to keep it warm while you were away? Not at all. When you return,
you are replacing heat lost by the house at an average temperature below the normal
setting, but if you had left the heater on, it would have supplied more heat, enough to
make up for heat lost by the house at its normal, higher temperature setting. (Perhaps
your instructor will demonstrate this with the analogy of leaking water buckets.)
31. If you wish to save fuel and you’re going to leave your cool house for a half hour or so on a very hot
day, should you turn your air conditioning thermostat up a bit, turn it off altogether, or let it remain
at the room temperature you desire?
Turn the air conditioner off altogether to keep ΔT small, as in the preceding answer.
Heat leaks at a greater rate into a cold house than into a not-so-cold house. The greater
the rate at which heat leaks into the house, the greater the amount of fuel consumed by
the air conditioner.
32. You can determine wind direction by wetting your finger and holding it up in the air. Explain.
When a wet finger is held to the wind, evaporation is greater on the windy side, which
feels cool. The cool side of your finger is windward.
33. If all the molecules in a liquid had the same speed, and some were able to evaporate, would the
remaining liquid be cooled? Explain.
In this hypothetical case evaporation would not cool the remaining liquid because the
energy of exiting molecules would be no different than the energy of molecules left
behind. Although internal energy of the liquid would decrease with evaporation,
energy per molecule would not change. No temperature change of the liquid would
occur. (The surrounding air, on the other hand, would be cooled in this hypothetical
case. Molecules flying away from the liquid surface would be slowed by the attractive
force of the liquid acting on them.)
34. Where does the energy come from that keeps the dunking bird in Figure 8.31 operating?
The energy that keeps the dunking duck in operation comes from the sun, lamps, or
whatever is heating the lower chamber where evaporation is taking place. To see this,
simply direct heat energy to the lower chamber of the duck and you’ll see an increase
in the number of times per minute the duck dunks.
35. Why will wrapping a bottle in a wet cloth at a picnic often produce a cooler bottle than placing the
bottle in a bucket of cold water?
A bottle wrapped in wet cloth will cool by the evaporation of liquid from the cloth. As
evaporation progresses, the average temperature of the liquid left behind in the cloth
can easily drop below the temperature of the cool water that wet it in the first place. So
to cool a bottle of beer, soda, or whatever at a picnic, wet a piece of cloth in a bucket of
cool water. Wrap the wet cloth around the bottle to be cooled. As evaporation
progresses, the temperature of the water in the cloth drops, and cools the bottle to a
temperature below that of the bucket of water.
36. Why does the temperature of boiling water remain the same as long as the heating and boiling
continue?
When water is boiling, it is being cooled by the boiling process as fast as it is being
heated by the stove. Hence its temperature remains the same—100°C.
Conceptual Physical Science—Third Edition 273
37. Why do vapor bubbles in a pot of boiling water get larger as they rise in the water?
As the bubbles rise, less pressure is exerted on them.
38. Why does the boiling temperature of water decrease when the water is under reduced pressure,
such as at higher altitude?
Decreased pressure lessens the squeezing of molecules, which favors their tendency to
separate and form vapor.
39. Place a jar of water on a small stand within a saucepan of water so the bottom of the jar is held
above the bottom of the pan. When the pan is put on a stove, the water in the pan will boil, but not
the water in the jar. Why?
When the jar reaches the boiling temperature, further heat does not enter it because it is
in thermal equilibrium with the surrounding 100°C water. This is the principle of the
“double boiler.”
40. Water will boil spontaneously in a vacuum—on the moon, for example. Could you cook an egg in
this boiling water? Explain.
You could not cook food in low-temperature water that is boiling by virtue of reduced
pressure. Food is cooked by the high temperature it is subjected to, not by the bubbling
of the surrounding water. For example, put room-temperature water in a vacuum and it
will boil. But this doesn’t mean the water will transfer more internal energy to an egg
than before boiling—an egg in this boiling water won’t cook at all!
41. Our inventor friend proposes a design of cookware that will allow boiling to take place at a
temperature less than 100°C so that food can be cooked with less energy consumption. Comment
on this idea.
As in the answer to the previous exercise, high temperature and the resulting internal
energy given to the food are responsible for cooking—if the water boils at a low
temperature (presumably under reduced pressure), the food isn’t hot enough to cook.
42. When you boil potatoes, will your cooking time be reduced with vigorously boiling water instead of
gently boiling water? (Directions for cooking spaghetti call for vigorously boiling water—not to
lessen cooking time, but to prevent something else. If you don’t know what it is, ask a cook.)
Cooking time will be no different for vigorously boiling water and gently boiling
water, for both have the same temperature. The reason spaghetti is cooked in
vigorously boiling water is simply to ensure the spaghetti doesn’t stick to itself and the
pan. For fuel economy, simply stir your spaghetti in gently boiling water.
43. Why does putting a lid over a pot of water on a stove shorten the time it takes for the water to come
to a boil, whereas after the water is boiling, use of the lid only slightly shortens the cooking time?
