Potential Energy and Fields

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 isDw = 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 which 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 magenta lines as the potential energy stored in the spring and the cyan lines 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 work-energy 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 which 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 can not 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 non-conservative 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 non-conservative 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 instant1/2*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 which will 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. SoDke + 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. Remember that the magenta lines trace out the curve of potential energy vs position. The slope of this 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 by the magenta lines in the preceeding 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.