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Energy is an idea that
runs through the entire gamut of science, and defines what is
meant by physical reality. Everything that exists has energy, and occupies
space and evolves in time. All of physical reality can be said to be the
manifestation of the different forms of energy. Energy is also central to the
functioning of human civilization; to reshape nature in accordance with human
needs involves the expenditure of energy. Hence, engineering and
technology are intimately concerned with the properties of energy
which make it amenable for human manipulation and utilization.
All of things that exist in nature, as encountered in physics, chemical
processes, biological entities and so on, regulate and transform energy
from one form into another. Consequently, being familiar with the
major forms of energy lays the basis for understanding the
underlying substratum of apparently diverse phenomena, and
prepares one to understand new and unforseen forms of energy.
Figure 3.1 shows the organization of this chapter on
energy. The various forms of energy are discussed, as these forms
are the basis of the physics of a diverse range of phenomena.
Every form of energy has its specific features, and although the
various may at first seem unrelated, as one progresses deeper into
nature's laws, the various forms energy turn out to inter-connected all sorts
of strange and
unexpected manners. The transformation of one form of energy
into another is a major focus in the study of physics, and reveals
the inner workings of the various forms of energy.
The concept of energy is pervasive not only in physics but in the
rest of the sciences as well. This was not always the case. Physics
in Issac Newton's time (1642-1727) started with the concept of
force and it was only after a few centuries that the concept of energy
was understood to be more central.
One of the most fundamental and useful laws of nature is the conservation of
energy. Energy is a concept that took many centuries of science to
discover and we briefly recount the reasoning which led to its emergence.
The mechanics of Newton emphasized the concept of force,
and the century that followed was one of the unchallenged victory
of Newtonian, also called classical, physics in all spheres that it was applied to. The
motion of planets, the mechanics of solid bodies, and so on were understood in
great detail within the framework of classical mechanics.
By the end of the eighteenth century, it was becoming clear that classical mechanics
was no longer able to address all the questions confronting
scientists. All that Newtonian mechanics could
tell them was that if they could generate a force, then it
would cause acceleration and so on. The pioneers of the industrial revolution, however,
needed to know more, in particular, how does one generate a force? Engineers
and scientists needed to have a prescription on how to move
things, build engines and so on, and Newtonian mechanics is silent on this
question for practical applications.
Newtonian mechanics was also
confronted by the phenomenon of heat, for which it had no
explanation. And even more embarrasing was the complete inability of Newtonian
physics to explain chemical elements and chemical processes that
were being vigorously investigated under the impetus of the
We briefly discuss Newton's laws to understand how the concept of energy
goes beyond the idea of force.
Newton was the first
physicist to try and predict the path that a particle would follow, once its
present position and velocity is known. The trajectory of the particle is determined by
what is called equations of motion. Newton's law makes a quantitative
prediction of the
future, something unique in the history of science. The prediction for the particle's trajectory
was based on the idea of the particle's momentum, as well as on the idea of
the forces acting on the particle.
Motion is usually associated with velocity, that is, how fast you are changing your position.
In general, velocity has both magnitude (are you exceeding the speed limit?) and direction
(are you heading South or North?). However, if a light body like a bullet moves very fast,
the same amount of motion as a slower moving body which is heavier.
Hence, a better measure of motion is not just velocity, but something which also
takes into account the mass of the object as well.
Momentum is such a measure of motion. Hence we expect that
Forms of Energy
Since we have not yet defined the units for mass, we can
absorb the proportionality constant into the definition of mass.
For a particle with mass and velocity ,
momentum is consequently defined to be the following.
The dimension of momentum is given by .
If there are many particles, the total momentum of the system is
the sum of the individual momenta.
Most of us have an intuitive idea of force. We know that for
instance, to move a heavy object, we have to exert a lot of
``force''. Newton took this experience of force, and gave it a
precise mathematical formulation.
Let a force act on a particle from time to time ,
and during this time interval let the momentum change from to
Then the change in momentum, in time interval
is given by
. Force is defined by Newton to be
that physical quantity which causes the momentum
of a particle to change.
In other words, the rate of change of momentum,
, is caused by a force
acting on the particle. In other words, Newton's Second Law of motion states
Given that in most circumstances mass is a constant, the second law is equivalent to
saying that for the object to experience acceleration, which is the rate of change in
velocity - be it in its magnitude
or direction - there must be a force acting on it.
Hence, for acceleration given by
, we have
The force equation looks very reasonable. The more the force, the greater the change
in the amount of motion of the particle. The dimension of force is given by
. Let us look at the units for above equation.
Mass is measured in
, acceleration is measured in ; hence unit of force,
called a Newton, is given by .
Initial position and velocity specified
Initial and final position specified
We can now state Newton's three laws of motion.
The first law took an enormous leap of imagination, since we know
from experience that nothing can continue to be in motion for ever. The
reason being that frictional forces slow down any moving body.
Newton, however, could imagine the idealized situation and
understood the underlying principle of motion. The famous
experiment of Galileo in dropping two different masses - and showing
that they fell at the same rate - also needed a leap of imagination,
since again friction due to air had hindered a clear understanding
of the workings of the force of gravity.
