- Importance of Units: The Gimli Glider:
HRW 5-69 (20 points)
Throughout this course, we will expect you to be careful
with units in your calculations. Yet some students tend to neglect
them and just trust that they always work out properly. Maybe this
real-world example will keep you from such a sloppy habit. The
incident is known in aviation as
the Gimli Glider
On July 23, 1983, Air Canada Flight 143 was being readied for its long
trip from Montreal to Edmonton. The fuel gauges on board were not
operational, and the flight crew asked the ground
crew to determine how much fuel was already on board. The flight crew
knew they needed to begin the trip with 22,300 kg of fuel. They knew
that amount in kilograms because Canada had recently switched to the
metric system; previously fuel had been measured in pounds. The ground
crew could measure the onboard fuel only in liters, which they
reported as 7,682 L. Thus, to determine how much fuel was on board and
how much additional fuel was needed, the flight crew asked the ground
crew for the conversion factor from liters to kilograms of fuel. The
response was 1.77, which the flight crew used (assuming that 1.77 kg
corresponds to 1 L).
(a) How many kilograms of fuel did the flight crew think they
had? (In this problem, take all given data as being exact.)
(b) How many liters did they ask to be added?
Unfortunately, the response from the ground crew was based on
pre-metric habits: 1.77 was the conversion factor not from liters to
kilograms but rather from liters to pounds of fuel (1.77 lb
corresponds to 1 L).
(c) How many kilograms of fuel were actually on
board, before the ground crew added some? (Except for the given 1.77,
use four significant figures for other conversion factors.)
(d) How
many liters of additional fuel were actually needed?
(e) When the
airplane left Montreal, what percentage of the required fuel did it
have?
On route to Edmonton, at an altitude of 41,000 feet, the Boeing
767 airplane ran out of fuel and became a $40M glider. Although the airplane
had no power, the
pilot managed to glide it toward an old airforce base in Gimli,
Manitoba.
Under the circumstances, the glide and the landing were impeccable.
Unfortunately, the runway at that airport had been converted to a
track for race cars, and a steel barrier had been constructed across
it. Fortunately, as the airplane hit the runway, the front landing
gear collapsed, dropping the nose of the airplane onto the runway. The
skidding slowed the airplane so that it stopped just short of the
steel barrier, with stunned race drivers and fans looking on. All on
board the airplane emerged safely. The point here is this: Take care
of the units.
- Champagne from United 97
(30 points)
This is another real life story; one in which yours truly was able to
gain an (unfair) advantage from his physics problem-solving skills.
United Airlines runs a little contest on every flight from the US
mainland to Hawaii. The passengers are asked to determine the time the
plane will reach the exact geographical midpoint of the journey
(e.g. the point exactly half distance between SFO and HNL airports). The
person who gets the closest answer wins a bottle of good French champagne
(typically reserved for those in First Class  ).
I won the bottle on my recent (and the only) trip to Honolulu; I was
off by 7 seconds (I do not mean to brag; as you may guess from the
calculation below, there is an uncertainty of a few minutes in the answer, and
hence some luck is involved). But let's see if you can reproduce the
calculation.
Shortly after takeoff, the flight attendant announced the game, and
gave the relevant (and some irrelevant) data:
- Total estimated flight time: 5 hrs 15 mins
- Departure (wheels up) time: 6:28pm PDT
- Flight distance SFO-HNL: 2128 nautical miles
- Cruising airspeed: 466 knots (nautical miles per
hour). This is the speed of the plane relative to the air.
- Estimated headwind: 33 knots for the first half of the journey,
40 knots for the second half.
We will split the calculation in a few parts:
(a) What is the ground speed of the airplane for the
first and second parts of the flight ? That is, how fast is the
airplane moving relative to the ground (or rather, ocean) ?
(b) If you assume that the airplane travels the entire distance
with the airspeed of 466 knots under the headwind conditions above,
how long would it take to fly between SFO and HNL ? Compare this
time to the flight time declared above: this tells you how good the
assumption is (and roughly how long it takes for takeoff and descent).
(c) Luckily for me, I noticed that we took off almost directly
westward, and spent about 13 minutes to reach the cruising
speed. What was the average acceleration and average ground speed
during that time ?
(d) Taking results from (c) into account and assuming that the
ground speed was constant after the first 13 minutes, how long did it
take to reach the midpoint of the journey ?
Good luck !
