Problem 1: Chapter 3, Case 9a-e (Pg. 155)
A submarine hovers below the surface of the water, supported by the buoyant force. Although it also pushes on the water to propel itself forward and to adjust its height, the submarine’s most important depth control is its average density.
a. If a submarine is hovering motionless, 50 m below the surface of
the water, what is the net force it’s experiencing?
Answer: Zero.
Why: Since the submarine is motionless, it isn't accelerating and must
be experiencing a net force of zero.
b. What is the motionless submarine’s average density?
Answer: The same as that of water.
Why: For the upward buoyant force on the submarine to exactly balance the
submarine's weight, the submarine has to have the same weight as the water
it displaces. Its average density will then be the same as that of the
water it displaces.
c. A submarine has ballast tanks that control its buoyancy. To start
descending, the submarine floods these tanks with outside sea water. How
does this flooding affect the submarine’s average density and the net force
on the submarine?
Answer: The submarine's average density increases and the submarine
begins to experience a downward net force.
Why: The submarine's mass increases while its volume remains constant.
As a result, its average density (its mass divided by its volume) increases.
Since it is now more dense than the water it displaces, the submarine experiences
a net downward force.
d. To ascend, the submarine uses compressed air to purge the sea water
from its ballast tanks. How does eliminating this sea water affect the
submarine’s average density and the net force on the submarine?
Answer: The submarine's average density decreases and the submarine's
net force becomes more upward.
Why: The submarine's mass decreases while its volume remains constant.
As a result, its average density (its mass dividied by its volume) decreases.
Since it is now less dense than it was before, it experiences a more upward
net force (either a weaker downward net force than before or perhaps even
an upward net force).
e. To control the submarine’s tilt, water is pumped between tanks in
the front and back of the vessel. Why doesn’t this transfer of water affect
the submarine’s buoyancy?
Answer: The submarine's average density doesn't change.
Why: The submarine's average density is its total mass divided by its total
volume. Since neither its mass nor its volume change, its average density
remains unchanged, too.
Problem 2: Chapter 4, Case 2a-e (Pg. 207)
A helicopter suspends itself from a spinning rotor. Each blade in the rotor is actually a wing, with an airfoil shape that creates lift as it slices through the air. To hover, the helicopter’s rotor creates just enough upward lift to exactly balance the helicopter’s downward weight.
a. The rotor keeps turning at a constant rate. To control the rotor’s
lift, the helicopter adjusts the angles of attack of the blades as they
turn. If a hovering helicopter increases the angles of attack of all of
its blades simultaneously, what will happen to the forces on the helicopter
and how will it move?
Answer: The helicopter's net force will become upward (its downward
weight will remain unchanged but its upward lift force will increase) and
it will accelerate upward.
Why: While the helicopter was hovering, its upward lift force exactly balanced
its downward weight. By increasing the angles of attack of all its blades,
it increases their upward lifts and thus its overall net force becomes
upward. It accelerates upwardin response to this upward net force.
b. The helicopter can also adjust the angle of attack of a single blade.
If it increases the angle of attack of whatever blade is currently on the
pilot’s right, the helicopter will experience a torque about its center
of mass. Which way will the helicopter begin to tilt?
Answers: Its right side will rise and its left side will descend.
Why: The blades to the helicopter's right will experience a larger upward
lift force than before. Thus lift is exerted to the right of the helicopter's
center of mass and exerts a torque on the helicopter about that center
of mass. The helicopter will begin to rotate about that center of mass
so that its right side will rise and its left side will descend.
c. If the helicopter is tilted nose down and tail high, the helicopter
will begin to accelerate forward. Why?
Answer: The lift force experienced by the blades will have a component
in the forward horizontal direction.
Why: While the helicopter is upright and its blades are turning in a horizontal
plane, the blades experience a lift force that is directly upward. But
while it's leaning forward, the blades are turning in a forward-tilted
plane and they experience a lift force that is both upward and forward.
The forward component of this lift force is the only horizontal force acting
on the helicopter, so the helicopter accelerates forward.
d. To turn the helicopter so that it faces a different direction, something
must exert a torque on it about a vertical axis. That’s one reason why
the helicopter has a small rotor at the end of its tail boom. This rotor
pushes air to the right or left, depending on the angles of attack of its
blades. Which way should the tail rotor push the air to turn the helicopter
to the right?
Answer: The tail rotor should push the air to the right.
