Resource Lesson
Potential Energy Functions
Printer Friendly Version
In physics, the use of the term mechanical energy usually involves three types of energies: potential gravitational energy, kinetic energy, and elastic potential energy. Although potential energy is often represented by the expression PE, in this lesson we will use the variable U; similarly, kinetic energy will be represented by the variable K.
In the absence of nonconservative, or dissipative forces, these energies obey the law of conservation of energy, or ΔU + ΔK = 0. That is, when a system is only acting under the influence of conservative forces its total energy content never changes, the energy just converts between forms.
Let's practice with an example of a problem using conservation of energy and a potential energy graph.
Refer to the following information for the next five questions.
A 5kg mass moving along the xaxis passes through the origin with an initial velocity of 3 m/sec. Its potential energy as a function of its position is given in the graph shown below.
How much total energy does the mass have as it passes through the origin?
Between 2.5 meters and 5 meters is the mass gaining speed or losing speed?
How fast is it moving at 7.5 meters?
How much potential energy would have to be present for the mass to stop moving?
At what position would this occur if the graph were to continue past 15 meters?
Force and Position Relationships
To look into this more carefully, let's reexamine some important graphs for a vibrating spring.
Notice that the position and acceleration/force graphs are 180º outofphase: when the spring's displacement from its equilibrium is UP, the restoring force and acceleration are DOWN and viceversa.
Our next graphs draw our attention to the spring's displacement, energy modes and restoring force.
Notice that
when the spring is either in a state of maximum extension or compression its potential energy is also a maximum
when the spring's displacement is DOWN the restoring force is UP
when the potential energy function has a negative slope, the restoring force is positive and viceversa
when the restoring force is zero, the potential energy is zero
at any point in the cycle, the total energy is constant, U + K = U
_{max}
= K
_{max}
Force Functions
Our next step will be to show that a function representing the instantaneous values of the restoring force can be expressed as the negative of the derivative of our potential energy function
Remember our two relationships involving work
The work done by a conservative force decreases an object's potential energy while it is increasing its kinetic energy
Defining the initial potential energy U
_{o}
= 0, gives us
Using the calculus, we see that our desired expression of the instantaneous restoring force being equal to the negative derivative of the potential energy function.
Let's practice this relationship with an example.
Refer to the following information for the next three questions.
Based on the following potential energy graph, calculate the force acting on the system during each segment.
0
£
x
<
1
1
<
x
<
3
3
<
x
£
6
segment
U(x)
0 ≤ x < 1
1 < x < 3
3 < x ≤ 6
4  3.5x
½(x  2)
^{2}
½x  1
States of Equilibrium
On a potential energy graph, when the function's derivative is equal to zero, then the net force acting on the system is equal to zero. When an object is located at one of these positions or in one of these regions it is said to be in a
state of equilibrium
: stable, unstable, dynamic, and static (or neutral).
On the following diagram,
x
_{3}
and
x
_{5}
are points of
stable equilibrium

bowls
. If the system is slightly displaced to either side the forces on either side will return the object back to these positions.
x
_{4}
is a position of
unstable equilibrium

a crest or peak
. If the object is displaced ever so slightly from this position, the internal forces on either side will act to encourage further displacement instead of returning it back to x
_{4}
.
x
_{6}
is a position of either
dynamic or static (neutral) equilibrium