The second law is the backbone of the predictive power of Newton's
laws. One may (correctly) object that this law has no predictive power, since the
moment one sees a particle accelerating, one can simply multiply it by the mass
of the particle and determine the force that is causing this acceleration.
And if one did this,
then Newton's second law would indeed have no predictive power. The
genius of Newton lies in understanding that nature has a quantity
called force, and once we can determine these forces independently from
Newton's second law, we can then
use this force as input to predict the motion of particles experiencing
this force. And sure enough, there are multifarious forces to be found in
nature, from gravitational and electrical to subnuclear forces such as
strong and weak interactions - which can be then be plugged into Newton's
second law to predict the future.
The third law is often stated in popular literature as action equals reaction, and its
formulation as given in III is more transparent.
There is a crucial concept buried in the third law, and that is
the concept of an isolated system. We will use this concept
time and again, and so it best to say a few words on it. No system
in the universe is perfectly isolated, since, if it was, it would
not be a part of our universe. What we mean by an isolated system,
be it in mechanics or thermodyamics, is that we can isolate and
shield the system from all the forces in the environment to any degree of
accuracy. This statement is correct except for one exception, and
that is gravity. We can never ``shield'' any system from the
effects of gravity. However, nature is kind in that the effects of
gravity are usually trillions of times weaker than all the other
forces of nature, and hence can safely be ignored!
A number of subtle assumptions have been made in stating Newton's three laws
For example, what is mass? In which
frame of reference are position, velocity and force being measured? If I am stationary
and you are moving very fast, and we both observe the same particle, clearly
we will observe very different
forces and velocities. Newton's response was that all observers
who are moving with constant velocity - called inertial observers - will
observe the same physics.
The question then arises is whether all inertial observers will
measure the same flow of time? Although Newton's answer was yes,
we know that this is not true.
The ideas of time, position, velocity, mass and acceleration were all radically
changed by Albert Einstein, in 1905, and form the basis of the Special
and General Theory of Relativity.
We now discuss the limitations of Newton's formulation of mechanics, which
hinges on the ideas of force and momentum.
Following are some examples to illustrate the limitations :
- First Law A body continues in uniform motion if no force acts on it.
- Second Law A force acting on a body causes it to accelerate (or decelerate).
- Third Law The total momentum of an isolated system is constant.
The forward force
requires that we expend fuel. The braking force costs no
fuel, and results in the heating of the brake pads. The heat in turn is
released into the air flowing over the brake pads. Newton's laws
make no distinction between the direction of the force, be it
causing acceleration or deceleration.
To keep the car moving at constant speed, some force is required
to overcome friction and air resistance. Newton's laws have very
little to say on this, since net acceleration is zero, although
to achieve constant speed the car must constantly burn fuel.
And lastly, one can turn a car around bends at a constant speed
and with high acceleration, but without hitting the accelerator
and hence without the expenditure of any fuel.
So clearly we need a new concept which makes up for the inadequacy
of the concept of force.
To more fully understand the science of
motion, we need to have a concept which has the following
- Consider a bullet fired from a gun. According to Newton's
third law, momentum is conserved, so that the recoil of the gun must
have the same momentum as that of the bullet, and hence their combined
momentum before and after remains zero.
But common sense tells us that something very significant and
irreversible has happened; something was taken from gunpowder and
transformed into motion. Physics must be able to capture the
transformation of gunpowder into motion, and Newton's laws are silent
on what this connection is.
- Car manufacturers need to know how to
build cars which can accelerate, coast at a constant speed, and, of
course, decelerate as well. Newton's laws are useful in telling us
how much force is required if we want to accelerate, and also how
much force the brakes must exert to decelerate. But Newton is
silent on the great difference between the force that accelerates
the car, and the brakes which decelerate.
The concept of energy fulfills these criteria precisely. We will explain later
how the concept of energy clarifies the physics of the examples given above.
Energy (in some units) is a pure number and is a measureable quantity that
we can assign to every physical system. One may ask, well what is energy?
The surprising answer is that we
do not know! To quote Richard Feynman [#!feynman!#] "It is important to
realize that in physics today, we have no knowledge of what energy
is. We do not have a picture that energy comes in little
blobs of a definite amount." In other words, energy is not a material
thing. Rather, it is a intrinsic property of a material thing.
Energy is intrinsic to a body in that it inherent to the very
physical existence of the body.
All we know at present is that for every physical object
there is an abstract quantity called energy which we can compute, and
which always remains constant no matter how many changes the object goes through.
In other words, the most significant
property of energy is the following: in every known experiment
performed to date, it is an empirical result that energy is
We will later discuss the subtleties of
energy conservation in quantum theory, and even though there is an
uncertainty relation involving energy and time, we will find that
energy is absolutely conserved not only in Newtonian physics, but
in quantum theory as well.
One of the most informative and useful quantity associated with
natural phenomenon is the energy involved in the process. In
Figure 3.4 we list a number of phenomena to have an idea of the vast
range of energy that occur in natural processes.
