Module 9  Potential Energy Graphs
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Introduction
Mechanical Energy is a scalar quantity (just a number that can be positive or negative). The Mechanical Energy of a system can be augmented or decreased by forces that do nonconservative work, e.g. any outside force, or a nonconservative internal force like friction or chemical reactions. Mechanical energy can be converted between potential and kinetic by conservative interactions internal to the system.
All of these considerations are embodied in the general formula,
where
.
In cases where the nonconservative work, , is zero, we say that "mechanical energy is conserved"  often when considering Newtonian Mechanics, people will sloppily say just "energy is conserved" instead.
Energy Bars
A simple way to visualize the above formula for the change of mechanical energy from an initial state to a final state is with energy bars. Bars are stacked up vertically, each representing one of the three terms on the right side of the equation. The total length of the stacked bars is the sum of the terms and is the total mechanical energy with the addition of the work (which could be negative).
The above figure is an example: on the left we show a case where the mechanical energy changes due to nonconservative work. On the right, energy is conserved because , so the only possible processes involve conservative work that shift mechanical energy between kinetic and potential. In both cases, the total (or final on the left} energy does not change.
Illustrative Example: Diagrams and Mechanical Energy
Part A: InitialState FinalState Diagrams  

Because the WorkEnergy Theorem and the principle Law of Change for the [Mechanical Energy, External Work, and Internal NonConservative Work] model involve only the initial and final energies of the system, it is useful to devote considerable attention to understanding the system's configuration at those times. It is customary to sketch the system in its initial and final configurations, labeling the quantities that are relevant for the kinetic and potential energies of the system.
Suppose we were asked to determine the speed of a 0.25 kg block launched vertically from rest by a spring of spring constant 500 N/m that was initially compressed 0.10 m from its natural length when the block reaches a height of 0.50 m above the natural position of the spring. How would you draw an initialstate finalstate diagram for this situation?

Part B: Energy Bar Graphs  

Drawing and labeling the [initialstate finalstate diagram] will give a good idea of where the [mechanical energy] is in the [system]. Thus, once the system is depicted, it is often useful to represent the energy distribution graphically as well. This is done using a *bar graph* that contains one bar for each distinct type of energy that will make up the mechanical energy of the system. This *bar graph* is not necessarily quantitative, but it should be drawn to reflect conservation of energy if the nonconservative work is zero.
Continuing with the example from Part A, we can see that in the initial state the energy is divided among spring potential and gravitational potential, with zero kinetic energy present. In the final state, the energy is divided among gravitational potential and kinetic energy, with the spring (returned to its natural position) contributing zero energy. Draw a bar graph representation of these distributions.

Part C: Dealing with NonConservative Work  

Initialstate finalstate diagrams and energy bar graphs are also useful for detecting problems in which mechanical energy is _not_ conserved. Consider the following example:
A 1500 kg roller coaster car is moving at a speed of 10 m/s moving with no friction when it crests the final hill at a height of 15 m above the end of the ride. It then descends to a level area of track that extends for 20 m before the finish. As soon as the car reaches the level track, it hits the brakes, coming to a stop exactly at the finish line. Determine the effective coefficient of friction experienced by the car during braking. Draw initialstate finalstate and energy bar diagrams for this situation.

Potential Energy Curves with Energy Conserved
Consider the motion of a single particle when it is acted on by conservative forces that are represented by the potential energy curve shown below:
Given that its total energy is fixed, the conservative forces transform potential energy to/from kinetic energy as it moves along. In the figure above, the potential energy is increasing for motion to the right, consequently the kinetic energy is decreasing. Assessment Question: How far will it move to the right in this case?
Turning Points, Equilibria
Given a potential energy curve U(x) (e.g. the one above) you can determine several important things about the motion of a single particle with total energy E_{t}ot.
Importantly, you also know the force on the particle at any point  it is determined by .
Turning Points and Allowed Regions of Motion
Since the kinetic energy goes to zero when U(x_{t}) = E_{tot}, the particle must come to a stop as it approaches x_{t}. In general the force will push in the direction it came from, so the particle will turn around there. Such points are therefore called classical turning points (or just turning points).
Equilibria
An equilibrium is where the force on a particle is zero. This means the particle has no acceleration, but in general it has finite kinetic energy so it will move beyond the equilibrium point. Zero force means that . Equilibria occcur whenever the potential has a horizontal region. These points x_{e} are called equilibrium points. There are two types of equilibria: stable and unstable.
Stable equilibrium: x_{e} is at a potential minimum, and therefore it will feel a force restoring it to x_{e} as it moves away from x_{e}. This is illustrated in the Figure:
Note that x_{e} is at a minimum of the potential. Be sure you can calculate the force curve that appears under the potential energy curve. (The positive derivative of the potential is shown dashed; hte force is its negative.)
Unstable equilibrium: x_{e} is at a potential maximum, and therefore a particle there will feel a force that pushes it away from x_{e} in the direction it has moved away already. Clearly the particle is not going to remain close to x_{e} for long (unless E_tot is such that x_{e} is also a turning point).