Download ENERGY CONSERVATION The Fisrt Law

A. La Rosa
Lecture Notes
PH 213
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ENERGY CONSERVATION
The Fisrt Law of Thermodynamics and the
Work/Kinetic-Energy Theorem
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ENERGY [This section taken from The Feynman Lectures Vol-1 Ch-4]
1. What is energy ?
2. Ideal machines and reversible processes
2.A Reversible process
2.B The principle of non-perpetual motion
2.C Comparing reversible and non-reversible machines.
2.D Universal behavior of reversible machines
3. Figuring out the potential energy
TRANSFER of ENERGY
Heat-transfer Q
Macroscopic external Work W ’ done on a system
ENERGY CONSERVATION LAW
The work/kinetic-energy theorem
Case: inelastic collision
Generalization of the work/kinetic-energy theorem
Fundamental Energy Conservation Law
Inelastic collision (revisited)
Case: Pure Thermodynamics
The First Law of Thermodynamics
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1
ENERGY [This section taken from The Feynman Lectures Vol-1 Ch-4]
1. What is energy ?
 There is a fact or law governing all natural phenomena that are known to
date, called the conservation of energy. It states that:
There is a certain quantity, which we call energy, that does
not change in the manifold changes which nature undergoes
It is not a description of a mechanism
It is a mathematical principle
It says that it is a numerical quantity which does not
change when something happens
A strange fact that we can calculate some numbers and
when we finish watching nature go through her tricks
and calculate the number again, it is the same
 Conservation of blocks
Little John plays with 28 indestructible blocks
At the beginning of the day, his mother puts him, with his blocks, into a room.
At the end of every day, she counts the blocks very carefully, and discovers a
law: No matter what little John does with the blocks, there are always 28
remaining.
One day she finds only 26. There is consternation. But when she looks through
the window, she find that 2 blocks lies on the garden.
Another day she finds 30! … Later she finds out that a friend came over to visit
John and left 2 block.
She starts enjoying the counting blocks game. But from now on she will close
the windows and will not allow extra blocks entering the room.
Everything goes all right …
Each block weights 20 grams
number of

blocks seen
(weight of the box) - 50 grams
 constant
20 grams
Original height of the buthtub is 40 cm. Each block raises the water by 2 cm
2
number of
blocks seen

(weight of the box) - 50 grams

20 grams
height of the water - 40 cm
 constant
2 cm
In the complexity of her world she finds a whole series of terms representing
ways of calculating how many blocks are in places where she is not allowed to
look. As a result she finds,
A complex formula, a quantitity that has to be computed, which
always stays the same.
What is the analogy of this to the conservation of energy?
The most remarkable aspect that must be abstracted from this picture
is that there are no block
The analogy has the following points:
When we are calculating the energy, sometimes it leaves the system
and goes away, or sometimes some comes in. To verify the
conservation, we have to track that we have not put any in nor taking
out.
The energy has a large number of different forms, and there is a
formula for each: gravitational, electrical, radiant energy, …
Quoting Feynman,“In physics, today, we have no knowledge of what energy is.”
2. Ideal machines and reversible processes
2.A Reversible process
m’
Consider a frictionless lever (an ideal machine indeed.)
Y
3
1 meter
1
0
Figure 1. A massless, friction free, weight lifting machine.
 We wish to lift a block of mass 3m.
3
g
For that purpose, the block of mass 3m is placed at the bottom on the right
side of the lever, and a mass m’ is placed at the left of the lever at a height
h= 1 m.
Here m’ is slightly smaller than m. [ m’ = m -  with   0 ]
Y
m’
Uinternal=  12m
3
1 meter
i
g
1
3m
0
Figure 2. Lever in its initial position.
Since m’ < m the system in Fig. 2 will remain stationary.
 The lever will be able to lift the block of mass 3m if some additional energy
K were added to the mass m’.
Y
K
m’
3
g
1 meter
1
3m
0
Figure 3. A little bit of kinetic energy is given to m’ so it can start lifting
the 3m block.
Let’s choose the magnitude of K such that , afterwards, m’ lowers down to
the ground level height=0 , and the lever becomes stationary. Thus the lever
machine has lifted the block 3m to a height X:
4
K
g
1 meter
3m
1
m’
3
X
0
Figure 4. The amount of kinetic energy given to m’ is just to make m’ to
go down to the bottom level (while lifting the 3m block.)
