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.
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.
1.1.1. momentum describes collisions, but not completely
1.1.2. heat can be integrated into and explained by mechanical theory
220.127.116.11. studied since Aristotle
18.104.22.168. little understanding of principles
22.214.171.124. nothing new added until 19th century using Newtonian paradigm
126.96.36.199. concept of particles in motion will unite mechanics and atomic theory
1.2.1. No single individual is responsible for theory
188.8.131.52. developed over 150 year period
1.2.2. Historical accident that Newton developed laws in terms of kinematics
184.108.40.206. Newton's efforts were directed towards explaining planetary motion
220.127.116.11. Newton's Laws can be derived from conservation laws
1.2.3. Conservation of momentum proposed by Hooke, Wren, Huygens
18.104.22.168. contemporaries of Newton
22.214.171.124. Newton's work provided framework (paradigm) used to formally derive concept later
126.96.36.199. speed depends on height
188.8.131.52. rolls up to same height as rolled from
184.108.40.206. inertia is special case of energy conservation
220.127.116.11. Newton defined "vis insita" or innate force of matter
18.104.22.168. Huygens: mv2 is conserved in certain collisions
22.214.171.124. Leibnitz called it "vis viva" (living force)
126.96.36.199. later combined with Newton's Laws and Galileo's Kinematic equations
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.
2.4.1. measured by work done
2.4.2. work has precise physical definition
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
2.6.1. more laws allow increased prediction
188.8.131.52. mathematical law = relationship
184.108.40.206. relationship = equation
220.127.116.11. more equations => more variables can be incorporated into systems
18.104.22.168. more variables => more complex and therefore more realistic situations
2.6.2. is especially powerful when combined with momentum conservation
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.
3.1.1. can double work by doubling either force or distance
3.1.2. precise and limited definition
22.214.171.124. note contrast with everyday usage
126.96.36.199. reason for definition is to relate to energy
188.8.131.52. must be consistent to be meaningful
3.1.3. work is only done if motion is involved
184.108.40.206. no work is done by stationary force
220.127.116.11. component of force in direction of motion does work
3.1.4. 1 joule = 1 Newton x 1 meter
18.104.22.168. 1 J = 1 N*M
3.2.1. inertia, gravity, friction, elasticity, electric force, magnetic force, etc.
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.
4.2.1. 1 W = 1 J/s
4.3.1. 1 watt = 1 volt x 1 ampere
22.214.171.124. 1 kilowatt = 1000 watts
126.96.36.199. 1 kilowatt hour = 1000 watts for one hour
188.8.131.52.1. 1000 J/s x 3600 s = 3,600,000 J
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.
5.1.1. Kinetic energy equals one half m v squared
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
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?
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.
5.6.1. not just force, but a certain relationship between force and distance
5.6.2. not the same as momentum
184.108.40.206. hard to see difference since both are involved in any change in motion
220.127.116.11. inertia (mass) in motion possesses both kinetic energy and momentum
18.104.22.168. Newton's vis in sita has two components
22.214.171.124. momentum is acquired by impulse (Ft)
126.96.36.199.1. second law: impulse = change in momentum
188.8.131.52. kinetic energy is acquired by work (Fd)
184.108.40.206. kinetic energy divided by momentum equals velocity
5.7.1. there is only one form of momentum
220.127.116.11. important difference between the two concepts
18.104.22.168. impulse changes momentum
5.7.2. Work can be done without changing kinetic energy
22.214.171.124. lifting against gravity
126.96.36.199. stretching or compressing spring
188.8.131.52. pushing or pulling against electric or magnetic force
184.108.40.206. sliding at constant speed against friction
5.7.3. Something changes when work is done
220.127.116.11. if not kinetic energy then what?
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.
6.1.1. gravity, springs, electricity, magnetism
6.2.1. this equation simply shows that work done equals energy gained
6.2.2. E = mgh
18.104.22.168. mg is the weight of a given mass and represents the force necessary to lift it
22.214.171.124. 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
6.3.1. can be shown that amount of kinetic energy gained exactly equals amount of potential energy lost (in the absence of friction)
Conservation of Energy
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.
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.
7.1.1. (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
7.2.1. 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
126.96.36.199. initial state need not be at rest
188.8.131.52.1. work is done in changing speed from 0 to 30 mph
184.108.40.206.2. work is also done in changing speed from 30 to 60 mph, but not the same amount as from 0 to 30
220.127.116.11. rock hits windshield vs. windshield hits rock: equivalent
18.104.22.168. carrying a box up one flight of stairs requires the same amount of energy regardless of which floor it started from
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.)
8.1.1. (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
22.214.171.124. different form of energy
126.96.36.199. energy given to another object
188.8.131.52. loss of potential energy equals gain in kinetic energy
184.108.40.206. the sum of kinetic and potential energy, called total mechanical energy, remains constant
220.127.116.11. mgh = 1/2 mv2 (see table above)
18.104.22.168. 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".
22.214.171.124.1. work done is the same for a given displacement in the absence of friction
126.96.36.199. strobe photo of pendulum
188.8.131.52.1. rises to same height on either side
184.108.40.206.2. like Galileo's ball on the incline
220.127.116.11. graph of energy of pendulum
18.104.22.168. ball in valley
22.214.171.124. planet in orbit
126.96.36.199. electron in atom
188.8.131.52. simple machines
184.108.40.206.1. lever, inclined plane, hydraulic jack
220.127.116.11. elastic collision <==> kinetic energy conserved
18.104.22.168. inelastic collisions involve frictional "losses"
22.214.171.124. certain collisions are nearly elastic
126.96.36.199. steel ball colliding with steel ball for example
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.
9.1.1. requires that energy be added from or lost to outside of system
9.2.1. where does energy go?
188.8.131.52. 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
184.108.40.206. transfer of money between savings and checking does not affect the total balance
220.127.116.11. deposits and withdrawals affect balance
18.104.22.168.1. this can be accounted for
22.214.171.124. interest also affects balance
126.96.36.199.1. books will not balance if it is ignored
188.8.131.52.2. money is not really disappearing
184.108.40.206.3. system must be expanded in order to account for "loss"
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
220.127.116.11. not at 100% efficiency
18.104.22.168. efficiency is ratio of work done to energy input
22.214.171.124. some is lost to the system in each transfer
126.96.36.199. still accountable, but no longer in the form of mechanical energy
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.
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.
10.2.1. 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.
10.3.1. THE ONLY COMBINATION WHICH CONSERVES BOTH ENERGY AND MOMENTUM IS:
10.3.1.1. THE NUMBER AND VELOCITIES OF BALLS OUT IS EQUAL TO THE NUMBER AND VELOCITIES OF BALLS IN
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