Download Conservation of energy∗

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Conservation of energy
∗
Sunil Kumar Singh
This work is produced by The Connexions Project and licensed under the
†
Creative Commons Attribution License
Abstract
When only conservative forces interact, the mechanical energy of an isolated system can not change.
Work - kinetic energy theorem is used to analyze motion of a particle. This theorem, as pointed out, is a
consideration of energy for describing motion of a particle in general, which may involve both conservative
and non-conservative force. However, work-kinetic energy theorem is limited in certain important aspects.
First, it is dicult to apply this theorem to many particle systems and second it is limited in application to
mechanical process involving motion.
Law of conservation of energy is an extension of this theorem that changes the context of analysis in
two important ways.
of particles.
First, it changes the context of energy from a single particle situation to a system
Second, law of energy conservation is extremely general that can be applied to situation or
processes other than that of motion.
It can be applied to thermal, chemical, electrical and all possible
processes that we can think about. Motion is just one of the processes.
Clearly, we are embarking on a new analysis system. The changes in analysis framework require us to
understand certain key concepts, which have not been used before. In this module, we shall develop these
concepts and subsequently conservation law itself in general and, then, see how this law can be applied in
mechanical context to analyze motion and processes, which are otherwise dicult to deal with. Along the
way, we shall highlight advantages and disadvantages of the energy analysis with the various other analysis
techniques, which have so far been used.
In this module, the detailed treatment of energy consideration will be restricted to process related to
motion only (mechanical process).
1 Mechanical energy
Mechanical energy of a system comprises of kinetic and potential energies. Signicantly, it excludes thermal
energy. Idea of mechanical energy is that it represents a base line (ideal) case, in which a required task is
completed with minimum energy. Consider the case of a ball, which is thrown upward with certain initial
kinetic energy. We analyze motion assuming that there is no air resistance i.e. drag on the ball. For a given
height, this assumption represents the baseline case, where requirement of initial kinetic energy for the given
height is least. Mechanical energy is expressed mathematically as :
EM = K + U
We can remind ourselves that potential energy arises due to position of the particle/ system, whereas
kinetic energy arises due to movement of the particle/ system.
∗ Version
1.10: Sep 3, 2009 6:13 am GMT-5
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1.1 Other forms of energy
Dierent forms of energy are subject of individual detailed studies. They are topics of great deliberation
in themselves. Here, we shall only briey describe characteristics of other forms of energy. One important
aspect of other forms of energy is that they are simply a macroscopic reection of the same mechanical
energy that we talk about in mechanics.
On a microscopic level or still smaller level, other forms of energy like thermal, chemical, electrical and
nuclear energies are actually the same potential and kinetic energy. It is seen that scale of dimension involved
with energy changes the ultimate or microscopic nature of mechanical energy as dierent forms of energy.
Hence, thermal energy is actually a macroscopic reection of kinetic energy of the atoms and molecules.
On the other hand, internal energy is potential and kinetic energy of the particles constituting the system.
Similarly, various forms of electromagnetic energy (electrical and magnetic energy) are actually potential
and kinetic energy. Besides, eld energy like radiation energy does not involve even matter.
Nevertheless, analysis of energy at macroscopic level requires that we treat these forms of energy as a
dierent energy with respect to mechanical energy, which we associate with the motion of the system
2 Process
In mechanics, we are concerned with motion a change in position. On the other hand, energy is a very
general concept that extends beyond change in position i.e.
motion.
It may involve thermal, chemical,
electric and such other changes called processes. For example, a body may not involve motion as a whole,
but atoms/molecules constituting the body may be undergoing motion all the time. For example, work for
gas compression does not involve locomotion of the gas mass. It brings about change in internal and heat
energy of the system.
Clearly, we need to change our terminology to suit the context of energy.