- A non-directional conservation law, that unlike
conservation of momentum, does not cancel out for motion in
- A quantity that takes into account the direction of
force relative to the motion, so that a force along the direction
of motion has a positive effect, force against the direction of
motion has a negative effect, and force perpendicular to motion
has no effect.
- And most importantly, the concept must connect apparently
unrelated things, such as the explosion of gunpowder with the
motion of the bullet, the burning of gasoline with the motion of
To illustrate the idea of energy conservation, let us start with the example
used by Feynman [#!feynman!#] in his famous Lectures on Physics.
Consider the case of a 10 year old who has 28 indestructible
playing blocks, the weight of each block being 0.1 kg.
Everyday, the boy's mother counts the blocks, and it always adds
up to 28. One day she finds only 27 blocks, and by looking around
she finds one lying under the bed.
A few days later she finds only
26 blocks. On searching the room she finds a toy box; when
she tries to open the toy box, the boy screams ``Don't open my toy
box''. Being a smart mother, she weighs the box when she has 28 blocks and
finds that the empty toy box weighs 0.5kg. Henceforth, when she cannot find
all the 28 blocks, she weighs the toy box to determine how many blocks
The mother now has the following formula for the number of blocks
Energy Scales in Nature
After a few days, she finds that the blocks no longer add up to 28,
and she finds that the bathtub has water full of soap, submerged
in which, she suspects, there are some more blocks. Given that the
original height of the bath-water was 0.5m, and that each block
raises the height of water by 0.1m, she now has a new formula for
computing the number of blocks, namely
We see that the calculation for the number of blocks is becoming
more and more abstract, and has less and less to do with counting
the blocks themselves. Measuring the weight of a toy box or the
height of bath water has no direct correlation with the blocks
The application of this example is straightforward to energy.
In other words, there are many, many forms of energy, that taken
together are conserved, but
there is no analogy of what is energy per se, that is,
there is no analogy for the ``blocks'' themselves. Just as
the total number of blocks was constant, energy is absolutely
conserved, that is energy can neither be created nor destroyed.
All that a physical process can do is to transform energy from one form into
another, analogous to, for example, the weight of the toy box being reduced at
the expense of the bath water rising in height, so that the total
number of blocks remains constant.
Hence, for energy we have the following expression.
There are various forms of energy such as solar energy, chemical energy,
nuclear energy, wind energy, and so on.
All the forms of energy
can be classified under two great headings: kinetic and
Energy due to motion
in space is called kinetic energy , and energy due to position
or the internal
configuration of the material body is called potential energy .
For example a stone on a mountain has potential energy due to its
position, and a stretched spring has potential energy due to its internal
Denoting total energy by we have the fundamental relation
Suppose the system has energy
at time and energy at a later time , then, the
change in energy is
. Conservation of energy implies that
Note an important fact that since all we know is that , the absolute value of has not been fixed. Hence, energy
is only defined upto a constant, since and constant would
both be equally conserved.
In both the example of the bullet being fired or the car being
accelerated, kinetic energy due to motion was created by expending
energy that was present as potential energy in the gunpowder and
in the gasoline, respectively.
Kinetic energy is that portion of total energy that is due to movement in space.
We all know that a
truck moving at high speed is something to be avoided; a head-on
collision is violent enough to demolish a wall. The violence
that a truck can do is due to the large kinetic energy
that it has, and comes both from the fact that it is heavy, and
depends on how fast it is moving. No matter in which direction the truck
moves in, for a given velocity it has the same amount of kinetic energy. If
is the mass of the truck and its velocity, its
kinetic energy is defined by
The dimension of energy is
. The units of energy can be deduced
from the above equation as
, and is called a Joule, abbreviated as .
Example 1 Revisited
Recall in Example 1, we had a bullet of mass being fired at some velocity
from a gun of mass with a recoil velocity given by
Momentum conservation implies that initial and final momentum must
be zero. Consequently
This doesn't tell us much, since we are not able to translate this
information into how much energy must be expended by the
However, the total energy expended in shooting the bullet is
Using eqn.(3.23) we have the result that to design a gun which specifies the
velocity of the bullet to be , we need to spend an amount of energy given
For most guns, the mass of the gun is much greater than that of the bullet,
, and hence above
reduces to the kinetic energy of the bullet.
There are only two kinds of kinetic motion. Namely, rectilinear, or linear, motion,
that is, motion in a straight line, and kinetic
motion associated with rotation. Rotational motion, in general, is also associated
with periodic motion.
Let a solid object be spinning, around some fixed axis say the direction perpendicular
to this piece of paper, with angular velocity
. Let the angular position of a point on the
solid body be given by the angular coordinate , with the range
. Suppose that at time the angular coordinate is
and at later time the angular coordinate is
as shown in Figure 3.5. Then, in analogy with
linear motion, we define angular
Solid body spinning. Coordinate
The angle is dimensionless. Angular velocity has dimension of ,
and has units of .