The lid on the pot traps heat which quickens boiling; the lid also slightly increases
pressure on the boiling water which raises its boiling temperature. The hotter water
correspondingly cooks food in a shorter time, although the effect is not significant
unless the lid is held down as on a pressure cooker.
44. In a nuclear submarine power plant, the temperature of the water in the reactor is above 100°C.
How is this possible?
The boiling point of water is higher in a nuclear reactor because of increased pressure.
The reactor behaves like a pressure cooker.
45. A piece of metal and an equal mass of wood are both removed from a hot oven at equal
temperatures and dropped onto blocks of ice. The metal has a lower specific heat capacity than the
wood. Which will melt more ice before cooling to 0°C?
The wood, because its greater specific heat capacity means it will release more energy
in cooling.
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46. Why is it that in cold winters a tub of water placed in a farmer’s canning cellar helps prevent canned
food from freezing?
Every gram of water that undergoes freezing releases 80 calories of thermal energy to
the cellar. This continual release of energy by the freezing water keeps the temperature
of the cellar from going below 0°C. Sugar and salts in the canned goods prevent them
from freezing at 0°C. Only when all the water in the tub freezes will the temperature of
the cellar go below 0°C and then freeze the canned goods. The farmer must, therefore,
replace the tub before or just as soon as all the water in it has frozen.
47. Why will spraying fruit trees with water before a frost help to protect the fruit from freezing?
The answer to this is similar to the previous answer, and also the fact that the coating of
ice acts as an insulating blanket. Every gram of water that freezes releases 80 calories,
much of it to the fruit; the thin layer of ice then acts as an insulating blanket against
further loss of heat.
48. Why does a hot dog pant?
Dogs have no sweat glands (except between the toes for most dogs) and therefore cool
by the evaporation of moisture from the mouth and the respiratory track. So dogs
literally cool from the inside out when they pant.
Solutions to Chapter 8 Problems
1. A 10-kg iron ball is dropped onto a pavement from a height of 100 m. If half of the heat generated
goes into warming the ball, find the temperature increase of the ball. (In SI units the specific heat
capacity of iron is 450 J/kg °C.) Why is the answer the same for any mass ball?
0.5mgh = cmΔT
ΔT = 0.5mgh/cm = 0.5gh/c = (0.5)(9.8 m/s2)(100 m)/450 J/kg = 1.1°C.
Again, note that the mass cancels, so the same temperature would hold for any mass
ball, assuming half the heat generated goes into warming the ball. As in the previous
problem, the units check because 1 J/kg = 1 m2/s2.
2. Calculate the height that a block of ice at 0°C must be dropped to completely melt upon impact.
Assume no air resistance and that all the energy goes into melting the ice. [Hint: Equate the joules of
gravitational potential energy to the product of the mass of ice and its heat of fusion (in SI units,
335,000 J/kg). Do you see why the answer doesn’t depend on mass?]
mgh = mL , so gh = L and h = L/g.
h = (334000 J/kg)/(9.8 m/s2) = 34000 m = 34 km.
Note that the mass cancels and that the unit J/kg is the same as the unit m2/s2. So in the
ideal case of no energy losses along the way, any piece of ice that freely falls 34 km
would completely melt upon impact. Taking air resistance into account, only partial
melting would occur.
3. The specific heat capacity of ice is about 0.5 cal/g°C. Supposing that it remains at that value all the
way to absolute zero, calculate the number of calories it would take to change a 1-gram ice cube at
absolute zero (−273°C) to 1 gram of boiling water. How does this number of calories compare to the
number of calories required to change the same gram of 100°C boiling water to 100°C steam?
From −273°C “ice” to 0°C ice requires (273)(0.5) = 140 calories.
From 0°C ice to 0°C water requires 80 calories.
From 0°C water to 100°C water requires 100 calories.
The total is 320 calories.
Boiling this water at 100°C takes 540 calories, considerably more energy than it took to
bring the water all the way from absolute zero to the boiling point! (In fact, at very low
temperature, the specific heat capacity of ice is less than 0.5 cal/g°C, so the true
difference is even greater than calculated here.)
Conceptual Physical Science—Third Edition 275
4. Find the mass of 0°C ice that 10 g of 100°C steam will completely melt.
First, find the number of calories that 10 g of 100°C steam will give in changing to 10 g
of 0°C water.
10 g of steam changing to 10 g of boiling water at 100°C releases 5400 calories.
10 g of 100°C water cooling to 0°C releases 1000 calories.
So 6400 calories are available for melting ice.
6400 cal
80 cal/g = 80 grams of ice.
5. If 50 grams of hot water at 80°C is poured into a cavity in a very large block of ice at 0°C, what will
be the final temperature of the water in the cavity? How much ice must melt in order to cool the hot
water down to this temperature?
The final temperature of the water will be the same as that of the ice, 0°C. The quantity
of heat given to the ice by the water is Q = cmΔT = (1 cal/g°C)(50 g)(80°C) = 4000 cal.
This heat melts ice. How much? From Q = mL, m = Q/L = (4000 cal)/(80 cal/g) =
50 grams. So water at 80°C will melt an equal mass of ice at 0°C.
