Science 122 Program 19 Work & Energy

©1998 RCBrill. All rights reserved

Work & Energy
Program 19
Lesson 3.4

Text References

Spielberg & Anderson 111-118
Booth & Bloom 143-152

Coming Up


1. Introduction

2. Energy

3. Work

4. Power

5. Kinetic Energy

6. Potential Energy

7. Work/Energy Theorem

8. Conservative Systems

9. Nonconservative Systems

10. Swinging Balls Revisited

11. Summary & Conclusions


Here are the objectives for today's lesson.

Before you begin to study the lesson, take a few minutes to read the objectives and the study questions for this lesson.

Look for key words and ideas as you read. Be sure to read these objectives in the study guide and refer to them as you study the lesson.

Focusing on the learning objectives will help you to study, to understand the important concepts and to synthesize.

Compare the objectives with the study questions for the lesson to be sure that you have the concepts under control.

1. Define the physical concept of work and apply it to various situations.
2. Distinguish between work and power.
3. Define kinetic energy and its variables.
4. Define potential energy and its variables.
5. State the work energy theorem and define its applications.
6. Define the characteristics of conservative and non conservative systems
7. Apply conservation of energy to various physical situations


1. Distinguish between work and power.
2. Define work as it is used in physical science and give an example of work.
3. Is work being done when a heavy weight is held motionless overhead? Explain.
4. How is power different from work or energy?
5. When we say that work is done by or against some agent, what do we mean? Name some "agents'>
6. What do we mean by conservative forces? Cite some examples.
7. What is the work/energy theorem?
8. How is energy different from momentum and inertia?
9. Define energy?
10. What do we mean when we say, "The concept of energy would be useless if it were not conserved?"
11. Compare kinetic and potential energy using the appropriate physical quantities?
12. How do we know whether or not an object has energy?
13. Discuss conservation of energy using the pendulum as an example.
14. What do we mean when we speak of a "conservative system?"
15. How do we know that energy is conserved when an object falls under the influence of gravity in the absence of air friction?

1. Introduction

In this lesson we will introduce the concept of work as the result of forces which cause motion. We will see that the result of these forces depends on the situation, whether the forces are applied horizontally or vertically and the degree to which friction interferes with the motion.

We will distinguish energy from momentum as we see that energy is what something acquires as the result of work being done, sometimes. With a precise physical definition of work and a concept of two different types of mechanical energy, we will explore the "sometimes".

The work/energy theorem shows a definite and specific relationship between work and energy which is useful in a bewildering variety of situations.

Near the end of the lesson we will examine the distinction between conservative and nonconservative forces and the apparent loss of energy in the latter.

Finally we will revisit the swinging balls and see that the missing requirement is the conservation of energy.

1.1. Newton's Laws are correct but inadequate for certain kinds of processes and interactions

1.1.1. momentum describes collisions, but not completely
1.1.2. heat can be integrated into and explained by mechanical theory studied since Aristotle little understanding of principles nothing new added until 19th century using Newtonian paradigm concept of particles in motion will unite mechanics and atomic theory

1.2. History is complex

1.2.1. No single individual is responsible for theory developed over 150 year period

1.2.2. Historical accident that Newton developed laws in terms of kinematics Newton's efforts were directed towards explaining planetary motion Newton's Laws can be derived from conservation laws

1.2.3. Conservation of momentum proposed by Hooke, Wren, Huygens contemporaries of Newton Newton's work provided framework (paradigm) used to formally derive concept later

1.2.4. Galileo speed depends on height rolls up to same height as rolled from inertia is special case of energy conservation

1.2.5. Others Newton defined "vis insita" or innate force of matter Huygens: mv2 is conserved in certain collisions Leibnitz called it "vis viva" (living force) later combined with Newton's Laws and Galileo's Kinematic equations

2. Energy

Energy is an abstract concept. Whether or not such a thing as energy "exists" outside the mind is one we will not address. We will note that it is a very useful concept, it provides the needed explanation for interactions such as the swinging ball, no violation of the conservation concept has never been observed, it leads to understanding other phenomena linked to heat which we will study in later programs.

Does it exist? Who cares. It is useful and consistent in describing certain interactions. It fits all of the definitions of a good theory as stated in Program 9.

Energy can be recognized and quantified by its effect on matter, and in fact is only useful because it is conserved. Not only that, but the concept, or something like it seems to be necessary to understand the universe.