We have , similar to kinetic energy for linear motion,
the angular component of kinetic energy given by
is the moment of inertia of the solid body. From
dimensional analysis has the dimensions of and has units .
Similar to (linear) momentum, a spinning body has angular momentum given by
Angular momentum measures the rotational motion of a rigid
body. If a point particle of mass m rotates with constant
velocity at a distance about, say, the -axis, its angular
momentum is given by
Similar to linear momentum, the total angular momentum of a
physical system is conserved.
In general, for a solid body that is moving with velocity and
spinning with angular velocity , the total kinetic energy
is given by
The moment of inertia of a sphere of radius and mass is given
Note to obtain the moment of inertia for a sphere, we did not specify around which
axis it is spinning since for a sphere all directions yield the
same moment of inertia
However, for a
cylinder of mass , radius and length , this is not the case, as
there are two inequivalent axis around which we can spin the
cylinder as can be seen from Figure 3.6. Spinning
the cylinder along the axis of the cylinder will have
a moment of inertia given by
, and orthogonal to
the axis of the cylinder will yield
We consequently will have two different moments of inertia, given
Axis of rotation for the cylinder and for the sphere
Work is a crucial concept which links energy to force. How does one
increase or decrease the energy of a particle? Intuitively, one
would think that by acting with a force on a body, we should be
able to change its energy. For example, by acting with a force we
can increase the velocity of a particle, and which in turn implies
that its (kinetic) energy has increased. So clearly, there is an
intimate connection between force and energy. Note an important
fact that force acts over
a finite period of time, and hence is a dynamic quantity, whereas
energy is a property inherent and intrinsic to a body which can be
time independent. Hence, the a concept of work distinct from
necessary to relate dynamic force to intrinsic energy.
Energy is defined in many books as the capacity to do
work. Work, denoted by is in turn defined as
The dimension of work
, which is the same as the
dimension of energy. Hence we can in principle add work to
energy, showing that there is a deep relationship between energy
and work. This is an instance where dimensional analysis yields
new insights into dimensionally related quantities.
Work is a measure of the action of force acting over a distance.
Think of pushing a car; the heavier the car, the more the force
required to move it. Also, the longer the distance the car needs to be
pushed, the larger the amount of work that needs to be done. So
the definition seems reasonable.
However, many students have an intuitive problem with this definition of
work. We all know that if we hold up a big piece of stone, even
though the stone is not moving, and consequently no work is being done,
we will soon
break out into a sweat from the exertion that we are undergoing.
So what's going on? How can there be no work done, even though we
have had to exert ourselves? The answer to this counter-intuitive
result lies in the physiology of human muscle.
There are two kinds of muscle cells, one which change over a long period of
time and the other which change over a short period.The clam, for
example, has a muscle cell which relaxes over a very long
interval, and hence a clam can be in an open position and support
a large weight without expending any energy. In contrast, human
muscle needs constant electrical impulses to hold its position,
and consequently to hold a piece of stone requires a large
expenditure of biological energy. Hence we are doing internal
work, and not work on the piece of stone, when we hold it in air.
To confirm that this is true, one can just place the stone on a
table, and it will sit on the table without any work being done, and with
no energy being used up for that purpose.
The fact that no work is done on a load put on a table is the same reason
why high rise buildings can stand without any work being done.
All the high floors are stationary, and hence do not require any
expenditure of work (energy) to hold them up. Of course overloading
may cause the floor to break, and this then becomes a problem of
material science rather than that of mechanics.
One cannot store work, since once the body ceases to move,
no more work is done. This is a reflection of the dynamic
nature of force. However, unlike work, we can store energy. When work is
done on, or by, a particle, the result is to increase, or to
decrease, its energy. In other
words, the deep connection between work and energy is that
force results in work , which in turn increases or
decreases the energy of a body. Of
course, energy conservation tells us that we are simply
transforming or transferring energy from one form to another.
Consider a moving particle with mass that at initial time has a
position of and speed of
. Let a constant force act on it from time to , during which
time it travels, along the direction of the force, to the final position of
At the end of time ,
it has increased its velocity to . The particle has only kinetic
energy, and the conservation of energy
then tells us that the increase in the (kinetic) energy of the
particle must be due to the work done on it by the action of the
constant force . Distance , given by
was covered during time given by
Hence, we have the conservation of energy
Power is defined as the rate at which work is done. For the case of
the particle moving under constant acceleration, its total energy
is equal to its kinetic energy, and work done on the particle
is equal to the change in its kinetic energy. Hence, in this case power is
equal to the rate of change of energy of the particle.
From eq.(3.40) we have that the work done in time
is given by
We rewrite (3.40) in the following manner. The
change in the particle's kinetic energy is seen to be the work done on it.
The change in comes about by constant force acting over a distance s.
Power, denoted by , is then defined as
Since the time interval over which the kinetic energy has changed
is , we have
, and hence
This is an important equation, since it tells us that the rate of
change of kinetic energy of a particle, that is the power being expended
on the particle, is equal to the force times the velocity of the
particle. The dimension of power is . One unit of power is
defined to be 1 Watt (W) defined to be 1 Joule per
second, and W has units of .