6. A 50-gram chunk of 80°C iron is dropped into a cavity in a very large block of ice at 0°C. How many
grams of ice will melt? (The specific heat capacity of iron is 0.11 cal/g°C.)
The quantity of heat lost by the iron is Q = cmΔT = (0.11 cal/g°C)(50 g)(80°C) = 440 cal.
The iron will lose a quantity of heat to the ice Q = mL. The mass of ice melted will
therefore be m = Q/L = (440 cal)/(80 cal/g) = 5.5 grams. (The lower specific of heat of iron
shows itself compared with the result of the previous problem.)
7. The heat of vaporization of ethyl alcohol is about 200 cal/g. If 2 kg of it were allowed to vaporize in
a refrigerator, how many grams of ice would be formed from 0°C water?
Note that the heat of vaporization of ethyl alcohol (200 cal/g) is 2.5 times more than the
heat of fusion of water (80 cal/g), so in a change of phase for both, 2.5 times as much ice
will change phase; 2.5 × 2 kg = 5 kg.
Or via formula, the refrigerant would draw away Q = mL = (2000 g)(200 cal/g) =
4 × 105 calories. The mass of ice formed is then (4 × 105 cal)/(80 cal/g) = 5000 g, or 5 kg.

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Chapter 9: Static and Current Electricity
Answers to Chapter 9 Review Questions
1. Nucleus (with protons) is positive;
electrons are negative.
2. Same.
3. Much less, almost 2000 times less
massive than protons.
4. Same. (When the numbers don’t
match, we have an ion.)
5. Positive.
6. That charge is rearranged and always
there. It is not created or destroyed,
similar to the way energy is
conserved.
7. Both are similar in form, both an
inverse-square law. Different in that
there are both attractive and repelling
forces in Coulomb’s law, but only
attractive in Newton’s law of
gravitation.
8. Very large. The charge of 6.25 billion
electrons = 1 coulomb.
9. Reduces to 1/4. Reduces to 1/9.
10. An electrically charged object has a
net charge. In a polarized object,
rather than a net charge, the charges
are rearranged with opposite charges
concentrated at opposite ends.
11. Gravitational, electrical (later, we’ll
see magnetic).
12. Field direction is defined to be the
same as the force on a positive charge
in the field.
13. Electric potential energy is measured
in joules. Electric potential is potential
energy per unit of charge—a ratio,
measured in volts.
14. No, it means it has several thousand
joules of energy per coulomb of charge.
Only if the charge is 1 coulomb is the
energy several thousand joules.
15. A temperature difference. A potential
difference.
16. A sustained difference in potential.
17. Six joules.
18. Protons are anchored in the nucleus.
Electrons, at least the outermost ones
in metals, are not strongly tied to
atoms and can easily flow.
19. Charge flows through a circuit.
Voltage is impressed across a circuit.
Just as water flows through a pipe
when a difference in pressure exists
across its ends, charge flows in a
circuit when a voltage difference
exists across it.
20. AC is alternating current, where
charges surge to and fro. DC is direct
current, where charges flow in only
one direction.
21. Battery, DC. Generators, AC.
22. Thin wire has more resistance, like a
thin pipe is to water flow.
23. The unit of electrical resistance is the
ohm (Ω).
24. Doubled also. If voltage and resistance
are doubled, then no change.
25. 1.5 A.
26. Dry skin has more resistance.
27. Voltage DIFFERENCE.
28. To ground the appliance by
conducting away unwanted current.
29. Your own body.
30. A circuit is a complete path for
electron travel. A gap breaks the path
and stops the flow of electrons.
31. Same, 1 A.
32. 4 V. (Sum of voltage in series =
total voltage.)
Conceptual Physical Science—Third Edition 277
33. Same, 6 V.
34. Same also (because voltage across
each branch is the same).
35. Adds up to equal the current in the
source.
36. As more branches are added in a
circuit, there’s less resistance to the
overall flow of charge (current).
37. Parallel. If it were wired in series and
one lamp burns out, all go out!
38. Current in a reading lamp comprises
only part of the overall current in a
home (unless the lamp is the only
electrical device turned on).
39. Currents in parallel circuits add. If
they add to more than is safe, a fuse
blows and interrupts the flow to
prevent overheating and fire.
40. Power = current × voltage.
41. A 100-W bulb draws more current.
Solutions to Chapter 9 Exercises
1. We do not feel the gravitational forces between ourselves and the objects around us because these
forces are extremely small. Electrical forces, in comparison, are extremely huge. Since we and the
objects around us are composed of charged particles, why don’t we usually feel electrical forces?
There are no positives and negatives in gravitation—the interactions between masses
are only attractive, whereas electrical interactions may be attractive as well as repulsive.
The mass of one particle cannot “cancel” the mass of another, whereas the charge of
one particle can cancel the effect of the opposite charge of another particle.