- Kickoff at the Oakland Coliseum
(30 points)
Sebastian Janikowski is kicking off for the Oakland Raiders. His
powerful left leg launches the ball from the 30-yard line with the
initial velocity of 30 m/s. Assuming no air
resistance, can the ball reach the end of the field, i.e. travel
at least 80 yards (73 m), resulting in touch-back ? Explain your answer
(i.e a simple yes or no answer is not
sufficient). Assuming Mr. Janikowski has picked an optimal angle
for the kick (he did graduate from college, after all), how long is
the ball in the air ? What is the highest elevation of the trajectory ?
Solutions to Problems 1-3
- Miracle above the Vatican
(30 points)
In the climax of the novel Angels and Demons by Dan Brown, the
hero, Robert Langdon, escapes the anti-matter bomb
explosion by jumping from a helicopter at about 10,000
feet with a makeshift parachute. He is saved by a 2x4 sq. yard
(approximately 8 m2) canopy, which he holds with his
bare hands. The canopy provides enough drag force so that Robert
could survive the fall into the Tiber river, recover, and
bring the bad guy to justice. I am not spoiling it for you too
much, am I ?
Here, we will analyze whether this miraculous escape is
plausible (never mind the radiation from the bomb).
(a)
If Robert's mass is 80 kg, the drag coefficient for the canopy
is C=1 and the air density is 1.2 kg/m3,
estimate Robert's terminal speed.
(b) Robert was an avid swimmer and diver, which helped
him survive the high-speed dip into Tiber. Ignoring air
resistance, what elevation would a human need to jump from to
hit the water with the same speed ?
(c)
What is the ratio of Robert's terminal speeds with and without
the canopy ? If Robert falls face-down, we can estimate that
his cross sectional area is about 1 m2. In the
beginning of the book, Robert learns from
a prominent physicist (director of CERN, the particle physics
lab in Geneva, Switzerland) that "One square yard of drag
reduces one's speed by 20%". How accurate is this number ?
(d)
If Robert reaches the terminal speed falling face-down without
the parachute first (i.e. with cross sectional area of 1
m2) and then spreads the canopy to cover 8
m2, what initial drag force would he experience ?
Would he be likely to hold on ?
- The Great Pyramid of Gizeh
(40 points)
The Great Pyramid (also known as Pyramid of Khufu or Cheops in
Greek) at Gizeh, Egypt, when first erected (it has since lost
a certain amount of its outermost layer) was about 150 m
high and had a square base of edge length 230 m. It is
effectively a solid block of stone of density about
2.5 g/cm3.
(a)
What is the minimum amount of work required to assemble
the pyramid, if the stone is initially at ground level?
(b)
Assume that a slave employed in the construction of
the pyramid had a food intake of about 1500 Cal/day (1 Cal =
4182 Joules). The Greek historian Herodotus reported that the
job took 100,000 slaves 20 years. How efficient were the
slaves (i.e. what is the ratio of work done toward
pyramid-building by energy consumed)?
(c)
The blocks were probably raised to the top of the pyramid
along the sides by a system of ropes, perhaps with a pulley
located at the top. The average mass of the block in the
pyramid is 2.5 tons. Suppose that to raise a block, the slaves
place it on a cart which reduces the coefficient of friction
between the block and the side of the pyramid to about
μ=0.1. Estimate the amount of
work needed to raise a single block to the top of the
pyramid. If each slave can pull with a force of 500 N (about
110 lbs), how many slaves are needed to raise a single block
without any mechanical advantage device (e.g. a
block-and-tackle) ?
Solutions to Problems 4-5
- NHL Slapshot
(20 points)
The hockey legend Bobby Hull could shoot the puck at the mind-boggling speed
of v=120 mph. The standard hockey puck is a disc of vulcanized rubber,
with the regulation weight of 6 oz (m=170 g). Imagine one of Hull's
pucks hits a 200-lb (M=90 kg) goaltender and gets absorbed in his
chestpad (the collision is fully inelastic).
(a)
If the goaltender is initially at rest and ignoring friction (the
goalies used to stand much more on their skates in Hull's days !),
what is the speed of the goaltender and the puck after the collision ?
(b)
How much kinetic energy is lost ? This energy presumably has gone into
deforming the goalie's gear (and warming it up in the process).
(c) Suppose the goalie's pad is 10 cm thick, and it's fully
compressed by the puck. What average force does the goalie experience ?