Although the concepts of work and energy were developed from Newton's laws, it is apparent that the Laws can be derived from the definition of work and energy. From the conservation law even Galileo's kinematic relationships can be derived. It is apparent that the two methods are equivalent, but different, ways of looking at the same quantities of distance, acceleration, velocity, force, mass, and time.

For practical purposes, we use whichever method suits itself to the problem at hand. For theoretical understanding we observe the mathematical equivalence and move on to higher order abstractions.

2.1. Concept is abstract
2.2. something with energy can exert force or move things
2.3. ENERGY cannot be seen or measured directly, but amount can be calculated
2.4. ENERGY is recognized by its effect on matter

2.4.1. measured by work done
2.4.2. work has precise physical definition

2.5. ENERGY is a useful concept only because of conservation

2.5.1. represents stored work
2.5.2. could not define unless conserved
2.5.3. exists in many different forms
2.5.4. objects exchange energy through forces when interacting
2.5.5. amount remains constant during change

2.6. ENERGY is necessary to understand universe

2.6.1. more laws allow increased prediction mathematical law = relationship relationship = equation more equations => more variables can be incorporated into systems more variables => more complex and therefore more realistic situations

2.6.2. is especially powerful when combined with momentum conservation

3. Work

Work is defined simply as a force which is applied through a displacement, or distance. This is one of the simplest definitions in all of physics, but also one of the most useful. We will see the result, or effect of work later in the lesson when we consider the relationships between work and energy.

Work is done against some agent, such as gravity, the stiffness of a spring, inertia, or friction. It is independent of time, meaning that there are no restrictions on how fast or slow the work is done.

3.1. Force times distance (Fd)

3.1.1. can double work by doubling either force or distance
3.1.2. precise and limited definition note contrast with everyday usage reason for definition is to relate to energy must be consistent to be meaningful

3.1.3. work is only done if motion is involved no work is done by stationary force component of force in direction of motion does work

3.1.4. 1 joule = 1 Newton x 1 meter 1 J = 1 N*M

3.2. Work is done by or against some agent

3.2.1. inertia, gravity, friction, elasticity, electric force, magnetic force, etc.

3.3. Work is independent of time

4. Power

Power and work are often confused with one another or used interchangeably. The difference is one of time. The rate at which work is done is power. Intuitively, the faster a weight is lifted overhead the more power is consumed. Not so obvious is that the force applied and the work done do not depend on the speed in any way.

Power is also an electrical term, providing an important connection between the mechanical world and the electrical world. That we can describe electricity in mechanical terms gives us faith that our techniques are indeed universal and apply to more than just a small sample of the world.

4.1. Rate at which work is done
4.2. 1 watt = 1 joule per second

4.2.1. 1 W = 1 J/s

4.3. also electrical term

4.3.1. 1 watt = 1 volt x 1 ampere 1 kilowatt = 1000 watts 1 kilowatt hour = 1000 watts for one hour 1000 J/s x 3600 s = 3,600,000 J

4.3.2. shows correspondence between mechanical and electrical systems
4.3.3. Newtonian paradigm extends into study of electricity

5. Kinetic Energy

Kinetic energy is the energy of motion. It is the result of work being done against inertia.

We will begin studying this and other forms of mechanical energy in the ideal state, that is in the absence of friction. Recall that Galileo used a similar idealization to arrive at the principle of inertia, so we are justified in thinking in "frictionless" terms as long as we don't forget that it is really there in all real situations.

It is easy to see how to calculate the kinetic energy possessed by a moving mass by combining Newton's second law with Galileo's kinematics.

We should begin to see that kinetic energy is one way of storing work in the form of motion. Although kinetic energy looks intuitively like momentum, it is not the same. Yes, both momentum and kinetic energy involve mass and velocity, but the relationships are different and they are really quite different thing. One important difference is that, while there is only one form of momentum, there are many forms of energy. This means that, although one moving object may transfer its momentum to another, the momentum cannot change form in the same object. Energy can do that, and it leads to all kinds of interesting results logically and mathematically.