From the energy point of view, a motion, besides involving a change in position, also involves heat due
to friction. For example, heat is produced, when a block slides down a rough incline. Thus, we see that even
a process involving motion (mechanical process) can involve energy other than mechanical energy (potential
and kinetic energy). The important point to underline is that though mechanical energy excludes thermal
energy, but mechanical process does not.
3 System
A system comprises of many particles, which are interacted by dierent kinds of force. It is characterized by
a boundary. We are at liberty to dene our system to suit analysis of a motion or process. Everything else
other than system is surrounding.
The boundary of the system, in turn, is characterized either to be open, closed or isolated. Accordingly, a system is open, closed or isolated. We shall dene each of these systems.
In an open system, the exchange of both matter and energy are permitted between the system and
its surrounding. In other words, nothing is barred from or to the system.
A closed system, however, permits exchange of energy, but no exchange of matter. The exchange of
energy can take place in two ways. It depends on the type of process. The energy can be exchanged in the
form of energy itself. Such may be the case in thermal process in which heat energy may ow in or out
of the system. Alternatively, energy can be transferred by work on the system or by the system. In the
nutshell, transfer of energy can take place either as energy or as work.
An isolated system neither permits exchange of energy nor that of matter. In other words, everything
is barred to and from the system.
Interestingly, there is no exchange of mass in both closed and isolated system. Barring system involving
nuclear reaction, the conservation of mass in these systems means that total numbers of atoms remain a
constant.
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3.1 Mechanical system
System types are dened with respect to energy and matter. However, we are required to know about
the role of external force on a system in mechanics. Remember internal forces within a system sum up to
zero. Does external force anyway change the system type?
In order to answer this question, we seek to know what does an external force do? For one, it tries to
change the motion of the system and hence does the work. So if we apply the force on an isolated system,
work will change energy of the system even if system is isolated! Actually, there is nothing wrong in the
denition of an isolated system. It says that energy and matter both are not exchanged. We know that work
is just a form of energy is transit. Hence, work on the system is also barred by the denition.
We have actually brought this context with certain purpose. Our mechanical system is more akin to the
description as given above. We need to know the role of force. Now, the question is what would we call such
a system a closed system or an isolated system.
Here, we consider an example of mechanics to bring out this point. Let us consider a block, which slides
down a rough incline. Here, Earth-incline-block system is an isolated system as shown in the gure with
an enclosed area.
Earth-incline-block system
Figure 1:
Earth-incline-block system plus an external force
Let us, now, consider an external force, which pushes the block up as shown in the gure. The work by
force increases potential energy of the system, may increase kinetic energy of the block (if brought up with
an acceleration) and produces heat in the isolated system.
In the nutshell, an external force on an isolated system is equivalent to closed system, which allows
exchange of energy with surrounding only via work by an external force. So it is a special case of closed
system. Remember that energy exchange in closed system takes place either as energy or work. In our
mechanical context, the closed system allows exchange of energy as work only.
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3.2 Interpretation of system type
The denitions of three system types are straight forward, but its physical visualization is not so easy. In
the following paragraphs, we bring out important points about a system type.
1:
Any system on Earth is linked with it whether we say so or not. We have seen that potential
energy of a particle as a matter of fact belongs to the system of Earth particle system not only to
the particle. Since Earth is comparatively very large so that its mechanical energy does not change due to
motion of a particle, we drop the Earth reference. It means if we draw a boundary for a system constituting
particles only we implicitly mean the reference to Earth and that Earth is part of the system boundary.
2:
We should emphasize that system is not bounded by a physical boundary. We can understand this
point by considering a particle, which is projected vertically up from the surface of Earth. What would be
the type of this Earth-particle system? It is an open system, if we think that particle is dragged by air.
Drag is a non-conservative force. It takes out kinetic energy and transfers the same as heat or sound energy.
The system, therefore, allows exchange of energy with the air, which physically lies in between particle and
Earth of the system. However, if we assert that there is negligible air resistance, then we are considering
the system as an isolated system.
