Problem 1: Chapter 5, Case 7a-e (Pg. 248)
The brakes of an automobile are controlled by hydraulic fluid. When you step on the brake pedal, a brake lever attached to that pedal pushes a piston into the brake system’s master cylinder. This cylinder is filled with brake fluid.
a. Because the piston of the master cylinder attaches to the brake lever
close to its pivot, the piston moves only half as far as your foot does
when you step on the brake pedal. If you exert 200 N of force on the brake
pedal, how much force does the brake lever exert on the piston?
Answer: 400 N
Why: If the brake lever is rotating at a constant angular speed (a pretty
safe assumption), then it's experiencing zero net torque. The torque you
exert on it with your foot is equal and opposite to the torque that the
piston is exerting on the brake lever. Since your force on the brake lever
is exerted twice as far from the pivot as the force exerted by the piston
on the brake lever, the piston must push twice as hard. Since your force
is 200 N, its force must be 400 N.
b. Assume for the moment that the brake fluid is trapped in the cylinder.
What happens to that fluid when you push downward on the brake pedal? How
far does the pedal move and what limits its motion?
Answer: The fluid's pressure rises. Because liquids are nearly incompressible,
the pedal barely moves before the fluid stops the piston from moving any
farther into the cylinder.
Why: You are squeezing liquid in the cylinder as you step on the brake
pedal. The liquid's pressure rises rapidly even though its density increases
only slightly. Before you have made much of a change in the liquid's volume,
its pressure stops the piston from moving any farther into the cylinder.
c. In reality, brake fluid can flow out of the master cylinder through
pipes that connect to brakes on the four wheels. In most cars there are
two separate master cylinders, one for the front brakes and one for the
rear brakes. That way, if one hydraulic system fails, you can still control
half of the brakes. Each brake contains a slave cylinder with a piston
inside it. As pressurized brake fluid flows into a slave cylinder, why
does that slave cylinder’s piston experience a force?
Answer: The rising pressure in the hydraulic fluid exerts a force on
the slave cylinder's piston.
Why: As the pressure in the master cylinder rises, fluid begins to flow
into the slave cylinder. The pressure in the slave cylinder rises to match
that of the master cylinder--if we neglect gravity, fluids flow until the
pressures are equal everywhere. The increased pressure in the slave cylinder
means that there is a net outward force on the piston, due to a pressure
imbalance across it--high pressure inside and atmospheric pressure outside.
d. If the brake fluid from a particular master cylinder is supplying
fluid to two slave cylinders and all of the pistons are the same diameter,
what is the relationship between the distance you push the piston into
the master cylinder and the distance a piston moves out of one of the slave
cylinders?
Answer: The pistons of the slave cylinders will move half as far as
the piston of the master cylinder.
Why: The two slave cylinders must share the fluid that's being squeezed
out of the master cylinder. If they didn't have to share, their pistons
would move exactly as far as the piston of the master cylinder. But since
they do share, each of these cylinders receives half the hydraulic fluid
that leaves the master cylinder and the piston in each slave cylinder moves
half as far as the piston in the master cylinder.
e. Occasionally cars develop an air bubble in their brake lines (the
pipes connecting the master cylinder to the slave cylinders). Air compresses
easily as the pressure inside the brake lines increases. When you push
the brake pedal down and squeeze brake fluid out of the master cylinder,
the slave cylinders don’t respond properly. Why do the pistons of the slave
cylinders move out less than they should?
Answer: As the pressure in the hydraulic system rises, the air bubble
will compress and its volume will shrink. Some of the hydraulic fluid flowing
out of the master cylinder will go to filling the space vacated by the
shrinking gas bubble, rather than to filling the slave cylinders.
Why: Without any air bubbles in the system, each liter of fluid that leaves
the master cylinder will flow into the slave cylinders. But with the air
bubble occupying a variable amount of volume in the brake lines, there
is another place for the brake fluid to go: into the space occupied by
the bubble.
Problem 2: Chapter 5, Case 9a-d (Pg. 248)
A one-wheeled cycle, or unicycle, is the ultimate in statically unstable vehicles.
a. Why does a unicycle always fall over when the rider doesn’t try to
keep it upright?
Answer: Its center of gravity descends whenever it tips (or it has an
unstable equilibrium).