Example. Consider a typical CD system that uses 250 W
of power per
hour. If you play the system for 3 h how much energy do you use?
If electricity costs $0.08/kWh, how much is the electric bill?
We know that Energy= Power x Time, and hence
Hence, the electric bill is
Example 2 Revisited
We can now answer the questions raised in Example 2 regarding the
acceleration of a car. When the car accelerates, and are
in the same direction, and hence required is
positive since power has to be supplied to the car , and that is
why the car has to burn fuel to provide this power. When we brake, and are
in opposite directions, with the brake causing acceleration in the
direction opposite to velocity . Hence the required is
seen to be
negative, since the car is providing power to the brake, causing it to heat,
and hence no fuel needs to be burnt.
For a typical passenger car moving at 25 on a level surface
requires a force of 1000 Newtons to keep it moving. Hence the
Since most car engines have much more power, the car can easily
coast at constant speed. Since the car is not accelerating, the
force of resistance is exactly equal to the power supplied by the
engine, that is
Consequently, we see that the velocity of the car
The frictional forces
dissipate fuel energy being expended by the car into heating. We
will see later that the conversion of low entropy fuel into high
entropy heat follows from the Second Law of Thermodynamics, and causes
the process to be irreversible.
When the car turns, say in a perfect circle, the force is
perpendicular to the displacement of the car, and the distance
moved by the car along the direction of the force is hence zero.
From eq.(3.36), we then have that no power needs to be expended for
making this turn.
Question.Why is the force perpendicular to displacement when
the car is moving in a circle?
We see from above that it is the idea of
power rather than force that is required to explain
the motion of a car.
Notice the questions
regarding the expenditure of power does involve the use of force
, but we will see later when we discuss potential energy, that
the concept of force will be replaced by concept of potential
and then eq.(3.48) will be seen to be a re-statement of the
principle of conversation of energy.
From Newton's second law, we know that a constant force causes
motion with constant acceleration, say a. That is
Hence, from (3.41) and above, we have
From eq.(3.51) we see that we have recovered the result of
mechanics of how velocity and acceleration are related to distance
covered. Since acceleration is a constant, we also have
Hence, combining eqs.(3.51) and (3.53), for , we have
We see, from the equations (3.51),(3.53) and (3.54), that all the
equations of motion for constant acceleration can be derived from the principle
of energy conservation.
Kinetic energy is easy to visualize. Potential energy, denoted by is a more
complicated concept, since potential energy means energy which is
in a latent (``hidden'') form, and capable of being ``released''. Potential energy
stored in the physical shape and configuration of the body, and is different
from kinetic energy in that it is present in the body without any motion or movement.
The dimension and units of potential energy is the same as
energy (of which it is one form).
Replacing the concept of force by that of the potential is the essence of why the idea
of energy is so useful. Can we
always replace by ? The answer is that if the force is
that energy is conserved, we can always replace by . For such forces,
if one compares the value of the potential at two neighboring positions
and , we obtain the force at that point given by
One may correctly object that since energy is always conserved, it
should always be possible to express force in terms of a potential.
In principle, this observation is true.
However, there are many non-conservative cases,
such as in the study of viscous fluids, that one uses an approximate
formulation of the problem where for example energy losses
due to friction are not included
in the problem. For such cases, we have to directly use the
concept of force, and it cannot be then replaced by a potential
Conservative and non-conservative
forces are characterizing by whether the work done in going between two points in
space is path dependent or not. Consider for example traveling between
two points, say and , by two different paths, one which is
a gravel road and the other which is a paved road. This situation is shown in Figure 3.7 with
being the label for say the gravel path and the label for the paved path.
via the gravel road will entail doing more work than the paved road in getting
from from to , and hence the work
dependent in that it depends on which path we take, and we
conclude that the force experienced in traveling is
non-conservative. On the other hand, if in Figure
3.7 the work done on the particle by a force which
takes it from point to point does not on the path taken,
for example the work along path 1
is equal to that of taking it along path 2, then the force is conservative.
Path dependence and path independence
A body can come to rest only if there is no force acting on it.
Since force is equal to the negative gradient of the potential,
we can conclude that no force acts on a particle that is located
at the minimum value of the potential. An example of this is
shown in Figure 3.9, where at the point is the
position of equilibrium. We have the important and general result
A body reaches equilibrium when its position is at the
minimum value of the potential
We all know that a stone dropped from a great height picks up a
lot of speed before hitting the ground. Since a fast moving piece
of stone has a lot of kinetic energy, and it started with zero
kinetic energy, clearly its position at a great height must have
endowed it with ``potential'' energy, and that becomes ``actual'' (kinetic)
energy before it hits the ground.
So what is this potential energy? The hint is already given by the
connection of potentiality with height. The higher the elevation,
the greater must be its potential energy. Also, the heavier the
body, the greater its potential energy. Both these intuitive expectations
reflect everyday experience when we see objects fall - if they fall
from a greater height, they have higher impact, and similarly if they are heavy.