a plateau
. Since there is no net force acting on the object it must either possess only potential energy and be at rest or, it also possesses kinetic energy and must be moving at a constant velocity.
If a mass were to be released from rest where the grey line meets our potential energy curve, why would the region marked off by the grey line be called an
energy well
?
Related Documents
Lab:
Labs 
2Meter Stick Readings
Labs 
A Battering Ram
Labs 
A Photoelectric Effect Analogy
Labs 
Acceleration Down an Inclined Plane
Labs 
Addition of Forces
Labs 
Air Track Collisions
Labs 
Ballistic Pendulum
Labs 
Ballistic Pendulum: Muzzle Velocity
Labs 
Bouncing Steel Spheres
Labs 
Circumference and Diameter
Labs 
Collision Pendulum: Muzzle Velocity
Labs 
Conservation of Energy and Vertical Circles
Labs 
Conservation of Momentum in TwoDimensions
Labs 
Cookie Sale Problem
Labs 
Density of a Paper Clip
Labs 
Determining the Distance to the Moon
Labs 
Determining the Distance to the Sun
Labs 
Eratosthenes' Measure of the Earth's Circumference
Labs 
Force Table  Force Vectors in Equilibrium
Labs 
Home to School
Labs 
Indirect Measurements: Height by Measuring The Length of a Shadow
Labs 
Indirect Measures: Inscribed Circles
Labs 
Inelastic Collision  Velocity of a Softball
Labs 
Inertial Mass
Labs 
Introductory Simple Pendulums
Labs 
Lab: Rectangle Measurements
Labs 
Lab: Triangle Measurements
Labs 
LooptheLoop
Labs 
Marble Tube Launcher
Labs 
Quantized Mass
Labs 
Ramps: Sliding vs Rolling
Labs 
Roller Coaster, Projectile Motion, and Energy
Labs 
Rotational Inertia
Labs 
Rube Goldberg Challenge
Labs 
Spring Carts
Labs 
Target Lab: Ball Bearing Rolling Down an Inclined Plane
Labs 
The Size of the Moon
Labs 
The Size of the Sun
Labs 
Video Lab: Blowdart Colliding with Cart
Labs 
Video LAB: Circular Motion
Labs 
Video Lab: Falling Coffee Filters
Labs 
Video Lab: M&M Collides with Pop Can
Labs 
Video Lab: Marble Collides with Ballistic Pendulum
Resource Lesson:
RL 
APC: Work Notation
RL 
Basic Trigonometry
RL 
Basic Trigonometry Table
RL 
Conservation of Energy and Springs
RL 
Curve Fitting Patterns
RL 
Dimensional Analysis
RL 
Energy Conservation in Simple Pendulums
RL 
Gravitational Energy Wells
RL 
Linear Regression and Data Analysis Methods
RL 
Mechanical Energy
RL 
Metric Prefixes, Scientific Notation, and Conversions
RL 
Metric System Definitions
RL 
Metric Units of Measurement
RL 
Momentum and Energy
RL 
Principal of Least Action
RL 
Properties of Lines
RL 
Properties of Vectors
RL 
Rotational Dynamics: Pivoting Rods
RL 
Rotational Kinetic Energy
RL 
Significant Figures and Scientific Notation
RL 
Springs and Blocks
RL 
Symmetries in Physics
RL 
Tension Cases: Four Special Situations
RL 
Vector Resultants: Average Velocity
RL 
Vectors and Scalars
RL 
Work
RL 
Work and Energy
Review:
REV 
Honors Review: Waves and Introductory Skills
REV 
Physics I Review: Waves and Introductory Skills
REV 
Test #1: APC Review Sheet
Worksheet:
APP 
Puppy Love
APP 
The Dognapping
APP 
The Jogger
APP 
The Pepsi Challenge
APP 
The Pet Rock
APP 
The Pool Game
APP 
War Games
CP 
Conservation of Energy
CP 
Inverse Square Relationships
CP 
Momentum and Energy
CP 
Momentum and Kinetic Energy
CP 
Power Production
CP 
Sailboats: A Vector Application
CP 
Satellites: Circular and Elliptical
CP 
Tensions and Equilibrium
CP 
Vectors and Components
CP 
Vectors and Resultants
CP 
Vectors and the Parallelogram Rule
CP 
Work and Energy
NT 
Cliffs
NT 
Elliptical Orbits
NT 
Escape Velocity
NT 
Gravitation #2
NT 
Ramps
NT 
Satellite Positions
WS 
Advanced Properties of Freely Falling Bodies #1
WS 
Advanced Properties of Freely Falling Bodies #2
WS 
Advanced Properties of Freely Falling Bodies #3
WS 
Calculating Vector Resultants
WS 
Charged Projectiles in Uniform Electric Fields
WS 
Circumference vs Diameter Lab Review
WS 
Data Analysis #1
WS 
Data Analysis #2
WS 
Data Analysis #3
WS 
Data Analysis #4
WS 
Data Analysis #5
WS 
Data Analysis #6
WS 
Data Analysis #7
WS 
Data Analysis #8
WS 
Density of a Paper Clip Lab Review
WS 
Dimensional Analysis
WS 
Energy Methods: More Practice with Projectiles
WS 
Energy Methods: Projectiles
WS 
Energy/Work Vocabulary
WS 
Force vs Displacement Graphs
WS 
Frames of Reference
WS 
Graphical Relationships and Curve Fitting
WS 
Indirect Measures
WS 
Introduction to Springs
WS 
Kinematics Along With Work/Energy
WS 
Lab Discussion: Inertial and Gravitational Mass
WS 
Mastery Review: Introductory Labs
WS 
Metric Conversions #1
WS 
Metric Conversions #2
WS 
Metric Conversions #3
WS 
Metric Conversions #4
WS 
Potential Energy Functions
WS 
Practice: Momentum and Energy #1
WS 
Practice: Momentum and Energy #2
WS 
Practice: Vertical Circular Motion
WS 
Properties of Lines #1
WS 
Properties of Lines #2
WS 
Rotational Kinetic Energy
WS 
Scientific Notation
WS 
Significant Figures and Scientific Notation
WS 
Static Springs: The Basics
WS 
Work and Energy Practice: An Assortment of Situations
WS 
Work and Energy Practice: Forces at Angles
TB 
Work, Power, Kinetic Energy
TB 
Working with Vectors
TB 
Working with Vectors
REV 
Math Pretest for Physics I
PhysicsLAB
Copyright © 19972020
Catharine H. Colwell
All rights reserved.
Application Programmer
Mark Acton