At this stage, we clamp the lever to, momentarily, avoid further motion.
K
 The motion could continue if we just unclamp the lever (this occur because m’
is slightly less than m, so a small torque acts on the lever).
When m’ reaches a height of 1 meter, it would have a kinetic energy K.
We can place a spring at the proper position such that , when the mass m’
reaches the height 1 m, the energy K gets stored into the spring. Right
after this occurs, we clamp the lever again.
Spring compressed a bit
Y
m’
3
g
1 meter
1
0
3m
X
Figure 5. Lever and masses m’ and 3m are back to their initial positions
(as in Figure 2).
 If we wanted to lift the 3m block again, all we need to do is to unclamp the
lever.
What we have, then, is a reversible process.
The work (energy) that the lever invested in lifting the
3m block by a distance X (Fig. 4) was at the expense of
lowering the mass m’ by 1 meter. By operating the
lever in reverse, mass m’ is lifted at the expense of
lowering the 3m block (Fig. 5).
5
Notice that for operating this reversible process,
we (the external agent) just need to clamp and
unclamp the lever during the process; i.e. we do
not input to, nor extract from, the machine any
energy.
(! )
2.B The principle of non-perpetual motion.
 Reversible and non-reversible machines
The machine described above is called a reversible machine. It is an ideal
machine (where friction is absent), which is in fact unattainable no matter
how careful our design. Its concept is however useful, for comparing it with
other non-reversible machines.
A non-reversible machine includes all real machines. They are subjected to
friction and other adverse factors that detriment their motion.
 Non-perpetual motion
Let’s now consider the following hypothesis:
There is not such a thing like perpetual motion.
(This is basically a general statement of the conservation of energy.)
For the case of lifting-weight machines:
If, when we have lifted and lowered a lot of weights and restored
the machine to the original condition, we find that the net result is
to have lifted a weight,
then we would have a perpetual motion machine,
(because we can use that lifted weight to run something else, which
can be repeated again and again.)
Accordingly, for the case of weight-lifting machines, in the absence
of perpetual motion, after bringing the machine to its initial state,
the net lifted weight should be zero.
2.C Comparing reversible and non-reversible machines.
Consider the reversible machine shown in Figures 1 and 2. Let’s call it
“machine-A”.
This machine lowers the mass m’ by 1 meter and lifts a 3m weight to a
maximum height X; and then it is run in reverse (as described above, Figures
1 to 5.)
Consider another machine-B which is not necessarily reversible.
This machine lifts a 3m weight to a maximum distance Y, while lowering a
mass m’ by one meter (m’  m). to its initial position (similar to the process as
6
described above, Figures 1 to 5.) The mechanism of how machine-B operates
is unknown. [Maybe it just drops the mass m’ from the 1 meter height, and at
the bottom it trampolines the mass 3m, the latter the shooting to a height Y.]
We now prove that Y cannot be greater than X
More general, we state that it is impossible to build a weight-lifting machine
that, by lowering a mass m’ by one meter, it will lift a weight any higher than
it will be lifted by a reversible machine.
K
Proof: Suppose Y >X.
That is, the special unknown design of machine-B allows lifting the 3m
block to a height Y while bringing the mass m’ one meter down. The
design is assumed to be so good that Y>X.
Y
Machine-B
m’
3
Unknown
arbitrary
design
1 meter
g
Y
1
3m
X
0
Figure 6.
K
Once the 3m block is at Y, we can let it free-fall to a height X and thus
obtain free energy.
We then use the machine-A (with the lever in figure 1 with the left
side located down. The mass m’ (being already at the lower position)
is placed at the left side of the lever; and the block of mass 3m (being
already at the height X) is placed at the right side of the lever (see
figure 7 below.)
Machine-A
g
1 meter
1
m’
3
3m
X
0
Figure 7.
We now run the the machine-A backwards (raising the mass m’). At
the end of this process we would have brought all the masses (m’ and
3m to their initial height (shown in Figure 6). In addition we would
have had an additional energy (when the 3m block was thrown from Y
to X). But this would constitute perpetual motion, which we
postulated is not possible. Therefore Y cannot be greater than X.
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Accordingly, among all the machines that can be designed, the
reversible machines are the best. They lift the 3m block to the
highest height.
2.D Universal behavior of reversible machines
 All reversible machines must lift the 3m block to exactly the same height.
Notice the proof of this statement is similar.
If one reversible machine-C were to lift the 3m weight to a height Z>X, we
could free fall that block to a height and then operate machine-A in reverse.