2. With respect to forces, how are electric charge and mass alike? How are they different?
Charge and mass are alike in that both determine the strength of a force between
objects. Both appear in an inverse-square law of force. They differ in that charge can be
positive or negative while mass is always positive. They differ also in the strength of
force they determine.
3. When combing your hair, you scuff electrons from your hair onto the comb. Is your hair then
positively or negatively charged? How about the comb?
Excess electrons rubbed from your hair leave it with a positive charge; excess electrons
on the comb give it a negative charge.
4. An electroscope is a simple device consisting of a metal ball that is attached by a conductor to two
thin leaves of metal foil protected from air disturbances in a jar, as shown. When the ball is touched
by a charged body, the leaves that normally hang straight down spread apart. Why? (Electroscopes
are useful not only as charge detectors, but also for measuring the quantity of charge: the more
charge transferred to the ball, the more the leaves diverge.)
The leaves, like the rest of the electroscope, acquire charge from the charged object and
repel each other because they both have the same sign of charge. The weight of the conducting
gold foil is so small that even tiny forces are clearly evident.
5. The leaves of a charged electroscope collapse in time. At higher altitudes they collapse more rapidly.
Why is this true? (Hint: The existence of cosmic rays was first indicated by this observation.)
Cosmic rays produce ions in air, which offer a conducting path for the discharge of
charged objects. Cosmic-ray particles streaming downward through the atmosphere are
attenuated by radioactive decay and by absorption, so the radiation and the ionization
are stronger at high altitude than at low altitude. Charged objects more quickly lose
their charge at higher altitudes.
6. Strictly speaking, will a penny be slightly more massive if it has a negative charge or a positive
charge? Explain.
The penny will be slightly more massive with a negative charge, for it will have more
electrons than when neutral. If it were positively charged, it would be slightly lighter
because of missing electrons.
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7. When one material is rubbed against another, electrons jump readily from one to the other but
protons do not. Why is this? (Think in atomic terms.)
Electrons are easily dislodged from the outer regions of atoms, but protons are held
tightly within the nucleus.
8. If electrons were positive and protons negative, would Coulomb’s law be written the same or differently?
The law would be written no differently.
9. The five thousand billion billion freely moving electrons in a penny repel one another. Why don’t
they fly out of the penny?
The electrons don’t fly out of the penny because they are attracted to the five thousand
billion billion positively charged protons in the atomic nuclei of atoms in the penny.
10. Two equal charges exert equal forces on each other. What if one charge has twice the magnitude of
the other? How do the forces they exert on each other compare?
The forces they exert on each other are still the same! Newton’s third law applies to
electrical forces as well as all forces.
11. How does the magnitude of electric force compare between a pair of charged particles when they
are brought to half their original distance of separation? To one-quarter their original distance? To
four times their original distance? (What law guides your answers?)
The inverse-square law is at play here. At half the distance the electric force field is
four times as strong; at 1/4 the distance, 16 times stronger. At four times the distance,
one-sixteenth as strong.
12. Suppose that the strength of the electric field about an isolated point charge has a certain value at a
distance of 1 m. How will the electric field strength compare at a distance of 2 m from the point
charge? What law guides your answer?
At twice the distance the field strength will be 1/4, in accord with the inverse-square law.
13. Why is a good conductor of electricity also a good conductor of heat?
For both electricity and heat, the conduction is via electrons, which in a metal are
loosely bound, easy flowing, and easy to get moving. (Many fewer electrons in metals
take part in heat conduction than in electric conduction, however.)
14. When the chassis of a car is moved into a painting chamber, a mist of paint is sprayed around the
chassis. When it is given a sudden electric charge and mist is attracted to it, presto—the car is
quickly and uniformly painted. What does the phenomenon of polarization have to do with this?
The paint particles in the mist are polarized and are therefore attracted to the charged
chassis.
15. If you place a free electron and a free proton in the same electric field, how will the forces acting on
them compare? Their accelerations? Their directions of travel?
The forces on the electron and proton will be equal in magnitude, but opposite in
direction. Because of the greater mass of the proton, its acceleration will be less than
that of the electron, and be in the direction of the electric field. How much less? Since
the mass of the proton is nearly 2000 times that of the electron, its acceleration will be
about 1/2000 that of the electron. The greater acceleration of the electron will be in the
direction opposite to the electric field. The electron and proton accelerate in opposite
directions.
16. If you put in 10 joules of work to push a 1 coulomb of charge against an electric field, what will be
its voltage with respect to its starting position? When released, what will be its kinetic energy if it
flies past its starting position?
10 joules per coulomb is 10 volts. When released, its 10 joules of potential energy will
become 10 joules of kinetic energy as it passes its starting point.
Conceptual Physical Science—Third Edition 279
17. You are not harmed by contact with a charged metal ball, even though its voltage may be very high.
Is the reason similar to why you are not harmed by the greater-than-1000°C sparks from a 4th-of-
July type sparkler? Defend your answer in terms of the energies that are involved.