5.1. energy of motion

5.1.1. Kinetic energy equals one half m v squared

5.2. work done against inertia
5.3. moving object is capable of exerting force

5.3.1. force is required to give motion to the cart
5.3.2. amount of force depends on speed and mass
5.3.3. distance over which force can be applied depends on the magnitude of force and the amount of work done on the cart

5.4. derivation (from Galileo's kinematics and Newton's laws)

5.4.1. assuming no friction
5.4.2. Do not panic if you don't get this

This is included only to show that the "formula" for kinetic energy is not arbitrary. It is derived from the question: what is the result of applying a certain force horizontally over a given distance if the object in question gains speed on a level surface according to Newton's laws and Galileo's kinematics?

5.5. in words

An object of a given mass will acquire a certain velocity when accelerated by a given force for some specified distance. When the force is no longer applied, the object behaves according to the first law, maintaining a constant speed in a straight line until another force acts to stop it. The stopping force may involve a different combination of force and distance than that required to give it its velocity in the first place.

The work done against the inertia of the cart is stored in the motion of the object, to be used at will, exerting an unspecified force for some unspecified distance as long as the product of force and distance does not exceed the amount of work done on the cart in the first place.

5.6. Kinetic energy is a way of storing work in the form of motion

5.6.1. not just force, but a certain relationship between force and distance
5.6.2. not the same as momentum hard to see difference since both are involved in any change in motion inertia (mass) in motion possesses both kinetic energy and momentum Newton's vis in sita has two components momentum is acquired by impulse (Ft) second law: impulse = change in momentum kinetic energy is acquired by work (Fd) kinetic energy divided by momentum equals velocity

5.7. Kinetic Energy is only one form of energy

5.7.1. there is only one form of momentum important difference between the two concepts impulse changes momentum

5.7.2. Work can be done without changing kinetic energy lifting against gravity stretching or compressing spring pushing or pulling against electric or magnetic force sliding at constant speed against friction

5.7.3. Something changes when work is done if not kinetic energy then what?

6. Potential Energy

Potential energy results when work is done against certain kinds of forces which are known as "conservative" forces. Gravity, elasticity, electric forces and magnetic forces are examples of conservative forces.

We define gravitational potential energy as the stored work done against a conservative force. Usually this is related to the change of location of one object in relation to the force required to move it a certain distance.

With this definition is it easy to note than an object acquires gravitational potential energy when a force (equal to its weight) is used to lift it to a certain height. It then possesses something it didn't have before, that is the ability to do work.

For example if a weight is allowed to settle slowly on the diameter of a wheel, the wheel can turn and do work. This is the principle of the water mill which has been used for centuries to grind grain.

The potential energy of the weight can be changed to kinetic energy is the weight is allowed to freefall. AS IT LOSES POTENTIAL ENERGY (gets closer to the ground) IT GAINS KINETIC ENERGY (gains speed in freefall). It is obvious in a quantitative sense that this is true. Even more interesting is that it can be demonstrated that the LOSS OF POTENTIAL ENERGY IS EXACTLY EQUAL TO THE GAIN OF KINETIC ENERGY. This is once again assuming the frictionless case, but we already know that the concept of freefall generalizes to the case of no air friction.

6.1. work done against conservative forces

6.1.1. gravity, springs, electricity, magnetism

6.2. Gravitational Potential Energy

6.2.1. this equation simply shows that work done equals energy gained
6.2.2. E = mgh mg is the weight of a given mass and represents the force necessary to lift it h is the vertical distance through which the upwards force is applied to counter the weight

6.2.3. work done against gravity is stored in position of object

6.3. an object in freefall will acquire kinetic energy as it loses potential energy

6.3.1. can be shown that amount of kinetic energy gained exactly equals amount of potential energy lost (in the absence of friction)

Conservation of Energy

Kinematic Equations

Note that we have used the equivalent symbols for acceleration ("a" for the general case, "g" for gravitational) and distance ("x" for the general case, "h" for "height") but the two relationships are identical. So we must conclude from this either:

Newton's laws and Galileo's kinematic equations are BOTH INCORRECT.


Energy is CONSERVED in falling objects.

Either we give up on Galileo's definitions of motion AND Newton's laws, or we accept that energy is conserved in falling objects. Since we don't want to reject the work of Galileo and Newton, we choose to state that ENERGY IS CONSERVED IN FALLING OBJECTS.