Why: For the helicopter to turn to the right, its tail and tail rotor must
move to the left. The tail rotor needs something to push it toward the
left to cause this shift, so it pushes the air toward the right. The air
pushes back and the tail rotor accelerates toward the left. The helicopter
thus turns toward the right.
e. The rotor blades are actually flexible, but they extend almost straight
outward when the rotor is spinning. Why?
Answer: When the blades are spinning around the central hub, they are
accelerating inward and must thus be pulled inward by tension in the blades.
This tension pulls the blades straight.
Why: Whenever a limp object experiences tension, it tends to straighten
out. In this case, the tension is exerted on the blades by the hub pulling
inward and the ends of the blades trying to go in straight lines. With
inertia causing the blade tips to pull the ends of the blades away from
the hub and the hub pulling the other ends of the blades toward the hub,
the blades are under enormous tension and they pull straight.
Problem 3: Chapter 5, Case 2a-e (Pg. 247)
Modern Olympic athletes are aided by a variety of scientific advances. Here are a few examples of how science enhances performance in athletics.
a. A ski-jumper is suspended in a wind-tunnel, where a device measures
the horizontal and vertical forces on him as air flows past him horizontally.
The jumper arranges his body to obtain the most favorable forces for a
long flight. Describe the most favorable horizontal and vertical forces.
Answer: The ski-jumper wants the least possible drag (or least backward
horizontal force) and the greatest possible lift (or most upward vertical
force).
Why: The ski-jumper flies through the air. By minimizing the slowing effects
of (mostly horizontal) drag and maximizing the flight-prolonging effects
of (mostly vertical) lift, the ski-jumper can extend this flight to great
distances down the hill.
b. In cross-country skiing, the skier slides forward first on one ski
and then on the other. The ski bottoms are coated with special waxes so
that they exhibit large static frictional forces on snow but small sliding
frictional forces. How does this difference in frictional forces allow
the skier to use a stationary ski to push herself forward on a sliding
ski?
Answer: The stationary ski experiences a stronger forward force of static
friction while the sliding ski experiences a weaker backward force of sliding
friction. The net force on the skier is thus forward and she accelerates
forward.
Why: To accelerate forward, the skier needs a net forward force. If both
skies experienced equal forces but in opposite directions, then she wouldn't
accelerate at all. By experiencing a strong forward force on one ski and
a weaker backward force on the other, she is able to accelerate forward.
c. Modern tracks are often constructed out of recycled rubber tires.
The track surface is very resilient, deforming slightly when you push down
on it and then bouncing back when you stop pushing. How does this surface
improve runners’ times as compared to the dirt, gravel, or cinder tracks
of long ago?
Answer: They return energy to the runners' feet as the feet rise upward
after coming to a stop.
Why: Granular tracks cushion the impact of landing, but they extract energy
from the runners' feet (the feet push downward on the dirt, gravel, or
cinders and the dirt, gravel, or cinders move downward). This energy isn't
returned to the feet as they rise upward. A rubber track still cushions
the runners' feet, but it also returns energy to those feet as they rebound
upward.
d. Bicyclists now use aerodynamically designed bicycles and clothing
to minimize air drag. Bicyclists also "draft" one another, a
strategy in which one bicycle follows immediately behind another bicycle,
easing the rear bicycle’s passage through the air. If the front bicycle
were so aerodynamic that it experienced no air drag at all, how would this
affect the strategy of drafting and why?
Answer: Drafting would be of no value because the front bicycle won't
have any affect on the air it passed through.
Why: The first bicycle experiences drag because the air pushes back on
any object trying to move forward through it. Since forces come in equal
but oppositely directed pairs, the bicycle must push forward on the air.
This effect leaves the air moving forward on average. When the second bicycle
encounters the forward moving air, it experiences less than the usual drag
and has an easier time moving forward. If, however, the first bicycle experiences
zero drag, the air behind that bicycle doesn't experience any forward force
and is left unaffected by the first bicycle's passage. The second bicycle
experiences no effect from the first bicycle's passage.
e. During a downhill ski race, the racer tries to keep her skies in
contact with the surface at all times. Even if she remains in a tight,
aerodynamic tuck position, she slows down whenever she becomes airborne
after hitting a bump. Why does contact with the ground keep her moving
forward as she races downhill?
Answer: She only experiences the downhill ramp force when she's in contact
with the ground.
Why: The only forward force the skier experiences in the downhill ramp
force that comes from her contact with the ground. This force is the result
of her downward weight and the forward-tilted support force that the mountain
slope exerts on her (it's at right angles to the slope, so it's upward
and forward). When she is in the air, the only force she experiences along
the slope is the backward force of drag. When when she is on the ground,
she also experiences the small backward force of frictional force and this
large forward ramp force.