Yes, in both cases we have a ratio of energy per something. In the case of temperature,
the ratio is energy/molecule. In the case of voltage it is energy/charge. Even with a
small numerator, the ratio can be large if the denominator is small enough. Such is the
case with the small energies involved to produce high-temperature sparklers and highvoltage
metal balls.
18. What is the voltage at the location of a 0.0001 C charge that has an electric potential energy of 0.5 J
(both measured relative to the same reference point)?
Voltage = 0.5 J
0.0001 C = 5000 V.
19. What happens to the brightness of light emitted by a lightbulb when the current that flows in it increases?
As the current in the filament of a light bulb increases, the bulb glows brighter.
20. One example of a water system is a garden hose that waters a garden. Another is the cooling system
of an automobile. Which of these exhibits behavior more analogous to that of an electric circuit? Why?
The cooling system of an automobile is a better analogy to an electric circuit because
like an electric system it is a closed system, and it contains a pump, analogous to the
battery or other voltage source in a circuit. The water hose does not re-circulate the
water as the auto cooling system does.
21. Is a current-carrying wire electrically charged?
No. The net charge in a wire, whether carrying current or not, is normally zero. The
number of electrons is ordinarily offset by an equal number of protons in the atomic
lattice. Thus current and charge are not the same thing: Many people think that saying
a wire carries current is the same thing as saying a wire is charged. But a wire that is
charged carries no current at all unless the charge moves in some uniform direction.
And a wire that carries a current is typically not electrically charged and won’t affect an
electroscope. (If the current consists of a beam of electrons in a vacuum, then the beam
would be charged. Current is not charge itself: Current is the flow of charge.)
22. Your tutor tells you that an ampere and a volt really measure the same thing, and the different terms
only serve to make a simple concept seem confusing. Why should you consider getting a different tutor?
Your tutor is wrong. An ampere measures current, and a volt measures electric potential
(electric pressure). They are entirely different concepts; voltage produces amperes in a
conductor.
23. In which of the circuits below does a current exist to light the bulb?
Only circuit number 5 is complete and will light the bulb. (Circuits 1 and 2 are
“shortcircuits” and will quickly drain the cell of its energy. In circuit 3 both ends of the
lamp filament are connected to the same terminal and are therefore at the same
potential. Only one end of the lamp filament is connected to the cell in circuit 4.)
24. Does more current flow out of a battery than into it? Does more current flow into a light bulb than
out of it? Explain.
Current flows through electrical devices, just as water flows through a plumbing circuit
of pipes. If a water pump produces water pressure, water flows through both the pump
and the circuit. Likewise with electric current in an electric circuit. For example, in a
simple circuit consisting of a battery and a lamp, the electric current that flows in the
lamp is the same electric current that flows through the wires that connect the lamp and
the same electric current that flows through the battery. Current flows through these
devices. (As a side point, it is common to speak of electric current flowing in a circuit,
but strictly speaking, it is electric charge that flows in an electric circuit; the flow of
charge is current. So if you want to be precisely correct grammatically, say that current
is in a circuit and charge flows in a circuit.)
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25. Sometimes you hear someone say that a particular appliance “uses up” electricity. What is it that the
appliance actually uses up and what becomes of it?
An electric device does not “use up” electricity, but rather energy. And strictly
speaking, it doesn’t “use up” energy, but transforms it from one form to another. It is
common to say that energy is used up when it is transformed to less concentrated
forms—when it is degraded. Electrical energy ultimately becomes heat energy. In this
sense it is used up.
26. A simple lie detector consists of an electric circuit, one part of which is part of your body—like from
one finger to another. A sensitive meter shows the current that flows when a small voltage is
applied. How does this technique indicate that a person is lying? (And when does this technique not
tell when someone is lying?)
A lie detector circuit relies on the likelihood that the resistivity of your body changes
when you tell a lie. Nervousness promotes perspiration, which lowers the body’s
electrical resistance, and increases whatever current flows. If a person is able to lie with
no emotional change and no change in perspiration, then such a lie detector will not be
effective. (Better lying indicators focus on the eyes.)
27. Only a small percentage of the electric energy fed into a common light bulb is transformed into
light. What happens to the rest?
Most of the energy, typically 95%, of the electrical energy in an incandescent lamp goes
directly to heat. Thermal energy is the graveyard of electrical energy.
28. Will a lamp with a thick filament draw more current or less current than a lamp with a thin filament?
The thick filament has less resistance and will draw (carry) more current than a thin
wire connected across the same potential difference. (Important point: It is common to
say that a certain resistor “draws” a certain current, but this may be misleading. A
resistor doesn’t “attract” or “draw” current, just as a pipe in a plumbing circuit doesn’t
“draw” water; it instead “allows” or “provides for” the passage of current when an
electrical pressure is established across it.)
29. A 1-mile long copper wire has a resistance of 10 ohms. What will be its new resistance when it is
shortened by (a) cutting it in half; (b) doubling it over and using it as “one” wire?
(a) The resistance will be half, 5 ohms, when cut in half. (b) The resistance will be half
again when the cross-sectional area is doubled, so it will be 2.5 ohms.
