The Work-Energy Relationship
Student Extras
Need to see it? View the Energy Transformation on a Roller Coaster animation from the Multimedia Physics Studios.
The Physics Classroom: Calculator Pad: Mechanics: Work, Energy, and Power problems
Improve your problem-solving skills with problems, answers and solutions from The Calculator Pad.
PhET Simulation: Energy Skate Park
Explore the transformation of energy between different forms with The Skate Park simulation from PHET.
Flickr PhysicsVisit The Physics Classroom's Flickr Galleries and enjoy a visual overview of the topic of work, energy and power.
Flickr PhysicsVisit The Physics Classroom's Flickr Galleries and take a virtual ride through the topic of roller coaster physics.
PBS TeachersThis Study Guide from PBS combines video and questions to guide students to an understanding of energy conservation.
Roller Coaster Model
Explore energy conservation and forces for a variety of roller coaster designs at with this downloadable applet from Open Source Physics (OSP).
Modular Approach to Physics: Simple Harmonic Motion - Weighted Spring
Explore the transformation of KE to PE (and back) for a vibrating mass on a spring. Manipulate the spring constant, amplitude and mass and observe the effect on the resulting motion.
Virtual Laboratory: Potential Energy
Explore the physics of a bouncing ball with this interactive Java applet of a bouncing ball.
Teacher's Guide
Engage your students with The Skate Park simulation from PHET.
Problem Based Learning: Bungee Jumping
Try something new with this problem based learning activity involving bunjee jumping.
PBS TeachersView a lesson plan from PBS Teachers on the topic of mechanical energy conservation.
The LaboratoryLooking for a lab that coordinates with this page? Try the Energy on an Incline Lab from The Laboratory. Requires motion detectors and force probes
The LaboratoryLooking for a lab that coordinates with this page? Try the Spring Energy Lab from The Laboratory.
The LaboratoryLooking for a lab that coordinates with this page? Try the Energy of a Pendulum Lab from The Laboratory.
Curriculum CornerPractice makes perfect with this work-energy problem-solving activity from The Curriculum Corner.
Treasures from TPFNeed ideas? Need help? Explore The Physics Front's treasure box of catalogued resources for teaching about energy.
Curriculum CornerLearning requires action. Give your students this sense-making activity from The Curriculum Corner.
Treasures from TPFThe Physics Front's has catalogued a collection of interactive energy activities from across the web. Try 'em; you'll like 'em.
Roller Coaster Model
Explore energy conservation and forces for a variety of roller coaster designs at Open Source Physics (OSP). Modify designs and observe the results. Comes with classroom-ready materials.
Pendulum Energy Model
This downloadable interactive simulation from Open Source Physics (OSP) models the motion of a pendulum. The simulation combines numerical outputs and bar chart illustrations to demonstrate energy conservation for a pendulum. Includes classroom-ready handouts.
Modular Approach to Physics: Simple Harmonic Motion - Weighted Spring
This highly interactive applet depicts energy conservation for a mass on a spring. The mass, amplitude and spring constant of the spring can be modified. Vector displays and energy bar charts accompany the animation.
Physics of JumpingA collection of information, theory, experiments and movies associated with the use of a Springbok to model the physics of jumping.
Virtual Laboratory: Potential Energy
An interactive Java applet of a bouncing ball. Initial height, ball mass and degree of elasticity can be altered. Accompanied by guiding questions.
Analysis of Situations in Which Mechanical Energy is Conserved
It has previously been mentioned that there is a relationship between work and mechanical energy change. Whenever work is done upon an object by an external force (or nonconservative force), there will be a change in the total mechanical energy of the object. If only internal forces are doing work (no work done by external forces), then there is no change in the total amount of mechanical energy. The total mechanical energy is said to be conserved. In this part of Lesson 2, we will further explore the quantitative relationship between work and mechanical energy in situations in which there are no external forces doing work.
The quantitative relationship between work and the two forms of mechanical energy is expressed by the following equation:
KEi +
PEi + Wext = KEf +
PEf
The equation illustrates that the total mechanical energy (KE + PE) of the object is changed as a result of work done by external forces. There are a host of other situations in which the only forces doing work are internal or conservative forces. In such situations, the total mechanical energy of the object is not changed. The external work term cancels from the above equation and mechanical energy is conserved. The previous equation is simplified to the following form:
KEi +
PEi = KEf +
PEf
In these situations, the sum of the kinetic and potential energy is everywhere the same. As the potential energy is increased due to the stretch/compression of a spring or an increase in its height above the earth, the kinetic energy is decreased due to the object slowing down. As the potential energy is decreased due to the return of a spring to its rest position or a decrease in height above the earth, the kinetic energy is increased due to the object speeding up. We would say that energy is transformed or changes its form from kinetic energy to potential energy (or vice versa); yet the total amount present is conserved - i.e., always the same.
