Download Absolute potential energy

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Absolute potential energy
∗
Sunil Kumar Singh
This work is produced by OpenStax-CNX and licensed under the
Creative Commons Attribution License 2.0†
Abstract
Absolute potential energy is dened with reference to innity.
There is a disconcerting aspect of potential energy. In the earlier module titled Potential energy, we
dened change in potential energy not the potential energy itself !
We assigned zero gravitational potential reference for Earth's gravitation to the ground level and zero
elastic potential energy to the neutral position of the spring. The consideration of zero reference potential
energy enabled us to dene and assign potential energy for a unique position not to the dierence of
positions. This was certainly an improvement towards giving meaning to absolute value of potential energy
of a system. In this module, we shall broaden the reference and aim to dene absolute potential energy for
a particular conguration of a system in general.
1 Reference at Innity
The references to ground for gravitation or a neutral position for a spring are essentially local context. For
example, gravitation is not conned to Earth system only. What if we want to refer potential energy value
to an object on the surface of our moon? Would we refer its potential energy in reference to Earth's ground?
We may argue that we can have moon's ground as reference for the object on its surface. But this will also
not serve purpose as there might be occasions (as always is in the study of the motions of celestial bodies)
where we would need to compare potential energies of systems belonging to Earth and moon simultaneously.
The point is that the general concept of potential energy can not be bounded to a local reference. We need
to expand the meaning of reference, which is valid everywhere.
Now, we have seen that change in potential energy is equal to negative of work by conservative force.
So existence of potential energy is related to existence of conservative force. Can we think a situation in
which this conservative force is guaranteed to be zero. There is no such physical reference, but there is a
theoretical possibility of such eventuality. Let us have a look at the Newton's law of gravitation (this law
will be discussed subsequently). The force of gravitation between two particles, m1
and m1
is given
by :
F =
As
r → ∞,
F → 0
.
Gm1 m2
r2
As there is no force on the particle, there is no work involved.
Hence, we
can conclude that a system of two particles at a large (innite) distance has zero potential. As innity is
undened, we can think of system of particles at innity, which are separated by innite distances and thus
have zero potential energy.
∗ Version
1.4: Aug 24, 2009 4:07 am -0500
† http://creativecommons.org/licenses/by/2.0/
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Theoretically, it is also considered that kinetic energy of the particle at innity is zero. Hence, mechanical
energy of the system of particles, being equal to the sum of potential and kinetic energy, is also zero at innity.
Innity appears to serve as universal zero reference. The measurement of potential energy of any system
with respect to this zero reference is a unique value for a specic conguration of the system. Importantly,
this is valid for all conservative force system and not conned to a particular force type like gravitation.
2 Denition of potential energy
Having decided the universal zero reference, we are now in position to dene potential energy, using the
expression obtained for the change in potential energy :
∆U = U2 − U1 = −WC
If we set initial position at innity, then
U1 = 0
. Let us denote potential energy of a system to be U
for a given conguration. Then,
⇒ U − 0 = −WC
Z0
⇒ U = −WC = −
FC dr
∞
Hence, we can now dene potential energy as given here :
Denition 1: Potential energy
The potential energy of a system of particles is equal to negative of the work by the conservative
force as a particle is brought from innity to its position in the presence of other particles of the
system.
We should understand that the work by conservative force is independent of path and hence no reference
is made about path in the denition.
This work has a unique value.
Hence, it gives a unique value of
potential to the system of particles.
3 Distribution of potential energy
There is a peculiar aspect of the denition of potential energy, presented above. It denes potential energy
of the system of particles in terms of work on a single particle. This peculiarity can be explained as it
denes work in the presence of other particles and as such accounts for the forces operating on the particle
due to their presence.
This denition, however, is not clear about how potential energy is distributed among the particles in
the system. The value of potential energy does not throw any light on this aspect. As a matter of fact, it
is not possible to segregate potential energy for the individual constituents of the system. Potential energy,
therefore, belongs to all of them not to any one of them.
4 Potential energy and external force
The potential energy is dened in terms of work by conservative force and zero reference potential at innity.
It is equal to the negative of work by conservative force :
Z0
⇒ U = −WC = −
FC dr
∞
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Can we think to express this denition of potential energy in terms of external force? In earlier module,
we have analyzed the motion of a body, which is raised by hand slowly to a certain vertical height. The
signicant point of this illustration was the manner in which body was raised. It was, if we can recall, raised
slowly without imparting kinetic energy to the body being raised. It was described so with a purpose. The
idea was to ensure that work by the external force (in this case, external force is equal to the normal force
applied by the hand) is equal to the work by gravity.
