Let’s pick up where we left off last time…..the topic was gravitational potential
energy
Now, let’s talk about a second form of energy
Potential energy
Imagine you are standing on top of half dome in
Yosemite valley, holding a rock in your hand.
The rock has no kinetic energy, but if you threw
it off the cliff it would have quite a bit of kinetic
energy by the time it hit the valley floor.
We say that the rock has potential energy. If m
is the mass of the rock and h the height above
ground, the potential energy of the rock is…
PE = mgh
Physics 190E: Energy & Society
Fall 2007
What is g here?
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Recall also the reading assignment….
Reading assignment in
textbook - chapter 3 - work,
energy & power
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g is known as the gravitational constant. It measures the strength of
the Earth’s gravitational pull on falling objects.
Galileo demonstrated that all objects fall the same way.
If two objects are dropped from the same height at the
same time, then they will hit the ground at the same time
(as long as other forces like air resistance are negligible).
Falling objects accelerate downwards at a rate of …
g = 9.8m / s 2
Acceleration is the rate of change of velocity with time.
So, the units of acceleration are the units for velocity
divided by another factor of time.
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More on acceleration & related physics…
2007 Ferrari F430
Weight: 3196 lb (1450 kg)
Acceleration: 0-62 mph in 4.0s
Top Speed:>196 mph (>315 km/h)
Fuel Economy city/highway 11/16 mpg
Let’s calculate its acceleration in meters/(second)2
Physics 190E: Energy & Society
Fall 2007
2007 Toyota Prius
0-60 mph in 10s
60mpg(city), 50mpg(hway)
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Basic physics result - if an object starts at rest at time t=0 and accelerates with a
constant acceleration, its velocity increases linearly with time….
v = a! t
acceleration
If we want to figure out the acceleration, we can rewrite this as
a = v /t
The car accelerates, reaching a velocity of v=62 mph = 28 m/s in t=4 s,
which gives
!1
!2
a = (28ms ) /(4s) = 7ms
Physics 190E: Energy & Society
Fall 2007
A little bit smaller than the
gravitational acceleration of
g=9.8m/s2
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While we’re talking about acceleration, let’s introduce another piece
of basic physics … Newton’s 2nd law.
F = m! a
Force equals mass times acceleration. If there is a net
force on an object, it will accelerate. Conversely, if
something is accelerating, there must be a force on it.
Back to gravity…the gravitational force (at the
earth’s surface) is
F = m! g
Setting these two expressions equal, we see that the mass cancels giving
a=g
independent of the mass of the object.
Physics 190E: Energy & Society
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This is your weight. The force
that a scale pushes up on your feet
with to counterbalance gravity.
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The fact that the masses are the same in these
two equations has very deep significance in
physics. This “equivalence principle” led
Einstein to his theory of gravity - general
relativity - in which the gravitational force is a
manifestation of the curvature of spacetime.
The mass in Newton’s 2nd law (F=ma) is
known as the inertial mass, while the mass
in the gravitational force law (F=mg) is
known as the gravitational mass.
The equivalence principle has been demonstrated
experimentally to one part in a trillion…
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Finally, back to gravitational potential energy
We can check that potential energy indeed has the units of energy…
PE = mgh
If the mass is measured in kilograms and the height in meters
then the units of potential energy work out to be…
"2
"2
kg ! (ms ) ! m = kg ! m ! s = Joules
2
Recall these units came out naturally from
the formula for Kinetic energy 1/2 mv2
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We can also check that falling objects satisfy conservation of energy.
If we drop something from a height D at time t=0, then it’s position and
velocities as functions of time are given by
1 2
h(t) = D ! gt
2
v(t) = !gt
Now, let’s calculate the total energy as a function of time.
1
E = KE + PE = mv(t) 2 + mgh(t)
2
The result is actually independent of time and equal to the initial potential energy,
demonstrating conservation of energy.
1
1 2
2
E = (!gt) + mg(D ! gt ) = mgD
2
2
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A practical application of gravitational potential energy …… How to store
energy without a battery?
We’ll see that one problem with electricity is that it’s difficult to
store. Batteries are only practical for relatively small amounts of
energy. How do you store more massive quantities? One way
is to use it to lift up water and convert the electrical energy to
gravitational potential energy. This is called pumped storage
hydroelectricity.
The Northfield Mountain pumped storage hydroelectric plant - operated by First
Light Power Resources - is located in Northfield, MA about 20 minutes north of
campus (up route 63).
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The 1080 MegaWatt plant at
Northfield Mountain facility
opened in 1972 and was the
largest in the world at that time.
During periods of low demand,
water is pumped 5.5 miles from
the Connecticut river to a 300
acre reservoir, 800 feet above the
river, which holds 5.6 billion
gallons of water.
In generating mode, water flows
downhill through 4 turbine
generators at a rate of 20,000
gallons per second.
Possible paper topic
Info from http://en.wikipedia.org/wiki/Northfield_Mountain
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Thermal Energy
So far, we’ve talked about two forms of energy - kinetic energy
and gravitational potential energy. Now, we’ll introduce a 3rd thermal energy. We will try to understand what it means for
something to be hot and how much energy it takes to heat
something up? We’ll see that for a gas, like the air in this room,
thermal energy is just the sum of the kinetic energies of the
individual gas molecules.
Read Chapter 7
Understanding the mechanical equivalence
of heat - that mechanical energy could be
transformed into heat and vice-versa - was a
major achievement of 19th century physics.
This effort was closely tied to the industrial
revolution and the need to understand how
things like steam engines (which convert heat
into mechanical energy) work …
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A good way to start into this subject is to talk
about the amount of energy it takes to heat
something up?