If we represent potential
energy by , for a body with mass at a height
we reasonably expect the simplest expression for
(gravitational) potential energy to be given by
There are almost unlimited forms of potential energy. We discuss a few
of these, in particular, the
gravitational and electrical potential energy, mass as potential energy as well as
elastic potential energy that is
stored in springs and other elastic objects. These examples are
chosen to give a flavor of great variety of forms that potential
energy can take.
Other forms of potential energy such as chemical energy, radiant energy, nuclear energy
and so on are more subtle, and require deeper study.
We have called the proportionality constant ; clearly
is linked to gravity, since if there was no
gravitational force, the particle would not fall towards the
earth to start with. By dimensional analysis, since is energy,
it has dimensions of , and hence the dimension of
is , which is the dimension for acceleration. From dimensional
analysis, must be proportional to acceleration due to gravity,
and in fact is precisely equal to acceleration arising from earth's
gravity, and is equal to .
The gravitational potential given in (3.64) is an
approximation, valid only for bodies that are close to the earth's
surface. At distances from the earth's surface that are comparable
to the earth's radius, the correct expression for is given by the ``inverse
Consider a particle falling under the force of gravity.
At a height of , it has velocity , and hence its
total energy that is given by
As the particle falls, its velocity increases as its height
decreases. Conservation of energy then requires that energy always
be a constant, that is, the change in be zero. Hence
Simplifying above equation, we have
To understand the content of above equation, at height let the velocity of
the particle be and its energy , and at height let it be and
respectively. Energy conservation requires that . We have
height that the particle has
fallen through. Hence, from (3.68) we have
We see that we have recovered (3.51), with the additional
information that the constant acceleration is due to gravity.
Note the significant fact that, unlike the case for (3.51),
we did not use the idea of force at any
stage in deriving (3.69). This is a reflection of a the
general procedure of replacing conservative force by potential .
Consider a spring with a ball, of mass attached to one end with the other
end of the spring fixed to a wall.
is in equilibrium when the ball is at position .
What do we expect for the potential energy of the spring?
Whether we compress or stretch
the spring, in both cases it gains energy. Hence the potential
energy should be equal if the new position of the ball is
compressed or stretched by the same distance from its resting position .
The simplest expression for such a potential is given by
Diagram of spring with ball
where is the spring constant, which is a measure of the stiffness
of the spring, and has the dimensions of and units of .
Hence the total energy
of the ball, moving at velocity , is given by
Potential Energy versus Position
Let the ball be at some position .
We can for example ask the following question:
What is the force required to displace it by a small distance ? Let the ball have
zero velocity. To change the energy of the ball by displacing it, an external force
has to do work on it. Hence we have
where in obtaining the last equation we have taken to be so small
that it can be completely ignored. For equilibrium, the restoring
force due to the spring has
to exactly cancel the external force. Hence, if the ball is moved to a position
, we have a restoring force
acting on the ball that is given by
In other words, the force required to stretch the spring is
proportional to the amount of stretching, namely . Note that
the spring is always a restoring entity since it acts against any change of
position of the ball. For the force is negative,
that is, it acts to pull back to , whereas if then
the force also acts to push the ball back to .
We solve for the motion of
the particle that is moving in the elastic potential. From
(3.72) we have for a particle moving with energy the
One can easily verify, by differentiating (3.87), that to satisfy the energy
equation given in (3.72), we have the following
Note has dimension of , has dimension of and the phase
is a dimensionless angle measured in radians.
Recall is the position of equilibrium of the particle.
Suppose the particle starts its oscillations at time from
the position . Then is fixed by
One can see from the behaviour of that if the ball is disturbed from
its equilibrium position at , it will undergo oscillations about this
position; note that if one increases time by the amount , that is,
, the particle returns to the position
that it occupied at time . In other words, it is undergoing periodic motion about its
equilibrium position, with period of oscillation given by
. The maximum departure that it has from is called the
amplitude of the oscillation, and is given by .
The phase tells us where the oscillations begin at , the
amplitude of the oscillation is fixed by the ratio of the total energy and
mass of the particle, and the period of oscillation of the particle, namely , is
given by the ratio of the spring constant and the mass of the particle.
In other words, the stiffer the spring is, the larger is the
value of and consequently the faster the particle oscillates.
The particle will oscillate forever if not for frictional
forces, which we have ignored in our discussion. Due to oscillatory
motion that the elastic potential energy term produces, it is
also called the harmonic oscillator potential. We studied the motion
due of the oscillator in some detail since it is of fundamental importance in
all kinds of diverse fields. In particular, as we will see later,
the underlying explanation of wave motion relies
on an understanding of oscillatory motion.
We discuss another form of potential energy. Albert Einstein showed
that mass itself is a form of potential energy. For a body with
mass , the formula for potential energy of the body at rest is given by
Diagram of x(t)
is the velocity of light. The
equivalence of mass and energy stated above is how the mass
equivalents to energy were arrived at in Figure 3.4.