30. Will the current in a light bulb connected to a 220-V source be greater or less than when the same
bulb is connected to a 110-V source?
Current will be greater in the bulb connected to the 220-volt source. Twice the voltage
would produce twice the current if the resistance of the filament remained the same.
(In practice, the greater current produces a higher temperature and greater resistance in
the lamp filament, so the current is greater than that produced by 110 volts, but
appreciably less than twice as much for 220 volts. A bulb rated for 110 volts has a very
short life when operated at 220 volts.)
31. Which will do less damage—plugging a 110-V appliance into a 220-V circuit or plugging a 220-V
appliance into a 110-V circuit? Explain.
Damage generally occurs by excess heating when too much current is driven through
an appliance. For an appliance that converts electrical energy directly to thermal energy
this happens when excess voltage is applied. So don’t connect a 110-volt iron, toaster, or
electric stove to a 220-volt circuit. Interestingly enough, if the appliance is an electric
motor, then applying too little voltage can result in overheating and burn up the motor
windings. (This is because the motor will spin at a low speed and the reverse
“generator effect” will be small and allow too great a current to flow in the motor.) So
don’t hook up a 220-volt power saw or any 220-volt motor-driven appliance to 110 volts.
To be safe use the recommended voltages with appliances of any kind.
Conceptual Physical Science—Third Edition 281
32. If a current of one- or two-tenths of an ampere flows into one of your hands and out the other, you
will probably be electrocuted. But if the same current flows into your hand and out the elbow above
the same hand, you can survive even though the current may be large enough to burn your flesh.
Explain.
In the first case the current passes through your chest; in the second case current passes
only through your arm. You can cut off your arm and survive, but you cannot survive
without your heart.
33. Would you expect to find DC or AC in the filament of a light bulb in your home? How about in the
headlight of an automobile?
Electric power in your home is likely supplied at 60 hertz and 110–120 volts via
electrical outlets. This is ac (and delivered to your home via transformers between the
power source and your home. We will see in Chapter 24 that transformers require AC
power for operation.) Electric power in your car must be able to be supplied by the
battery. Since the + and − terminals of the battery do not alternate, the current they
produce does not alternate either. It flows in one direction and is DC.
34. Electric current is generated at 50 Hz in Europe, and 60 Hz in the United States. If you viewed the signals on
an oscilloscope screen, which peaks would be closer together?
The 60-Hz peaks would be closer together.
35. Are automobile headlights wired in parallel or in series? What is your evidence?
Auto headlights are wired in parallel. Then when one burns out, the other remains lit.
If you’ve ever seen an automobile with one burned out headlight, you have evidence
they’re wired in parallel.
36. A car’s headlights dissipate 40 W on low beam, and 50 W on high beam. Is there more or less resistance in the
high beam filament?
There is less resistance in the higher wattage lamp. Since power = current × voltage,
more power for the same voltage means more current. And by Ohm’s law, more current
for the same voltage means less resistance. (Algebraic manipulation of the equations
P = IV and I = V/R leads to P = V2/R.)
37. What unit is represented by (a) joule per coulomb, (b) coulomb per second, (c) watt.second?
(a) volt, (b) ampere, (c) joule.
38. To connect a pair of resistors so their equivalent resistance will be more than the resistance of either
one, should you connect them in series or in parallel?
The equivalent resistance of resistors in series is their sum, so connect a pair of
resistors in series for more resistance.
39. To connect a pair of resistors so their equivalent resistance will be less than the resistance of either
one, should you connect them in series or in parallel?
The equivalent resistance of resistors in parallel is less than the smaller resistance of
the two. So connect a pair of resistors in parallel for less resistance.
40. Why is the wingspan of birds a consideration in determining the spacing between parallel wires in a
power line?
If the parallel wires are closer than the wingspan of birds, a bird could short circuit the
wires by contact with its wings, be killed in the process, and possibly interrupt the
delivery of power.
41. Estimate the number of electrons that a power company delivers annually to the homes of a typical
city of 50,000 people.
Zero. Power companies do not sell electrons; they sell energy. Whatever number of
electrons flow into a home, the same number flow out.
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42. If electrons flow very slowly through a circuit, why does it not take a noticeably long time for a
lamp to glow when you turn on a distant switch?
How quickly a lamp glows after an electrical switch is closed does not depend on the
drift velocity of the conduction electrons, but depends on the speed at which the
electric field propagates through the circuit—about the speed of light.
43. Consider a pair of flashlight bulbs connected to a battery. Will they glow brighter connected in
series or in parallel? Will the battery run down faster if they are connected in series or in parallel?
Bulbs will glow brighter when connected in parallel, for the voltage of the battery is
impressed across each bulb. When two identical bulbs are connected in series, half the
voltage of the battery is impressed across each bulb. The battery will run down faster
when the bulbs are in parallel.
44. If several bulbs are connected in series to a battery, they may feel warm to the touch but not visibly
glow. What is your explanation?
Most of the electric energy in a lamp filament is transformed to heat. For low currents
in the bulb, the heat that is produced may be enough to feel but not enough to make
the filament glow red or white hot.
