Download Notes for Class Meeting 5: Energy

Notes for Class Meeting 5: Energy
Introduction
Why are we studying energy in a course on time? There are a couple of reasons:
(1) There is a connection between time and energy in both relativity and quantum
mechanics. (There is also one in classical physics, but it is subtler and we will not study
it.)
(2) Energy will be important in our understanding of the direction of time.
Why Do We Need Energy?
When we examined the conservation of momentum, we discovered that it was not
sufficient to determine the outcome of events. For example, in the demonstration of
Newton’s cradle, we saw that when one ball is pulled back and released, it causes one
ball to be ejected on the other side with the same speed. However, the conservation of
momentum would be also be satisfied if two balls were ejected, each with half the speed
of the incident ball.
Energy will be the quantity that fills this void. Like momentum, it is always
conserved, but unlike momentum it is a scalar, rather than a vector quantity. In other
words, it is simply a number.
I have mentioned that force is not an intuitive concept because it does not cost
anything. An inanimate object can create a force and a small force can be magnified to a
large force. We will see that energy will give force a cost.
Why Did Newton Not Discuss Energy?
Newton does not mention energy in the Principia. The reason is that energy is only a
useful concept if it is a conserved quantity. However, to understand the conservation of
energy, one has to take into account all of the different forms of energy, and these were
not adequately understood until about 150 years after the Principia was published.
Forms of Energy
At the most fundamental level there are two types of energy, kinetic and potential.
Kinetic energy is the energy stored in the motion of particles. However, it will take two
forms that are quite different in their appearance and consequences. One form is the
coherent motion of particles, which is the motion of a large number of atoms moving in
the same direction, such as in a moving ball. The other form is a hidden form caused by
the incoherent motion of atoms. We normally call the first form “kinetic energy” and the
second form “thermal energy” or “heat.”1
1
The colloquial use of the word “heat” to mean “thermal energy” is technically incorrect. Heat technically
refers to the transfer of thermal energy. March uses the “word” heat in the colloquial sense and I am happy
to do the same. The context will make it clear what is being referred to.
The second type of energy, potential energy, arises from energy stored by a force.
We will examine gravitational potential energy. The other common, but more
complicated, form of potential energy is that stored by electromagnetic forces. It can take
several forms, which vary from the energy stored by a compressed spring to energy
released in a chemical reaction.
Gravitational Potential Energy
If a set of objects are moving at constant velocity in the presence of gravity, then
there is no net force on them and we can assume that potential energy is being conserved.
By doing a number of experiments (which we will demonstrate in class) in which we
observe how these motions occur, we can deduce the nature of gravitational potential
energy. We will consider three simple experiments.
The first is a block and tackle as shown to the right. Weights are
attached to massless and frictionless pulleys as shown.2 The single
weight to the left can move up while the double weights to the right
move down, but the double weights move only half as far as the
single weight. If we designate the vertical position by y, then a
conserved quantity is the force of gravity times the vertical position,
m1gy1 + m2 gy2 .
A second example is a seesaw shown below. Three masses can
balance one mass if the distance from the pivots is one to three. Then the three masses
will move one third as far as the single mass. Again, the conserved quantity is
m1gy1 + m2 gy2 .
The final example is a massless, frictionless cart on an inclined plane of 30 degrees.
The cart holds two weights and it is connected to a single weight through a massless,
frictionless pulley. As the single weight falls a given
distance, the double weights in the cart rise vertically
half that distance. Again, the conserved quantity is
m1gy1 + m2 gy2 .
These examples lead us to conclude that the
gravitational potential energy can be defined as
2
My request for such pulleys from the demonstration group has not been fully met, making the
demonstrations less than perfect.
U = mgy
(5.1)
Kinetic Energy
We can now deduce the formula for kinetic energy by requiring the sum of
gravitational and potential energy to be conserved in some simple situation such as the
dropping of a ball over a distance d.