Clearly, it all depends what attributes we assign to the boundary in
accordance with physical process.
3:
We need to quickly visualize system type in accordance with our requirement. To understand this,
let us get back to the example of Earth-incline-block system. We can interpret boundary and hence system
in accordance with the situation and objective of analysis.
In an all inclusive scenario, we can consider Earth-incline-block system plus the agent applying external
force as part of an isolated system.
Isolated system
Figure 2:
Earth-incline-block system plus an external force
We can relax the boundary condition a bit and dene a closed system, which allows transfer of energy
via work only. In that case, Earth-incline-block system is closed system, which allows transfer of energy
via work not via any other form of energy. In this case, we exclude the agent applying external force
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from the system denition.
Work on isolated system
Figure 3:
Earth-incline-block system plus an external force
In the next step, we can dene a proper closed system, which allows exchange of energy via both work
and energy. In that case, Earth block system constitutes the closed system. External force transfers
energy by doing work, whereas friction, between block and incline, produces heat, which is distributed
between the dened system of Earth-block and the incline, which is ,now, part of the surrounding. This
system is shown below :
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Closed system
Figure 4:
4:
Earth-block system
We see that energy is transferred between system and surrounding, if the system is either open
or closed. Isolated system does not allow energy transfer. Clearly, system denition regulates transfer of
energy between system and surrounding not the transfer that takes place within the system from one
form of energy to another.
Energy transfers from one form to another can take place within the system
irrespective of system types.
4 Work kinetic energy theorem for a system
The mathematical statement of work kinetic energy theorem for a particle is concise and straight forward
:
W = ∆K
The forces on the particle are external forces as we are dealing with a single particle.
be anything internal to a particle.
There can not
For calculation of work, we consider all external forces acting on the
particle. The forces include both conservative (subscripted with C) and non-conservative (subscripted with
NC) forces.
The change in the kinetic energy of the particle is written explicitly to be equal to work by
external force (subscripted with E) :
WE = ∆K
When the context changes from single particle to many particles system, we need to redene the context
of the theorem. The forces on the particle are both internal (subscripted with I)and external (subscripted
with E) forces. In this case, we need to calculate work and kinetic energy for each of the particles. The
theorem takes the following form :
⇒
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X
WE +
X
WI =
X
∆K
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We may be tempted here to say that internal forces sum to zero.
Hence, net work by internal forces
is zero. But work is not force. Work also involves displacement. The displacement of particles within the
system may be dierent. As such, we need to keep the work by internal forces as well. Now, internal work can
be divided into two groups : (i) work by conservative force (example : gravity) and work by non-conservative
force (example : friction). We, therefore, expand work-kinetic energy theorem as :
⇒
X
WE +
X
WC +
X
WN C =
X
∆K
Generally, we drop the summation sign with the understanding that we mean summation. We simply
say that both work and kinetic energy refers to the system not to a particle :
⇒ WE + WC + WN C = ∆K
However, work by conservative force is equal to negative of change in potential energy of the system.
WC = −∆U
Substituting in the equation above,
⇒ WE − ∆U + WN C = ∆K
⇒ WE + WN C = ∆K + ∆U = ∆Emech
4.1 Work by non-conservative force
One of the familiar non-conservative force is friction.
Work by friction is not path independent.
One
important consequence is that it does not transfer kinetic energy as potential energy as is the case of
conservative force. Further, it only transfers kinetic energy of the particle into heat energy, but not in the
opposite direction. What it means that friction is incapable to transfer heat energy into kinetic energy.
Nonetheless, friction converts kinetic energy of the particle into heat energy. A very sophisticated and
precise set up measures this energy equal to the work done by the friction. We have seen that gravitational
potential energy remains in the system.
What about heat energy?
Where does it go?
If we consider a
Earth-incline-block system, in which the block is released from the top, then we can visualize that heat so
produced is distributed between block and incline (consider that there is no radiation loss). Next thing
that we need to answer is what does this heat energy do to the system? We can infer that heat so produced
raises the thermal energy of the system.