Since speed of the body is zero at the end points, work-kinetic energy theorem reduces to :
W = Kf − Ki = 0
W =0
This means that work by net force is zero. It follows, then, that works by gravity (conservative force)
and external force are equal in magnitude, but opposite in sign.
⇒ W = W C + WF = 0
WF = −WC
Under this condition, the work by external force is equal to negative of work by conservative force :
Z
⇒ WF = −
where FC
FC dr
is conservative force. It means that if we work on the particle slowly without imparting
it kinetic energy, then work by the external force is equal to negative of the work by conservative force. In
other words, work by external force without a change in kinetic energy of the particle is equal to change in
potential energy only. Equipped with this knowledge, we can dene potential energy in terms of external
force as :
Denition 2: Potential energy
The potential energy of a system of particles is equal to the work by the external force as a particle
is brought from innity slowly to its position in the presence of other particles of the system.
The context of work in dening potential energy is always confusing. There is, however, few distinguishing
aspects that we should keep in mind to be correct. If we dene potential energy in terms of conservative
force, then potential energy is equal to negative of work by conservative force. If we dene potential energy
in terms of external force, then potential energy is simply equal to work by external force, which does not
impart kinetic energy to the particle.
5 Potential energy and conservative force
Potential energy is unique in yet another important respect. Unlike other forms of energy, potential energy
is directly related to conservative force. We shall establish this relation here. We know that a change in
potential energy is equal to the negative of work by gravity,
∆U = −FC ∆r
For innitesimal change, we can write the equation as,
⇒ dU = −F cdr
⇒ FC = −
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dU
dr
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Thus, if we know potential energy function, we can nd corresponding conservative force at a given
position. Further, we can see here that force a vector is related to potential energy (scalar) and position
in scalar form. We need to resolve this so that evaluation of the dierentiation on the right yields the desired
vector force.
As a matter of fact, we handle this situation in a very unique way. Here, the dierentiation in itself yields
a vector. In three dimensions, we dene an operator called grad as :
grad =
∂
∂
∂
i+
j+
k
∂x
∂y
∂z
where "
∂
∂x
" is partial dierentiation operator. This is same like normal dierentiation except that it considers other
dimensions (y,z) constant. In terms of grad,
⇒ F = −grad U
The example given here illustrates the operation of grad.
5.1 Example
Problem 1: Gravitational potential energy in a region is given by :
U (x, y, z) = − x2 y + yz 2
Find gravitational force function.
Solution :
We can obtain gravitational force in each of three mutually perpendicular directions of a
rectangular coordinate system by dierentiating given potential function with respect to coordinate in that
direction. While dierentiating with respect to a given coordinate, we consider other coordinates as constant.
This type of dierentiation is known as partial dierentiation.
Thus,
Fx = −
Fy = −
∂
∂
=−
− x2 y + yz 2 = 2xy
∂x
∂x
∂
∂
=−
− x2 y + yz 2 = x2 + y 2
∂y
∂y
Fz = −
∂
∂
=−
− x2 y + yz 2 = 2yz
∂z
∂z
Hence, required gravitational force is given as :
⇒ F = −grad U
F =−
∂
∂
∂
i+
j+
k U
∂x
∂y
∂z
F == 2xyi + x2 + y 2 j + 2yzk
This example illustrates how a scalar quantity (potential energy) is related to a vector quantity (force).
In order to implement partial dierentiation by a single operator, we dene a dierential vector operator
grad a short name for gradient as above.
gradient of potential energy.
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For this reason, we say that conservative force is equal to
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6 Potential energy values
Evaluation of the integral of potential energy is positive or negative, depending on the nature of work by
conservative force.
Z0
U = −WC = −
FC dr
∞
The nature of work by the conservative force, on the other hand, depends on whether force is attractive or
repulsive. The work by attractive force like gravitation and electrostatic force between negative and positive
charges do positive work. In these cases, component of force and displacement are in the same direction
as the particle is brought from innity.
However, as a negative sign precedes the right hand expression,
potential energy of the system operated by attractive force is ultimately negative.
It means that potential energy for these conservative forces would be always a negative value.
The
important thing is to realize that maximum potential energy of such system is zero ay innity.
On the other hand, potential energy of a system interacted by repulsive force is positive. Its minimum
value is zero at innity.
note: We shall not work with numerical examples or illustrate working of dierent contexts pre-
sented in this module. The discussion, here, is limited to general theoretical development of the
concept of potential energy for any conservative force. We shall work with appropriate examples
in the specic contexts (gravitation, electrostatic force etc.) in separate modules.
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