Why: The unicycle's equilibrium is unstable because whenever it tips, its
center of gravity descends. As a result, its gravitational potential energy
decreases and an object naturally accelerates in the direction that reduces
its potential energy as quickly as possible. Once the unicycle starts to
tip, it accelerates further into that tip and over it goes.
b. To keep the unicycle from falling over, the rider continually tries
to position the wheel so that the force the ground exerts on the wheel
points toward a particular point. What is that point?
Answer: The center of mass.
Why: If the force the ground exerts on the wheel points directly toward
the center of mass, it exerts no torque about the center of mass and the
unicycle doesn't undergo angular acceleration (it doesn't tip over).
c. A person riding a two-wheeled bicycle must lean left while making
a turn toward the left. Does a unicycle rider also have to lean left during
a left turn?
Answer: Yes.
Why: To turn left, the unicycle needs a force to the left. Since that force
is exerted by friction at the bottom of the wheel, it also exerts a torque
on the unicycle about its center of mass. To keep the unicycle from beginning
to rotate, the rider must lean the unicycle to the left. That way, the
ground's support force will exert a torque on the unicycle about its center
of mass and this additional torque will cancel the one caused by friction.
Overall, there will be no net torque on the unicycle and it won't tip over.
d. While a unicycle doesn’t exhibit dynamic stability the way a two-wheeled
bicycle does, there is a way to give it dynamic stability. If you spin
the unicycle extremely rapidly about its vertical axis, it will act like
a toy top and won’t fall over for a very long time. (Unfortunately, it’s
hard to ride this way.) While gravity will exert a torque on this spinning
unicycle if its axis isn’t perfectly vertical, the unicycle doesn’t simply
fall over. Instead, its axis of rotation changes directions. What is this
behavior an example of?
Answer: Precession.
Why: When the unicycle is spinning, it has angular moment and a particular
axis of rotation. The torque that it experiences about the bottom of its
wheel will cause it to undergo angular acceleration and will change its
axis of rotation (a process called precession). The unicycle will precess
rather than simply falling over.
Problem 3: Chapter 6, Case 1a-e (Pg. 291)
Glass transmits the visible light radiated by the white hot sun but it absorbs the infrared light radiated by objects near room temperature. This selective absorption and transmission gives rise to the "greenhouse effect."
a. A small patch of garden is bathed in sunlight. The sun is transferring
heat to this garden by radiation but the garden’s temperature remains essentially
constant. Why doesn’t the garden get hotter and hotter?
Answer: The garden transfers heat elsewhere exactly as fast as heat
arrives.
Why: The garden warms up until its rate of heat loss is exactly equal to
its rate of heat gain. With as much heat leaving as arriving, there is
no further change in the thermal energy content of the garden and its temperature
remains constant.
b. If you cover the garden with a glass dome, what happens to the garden’s
ability to get rid of heat via radiation?
Answer: The garden has more difficulty getting rid of heat.
Why: Some of the thermal radiation emitted by the garden is absorbed by
the glass. The glass gets hotter and it also emits thermal radiation. Some
of this thermal radiation is sent back toward the garden.
c. How does the glass dome affect the garden’s temperature?
Answer: The garden's temperature rises.
Why: Since the garden is now receiving more thermal radiation than before,
its temperature rises until it's able to eliminate heat as fast as that
heat is arriving.
d. Like glass, molecules such as water vapor, carbon dioxide, and methane
transmit visible light while absorbing some infrared light. How does the
presence of these "greenhouse gases" in the earth’s atmosphere
affect the earth’s ability to get rid of heat via radiation?
Answer: The earth has more difficulty getting rid of heat.
Why: Some of the thermal radiation emitted by the earth is absorbed by
the gases. The gases get hotter and they also emits thermal radiation.
Some of this thermal radiation is sent back toward the earth.
e. How do these greenhouse gases affect the earth’s temperature?
Answer: The earth's temperature rises.
Why: Since the earth is now receiving more thermal radiation than before,
its temperature rises until it's able to eliminate heat as fast as that
heat is arriving.
Problem 4: Chapter 5, Case 8a-e (Pg. 248)
An electric hot water heater takes cold water at its inlet pipe and delivers hot water at its outlet pipe. Between these two pipes is a large water container. Many electric hot water heaters have two separate heating elements: one near the top of the water container and one near the bottom. These elements use a lot of electric power so they don’t operate simultaneously. Instead, the top element operates until the water near the top of the hot water heater reaches the desired temperature and then the bottom element operates until the water near the bottom is also at the desired temperature. The water heater then turns itself off and waits until more heating is required.
a. Both heating elements project into the same hot water container inside
the water heater. When the top element is heating water at the top of the
container, why doesn’t convection occur and cause water at the bottom of
the container to also become hot?