45. In the circuit shown, how do the brightnesses of the identical lightbulbs compare? Which lightbulb
draws the most current? What will happen if bulb A is unscrewed? If C is unscrewed?
Bulb C is the brightest because the voltage across it equals that of the battery. Bulbs A
and B share the voltage of the parallel branch of the circuit and have half the current of
bulb C (assuming resistances are independent of voltages). If bulb A is unscrewed, the
top branch is no longer part of the circuit and current ceases in both A and B. They no
longer give light, while bulb C glows as before. If bulb C is instead unscrewed, then it
goes out and bulbs A and B glow as before.
46. As more and more bulbs are connected in series to a flashlight battery, what happens to the
brightness of each bulb? Assuming heating inside the battery is negligible, what happens to the
brightness of each bulb when more and more bulbs are connected in parallel?
As more bulbs are connected in series, more resistance is added to the single circuit
path and the resulting current produced by the battery is diminished. This is evident in
the dimmer light from the bulbs. On the other hand, when more bulbs are connected to
the battery in parallel, the brightness of the bulbs is practically unchanged. This is
because each bulb in effect is connected directly to the battery with no other bulbs in
its electrical path to add to its resistance. Each bulb has its own current path.
47. Why is there no effect on other branches in a parallel circuit when one branch of the circuit is
opened or closed?
What affects the other branches is the voltage impressed across them, and their own
resistance—period. Opening or closing a branch doesn’t alter either of these.
48. A battery has internal resistance, so if the current it supplies goes up, the voltage it supplies goes
down. If too many bulbs are connected in parallel across a battery, will their brightness diminish?
Explain.
Yes, there will be a decrease in brightness if too many lamps are connected in parallel
because of the increased current that flows through the battery. Internal voltage drop
increases with current in the battery, which means reduced voltage supplied at its
terminals to the circuit it powers. (If the parallel circuit is powered by a stronger source
such as the power utility provides via common wall sockets, no dimming of bulbs will
be seen as more and more parallel paths are added.)
49. Why are devices in household circuits almost never connected in series?
Household appliances are not connected in series for at least two reasons. First, the
voltage, current, and power for each appliance would vary with the introduction of
other appliances. Second, if one device burns out, the current in the whole circuit
Conceptual Physical Science—Third Edition 283
ceases. Only if each appliance is connected in parallel to the voltage source can the
voltage and current through each appliance be independent of the others.
50. If a 60-W bulb and a 100-W bulb are connected in series in a circuit, across which bulb will there be
the greater voltage drop? How about if they are connected in parallel?
The 100-watt bulb has the thicker filament and lower resistance (more current through
the filament answer) so in series where the current is the same in each bulb, less energy
is dissipated in going through the lower resistance. This corresponds to lower voltage
across the resistance—a lower voltage drop. So the greater voltage drop is across the 60-
watt bulb in series. Interestingly, in series the 60-watt bulb is brighter than the 100-watt
bulb! When connected in parallel, the voltage across each bulb is the same, and the
current is greater in the lower resistance 100-watt bulb, which glows brighter than the
60-watt bulb.
Solutions to Chapter 9 Problems
1. Two point charges are separated by 6 cm. The attractive force between them is 20 N. Find the force
between them when they are separated by 12 cm. (Why can you solve this problem without
knowing the magnitudes of the charges?)
By the inverse-square law, twice as far is 1/4 the force; 5 N.
The solution involves relative distance only, so the magnitude of charges is irrelevant.
2. If the charges attracting each other in the problem above have equal magnitude, what is the magnitude of
each charge?
From Coulomb’s law, the force is given by F = kq2
d2 , so the square of the charge is
q2 = Fd2
k =
(20 N)(0.06 m)2
9 x 109 N m2/C2 = 8.0 × 10–12 C2. Taking the square root of this gives
q = 2.8 × 10–6 C, or 2.8 microcoulombs.
3. Two pellets, each with a charge of 1 microcoulomb (10-6 C), are located 3 cm (0.03 m) apart. What is
the electric force between them? What mass object would experience this same force in the Earth’s
gravitational field?
From Coulomb’s law, F = k
q1q2
d2 = (9 × 109)
(1.0 x 10–6)2
(0.03)2 = 10 N.
This is the same as the weight of a 1-kg mass.
4. A droplet of ink in an industrial ink-jet printer carries a charge of 1.6 × 10–10 C and is deflected onto
paper by a force of 3.2 × 10–4 N. Find the strength of the electric field to produce this force.
Electric field is force divided by charge: E = Fq
=
3.2 x 10-4N
1.6 x 10-10C
= 2 × 106 N/C. (The
unit N/C is the same as the unit V/m, so the field can be expressed as 2 million volts
per meter.)
5. Find the voltage change when (a) an electric field does 12 J of work on a 0.0001-C charge, and (b) the
same electric field does 24 J of work on a 0.0002-C.
a. ΔV =
energy
charge =
12 J
0.0001 C = 120,000 volts.
b. ΔV for twice the charge is
24 J
0.0002 = same 120 kV.