This would constitute perpetual generation of free energy, which is not
possible. Therefore Z cannot be higher than X.
But X cannot be smaller than Z either. (The same argument used above with Z
and X interchanged).
Therefore, all reversible machines lift the 3m block up to the same height.
This is a remarkable observation because it permits us to
analyze the height at which different machines are going to
lift something without looking at the interior mechanism.
If somebody makes an enormous elaborated series of lever that lift
a 3m block a certain distance X by lowering a mass m by one unit
distance, and we compare it with a simple lever which does the
same thing and is fundamentally reversible, his machine will lift it
no higher, but perhaps less height. If his machine is reversible, we
also know exactly how high it will lift.
 In summary , we have a universal law:
Every reversible machine, no matter how it operates,
which drops one kilogram by one meter and lift a 3-Kg
weight always lifts it the same distance X.
The question is, what is the value of X?
3
m’
3
3. Figuring out the potential
energy
K
Y
m’
Y
1 meter
0
1 meter
0
8
3
3
K
m’
m’
1 meter 1 meter
1 meter
3
K
0
m’
X
3
X
0
m’
1 meter
1 meter
0
1 meter
1 meter
1 meter
X
3X
X
0
By design, the height of the boxes is X.
We claim that 3X has to be equal to 1 meter. So X= 1/3 meter.
Notice two blocks practically were not lifted. The net effect on the right is to
lift one ball by 3X, while on the right one block was down 1 meter.
(equivalent to lowering one ball by 1 meter)
Macroscopic and microscopic contributions to the
energy
The total energy of a system has two distinct contributions:
Uinternal=  12mi ui 2   12 Pij
i
RCM
i, j
VCM
Emacro = (1/2) MvCM2 +
+ (1/2) I 2 + Mgz
9
“Disordered” energy
(gas molecules inside
the container.)
ui = velocities
Pij = potential energies
Ordered energy
(of the can cylinder)
Fig. 1
A. MACROSCOPIC COMPONENT (“Ordered energy.”)
The total mechanical energy of the system, associated with the
macroscopic position and motion of the system as a whole.
This mechanical energy comprises:
i) Translational kinetic energy of the center of mass (CM) +
+ the rotational kinetic energy calculated with respect to the
CM.
ii) Potential energies associated to the position of the center of
mass (gravitational potential energy, electrical potential
energy, potential energy associated to the spring force, etc.)
B. MICROSCOPIC COMPONENT (“Disordered energy.”)
The other contribution to the energy is a vast collection of
microscopic energies, known collectively as the internal
energy U of the system.
U comprises:
The sum of individual kinetic and potential energies associated
with the motion of, and interactions between, the individual
particles (atoms and/or molecules) that constitute the system.
These interactions involve complicated potential energy
functions on a microscopic distance scale. In principle, after an
appropriate choice of the zeros of the potential energy functions,
one can talk about a definite value of U of the system (when the
latter is in a state of thermodynamic equilibrium.) But such
calculation of U can be a complicated endeavor. It is relatively
simpler to calculate the changes of U.
U:
When a system changes its state of thermodynamic equilibrium,
it is only the changes in the internal energy U that are
physically significantly.
10
TRANSFER of ENERGY
Different systems can transfer energy among themselves by two
processes:
(1) Via heat-transfer, driven by temperature differences
(2) Via work, driven by external macroscopic forces
We will see that,
Heat-transfer to a system is fundamentally a microscopic
mechanism for transferring energy to a system.
Work done on a system is a macroscopic mechanism for
transferring energy to a system
Heat-transfer Q
It can occur via conduction, convection, and radiation
The mechanism is fundamentally microscopic (at the atomic and
molecular level.) heat transfer is accomplished by random
molecular collisions and other molecular interactions.
The direction is from the higher to lower temperature (an aspect
better explained in the context of the second law of
thermodynamics.)
Warning: Do not confuse heat-transfer Q with the internal energy
U.
Heat transfer is not a property of the state of a system
(a system in thermal equilibrium does not have an amount of
heat or heat-transfer.)
On the other hand, a system in thermal equilibrium does have
(in principle) a specific internal energy.
That is,
Q is not a state variable
U is a state variable
11
Macroscopic external Work W ’ done on a system
The macroscopic external work W’ done on a system can cause a
change in either
the internal energy U of the system, or
the total mechanical energy E of the system
Example where the external work causes a change of purely
internal energy
Insulation
(no heat transfer
F
External
Q=0)
non conservative
force)
Fig. 2
Gas enclosed in an insulating container. The insulated walls
ensure an absence of heat transfer from or toward the system
(the gas.)