Problem 4: Chapter 5, Case 8a-e (Pg. 248)
Fire extinguishers that use carbon dioxide gas can be used to fight all kinds of fires because carbon dioxide displaces air, doesn’t support combustion, settles downward on surfaces, and is non-toxic.
a. Each carbon dioxide molecule is heavier than an average air molecule.
Why does carbon dioxide gas sink in air?
Answer: It is more dense than air when the two are at the same temperature
and pressure.
Why: A carbon dioxide molecule weights more than the average air molecule.
As a result, carbon dioxide gas that's at the same temperature and pressure
as the air it displaces weighs more than the air it displaces and sinks.
b. The pressure inside the bottle of the extinguisher is so high—about
60 times atmospheric pressure—that most of the carbon dioxide inside it
is liquid. This liquid turns to gas as you use the fire extinguisher. If
you were to release the gas directly into the air through a narrow opening,
why would the gas suddenly acquire a large forward velocity?
Answer: The air would accelerate toward the lower pressure(, so that
it's velocity would increase in the forward direction).
Why: Gases and other fluids accelerate toward lower pressure. In this case,
the lower pressure is on the far side of the narow opening. The carbon
dioxide in the extinguisher accelerates through the narrow opening and
leaves with a high forward velocity.
c. If you were to hold the extinguisher when the gas rushed out its
nozzle, you would feel a strong backward force. Explain the origin of this
force.
Answer: The fire extinguisher pushes the gas forward to make that gas
accelerate out its nozzle and the gas pushes back on the fire extinguisher.
Why: Since the gas ends up with a large forward velocity, the fire extinguish
must have pushed it forward. It, in turn, must have pushed backward on
the fire extinguisher.
d. To reduce the force noted in c, a carbon dioxide fire extinguisher
has a deflecting surface directly in the path of the carbon dioxide as
it leaves the bottle. The carbon dioxide hits this surface and turns toward
the right or left. Why does this surface allow you to use the fire extinguisher
without being pushed backward?
Answer: Half the gas leaves the fire extinguisher heading to the right
and half to the left. This gas has zero overall momentum. Since the overall
momentum of the gas starts at zero and ends at zero, there has been no
transfer of momentum between the fire extinguisher and the gas. They haven't
pushed on one another on average!
Why: While the fire extinguish may move the gas around through its nozzle
and deflecting plate, it eventually sends half of the gas in each of two
opposite directions. These two streams of gas end up pushing against one
another equally and there is no net force exerted on the fire extinguisher
itself.
e. The fire extinguisher collects the swirling gas after it has been
deflected by the surface and directs that gas toward the fire. Because
the gas travels slowly after hitting the deflecting surface, it carries
much less momentum and kinetic energy. What has become of the energy that
was stored in the compressed carbon dioxide before it left the extinguisher?
Answer: It has become thermal energy.
Why: By allowing the gas to swirl around in turbulent flow and rub against
itself in the process, the fire extinguisher has converted the gas's total
energy (pressure potential, kinetic, and gravitational energies) into thermal
energy.
Problem 5: Chapter 6, Case 6a-d (Pg. 292)
A typical electric oven has two separate heating elements: one on top and one on the bottom. The bottom element is used for baking while the top element is used to broil foods.
a. When only the bottom element is active and glowing red hot, what
heat transfer mechanism carries most of the heat to the food in the oven?
Answer: Convection (or Radiation--they're both active).
Why: Convection proceeds strongy as heated air rises from the hot element.Radiation
is also strong, although the surface character of the cooking pots determines
how effective it is.
b. When only the top element is on, what heat transfer mechanism carries
most of the heat to the food?
Answer: Radiation.
Why: Convection doesn't work when air is heated at the top. Only radiation
carries heat from the top element to the food beneath it.
c. How does wrapping the food in shiny aluminum foil affect the rate
of heat transfer to the food in these two cases?
Answer: The aluminum foil has little effect on convective heating, but
it dramatically reduces radiative heating.
Why: Heat flows easily through aluminum foil so that its presence on the
food doesn't slow convective heating much. But the aluminum foil reflects
thermal radiation pretty well so that radiative heat transfer is very weak.
d. A convection oven has a fan that circulates air in the oven. How
does the fan affect the rate of heat transfer to the food?
Answer: The forced circulation of air speeds up convective heat transfer
dramatically.
Why: Rather than waiting for hot air to rise and carry heat to the food,
a convection oven uses a fan to stir the air rapidly. The heated air carries
its thermal energy quickly from heating element to food and the heat transfer
is quite rapid.