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6. The wattage marked on a light bulb is not an inherent property of the bulb but depends on the
voltage to which it is connected, usually 110 or 120 V. How many amperes flow through a 60-W
bulb connected in a 120-V circuit?
From “Power = current × voltage,” 60 watts = current × 120 volts, current = 60W
120V = 0.5 A.
7. Rearrange the equation Current = voltage/resistance to express resistance in terms of current and
voltage. Then solve the following: A certain device in a 120-V circuit has a current rating of 20 A.
What is the resistance of the device (how many ohms)?
From current = voltage
resistance , resistance = voltage
current = 120V
20A = 6 Ω.
8. Using the formula Power = current x voltage, find the current drawn by a 1200-W hair dryer
connected to 120 V. Then using the method you used in the previous problem, find the resistance of
the hair dryer.
From power = current × voltage, current = power
voltage = 1200W
120V = 10 A.
From the formula derived above, resistance = voltage
current = 120V
10A = 12 Ω.
9. The total charge that an automobile battery can supply without being recharged is given in terms of
ampere-hours. A typical 12-V battery has a rating of 60 ampere-hours (60 A for 1 h, 30 A for 2 h, and
so on). Suppose you forget to turn off the headlights in your parked automobile. If each of the two
headlight draws 3 A, how long will it be before your battery is “dead”?
Two headlights draw 6 amps, so the 60 ampere-hour battery will last for about 10 hours.
10. How much does it cost to operate a 100-W lamp continuously for 1 week if the power utility rate is
20¢/kWh?
$3.36. First, 100 watts = 0.1 kilowatt. Second, there are 168 hours in one week (7 days ×
24 hours/day = 168 hours). So 168 hours × 0.1 kilowatt = 16.8 kilowatt-hours, which at
20 cents per kWh comes to $3.36.
11. A 4-W night light is plugged into a 120-V circuit and operates continuously for 1 year. Find the
following: (a) the current it draws, (b) the resistance of its filament, (c) the energy consumed in a
year, and (d) the cost of its operation for a year at the utility rate of 20¢/kWh.
a. From power = current × voltage, current = power/voltage = 4W/120V = 1/30 A.
b. From current = voltage/resistance (Ohm’s law), resistance = voltage/current =
120 V/(1/30 A) = 3600 Ω.
c. First, 4 watts = 0.004 kilowatt. Second, there are 8760 hours in a year (24 hours/day ×
365 days = 8760 hours). So 8760 hours × 0.004 kilowatt = 35.0 kWh.
d. At the rate of 20 cents per kWh, the annual cost is 35.0 kWh × $0.20/kWh = $7.00.
12. An electric iron connected to a 110-V source draws 9 A of current. How much heat (in joules) does it
generate in a minute?
The iron’s power is P = IV = (110 V)(9 A) = 990 W = 990 J/s. The heat energy generated
in 1 minute is E = power × time = (990 J/s)(60 s) = 59,400 J.
13. How many coulombs of charge flow through the iron in the previous problem in one minute?
Since current is charge per unit time, charge is current × time: q = It = (9 A)(60 s) =
(9 C/s)(60 s) = 540 C. (Charges of this magnitude on the move are commonplace, but this
quantity of charge accumulated in one place would be incredibly large.)
14. A certain light bulb with a resistance of 95 ohms is labeled “150 W.” Was this bulb designed for use
in a 120-V circuit or a 220-V circuit?
It was designed for use in a 120-V circuit. With an applied voltage of 120 V, the current
in the bulb is I = V/R = (120 V)/(95 W) = 1.26 A. The power dissipated by the bulb is
then P = IV = (1.26 A)(120 V) = 151 W, close to the rated value. If this bulb is connected
to 220 V, it would carry twice as much current and would dissipate four times as much
power (twice the current × twice the voltage), more than 600 W. It would likely burn
Conceptual Physical Science—Third Edition 285
out. (This problem can also be solved by first carrying out some algebraic
manipulation. Since current = voltage/resistance, we can write the formula for power as
P = IV = (V/R)V = V2/R. Solving for V gives V = √PR. Substituting for the power and the
resistance gives V = √(150)(95) = 119 V.)
15. In periods of peak demand, power companies lower their voltage. This saves them power (and
saves you money!). To see the effect, consider a 1200-W toaster that draws 10 A when connected to
120 V. Suppose the voltage is lowered by 10 percent to 108 V. By how much does the current
decrease? By how much does the power decrease? (Caution: The 1200-W label is valid only when
120 V is applied. When the voltage is lowered, it is the resistance of the toaster, not its power, that
remains constant.)
The resistance of the toaster is R = V/I = (120 V)/(10 A) = 12 W. So when 108 V is
applied, the current is I = V/R = (108 V)/(12 W) = 9.0 A and the power is P = IV =
(9.0 A)(108V) = 972 W, only 81 percent of the normal power. (Can you see the reason for
81 percent? Current and voltage are both decreased by 10 percent, and 0.9 × 0.9 = 0.81.)