7. Work/Energy Theorem

The work/energy theorem is a simple statement which relates work to energy in a simple way. The work/energy theorem simply states that the total amount of work done is equal to all changes in energy. This is easy when the only two forms of energy are potential and kinetic. It is even easy to include real-life friction in this statement. When doing so, we can simply see friction as "eating" some of the energy and therefore reducing the amount available to be converted from potential to kinetic or vise-versa.

It is important to not that it is CHANGES in energy which are significant, not the absolute amount of energy possessed by an object. By CHANGES we mean "gains or losses". Here we see that the work energy theorem is telling us that for every gain or loss in energy the re is a complimentary loss or gain somewhere else, or else work is done as a result.

7.1. work done = total change in energy

7.1.1. (read "the change in energy equals work done"). Work done results in a change in energy somewhere in the system and a change in energy requires work to be done by or against some agent..

7.1.2. work done causes an change in the total energy of all kinds
7.1.3. theorem defines an equation which accounts for all types of energy

7.2. changes or differences in energy are important, not the absolute amount

7.2.1. speed doesn't kill, it's the sudden stop (rapid change in energy)
7.2.2. being on top of a tall building (having lots of potential energy) won't hurt you unless you fall
7.2.3. initial and final states are important initial state need not be at rest work is done in changing speed from 0 to 30 mph work is also done in changing speed from 30 to 60 mph, but not the same amount as from 0 to 30 rock hits windshield vs. windshield hits rock: equivalent carrying a box up one flight of stairs requires the same amount of energy regardless of which floor it started from

8. Conservative Systems:

A conservative system is an idealized system in which no work is done. Specifically it is a system in which the total energy change is zero.

Remember that energy can be transferred between objects, but can also be transformed within a single object. Although there are no truly conservative systems, many systems in nature approximate the conservative system closely enough to deserve consideration.

We want to look at two different kinds of conservative systems, those in which energy is transformed (changed from one form to another) in a single object, and those in which energy is transferred (from one object to another.)

8.1. defined as a system in which the total energy change is zero

8.1.1. (read as "delta" E equals zero meaning "the change in E is zero)
8.1.2. loss of one form of energy results in a gain in energy somewhere else within the system different form of energy energy given to another object

8.2. non collisions

8.2.1. falling objects without friction loss of potential energy equals gain in kinetic energy the sum of kinetic and potential energy, called total mechanical energy, remains constant mgh = 1/2 mv2 (see table above)

8.2.2. pendulum

The pendulum may be visualized as a mechanical system which continually converts between kinetic and potential energy. In the ideal case no energy is lost and so at any given time the total energy is the sum of the kinetic and potential energy. It should be obvious that the pendulum gains kinetic energy (related to speed) as it loses potential energy (related to height) and vice-versa. At its highest point all the energy is in the form of potential while at its lowest point all the pendulum's energy is in the form of kinetic. work is not path dependent
In the absence of friction no work is done in moving an object horizontally. The only work that is done is related to the change in potential energy. Path independence means that we can move an object such as the pendulum in a series of short horizontal steps (as in the "infinitely small" horizontal movements of the circle) or as a single horizontal displacement followed by a single vertical displacement.

In the diagram at left no work is done moving an object along a horizontal direction when there is no friction (recall Galileo's principle of inertia. No force is required to keep an object moving. A small amount of work is necessary to start it moving and an equal amount is "given back" when it is stopped.)

Whether the motion is circular (as with the pendulum), up a series of steps, or in one horizontal movement followed by lifting the height h, the work done is the same to raise the object to a height h.

This is what we mean by "Path Independence". work done is the same for a given displacement in the absence of friction strobe photo of pendulum rises to same height on either side like Galileo's ball on the incline graph of energy of pendulum

8.2.3. others mass/spring ball in valley planet in orbit electron in atom simple machines lever, inclined plane, hydraulic jack

8.3. collisions

8.3.1. kinetic energy conserved only in elastic collisions elastic collision <==> kinetic energy conserved inelastic collisions involve frictional "losses" certain collisions are nearly elastic steel ball colliding with steel ball for example

8.3.2. can use with conservation of momentum to predict final state of motion

9. Nonconservative systems:

All real systems are actually nonconservative, that is they are systems where mechanical energy does not remain constant. In these systems energy is added from outside the system or escapes from the system. Mechanical energy refers to kinetic and potential energy specifically.