WN C = −∆Ethermal
Note that work by friction is negative.
Hence, we should put a negative sign to a positive change in
thermal energy in order to equate it to work by friction.
Putting this in the equation of Work-kinetic energy expression, we have :
⇒ WE = ∆K + ∆U + ∆Ethermal = ∆Emech + ∆Ethermal
5 Law of conservation of energy
We now turn to something which we have not studied so far, but we shall employ those concepts to complete
the picture of conservation of energy in the most general case.
Without going into detail, we shall refer to a consideration of thermodynamics. Work on the system,
besides bringing change in the kinetic energy, also brings about change in the internal energy of the system.
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Similarly, combination of internal and external forces can bring about change in other forms of energy as
well. Hence, we can rewrite Work-kinetic energy expression as :
⇒ WE = ∆Emech + ∆Ethermal + ∆Eothers
This equation brings us close to the formulation of conservation of energy in general.
We need to
interpret this equation in the suitable context of system type. We can easily see here that we have developed
this equation for a system, which allows energy transfers through work by external force. Hence, context
here is that of special closed system, which allows transfer of energy only through work by external force.
What if we choose a system boundary such that there is no external force.
In that case, closed system
becomes isolated system and
WE = 0
Putting this in work kinetic energy expression, we have :
⇒ ∆Emech + ∆Ethermal + ∆Eothers = 0
If we denote E to represent all types of energy, then :
⇒ ∆E = 0
⇒ E = constant
Above two equations are the mathematical expressions of conservation of energy in the most general case.
We read this law in words in two ways corresponding to above two equations.
Denition 1: The change in the total energy of an isolated system is zero.
Denition 2: The total energy of an isolated system can not change.
From above two interpretations, it emerges that energy can neither be created nor destroyed.
5.1 Relativity and conservation of energy
Einstien's special theory of relativity establishes equivalence of mass and energy. This equivalence is stated
in following mathematical form.
E = mc2
Where c is the speed of light in the vacuum. An amount of mass, m can be converted into energy E
and vice versa. In a process known as mass annihilation of a positron and electron, an amount of energy
is released as,
e+ + e− = hv
The energy released as a result is given by :
E = 2 X 9.11 X 10−31 X 3 X
108
2
= 163.98 X 10−15 = 1.64 X 10−13 J
Similarly, in a process known as pair production, energy is converted into a pair of positron and electron
as :
hv = e+ + e−
This revelation of equivalence of mass and energy needs to be appropriately interpreted in the context
of conservation of energy. The mass energy equivalence appears to contradict the statement that energy
can neither be created nor destroyed.
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There is, however, another perspective. We may consider mass and energy completely exchangeable
with each other in accordance with the relation given by Einstein. When we say that energy can neither be
created nor destroyed, we mean to include mass also as energy. However, if we want to be completely
explicit, then we can avoid the statement. Additionally, we need to modify the statements of conservation
law as :
Denition 1: The change in the total equivalent mass - energy of an isolated system is zero.
Denition 2: The total equivalent mass - energy of an isolated system can not change.
Alternatively, we have yet another option to exclude nuclear reaction or any other process involving
mass-energy conversion within the system. In that case, we can retain the earlier statements of conservation
law with the qualication that process excludes mass-energy conversion.
In order to maintain semblance to pre-relativistic revelation, we generally retain the form of conservation
law, which is stated in terms of energy with an implicit understanding that we mean to include mass as
just another form of energy.
5.2 Nature of motion and conservation law
So far we have studied motion in translation. We need to clarify the context of conservation of energy with
respect to other motion type i.e. rotational motion. In subsequent modules, we shall learn that there is a
corresponding work-energy theorem, potential energy concept and actually a parallel system for analysis
for rotational motion. It is, therefore, logical to think that constituents of the isolated system that we have
considered for development of conservation of energy can possess rotational kinetic and potential energy
as well. Thus, conservation law is all inclusive of motion types and associated energy. When we consider
potential energy of the system, we mean to incorporate both translational and rotational potential energy.
