Physics 1110 - Energy On This World and Elsewhere - Fall 2013
Problem Set #2 with solutions
When doing unit conversions, for full credit, you must explicitly show how units cancel.
Also, you may need to look up certain equivalence relations on the internet. Show your
work!
Useful information
1 gallon of gasoline = 1.3 × 108 J
1, 054 J = 1 BTU
1. With regard to finding and producing the necessary energy, why does the transportation sector (as opposed to other sectors) present a particular challenge to our society?
Whereas we can use many different primary energy sources for needs such as electricity
and heating, oil is, at present, the only option that is simultaneously practical and cost
efficient, given our current infrastructure. Additional comments: Natural gas is not an
unreasonable alternative, and is becoming increasingly prevalent, but its widespread use
would require considerable changes in infrastructure. Natural gas also requires considerably
more volume for storage on the vehicle. Biofuels, at least in the United States, are not
yet sufficiently scalable to replace oil, and have other issues as well. Both hybrid and and
fully-electric vehicles are also not yet in a position to reduce our reliance on oil, although
like biofuels, they can certainly help.
2. A truck weighing 2000 kg is climbing a hill with a height of 8 meters.
h = 8m
a. What is the change in the gravitational potential energy of the truck as it goes
from the bottom of the hill to the top of the hill? Express your answer in both
Joules and BTU’s.
m g h = (2000 kg)(9.81 m/s2 )(8 m) = 1.57 × 105 Joules
1 BTU
5
= 1.57 × 10 J
= 148.8 BTU0 s
0
1055 J s
b. Assuming that the truck’s engine is 100% efficient in converting gasoline into the
mechanical work that it takes to drive up the hill, how much gasoline is needed
(in gallons) to climb the hill? In other words, what is the “gasoline equivalent”
of your answer to part “a” in gallons of gasoline?
1 gallon gasoline
0
148.8 BTU s
= 0.0012 gallons
1.24 × 105 BTU0 s
c. Assuming that the truck takes 2.5 seconds to climb the hill, what is the power
output of the engine? Give your answer both in Watts and in horsepower.
Power =
energy
1.57 × 105 J
=
= 62, 800 Watts = 62.8 kW
time
2.5 seconds
Physics 1110 - Energy On This World and Elsewhere - Fall 2013
Problem Set #2 with solutions – continued
3. A household furnace has an output of 100,000 Btu/hour. What size electrical heating
unit (in kW) would be needed to replace it? Use the “substitution method” described
in the class notes, and find an appropriate unit to substitute for both “Btu’s” and
“hour”. You might find it most convenient to first convert to Watts (joules/sec), and
then to kW.
1, 055 J
BTU0 s
= 100, 000
= 29, 306 W = 29.3 kW
100, 000
hour
3600 seconds
4. A particular household draws an average of 2.5 kilowatts of electrical power from the
electrical grid. How many joules does the house use in one day?
Energy = Power × Time =
2.5 × 10
3J
s
×
60 s 60 min 24 hrs
1 day
1 min hr
day
= 2.16 × 108 J
5. Why is the United States particularly well situated to take advantage of renewable
energy sources? (Hint: this issue is specifically discussed in one of the readings.)
Renewable resources, such as wind and solar, have low energy density. In this
context, this means that a large amount of land area is needed to “harvest” meaningful amounts of energy at the utility scale. The population density of the United
States is significantly lower than is the case in many places in the world, including
Europe, Japan and China.
6. What is the relationship between brownian motion and the internal energy of gases.
Brownian motion refers to the random motion that can be observed in microscopic
particles of dust or smoke that are suspended in a gas. It is due to the collisions
these microscopic particles have with the atoms and molecules of the gas, which
possess the random kinetic energy that we refer to as internal energy. Brownian motion is thus visible evidence (albeit requiring a microscope) of the internal
energy of gases.
7. When the price of oil went up significantly in the early 1970’s, power companies were
able to shift away from generating electricity using oil relatively rapidly. What were
some of the factors that contributed to their ability to respond in a timely manner?
The largest factor is that electricity can be produced using many different primary
energy sources. Closely related to this is the fact that the particular mix of primary
energy sources used to produce electricity is invisible to the consumer. Thus, as the
price of oil went up, utility companies quickly tried to shift their production away
from oil, and except possibly for a rise in the price of electricity, the consumer was
not directly affected. Another contributing factor is that certain types of power
plants were not difficult to convert from oil to coal.
Physics 1110 - Energy On This World and Elsewhere - Fall 2013
Problem Set #2 with solutions – continued
8. Convert the following quantities into the indicated units. In all cases show your work,
being certain to show explicitly that your units cancel in the conversion to give the
desired final units.
a. What is the equivalent of 570 kW − hrs in BTUs?