Movable piston allows an external agent to compress the gas
(by pushing the piston), thus doing work on the system.
The work on the gas by the external agent results in an
increase of the gas temperature (indicative of an increase in
the internal energy U.)
On the other hand, simply lifting the gas container (described
above) would be an example of increasing the mechanical energy
of the system, without changing the internal energy.
12
ENERGY CONSERVATION LAW
The work/kinetic-energy theorem
We are already familiar with the work/kinetic-energy theorem,
which establishes the source (work) that causes a change in the
kinetic energy of a system. We illustrated this theorem for the case
of an individual particle, as well as for a system of particles
constituting a rigid body. The later allowed us to solve, in a very
straightforward manner, problems involving bodies rolling down
an inclined plane, for example.[ But cases involving work done by
internal forces in non-rigid bodies were not considered. We will
encounter such cases in this section.]
Case: Inelastic collision
In what follows, we illustrate the need for generalizing the
work/kinetic-energy theorem, in order to include cases in which
disordered (microscopic) energy is involved. To that effect, let’s
start consider an inelastic collision.
Before the collision
Both particles initially at the same temperature
and in thermal equilibrium
v
m
At rest
frictionless
i
M
X
Kinetic energy: Kbefore = ½ m v2
After the collision
V’
(m+M)
frictionless
i
X
13
Kinetic energy Kafter = ½ (m +M) V’2
Since the linear momentum is conserved
mv = ( m + M ) V’
Kafter = ½ (m +M) [ ( m / (m + M) v ]2
Kafter = ½ [ ( m2 / (m + M) v2 ]
= ½ m v2 [ ( m / (m + M) ]
The change in kinetic energy is given by,
K = Kafter - Kbefore = [( m / (m + M) -1 ] ½ m v2
= - [( M / (m + M) ] ½ m v2
that is, the kinetic energy is less after the collision than before.
According to the work/kinetic-energy theorem this change should
have resulted from the work done by the forces acting on the
system. But notice, all the external forces acting on the system
(normal forces and weight) are perpendicular to the displacement
of the particles, hence, their work on the system is zero (WN = 0,
WW = 0.)
NM
Nm
frictionless
i
(m+M)
X
W1
W2
Apparently, then, the work/kinetic-energy theorem Wtotal = K
appears not to be valid here.
14
The explanation lies in the fact that we are not including the
work done by the internal friction forces. Such forces act
during the inelastic collision. We say then,
Winternal-friction = K
Thus, in this particular example, we identify the decrease in the
kinetic energy in the negative work done by the microscopic
internal forces.
We would like to highlight that the change in kinetic energy K
may include not only the macroscopic kinetic energy (of the center
of mass) but also (presumably) an increase also of the microscopic
kinetic energy; that is,
Winternal-friction = Kmacroscopic + Kmicroscopic
(case for the inelastic collision depicted in the figure
above)
The work/kinetic-energy theorem
While we can in principle understand what is going on in the
particular example of inelastic collision (where the system under
study does not receive external heat-transfer), we would like to
explore reformulating the work/kinetic-energy such that include
also cases where thermal interaction (heat transfer) from the
surrounding environment is allowed.
As a firs step , let’s express the work/kinetic energy theorem as
follows,
Winternal + W’external-non-conservative + Wexternal-conservative =
= KCM + Kmicroscopic
(generalization of the work/kinetic-energy theorem)
Here W’external-non-conservative refers to the work done by forces like
the one pushing the piston in Fig. 2 above. Wexternal-conservative could
be the work done, for example, the gravitational force.
15
That is, we are explicitly separating out the macroscopic work
(done by external macroscopic forces, conservative and nonconservative) from the work done by microscopic forces.