Answer: Hot water floats on cold water, so there is no convection.
Why: When the heating element warms the water near the top of the container,
that hot water stays at the top of the container. The cold water near the
bottom of the container is essentially unaffected.
b. Convection dictates where the water pipes attach to the container.
Where should hot water be extracted from the container and where should
cold water be introduced into the container to replace that hot water?
Answer: How water should be extracted from the top of the container
and cold water should be added to the bottom of the container.
Why: If there is any hot water at all in the container, it will be at the
top so that's where the extraction pipe should be located. To replace this
hot water, cold water should be added to the bottom of the tank, where
it won't mix with any hot water that might still be present in the container.
c. Why does the water heater’s bottom heating element operate more often
and age more quickly than its top heating element?
Answer: When a small amount of hot water is removed from the tank, cold
water is added to the bottom of the tank. This water is then heated by
the bottom heating element. Thus the bottom heating element runs more often
and is more likely to burn out than the top heating element.
Why: Since small uses of hot water cause the bottom element to turn on,
it's the bottom element that wears out fastest.
d. Why is it important to thermally insulate the hot water heater’s
container?
Answer: Insulation keeps heat from flowing out of the hot water and
into the room.
Why: If heat leaks out of the hot water and into the room, the hot water
heater will have to keep warming the water. The heat that leaks into the
room may also overheat the room, requiring additional air conditioning.
e. Energy flows into the hot water heater as electricity. Since energy
is conserved and can’t accumulate forever inside the heater, it must leave
somehow. At a time when the hot water is being used, how does most of the
energy leave the hot water heater?
Answer: This energy leaves as thermal energy in the hot water.
Why: The energy that arrives as electric energy becomes thermal energy
in the water. When hot water flows out of the hot water heater, this energy
leaves with it.
Problem 5: Chapter 7, Case 5a-f (Pg. 318)
Decompression sickness was first observed in workers preparing underwater foundations or caissons for bridges. The work had to be done in pressurized air to keep the water out. When the pressure was released, the workers would experience "caisson disease," a painful or even fatal disorder in which gas bubbles formed inside their tissues. Because they bent over in pain, their illness was called "the bends."
a. The workers would enter the caisson while it was at atmospheric pressure.
The doors were then sealed and air was pumped in. The temperature in the
caisson rose rapidly. What caused this temperature rise?
Answer: They did work on the air as they pumped it into the caisson,
so the air's thermal energy rose and so did its temperature.
Why: Whenever you compress a gas, you do work on that gas and its thermal
energy and temperature rise.
b. A few minutes after the caisson was brought up to full pressure,
the air inside the caisson would return to normal temperature. What had
happened to its thermal energy?
Answer: The thermal energy flowed as heat into the walls of the caisson.
Why: Since the air was then warmer than anything else in the caisson, heat
flowed out of the air and into everything else, particularly the walls
of the caisson.
c. When the shift was over, the doors to the caisson were opened and
the air was allowed to rush out. That outgoing air was ice cold. The air
inside the caisson was also so cold that sometimes it would begin to snow.
What caused this temperature drop?
Answer: The expanding air did work on the air outside the caisson and
used up some of its thermal energy in the process. With less thermal energy
than before, the air became colder.
Why: Whenever a gas expands by pushing something out of the way, it does
work on whatever it pushes out of the way and the gas's thermal energy
and temperature decrease.
d. When during this cycle of pressurizing and depressurizing was work
done on the air inside the caisson?
Answer: During the pressurization process.
Why: Pressurizing the air requires an inward force and the air moves inward,
so work is done on the air.
e. When was work done by the air inside the caisson?
Answer: During the expansion (or depressurization) process.
Why: When the air inside the caisson expands, it pushes the air outside
the caisson outward and that air moves outward, so work is done by the
air in the caisson on that outside air.
f. What heat transfer took place?
Answer: Heat is transferred from things outside the caisson (including
air and objects) to things inside the caisson (particularly the walls of
the caisson).
Why: The heat that flows into the caisson walls after the pressurization
process is greater than the heat that flows out of them during the depressurization
process, so heat is transferred to the walls overall. The area outside
the caisson experiences only a flow of chilled air during the depressurization
process, so it gives up heat overall. In summary, heat is extracted from
the areas outside the caisson and delivered to the areas inside the caisson.