In real mechanical systems there is friction, which creates forces. Those friction forces, when applied to moving objects, amount to small amounts of work being done at the expense of the mechanical energy of the system.

At first glance this would appear to violate the conservation principle, but it doesn't. In every case where energy seems to disappear, the missing energy can be located by looking outside the system. In other words, if we look at the surroundings in which the systems exists, we will find something there which has gained or lost energy.

From within the mechanical system it appears that energy slowly leaks away. Be sure to study the graphs in this section and compare them with the graphs of conservative systems. In that comparison you will begin to see the genius in Galileo's method of isolating the main features while eliminating the complications such as friction.

We are getting a little ahead of the story and we will see in later programs how the concept of energy conservation was exonerated by James Joule.

9.1. A system where mechanical energy does not remain constant

9.1.1. requires that energy be added from or lost to outside of system

9.2. Real mechanical systems have friction

9.2.1. where does energy go? Could it be energy is not really conserved?

9.2.2. Energy is still conserved if system is enlarged to include surroundings
9.2.3. like a bank account transfer of money between savings and checking does not affect the total balance deposits and withdrawals affect balance this can be accounted for interest also affects balance books will not balance if it is ignored money is not really disappearing system must be expanded in order to account for "loss"

9.3. Energy slowly "leaks away" from mechanical system

9.3.1. graph with low friction

9.3.2. graph with high friction

9.3.3. Energy can be transformed back and forth between many different types not at 100% efficiency efficiency is ratio of work done to energy input some is lost to the system in each transfer still accountable, but no longer in the form of mechanical energy

10. Swinging Balls Revisited

Now it is time to revisit the swinging balls and see what role conservation of energy plays in describing their motion. It is clear that energy is conserved when the number of balls out equals the number of balls in. Recall that the balls are all of identical mass.

10.1. Conservation of Energy

We can reconsider those combination of balls in and out which satisfied the conservation of momentum requirement. You will recall that there are many combinations of number of balls and speed of balls which still conserves momentum. In general conservation of momentum is satisfied whenever the velocity is inversely proportional to the number of balls, like this:

Don't panic. This just says what we saw with the balls in program 18. As long as the speed of the balls is reduced by the same factor as the number of ball increases momentum is conserved.

The same is not true for energy. In fact there are no other combinations of speeds and number of balls which conserve energy. The only condition that conserves both energy and momentum is the one that happen reliably time after time. The number of balls out equals the number of balls in.

When you study the illustrations of the situations which do not conserve energy, pay special attention to the squared proportion in the kinetic energy expression. It is that proportion which prevents the balls from coming out in various combinations. Here's why:

Squaring a quantity is not the same as squaring a number. Try it. For example

because when you square the quantity 2v the two becomes a four.

Now the two on the left side and the four on the right side don't cancel out anymore and so we see that it is not an equation.

10.2. Other possibilities which do not conserve energy

10.2.1. the video program shows the graphics which are the same ones used for conservation of momentum in program 18. Do not panic here. Take some time to calculate the kinetic energy for each ball before and after the collisions. If you need numbers, use m = 1 and v = 2. Try to convince yourself that energy is only conserved if the number and speeds of balls in exactly equals the number and speeds of balls out.

10.2.2. two balls in, one ball out at twice the speed

10.2.3. two balls in, four balls out at half the speed

10.3. In fact there are no other combinations which conserve energy


11. Summary & Conclusions

In this lesson we have defined work, power, and energy. We have specified two types of mechanical energy and their relationship with work through the work/energy theorem.

Then we considered conservative and nonconservative systems before returning to the swinging balls to see the necessity for conservation of energy in explaining their behavior. We observe that invoking conservation of both momentum and energy provides the necessary and sufficient conditions for understanding their behavior.

We saw that the balls are restrained in their behavior by the requirements that both momentum and energy are conserved. This is a different view of things. It is our first encounter with the concept of restraints, but it wouldn't hurt for you to think back and relate this to previous concepts. For example, isn't the motion of the planets restrained by their gravitational interaction with the sun?

This programs sets the stage for our understanding of heat and temperature in terms of the properties of matter. These are the subject of the next three lessons.

11.1. Definitions of work, power, kinetic and potential energy
11.2. Conservative and nonconservative systems
11.3. Behavior of swinging balls requires energy conservation
11.4. Energy conservation is one of the most important principles in physical science
11.5. Later additions complete the concept