The Example of Pendulum Motion
The
tendency of an object to conserve its
mechanical energy is observed whenever external forces are
not doing any overall work. If the influence of
friction and air resistance can be ignored (or assumed to be
negligible) and all other external forces are absent or
merely not doing work, then the object is often said to
conserve its energy. Consider a pendulum bob swinging to
and fro on the end of a string. There are only two
forces acting upon the pendulum bob. Gravity (an internal
force) acts downward and the tensional force (an external
force) pulls upwards towards the pivot point. The external
force does not do work since at all times it is
directed at a 90-degree angle to the motion. (Review
a previous page to convince yourself that F•d•cosine angle =
0 J for the force of tension.)
As the pendulum bob swings to and fro, its height above the tabletop (and in turn its speed) is constantly changing. As the height decreases, potential energy is lost; and simultaneously the kinetic energy is gained. Yet at all times, the sum of the potential and kinetic energies of the bob remains constant. The total mechanical energy is 6 J. There is no loss or gain of mechanical energy, only a transformation from kinetic energy to potential energy (and vice versa). This is depicted in the diagram below.
As
the 2.0-kg pendulum bob in the above diagram swings to and
fro, its height and speed change. Use energy equations and
the above data to determine the blanks in the above diagram.
Click to view answers.
A common Physics lab involves the analysis
of a pendulum in its back and forth motion. The
transformation and conservation of mechanical
energy is the focus of the lab. 
A
0.200-kg (200 gram) pendulum is typically released from rest
at location A. The bob passes through a photogate at
location B and another photogate at location C. The speed of
the pendulum bob can be determined from the width of the bob
and the photogate times. The speed and mass can be used to
determine the kinetic energy of the
bob at each of the three locations. The heights of the bob
above the tabletop at each of the three locations can be
measured and used to determine the potential energy of the
bob. The data should reflect that the mechanical
energy changes its form as the bob passes from
location A to B to C. Yet the total mechanical
energy should remain relativity constant. Sample data
for such a lab are shown below.
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The sample data show that the pendulum bob loses potential energy as it swings from the more elevated location at A to the lower location at B and at C. As this loss of potential energy occurs, the pendulum bob gains kinetic energy. Yet the total mechanical energy remains approximately 0.785 Joules. We would say that total mechanical energy is conserved as the potential energy is transformed into kinetic energy.
The Example of a Roller Coaster
A roller coaster operates on this same principle of energy transformation. Work is initially done on a roller coaster car to lift to its initial summit. Once lifted to the top of the summit, the roller coaster car has a large quantity of potential energy and virtually no kinetic energy (the car is almost at rest). If it can be assumed that no external forces are doing work upon the car as it travels from the initial summit to the end of the track (where finally an external braking system is employed), then the total mechanical energy of the roller coaster car is conserved. As the car descends hills and loops, its potential energy is transformed into kinetic energy as the car speeds up. As the car climbs up hills and loops, its kinetic energy is transformed into potential energy as the car slows down. Yet in the absence of external forces doing work, the total mechanical energy of the car is conserved.
Conservation of energy on a roller coaster ride means that the total amount of mechanical energy is the same at every location along the track. The amount of kinetic energy and the amount of potential energy is constantly changing. Yet the sum of the kinetic and potential energies is everywhere the same. This is illustrated in the diagram below. The total mechanical energy of the roller coaster car is a constant value of 40 000 Joules.
The motion of a ski jumper is also governed by the transformation of energy. As a ski jumper glides down the hill towards the jump ramp and off the jump ramp towards the ground, potential energy is transformed into kinetic energy. If it can be assumed that no external forces are doing work upon the ski jumper as it travels from the top of the hill to the completion of the jump, then the total mechanical energy of the ski jumper is conserved. Consider Li Ping Phar, the esteemed Chinese ski jumper. She starts at rest on top of a 100-meter hill, skis down the 45-degree incline and makes a world record setting jump. Assuming that friction and air resistance have a negligible affect upon Li's motion and assuming that Li never uses her poles for propulsion, her total mechanical energy would never change.
Of course it should be noted that the original assumption that was made for both the roller coaster car and the ski jumper is that there were no external forces doing work. In actuality, there are external forces doing work. Both the roller coaster car and the ski jumper experience the force of friction and the force of air resistance during the course of their motion. Friction and air resistance are both external forces and would do work upon the moving object. In fact, the presence of friction and air resistance would do negative work and cause the total mechanical energy to decrease during the course of the motion. While the assumption that mechanical energy is conserved is an invalid assumption, it is a useful approximation that assists in the analysis of an otherwise complex motion.