Energy unrealized and fields unplowed...
In the previous section we introduced a block an spring
arrangement to illustrate the work done by a varying force and
the kinetic energy arising from the spring's work. We assume
that there is no friction in the system. Take
another look at the Work by Spring
display and notice the curve of work vs. position formed by the
lines drawn in the plotting area of the display. You should
recognize the quadratic shape. It seems like the work done by the
spring drops off as the square of the distance traveled on either
side of the neutral position.



Perhaps it would be useful to look at the work done on the
spring. When you drag the block off neutral against the force of
the spring you are doing work. First you must do enough work to
change the kinetic energy of the block so as to give it some
constant velocity. Once the velocity is established, the force
required to keep the block moving at a constant speed is exactly
equal and opposite to the force of the spring. In other words the
resultant force once the block stops accelerating must be
zero.
Even though you are no longer accelerating the block, you are
still applying a force and the block is still moving, therefore
work is being done on the block/spring system. We know that the
spring force is k*x so the force doing the work is k*x. The work
done in moving the block the tiny distance dx, from x to x+Dx is
Dw = k * x * Dx.
This should begin to sound like the time slice discussion we
had
earlier.
Here instead of working
with slices of time, we are working with slices of displacement.
The change in work (Dw) is just the
area of a displacement slice of height k*x and width Dx. By adding up the area of all the slices
between 0 and x, we get the work as a function of x. Run the Graph of Work on Spring display to see
this in action. Notice that this display allows a much longer
spring travel so as to more clearly illustrate the similarity to
the time slicing technique.

Now, returning to the original block and spring, we get a
slightly different view of the work done on the spring in the
Work on Spring
display.
Think about what happens when after compressing the spring we
turn the block loose. The force on the block is now coming from
the spring. The work done on the block as a function of x is
again the sum of all the displacement slices from the starting
point to x. By the time the block moves back to the original
position, x=0, the work done by the spring is exactly equal to
the work originally put into the system by pulling the block to
one side. It is evident from this discussion that the sum of the
work done on the spring and the work done by the spring on the
block is a constant, independent of the position, x.



Now take a look at the
Combined Spring Work display.
The work expended in compressing the spring seems to have been
stored in the spring until we let the block go. Then the work was
returned to the block. What the original work did was to change
the system by altering the system "configuration",
meaning that the location of the system parts relative to each
other changed. This gave the system the ability to do work. If
work is required to change a system's configuration, energy
will be stored in the system. Energy will remain stored in the
system until the system is allowed to return to its original
configuration. Potential energy is energy stored in a system by
virtue of its configuration.
The Newton's law view of the block and spring system is
this. An initial displacement produced a force which caused an
acceleration resulting in a velocity that required an opposite
force and acceleration to bring the block to a stop. By the time
the block was stopped, the spring was displaced in the opposite
direction so the other half cycle happened. And so on and so
forth.
Looking at it from the standpoint of work and energy, the work
of the initial displacement stored potential energy. After the
system returned to its original, low potential energy,
configuration the block continued to move and eventually made it
all the way to a configuration having the same amount of
potential energy as before. Where did the work come from, to
create the reconfiguration of the system in the second half
cycle? We said energy was the ability to do work, so evidently
the system still had the same amount of energy in it when the
block was at position x=0 as was stored by the initial
displacement.
The system configuration as the block passes through x=0 is
identical to the configuration before we touched it at all, so
the potential energy is returned to zero. The difference is that
in the current, cycling, situation the block has some velocity
and therefore some kinetic energy. What happens is that the work
of the initial displacement sort of charges up the system by
creating potential energy. Then when the block is released, the
potential energy is converted to kinetic energy. In the second
half of the cycle the kinetic energy does work on the spring
which shows up as potential energy and so on forever.

Run the
Combined Spring Work
display again but this time think of the
green curve as the potential energy stored in the spring and
the red curve as the kinetic energy of the block. This
interpretation is valid because the work done on the spring is
equal to the potential energy stored and the work done on the
block is equal to the kinetic energy of the block.
To summarize what we are seeing here, the energy of the system
when the spring is stretched and the block is motionless is all
potential energy, derived from the configuration of the system.
The source of that energy was the work done in initially moving
the block from x=0 to the far right position. When the block has
moved back to the x=0 position, the energy of the system is all
kinetic energy, derived from the velocity of the moving parts.
The total energy of the system after the initial displacement of
the block is complete remains constant until some outside agent,
like yourself, again interacts with the system to add or remove
energy.
Now we can expand the workenergy theorem to include potential
energy. The work done on a system is equal to the total change in
the system energy. The energy of a system may manifest itself as
potential energy, kinetic energy or some combination of the
two.
Wait a minute! Who said anything about a change in energy?
Weren't we talking about total energy here? Well, yes we
were, but the system we picked for our example began with the
spring unstressed and the block not moving. In other systems, the
'as found' state may include some potential energy and/or
kinetic energy. The choice of where is the zero potential energy
state is arbitrary. I can pick any configuration and measure from
there, plus or minus. The zero kinetic energy state depends not
only on the system but also on the frame of reference in which we
choose to measure velocities. It is differences in energy which
relate to the amount of work done on or by the system.