If the particle is moving at velocity with respect to an
inertial frame, its mass increases. As its velocity approaches the mass tends to
infinity, as shown in Figure 3.11, reflecting that a finite mass particle
can never reach the velocity of light. The mass of the particle increases
indefinitely with respect to the said
inertial frame, and is given by
If one thinks of the atomic nature of matter, one might be tempted to think that
all forms of energy are ultimately kinetic energy. This example is important
to realize that this is not so, and that there are some forms of energy which
are irreducibly potential in nature.
Mass as a Function of Velocity
Two of the most important forms of potential energy in
physics are the gravitational and electrostatic potential energy.
Gravitational and Electrostatic Potential Energy
Suppose two particles of masses and charges
are separated by a distance . The gravitational potential energy and
the potential energy between static electrical charges -
electrostatic potential energy - also known as the Coulomb potential, are given
Gravity and electromagnetism
where is Newton's gravitational constant with dimension of
, and its numerical value is given by
Electrical charge is measured in units of coulomb, labeled C, and
by an abuse of notation we will denote the dimension and the SI unit
for charge by . The Coulomb constant
has a dimension of
its numerical value being given by
The Coulomb constant is also defined in terms of the so called
permittivity constant by the relation
Question. Show that the gravitational potential energy given
in (3.64) can be derived from (3.94).
Determine the value of in terms of the gravitational constant
. [Hint: Consider the whole mass of the earth to be as if it
were located at the centre of the earth.]
An important similarity of the two forms of potential energy is
that mass and charge are the
source of the gravitational and electrostatic potential energies, respectively. In
other words, a mass or a charge at some point in
space creates, in the space around it, a gravitational or an electrostatic potential,
and other masses or charges gain potential energy as they enter the field created by
the mass or charge source.
One may wonder as to what is the potential ``made out of'', and
by what mechanism does the source exert an influence in space
around it. The answer to this question leads one to concept of a
field. Masses and charges are the sources of fields, and the
gravitational and electrostatic potentials are the physical
manifestation of the gravitational and electrostatic forces in nature. We
will see later that a field is a physical entity as real as the
source, and leads to many new insights into the workings of
Although the gravitational and electrostatic potential energy are similar in the
sense of having a common mathematical form and are generated by sources, they
are also profoundly different as well. The differences are the
An important extension of the idea of energy is the case
when a particle has a net negative energy. For all cases where the potential
is well defined, the concept of negative energy, that is
, can be properly defined, and the state with negative energy is said to be
a bound state. Bound states are the norm in nature. Neutrons and protons
are in a bound state forming the nucleus, the
electron is bound to the nucleus forming the atom, atoms bind together to form
molecules, all the bodies on the earth are
bound to it including ourselves, the earth is in a bound state
with the Sun, the Sun itself is bound to the Milky Way galaxy and
Since energy is
defined only up to a constant, what does it mean to talk of
negative net energy? Recall from (3.19) that
- Mass is always a positive quantity, so gravitational potential
always gives an attractive force of attraction between any two masses.
This is the reason that one can never 'shield' a system from gravity,
since there is no way of cancelling it out. So although gravity is
an extremely weak force, as we will soon show, its effects keep
piling up. Hence for celestial bodies, solar systems, stars, galaxies and so
on, the effects of gravity are the most important. Due to the fact
that gravity is always attractive, one can imagine that if one
gathers together a large enough mass, there could be no force which
could be strong enough to stop the inward pull of gravity. This is precisely what
happens when we have a star with mass three times bigger than that of our
Sun : for such stars, the force of gravitational attraction is so strong, that
the star - under the inward pull of gravity -
undergoes gravitational collapse, and results in the formation of a black
- Another difference between electrical and gravitational
potential energy is that every physical entity feels the force of gravity
- there are no gravitationally ``neutral'' entities. On the other
hand, there are many fundamental entities like the neutron that
are electrically neutral, and do not feel the effect of electrical forces.
- On the other hand, electrical
charge can be positive or negative. The gravitational and Coulomb
potential also differ by a minus sign, which implies that although
masses attract gravitationally, two particles with like charges
repel whereas as opposite charges attract. Unlike
gravitational forces, we can completely shield a system from
electrical forces by using negative charge to cancel positive
charge, since having a net charge of zero means we have
effectively cancelled out electrical forces. Recall all elements
are made out of electrically neutral atoms. The atoms of each element are made out
of a nucleus consisting of an
equal number of electrically neutral neutrons and positively charged protons,
which is surrounded by a 'cloud' of an equal number
of electrons. The Coulomb potential holds the electrons in a bound
state with the nucleus.
The repulsive Coulomb potential - inside the nucleus - between like charges
is overcome by strong nuclear forces which glue together the protons inside
the nucleus. The nuclear force acts over very short
distances, it can hence cancel Coulomb repulsion only over short distances.
However, since the Coulomb potential acts over long distances, it finally
wins over strong nuclear forces. Namely, if the number of protons exceeds a certain
value close to 140, the Coulomb repulsion becomes so large that
the nuclear force can no longer overcome it, and the nucleus falls
apart (disintegrates). This repulsion due to the Coulomb potential
is the reason that there are only a finite number of stable elements
in the universe, numbering about 100 only.
