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 non-conservative work, e.g. any outside force, or a non-conservative 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,


E^{mech}_{f} = K_f + U_f = K_i + U_i + W^{NC}_{f,i}

where

 W_{f,i}^{NC} = \sum_{int} W^{NC} + \sum_{ext} W .

In cases where the non-conservative work, W_{f,i}^{NC}, 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).

EnergyBars.png

The above figure is an example: on the left we show a case where the mechanical energy changes due to non-conservative work. On the right, energy is conserved because W_{f,i}^{NC} = 0, 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

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:

PECurveWithBars.png

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 Etot.

Importantly, you also know the force on the particle at any point - it is determined by F_x=-\frac{dU}{dx}.

Turning Points and Allowed Regions of Motion

Since the kinetic energy goes to zero when U(xt) = Etot, the particle must come to a stop as it approaches xt. 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 0 = F(x_e) = -\frac{dU}{dx}. Equilibria occcur whenever the potential has a horizontal region. These points xe are called equilibrium points. There are two types of equilibria: stable and unstable.

Stable equilibrium: xe is at a potential minimum, and therefore it will feel a force restoring it to xe as it moves away from xe. This is illustrated in the Figure:


PEForceStableEquilibrium.png

Note that xe 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: xe is at a potential maximum, and therefore a particle there will feel a force that pushes it away from xe in the direction it has moved away already. Clearly the particle is not going to remain close to xe for long (unless E_tot is such that xe is also a turning point).

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