A
key assumption in our discussion of work and energy so far was
that there was no friction involved. The forces operating on the
system components were such that net amount of work done in any
round trip was zero. If I start with a particle at rest, exert a
force on it that accelerates it to the right, turn around and
exert a force that stops its rightward motion and accelerates it
to the left and finally bring it to a stop at its initial
position, the total amount of work done will be zero.
This "zero sum" condition follows from the fact that
work is the dot product of force and displacement. The sign of
the work will be positive whenever the force and displacement
vectors are separated by an angle of less than 90 degrees and
negative when they are separated by more than 90 degrees. Refer
to the dot product display if you need
to.
Forces that meet the zero sum criteria are called
"conservative forces". Forces like the force of
friction clearly do not meet this criteria because the direction
of the friction force is always opposite from the direction of
motion so the work of frictional forces is always negative.
Therefore the work cannot add up to zero in any round trip. Only
when we are dealing with conservative forces does it makes sense
to define a potential energy based on system configuration. In
the presence of nonconservative forces the same system
configuration might have different energies associated with it
depending on how many and what kind of excursions to other
configurations and back (round trips) had occurred.
So why call forces meeting the zero sum criteria conservative?
Another way to express the zero sum criteria is to say that after
the initial interaction, setting the system into motion, the
transfer of work across the system boundary is zero. In such a
system neither potential energy not kinetic energy may be added
to or removed from the system. Therefore the change in kinetic
energy plus the change in potential energy of the system is zero.
This may be stated in another way by ke+pe=constant. So the sum
of potential and kinetic energy is conserved. We refer to the sum
of potential and kinetic energy as the "mechanical
energy" of the system.
In the case of the block and spring thing, if we define our
system to be just the block and the spring, is it a conservative
system? If there were friction between the block and the rails,
the work of the frictional force would heat the rails,
transferring energy across the system boundary so it would not be
conservative. During the time we are applying a force to
initially stretch the spring we are transferring energy across
the system boundary so the system is not conservative then. Once
we turn the block loose there is no further transfer of energy
into or out of the system so the system is conservative. In many
dynamical systems the nonconservative forces are so small
compared to the conservative ones that they may be neglected. It
is in these almost conservative system which we may predict the
future using only energy considerations.

Next let's
take a look at another simple dynamical system from the point of
view of energy. Gravity near the surface of a planet causes
bodies to experience a force (have a weight) equal to their mass
m times the acceleration due to gravity g. The work to move a
particle from at rest on the ground to at rest at a higher
elevation is the force applied (weight) times the distance moved
(y). Since we are talking about no kinetic energy change in this
case, the work is the potential energy. So pe as a function of x
for this planet  particle system is pe(x)=m*g*y.
So let's go back to the
situation where we lifted a 20kg box of rocks to a height of
1.5 meters. In this case m*g*y = 20*9.8*1.5 = 294J. What will be
the speed of the box when it hits the ground if we drop it? To
answer that question on the basis of Newton's laws of motion
is quite an involved process. Based on what we know now about
energy, the potential energy due to position will be converted
entirely to kinetic energy at the instant the box hits the ground
so at that instant
1/2*m*v^{2} = 294J, or
v^{2} = 29.4(m/s)^{2}, or v =
5.42m/s
This is one example of the convenience of the energy approach to
dynamics problems.
Notice that in the previous problem we assumed conservative
forces only were in effect. That means that the path taken by the
box on its way to the ground can be anything, as long as the
conservative force rule is not violated. We could drop the box
into a frictionless chute that was shaped like cork screw and
when it came out at ground level, the velocity would be 5.24m/s,
be it a vertical velocity, or horizontal or some combination. A
complication of that nature would make the Newton's law
approach to the question impossible, or at least totally
impractical, but the energy balance view of the problem remains
as simple as ever.
When we write pe
as a function of y, as in pe=m*g*y, we are saying in effect that
associated with every point along the y dimension there is a
scalar number, pe. When each point in a space is associated with
a mathematical object, that space is said to have a
"field" in it, or on it. You may hear either way of
expressing it. In this case we have a scalar potential field.



Another relationship that might improve your intuition in
this area is that between the rate of change of pe with respect
to displacement and the force a system exerts on a particle.
Consider a conservative system, one in which no energy is
transferred across the system boundary. This means that the
change in kinetic energy, Dke, plus
the change in potential energy, Dpe,
equals zero. So
Dke + Dpe = 0 or Dke =
Dpe.
But we know that Dke = work done by
the system so Dpe = work done by the
system = f*Dx, where x is the
displacement and f is the force exerted on a particle by the
system. The change in pe then is
Dpe = f*Dx
where f is the conservative force acting on the particle. This
leads us to the relationship
f = Dpe(x) /
Dx
.
The force exerted on a particle is the negative of the rate of
change of potential energy with respect to x. The significance of
the minus sign is that f and Dx are in
the same direction pe decreases. When they are opposed, pe
increases. Remember here that f is the force of the system on the
particle. Run
Work on Spring
display
again with these ideas in mind. The negative of the slope of
the curve is the force on the block.
Now imagine that all we knew about a system was the values of
the potential energy at each point in the space. In other words
we had a field map but no knowledge of the mechanisms applying
forces to a particle. Could we analyze the motion on that basis
alone? Well of course we could or I would not be going on about
it. Let's take for example the spring and block affair. If
the spring was hidden from view and the only measurement we could
get was the amount of work required to move the block to any
position we could come up with the curve defined in the preceding
display. From that and the knowledge that
the force on the block at any x was 
Dpe / Dx we could recreate the
motion of the block. Or if we choose we could get the change in
ke directly from the change in pe and calculate the velocity
change from that.
In the next section of this course we will begin to look at
objects other than single particles.
Are there any questions?

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