Science 122 Program 19 Work & Energy

# Work & EnergyProgram 19Lesson 3.4

## Objectives

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. 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.

## 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.

## 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.

## 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.

## 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.

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

#### 5.4.1. assuming no friction5.4.2. 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.

## 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.

### 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. OR 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

## 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.)

### 8.2. non collisions

#### 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.

##### 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".

## 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.

## 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.

## 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.