Suppose the strong attractive force between protons acts over a
distance of , and that the electrostatic potential acts
over an infinite distance. Estimate how many protons would make the
- The greatest difference in the gravitational and electrical
potential energy lies in their relative strengths, characterized by
the empirical values of the two constants
and . To determine the relative strengths of the two
potential energies, we evaluate the ratio of their values for the case of
an electron and proton separated by a distance .
Consider an electron, having mass
and a charge
, and a proton having mass
and a charge
We hence have
We see that, for any distance of separation of two charged bodies,
the gravitational potential energy is almost
smaller in magnitude than the electrical potential energy. Even if we consider bodies with
much bigger masses than an electron or a proton, the effect of gravity is extremely small.
So for all
cases of interest, we can ignore the effects due to gravity
compared to other forces.
One may object to the thought of ignoring gravity,
considering that we always feel the force of
gravity in our daily lives. This objection is correct if the mass
involved is immense; for example, what we experience in daily life
is the gravitational effect of the earth, which is an immense mass
indeed. Compared to the gravitational effect of the earth's mass, the
gravitational effect of any other massive body on the earth's surface,
on say an object undergoing experiments
in the laboratory, are negligible. The weakness of the gravitational force
is ultimately the reason why the idea of an isolated system is possible.
We have to first remove the arbitrariness in energy, which is
only defined upto a constant. Only then can we talk of positive and negative
Since there is no arbitrariness in , we have to first define what we mean
by zero potential energy .
Take the case of the gravitational potential. We consider the
approximate expression given in eq.(3.64), namely
It is natural to measure the height from the surface of the earth,
and that is what has been implicitly assumed in the equation for
above. In other words, we choose the surface of the earth as the
height for which . Hence, if , it has positive potential energy,
and if we can drop a particle from this height, it will get
transformed into kinetic energy as the particle loses height. All this we
have already discussed.
We may now ask: what happens if we dig a
ditch or a well which has depth , and put a particle inside the well?
The height for
a particle in a well is negative, that is , and hence
this particle has negative potential energy, that is .
This configuration is shown in Figure 3.13/
If the particle has velocity , the
total energy is then given, for (particle inside the well), by
Particle in a Well
Hence, for a bound state
What is the physics of this result? Negative energy, ,
means that the particle cannot reach the surface, since
on the surface its energy is at least zero (and higher if it has
non-zero velocity on the surface). Hence, as long as , the particle
can move inside the well, but cannot escape from it. The particle is consequently
bound to the
well, and the particle is said to be in a bound state.
To free the
particle from the well, we would have to give it enough energy so that its net
energy would then be at least zero. For example, suppose the particle is
stationary at the bottom of the well. Then and we would
have to supply the particle with a velocity of to
make its total energy equal to zero, and hence allow it to escape
from the well.
Can we escape from a bound state? Since the bound state is a result of the attractive
force due to some attractive potential, the only way one can escape from a bound
state is by acquiring enough kinetic energy so that the
total energy becomes at least equal to zero. To
understand the mechanism of escape, let us study a particle bound to the earth by gravity.
The particle in a ditch is identical to a particle held to the
surface by earth's gravitational force. We can find the minimum velocity
required for the particle to escape from the earth's gravity.
We have to use the correct gravitational potential given by (3.94).
Let the mass of the earth be and radius . Let the be the energy of a particle
of mass moving with vertical velocity . Hence, we have
For a bound state, we have
The escape velocity
of the particle is achieved when . Hence
The mass and radius of the earth
respectively. The resultant escape velocity is given by
From eq.(3.111), we see that the escape velocity depends
inversely on the radius of the earth . What does this mean?
Note we can reduce the radius of the earth by
squeezing the mass of the earth into a smaller volume, in other words,
by increasing the density of the earth. Suppose we compress
the mass of the earth to a new radius ; given that the volume of a
the new density of the earth is given by
The escape velocity from a body with the earth's mass but smaller
radius is given, from (3.111), by
Note that we can keep on increasing the value of by reducing the value
of . What is the maximum value that can have? We know that
the maximum velocity that any object in the universe can have is the velocity of
. When the escape velocity approaches
is an indication that nothing, not even light, can escape from the
gravitational pull of a highly dense object. Such an object is called a
black hole. The radius into which
the mass of the earth has to be squeezed so that it forms a black
hole is called the Schwarzchild radius of the earth, called , and is given by
Surprisingly, a much more complicated calculation based
on General Relativity yields exactly the same result for the
Schwarzchild radius as given in eqn.(3.122).
So the Schwarzchild radius of the earth is really tiny. For the
Sun the Schwarzchild radius is km. As mentioned earlier, it is know from
astrophysics that stars with masses greater than 3 times the
mass of our Sun all end their stellar evolution by undergoing
gravitational collapse and forming black holes.
Question. You may have noticed that light in a room is
spread over all space, whereas a piece of stone is what we
usually have in mind when we talk of a particle. Can we extend the
notion of energy to the phenomenon of light? Would energy be then
distributed all over space instead of being a property of a
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