Initially, the ball has potential energy of mgd , but no
v=0
kinetic energy. After it has fallen a distance d, all of its
potential energy has been converted to kinetic energy, K.
d
2
1
Thus, K = mgd. From Eqs. 2.4 and 2.5, d = 2 gt and
v = gt
v = gt ⇒ t = v / g . Substituting these values into the
expression for K, we obtain
 1 v2  1 2
 1 2
(5.2)
K = mg  gt  = mg  g 2  = mv .
2

2 g  2
This expression for K, K = 12 mv 2 , must be correct for all cases of motion, not just the
case of a dropping ball. And our conservation law now reads that U + K is conserved.
Work
These investigations allow us to now attribute a “cost” to a force. From the falling
ball example, we see that if the force acts in the direction of motion, it increases the
kinetic energy. If we allow the ball to bounce, the force of gravity will act in the opposite
direction of the motion, and the kinetic energy of the ball will decrease. In the case of
circular motion at constant speed, the force that keeps the object on the circular path
works at right angles to the velocity, and the kinetic energy does not change. We can
summarize this by
(5.3)
ΔK = Fd cosθ ,
where ΔK is the change in the kinetic energy, F is the magnitude of the force, d is the
distance through which it acts, and θ is the angle between the force and the velocity of
the object. The technical name for the right side of Eq. 5.3 is “work.”
Friction
You are all familiar with the force of friction. If you push a block of wood across a
table at constant velocity, you have to apply a force in the direction of the velocity. Since
the block is not accelerating, Newton’s second law tells us that there can be no net force
on the block. Therefore there must be an equal force on the block acting in the opposite
direction. This is the force of friction (and it always acts so as to oppose the motion).
Eq. 5.3 says that you have done work on the block and thus you should have increased its
kinetic energy. Yet its velocity is unchanged, so it does not appear that its kinetic energy
has changed. However, in at least some sense it has. What has happened is that the
molecules in the wooden block have hit against the molecules in the table and caused
both sets of molecules to vibrate more vigorously. The momentum of the block does not
change since the motion of the molecules is random and cancels in a vector quantity like
momentum. However, they all add in a scalar quantity, like energy and maintain the
conservation of energy. We call this form of energy “thermal energy” and it is measured
by the temperature of objects. For our purposes, we can consider the temperature as a
measure of the average kinetic energy of the molecules or the atoms of a material.3
Consider another example. Suppose a 2 kg ball of putty traveling 10 m/s collides
with a stationary 3 kg ball of putty. The balls stick together, and by conservation of
momentum, continue as a 5 kg ball of putty traveling 4 k m/s. (There is 20 kg m/s of
momentum both before and after the collision.)
10 m/s
4 m/s
2 kg
3 kg
5 kg
The kinetic energy before the collision is
1 2 1
kg m 2
mv = (2 kg)(10 m/s)2 = 100
2
2
s2
The kinetic energy after the collision is
K=
1 2 1
kg m 2
2
K = mv = (5 kg)(4 m/s) = 40
2
2
s2
2
4
Where have the missing 60 kg m/s gone?
(5.4)
(5.5)
James Joule answered this question in 1842 by showing that missing mechanical energy
turns up as heat. A kilocalorie (kcal) is the amount of heat required to raise the
temperature of 1 kg of water 1o C. Joule showed that
kg m 2
1 cal = 4.184
= 4.184 J ,
s2
where J stands for joule, the appropriately named unit of energy in the SI system.5
It was this demonstration of the equivalence of mechanical and thermal energy that made
the conservation of energy a useful concept.
Binding Energy
We will consider an example of binding energy by considering a satellite orbiting the
earth in a circle. The satellite will have both kinetic energy and gravitational potential
3
Technically, it is somewhat more complicated, but that need not concern us.
A collision in which kinetic energy is conserved is call “elastic” and one, like this one, in which kinetic
energy is not conserved, is call “inelastic.” The action of steel balls in Newton’s cradle are almost elastic,
which is why the same number of balls get ejected as are incident.