- Centripetal acceleration on Earth
(20 points)
Since Earth is rotating around its axis, it is not really an
inertial frame. That is, every object on the Earth
surface experiences centripetal acceleration,
which points toward the axis of rotation. At the Equator, the
centripetal acceleration, naturally, points directly downward (toward
the center of the Earth) and coincides in direction with the
gravitational acceleration. In very precise ballistics
calculations, we have to take that into account. (In weather
predictions, a similar effect, a fictitious Coriolis force,
plays a very important role).
(a) (5 pts) How
large is the centripetal acceleration at the Equator, compared to
acceleration of free fall g ? Circumference of the
Equator is approximately 40,000 km.
(b) (5 pts) How large is the centripetal acceleration
at the North Pole ?
(c) (10 pts) Now imagine two people: one at the North
Pole, and another at the Equator. Each drops a ball from the same
height h=1 m. Which ball would be falling longer ?
- Gymnast on a Horizontal Bar
(25 points)
A young gymnast is trying to learn a skill called the giants,
in which he would make a complete revolution around a Horizontal Bar
(an apparatus in men's artistic gymnastics). See animation below.
To aide the young gymnast, his coach tells him to kick at the
bottom of the swing (i.e. increase his angular velocity at the
bottom), and to push the bar at the top. Is the coach right ?
In the following, we will approximate the gymnast as a rigid solid
rod, with mass m=40 kg, and length (fingers to toes, with arms
stretched up, as in the rotating picture) L=1.5 m. As a
reminder, the moment of inertia of the solid rod rotating around its
end is I=mL2/3. Ignore friction between the hands and the
bar (chalk on the hands reduces that).
(a) (10 pts) As the gymnast rotates around the bar, his
angular velocity is not constant, since the gravity provides
torque, and changes the potential energy of his center of mass. Estimate
his minimal angular velocity at the bottom of the swing,
to barely make it around the top (i.e. so that he has zero
angular velocity at the top) ?
(b) (5 pts) What is the force the gymnast applies to the
bar on the top ? Is he pushing himself away from the bar, or pulling
toward it ?
(c) (10 pts) What is the force the gymnast applies to the
bar on the bottom ? Is he pushing himself away from the bar, or
pulling toward it ?
Solutions to Problems 6-8
- Practice problems for MT2 supplement
problems for this week. In particular, check out problems
on the rigid body equilibrium (#2b), angular momentum conservation
(#3), buoyancy (#4) and Bernoulli's law (#5).
Also see
solutions.
- A couple of oscillators
(25 points)
Here are two (unrelated) examples of unusual oscillators.
(a) (15 points) Imagine a straight tunnel bored through
the center of the Earth from one point on the surface (say, near
Berkeley, CA) to the other side (alas, this would be in the Indian
Ocean somewhere!). We'll assume that the tunnel is carefully built to
withstand large temperatures and pressures in the middle of the Earth,
and it is evacuated well enough to ignore air resistance inside the tunnel. If
you drop a rock into
this tunnel, it will experience a gravity force directed straight
toward the center of the Earth. Calculations show that the weight of
the rock inside the
tunnel depends linearly on the distance r from the center of
the Earth: W(r)=mgr/R where m is the mass of the rock,
g=9.8 m/s2, and
R=6400 km is the radius of the Earth. Prove that the motion of the
rock inside the tunnel is periodic, and find the period.
(b) (10 points) Now imagine a wooden cube, 10x10x10
cm3 in size. The density of wood is 500 kg/m3,
so this cube floats in water. Suppose that it floats with one of its
faces parallel to the water surface. You push this cube slightly
down, to submerge 3/4 of its volume in the water, and then let go. The
cube starts oscillating vertically around its equilibrium
position. What is the frequency and amplitude of this oscillation ?
(Unlike the previous question, you can do this experiment at home !)
The key to these problems is recognizing that they describe
motion of an object under the influence of restoring
force. If the force is proportional to displacement from the
equilibrium position (like in the spring), and is opposite in sign,
the result is the oscillating motion. This can be seen in many
physical systems. In fact, oscillations are probably the most common
type of motion you will see.
Solution (courtesy Eunhwa
Jeong). Notice that Eunhwa uses a special short-hand notation: x with
two dots on top. Don't be thrown off: it is just another symbol for
acceleration (or second derivative of position).
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