One way to talk about this is just to give it a name …..
The British Thermal Unit (or BTU) is defined as
1 BTU = amount of energy required to raise the
temperature of 1 pound of water by 1 degree
farenheit.
We already have another unit of energy - the
Joule. We need to know how many Joules
does a BTU correspond to? This is a question
for experimentalists?
Physics 190E: Energy & Society
Fall 2007
Frigidaire 6000 BTU Air Conditioner
Really this means BTU/hour - a
measure of the cooling capacity of the
air conditioner
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This was a question that interested James Joule, himself….
Of course, at the time the mechanical unit of energy in use was
not the Joule.
It was the foot-pound. Before coming back to Joule’s work,
let’s take yet another detour into units and talk about the footpound as a measure of energy…. This will allow us to bring up
another important point about energy.
James Joule, 1818-1889
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The usual definition of energy given in introductory physics textbooks
is ….
energy = capacity to do work
Of course, to complete the definition we need to ask what physicists
mean by work?
If you sit in the library reading a book for a
course, are you doing work in the physics
sense?
No
In physics work means very specifically exerting a force through a distance, with
the direction of motion in the same direction as the force.
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This part about directions is important ….
An elevator does work when it takes us between
floors, because the force it exerts is in the same
direction as its motion - up.
However, if someone is walking along carrying
something, they are not doing any work (at least on
the object they are carrying) in the physics sense there is no force in the direction of motion. The
force is upwards, while the direction of motion is
forwards.
This makes sense because in the first case, the elevator goes up and the work it does
increases the potential energy of itself and whoever is inside. However, in carrying
water, the water is always staying at the same height.
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So long as the force and motion are in the same direction, the formula
for work is
W = (Force)(Distance) = F D
The foot-pound combines a unit of force - a pound -with a unit of
distance - a foot - and is thereby a unit of work or energy. 1 foot-pound
is the amount of work that must be done to raise a 1 pound weight by 1
foot. This also gives the change in potential energy of the 1 pound
weight.
Note: Pounds are used both as a measure of force and of mass, which
can be confusing. A pound-mass is the amount of mass that weighs 1
pound on the surface of the Earth. However, on the surface of the moon
it would weigh something less than a pound…. When we make a
conversion 1 pound = 2.2 kilograms, we are really talking about the
pound-mass.
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Yet another unit…..the SI unit for force is called the Newton.
1Newton = 1N = 1(ki log ram)(meter) / (sec ond)2
This makes sense, based on the equation
F = m! a
The unit of force is the unit of mass times the unit of acceleration.
Let’s check that work has the same dimensions as energy. Work equals force
times distance. So a unit of work is a Newton-meter.
1 Newton-meter = 1 (kg m s-2) m = 1 kg m2/s2=1 Joule
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Back to Joule and the mechanical
equivalent of heat
Joule built an apparatus in which
water was heated by mechanical
agitation. He could measure both
the temperature change in the
water and the amount of work
done by the agitator.
Recall that 1 BTU is the amount of energy needed to raise the temperature
of a pound of water by 1 degree farenheit. Joule found that
1 BTU = 773 foot-pounds.
The modern measurement is 1 BTU = 778.3 foot pounds. So Joule’s
measurement was quite good.
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Let’s focus on this relation between heat and mechanical work
1 BTU = 778.3 foot-pounds
and note that 1 BTU is approximately the amount of energy
released by burning a match. Burning releases the stored
chemical energy in the wood.
We see that this same amount of energy with lift a 1 pound weight nearly 800 feet
in the air, or equivalently a 100 pound weight up to a height of 8 feet.
It is quite remarkable that the chemical energy stored in
such a small piece of material could accomplish such a feat!
Indeed, the fact that burning fossil fuels yields quite
useful amounts of mechanical energy is what made the
industrial revolution possible….. We’ll return to this
in much more detail later.
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Not only are the capacities of refrigerators usually stated in BTU’s. Total
annual world energy usage is often stated in terms of “Quads”.
1 Quad = 1 quadrillion BTU = 1 x 1015 BTU
In 2004, total world energy
consumption was 447 Quads…
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Back to thermal energy
Thanks to Joule we can measure the amount of energy that it takes to heat
something up. Can we also understand the nature of the energy contained in a
hot object via the basic laws of physics? What is thermal energy?
Matter comes in 3 basic phases solid, liquid & gas. The easiest to
understand are gases and that’s
where we’ll start.
In the case of simple gases, there is a
simple formula relating the thermal
energy of the gas to its temperature.
Solids and liquids are more complicated
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We’ll consider a gas that’s made up of single
atoms in a container. The atoms travel around
in straight lines colliding occasionally with
each other and with the walls of the container.
The number of atoms in a gas is immense
(approximately 3x1022 in a liter container at room
temperature and pressure).
The pressure of a gas comes from collisions of the atoms
with the walls of the container. The faster the atoms in the
gas are moving, the higher the pressure.
The speed of the atoms is in turn related to
temperature.
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The average kinetic energy of gas atoms is …
1 2 3
mv = k B T
2
2
m = mass of atoms
T = temperature
where kB is known as Boltzmann’s constant and is given by
kB = 1.4 !10"23 J /K
and temperature is measured using degrees Kelvin.
(This is the K in the units of Boltzmann’s constant)
The total energy of the gas is just the sum for all the gas atoms.
3
E gas = NkB T
2
Physics 190E: Energy & Society
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N = number of gas atoms
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8 quantitative problems
1 short answer problem
To be handed in in class
Be sure to show your
work on the quantitative
problems. Don’t just
write down the answer.
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