5
Warning: The “calorie” used as a measure of energy in food is actually a kilocalorie and is usually written
Calorie (with a capital C). Next time you are tempted to eat a doughnut, remember that it contains a
million joules of energy!
4
energy. Its total energy, the sum of the kinetic and potential energy, will be its binding
energy. We want to calculate this energy and understand why it is called a binding
energy.
The first thing to note is that the zero of an energy is arbitrary since it is only
differences in energies that will be of significance. Thus we can choose any convenient
point to be the zero of the gravitational potential energy. For systems like this one, the
convention is to choose the gravitational potential energy to be zero when the satellite is
infinitely far from the earth. If we think of releasing an object initially at rest and
infinitely far from the earth, it will fall toward the earth gaining kinetic energy. Since the
sum of its kinetic and potential energy must be conserved and was initially declared to be
zero, its potential energy must be negative at any finite distance from the earth.
It takes a bit of elementary calculus to calculate the formula for the potential energy of
the satellite. For those who are interested, the derivation can be found in the Appendix at
the end of this note. Others may simply accept the formula, which also appears in March
on page 56, (4th ed, footnote on p. 62)
Gme ms
U =−
,
(5.6)
r
where me is the mass of the earth, ms is the mass of the satellite, and r is distance
between the satellite and the center of the earth.
Now we can calculate the satellite’s total energy (the binding energy). In its stable
circular orbit at a distance R from the center of the earth, it is being accelerated toward
the earth with a force (Eq. 3.2),
m v2
F= s
(5.7).
R
and that force is just the force of gravitational attraction between the earth and the
satellite,
Gme ms
F=
.
(5.8)
R2
Equating these two forces and solving for v 2 , we obtain,
ms v 2 Gme ms
Gme
=
⇒ v2 =
.
(5.9)
2
R
R
R
Substituting Eq. 5.9 into the expression for the binding energy, Eb , we obtain,
Gme ms 1
Gme ms 1 Gme ms
1 Gme ms
+ ms v 2 = −
+
=−
.
(5.10)
R
2
R
2 R
2 R
We see from Eq. 5.10 that the satellite’s binding energy is negative and its magnitude is
half its potential energy. Thus, the satellite is bound to earth since it will require at least
Eb worth of energy to allow it to escape the earth’s gravity, i.e., get infinitely far from
the earth.
Eb = U + K = −
Energy Use in the United States
The figure below shows the energy use by source of energy and use of energy in the
United States for the year 2014. It will serve as a useful starting point for any exploration
of energy and environmental policy. This plot comes from https://flowcharts.llnl.gov/.
The units of this figure are “quads,” which is shorthand for a quadrillion (1015) British
Thermal Units (BTUs). 1 quad = 1.055 × 1018 joules or 2.93 × 1011 kilowatt-hours. 1
quad is also approximately equal to 8 × 109 gallons of oil, 4 × 107 tons of coal, or
9.7 × 1011 cubic feet of natural gas.
Total energy production was up 3.3% from 2012. Although they are quite small, the
three biggest percentage gainers from 2012 were solar, up 82%, wind, up 27%, and
biomass, up 11%.
You will notice that most of the energy used to produce electricity is not used (called
“rejected” in this plot. This is even more so for petroleum used in cars and trucks. We
will discover the reason for this when we discuss the direction of time.
Appendix: Derivation of Eq. 5.6
A body at rest falling from infinity to some distance r from the center of the earth will
gain kinetic energy equal to the work done on it by the gravitational force. Since it starts
with zero energy and energy is conserved, its potential energy at r must be the negative of
the work. Thus, we need only to integrate the work from infinity to r,
r
r
r
 Gme ms 
Gme ms
Gme ms
.
(5.6)
U = − ∫ Fdr ′ = − ∫  −
dr ′ = −
=−

2
r′ ∞
r
r′ 
∞
∞
Note that the force is taken to be negative because it points in the negative direction, i.e.,
from large r ′ to small r ′ .