Similarly, we assume also that the kinetic energy changes in both
macroscopic (the CM kinetic energy) and microscopic forms
i) For the conservative forces component, the work can be derived
from a potential energy function Ep,
Wexternal-conservative = - Ep
which gives,
Winternal + W’external-non-conservative + (-Ep ) =
= KCM + Kmicroscopic
Winternal + W’ external-non-conservative = KCM + Ep) + Kmicroscopic
Calling
KCM + Ep ≡ Emacro
the macroscopic mechanical
energy,
the work/kinetic-theorem can be written as,
Winternal + W’external-non-conservative
= Emacro + Kmicroscopic
ii) We can envision that, ultimately, Winternal causes a change in
microscopic potential energies of the interacting microscopic
particles that constitute the system. That is, Winternal =  -Pij.
i j
Hence,
 -Pij + W’external-non-conservative = Emacro + Kmicroscopic
i j
16
W’external-non-conservative = Emacro + Kmicroscopic +  Pij
i j
Change in
macroscopic
mechanical
energy
Change in the
internal
energy U
The last two terms in the right side of the expression above
constitute what we called, at the beginning of this section, the
disordered Internal Energy U of the system.
Through the derivation process followed above, we notice that
the work energy is deposited (transformed) into the system as
either,
macroscopic mechanical energy, or
internal energy.
The work Wexternal-conservative done by conservative macroscopic
external forces has been assimilated into the mechanical
energy, while the work Winternal done by microscopic forces
ended up being grouped into the internal energy term.
The expression above also shows that the work energy
W’external-non-conservative done by external non-conservative
forces could end up either as macroscopic mechanical energy
or internal energy (that the latter case can occur was
illustrated in the example above where a gas was compressed
by a piston; the force acting on the piston was the nonconservative force.)
Generalization of the work/kinetic-energy theorem
As illustrative as the expression above might be, it also reveals
its limitations for dealing with cases in which the system is in
thermal contact with a body at different temperature. Indeed, in
such a case, the system can also receive energy via heat-transfer, a
17
process driven by temperature differences.) Accordingly the
expression above needs to be modified or generalized.
Q + W’external-non-conservative= Emacro + Kmicroscopic +  Pij
i j
Change in
macroscopic
mechanical
energy
Heat-transfer
into the
system
Change in the
internal
energy U
In a simplified form
Q
+
W’external-non-conservative =
Heat-transfer
into the system
caused by
temperature
difference
Work done on the
system by a nonconservative
macroscopic
external force
Emacro +
U
Change in
macroscopic
mechanical
energy of the
system
Change in the
internal
energy U
of the system
which constitutes our Fundamental Energy Conservation
Law.
________________________________________
Inelastic collision (revisited)
Both particles initially at the same temperature
and in thermal equilibrium
v
m
At rest
frictionless
i
M
X
Kinetic energy Kbefore = ½ m v2
After the collision
V’
(m+M)
frictionless
i
X
18
 Here Q is the flow of energy by heat transfer, caused by
temperature differences. In our case is zero.)
 W’ is the work done by external forces. In our case it is zero.
 U change in the internal energy
 E change in the mechanical energy
In our case E = - [( M / (m + M) ] ½ m v2
Accordingly,
0 + 0 =  - [( M / (m + M) ] ½ m v2 ) + U
which gives,
U =  [( M / (m + M) ] ½ m v2 )
That is, the missing (ordered) kinetic energy appears as an increase
in the internal (disordered)s energy U of the system.
(The increase in the internal energy of the system typically
manifest itself in an increase in the temperature of the system.
As the temperature of the system increases above the ambient
environment because of the increase in the internal energy, heattransfer subsequently occurs from the system to the environment
until the system-ambient reach a common temperature.
Case: Pure Thermodynamics
In pure thermodynamics, one typically considers only systems
whose total mechanical energy does not change, Emacro = 0. The
general statement of the energy conservation becomes,
Q + W’external-non-conservative = U
19
nsulation
no heat transfer
Q=0)
nsulation
no heat transfer
Q=0)
Notice
Before
After
F
F
Work done by the external force F > 0
W’external-non-conservative > 0
Before
After
F
F
Work done by the external force F < 0
W’external-non-conservative < 0
It is typical to consider the work done by the system (no the
work done on the system by the external non-conservative
forces.) Since, according to the Newton’s third law, the
force exerted by the system is equal in magnitude but
opposite in direction, then
W’external non-conservative = - W done-by-the-system
Thus, for pure thermodynamic systems
QHeat-transfer-into-the-system
- Wdone-by-the-system = U
When all the terminology is understood, the subscripts are
omitted and one simply writes
Q - W=
U
The First Law of Thermodynamics
20
on
transfer
0)
on
transfer
0)
Notice
Before
After
F
F
Work done by the gas < 0
W <0
Before
After
F
F
Work done by the gas > 0
W >0
Question:
wedge
Before
W=?
21
After