Similar is the case with other forms of energy wherever applicable.
However, if we a have a system involving translational energy only, then it allows us to consider rigid
body as point mass equivalent to particle of the system. This is a signicant simplication as we are not
required to consider angular aspect of motion and hence energy associated with angular motion.
5.3 System types and conservation law
It may appear that conservation law is subject to system denition. Certainly it is not. We state conservation
law in the context of an isolated system for our convenience. We can as well state the law for open and
closed system. Not only that we can have a statement of conservation law considering universe as the
only system.
Actually, the statement of conservation law as energy can neither be created nor destroyed, applies to
all systems including universe. For system like open or closed systems, which allow exchange of energy,
we can think in terms of transfer of energy. A statement may be phrased like change in the energy of the
system is equal to the energy transferred to or from the system.
We can be quite exible in the application of conservation law with the help of accounting concept.
We can consider energy as money in our account. Our account is credited or debited by the amount we
deposit or withdraw money. Similarly, the energy of the system increases by the amount of energy supplied
to the system and decreases by the amount of energy withdrawn form the system.
5.4 Example
Problem 1: An ice cube of 10 cm oats in a partially lled water tank. What is the change in gravitational
potential energy (in Joule) when ice completely melts (sp density of ice is 0.9) ?
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Ice cube in a tank
Figure 5:
The ice cube is 90% submerged in the tank.
Solution :
This question has been included with certain purpose. Though we have not studied phase change in
the course up to this point, but we can apply our understanding broadly to understand this question. Along
the way, we shall point out relevance of this question for the conservation of energy.
Now, gravitational potential energy will change if there is change in the water level or the level of center
of mass of ice mass.
The ice cube is 90 % submerged in the water body as its specic density is 0.9.
When it melts, the
volume of water is 90 % of the volume of ice. Clearly, the melted ice occupies volume equal to the volume
of submerged ice. It means that level of water in the tank does not change. Hence, there is no change in
potential energy, as far as the water body is concerned.
However, the level of ice body changes after being converted into water. Its center of mass was 4.0 cm
below the water level in the beginning, as shown in the gure.
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Ice cube in a tank
Figure 6:
The center of mass of the ice cube is 4 cm below water level.
When ice converts in to water, the center of the converted water body is 4.5 cm below the same water
level. Thus, there is a change of level by 0.5 cm. The potential energy of the ice, therefore, decreases :
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Ice cube in a tank
Figure 7:
The center of mass of the ice cube is 4.5 cm below water level.
∆U = −mg∆h = −V ρg∆h
⇒ ∆U = −0.13 x0.9 X 10 X 0.5 = −0.045 J
We need to account for this energy change. The gravitational energy of the system of water-ice can not
decrease on its own. We shall come to know that phase change is accompanied by exchange of heat energy.
The ice cube absorbs this heat mostly from the water body and a little from surrounding atmosphere. If
we neglect energy withdrawn from the atmosphere (only 10 % is exposed), then we can say that energy is
transferred from potential energy of ice-water system to the internal energy of the system. As such, there
is a corresponding increase in the internal energy of the system. This transfer of energy forms take place as
heat to the ice body. Hence, this is merely a transfer of energy of the system from one form to another.
Here, we do not intend to prove the exactness of change in potential energy with the change in the
internal energy in the system. But, the point about accounting of energy, in general, is illustrated by this
example.
note: We shall not work additional problems involving other forms of energy at this juncture. We
shall, however, work with the application of conservation of energy in the mechanical context in
a separate module. Also, we should know that rst law of thermodynamics is a statement of law
of conservation energy that includes heat as well. Therefore, study of rst law of thermodynamics
provides adequate opportunity to work with situations in non-mechanical context.
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