60 s 60 min
1 hr = 2.052 × 109 J
570 kW hrs =
1 min 1 hr
1 BTU
9
= 1.947 × 106 BTUs
= 2.052 × 10 J
1054 J
J
570 × 10
s
3
I note here that the “equivalence relation” that I provided at the top of this homework, that 1054 J = 1 BTU, is incorrect. The correct relation is 1055 J = 1 BTU.
Using this, the answer comes out to be 1.945 × 106 BTUs
b. What is 50 m/s in miles per hour?
If we want to use the substitution method, we need a relationship of the form
1 m = X miles. Using Google, I just found that 1 m = 6.214 × 10−4 mi. Perhaps
a more common “equivalence relationship” that you see around is that 1 mi =
1, 609.3 m. It is important to remember how to use something like this to get the
relationship we need. That would go as follows:
If 1 mi = 1, 609.3 m, then
1 mi
1, 609.3 m
=
, or 6.214 × 10−4 mi = 1 m
1, 609.3
1, 609.3
You can similarly find that 1 s = 2.778 × 10−4 hr. Okay, having considered that
aside, we get to the problem itself:
m
50
= 50
s
6.214 × 10−4 mi
2.778 × 10−4 hr
= 111.8 mph
c. How much gasoline is the equivalent of 15 kW-hrs?
We start by noting that
15 kW hrs = (15 × 103 J/s) × (3600 s) = 5.4 × 107 J .
From this, we find that:
5.4 × 107 J
= 0.42 gal
1.3 × 108 J/gal
It is interesting to note that the battery in a Chevy Volt holds roughly 16 kW hrs
of energy. If we think of this as the equivalent of around a half gallon of gasoline,
it doesn’t sound like much. We need to remind ourselves, however, that electric motors are around four to five times more efficient than a typical internal
combustion engine.
Physics 1110 - Energy On This World and Elsewhere - Fall 2013
Problem Set #2 with solutions – continued
9. Consider the diagram at right illustrating a heat
engine. Assume that the efficiency of the engine
20%, and that it performs 500 J of useful work during each cycle.
a. How much heat energy does the engine consume while doing the work?
Efficiency =
Work
QH
Hot reservoir, TH
QH
Rearranging this, we have:
work
Work
500 J
QH =
=
= 2, 500 J
Efficiency
0.20
b. How much heat, QC , is expelled during each
cycle.
From conservation of energy we know that:
QC
Cold reservoir, TC
QH = Work + QC
so it follows that:
QC = QH − Work = 2, 500 J − 500 J = 2, 000 J
c. If TH = 700 K and TC = 300 K, what is the
maximum efficiency allowed by the 2nd law of
thermodynamics? How does this engine’s efficiency compare with its theoretical maximum?
The maximum possible thermodynamic efficiency is given by the Carnot efficiency:
TC
300 K
εCarnot = 1 −
=1−
= 0.571 = 57.1%
TH
700 K
Physics 1110 - Energy On This World and Elsewhere - Fall 2013
Problem Set #2 with solutions – continued
10. Consider the seesaw below, which you can assume is in a balanced condition. Assume
the person sitting on the left (Mbig ) has a mass of 80 kg, and is sitting 1.0 meters
from the pivot, and the person sitting on the right (Msmall ) is sitting 2.5 m from the
pivot.
lshort
llong
msmall
mbig
a. What is the mass of the the person on the right (msmall )?
We know that
mbig
llong
=
msmall
lshort
from which it follows that
msmall = mbig
lshort
1m
= 32 kg
= 80 kg
llong
2.5 m
b. If the seesaw tips by some angle so that the larger person is closer to the ground,
what can you say about their change in gravitational potential energy with respect
to the change in gravitational potential energy of the smaller person.
When the seesaw is in a balanced condition, the change in potential energy of
the larger person will be exactly equal in magnitude, and opposite in sign, to the
change in potential energy of the smaller person. In short, the sum of the potential
energies of the two people will remain the same.
11. What are the two laws or physics principles that lead to the Carnot’s maximum
efficiency for a heat engine.
The first and second laws of thermodynamics. Equivalently, one could also say
the conservation of energy, and the fact that, during any thermodynamic process,
entropy always increases, or at best, stays the same.
Physics 1110 - Energy On This World and Elsewhere - Fall 2013
Problem Set #2 with solutions – continued
12. You and a friend take a drive in the Blue Ridge Mountains. When you return, you
park in the same space from which you initially left. What happened to the energy
in the gasoline you used? Be specific. For example, don’t just say “heat”.
Since the kinetic energy of the car and the gravitational energy of the car are
both the same before and after the trip, it is certainly the case that the energy in
the gasoline went to heat. But how? The engine in the car expelled waste heat
(in the exhaust), because a car engine is a heat engine. The mechanical energy
from the engine was delivered to the wheels, with frictional losses on the way,
and the wheels pushed the car forward, again with frictional losses. When the car
needed to slow down, such as going down the mountain, some of the car’s kinetic
energy was converted into heat by the brakes. Finally, the air resistance of the
car throughout the trip caused heating of the surrounding air.