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Chapter Six Lecture
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What is Work?
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Formulas:
Main Ideas:
- Definition of Work.
- Kinetic Energy and the Work-Energy Theorem
- Potential Energy
- Conservative and Non-conservative Forces
- Conservation of Mechanical Energy
- Non-conservative Forces and Conservation of Energy
- Power
This chapter will deal with one of the most important concepts
introduced this semester. It is the concept of energy. We will find that total energy is always conserved in every reaction.
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Definition of Work
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In physics, the term work has a very specific definition. Work is defined as the product of the component of force along the direction
of displacement times the magnitude of the displacement. We write it as
where q is the angle between the direction of the force and the direction
of the motion. The SI unit of work is Newton-meter (N×m) which is given the name Joule.
1 Newton×meter = 1 Joule
Note that to do work, you need a displacement and force acting with some component in the direction of motion.
There will always be an object or objects doing work on another object. It is always necessary to determine what is doing the work on what
object. Work is only defined in relation to one thing doing work on another object. To solve problems, you must first determine this, or you will not solve for the work that is being asked for.
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 Problem: Suppose I pull a package with a force of 98 N
at an angle of 55° above the horizontal ground for a distance of 60m. What is the total work done by me on the package?
Note that Fcosq is the component of the force along the direction of motion. (Along the
direction of the package's displacement.)
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Note: If there is no force in the direction of motion, then there is no work done on that object.
When we carry a package at a constant velocity a long distance, we may say that we have done a lot of work, but using the definition of work
here, we have done no work, because the force was not parallel to the displacement. The angle was 90° and the cosine of 90° is 0, so
If I push on a wall and the wall does not move (no displacement), the work is 0J because the displacement is 0m.
Work is a scalar but the sign of the work is important. It is very important to ask who or what is doing the work? We can either ask how
much work is done on an object (all the forces on the object), or how much work does a certain object do?
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 Problem
: A snail massing .23 kg slides down an incline of 25° at a constant velocity. The snail slides 1.5 m. What is the work
done by the normal force, by gravity, and by friction? What is the total work done on the snail?
We can start by applying Newton's Second Law and determining the force of friction.
Now we need to determine the Normal Force
Now what is the work done by each individual force.
Work done by theNormal Force:
Work done by the Frictional Force:
Work done by the Gravitational Force:
Now, to finally determine the "net-Work" we add them all up!
We could have really determined this from the fact that the snail was
sliding at "constant velocity" thus there is no "net-force" acting on the snail, thus Wnet=0J.
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To solve a problem with work:
- First determine the force or forces that are doing the work.
- Look at the question carefully. Is it asking how much work a certain object is doing, or how much total work is being done on an object?
- Then determine the force or forces doing the work, and the angle between the displacement and the force.
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Work-Energy Theorem
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Energy is one of the most important concepts in physics. Yet it is not
an easy concept to describe simply. There are many different kinds of energy. Any one of those can be described. In any process we find that
the total energy of the system never changes. It is conserved. However, energy can change from one form to another. When two cars collide the
energy they had from moving (called kinetic energy) is changed primarily into heat.
In this chapter we will look at a number of types of energy. The first will be the energy of motion, called kinetic energy. We look first at the Kinematic Equations and Newton's 2nd Law:
we can combine these with work.
The term
 is called the Kinetic Energy.
We say the "net work" done on an object is equal to its change in Kinetic Energy. Note that this is the "net work" done on an object from all the forces acting on it. This is known as the Work Energy Theorem.
The work-energy theorem only applies to all of the forces acting on a single object. It does not apply to any single force. Note that kinetic
energy is related to the speed squared of an object.
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 Problem: How much work does it take to
stop a 1000 kg car traveling at 28 m/s?
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 Problem: A 58-kg skier is coasting
down a slope inclined at 25° above the horizontal. A kinetic frictional force of 70 N opposes her motion. Near the top of the slope the
skier's speed is 3.6 m/s. What is her speed at a point which is 57 m downhill?
Find the net-work:
Now use the Work-Energy Theorem:
So,
Therefore,
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Potential Energy
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Another form of energy, other than kinetic energy, is potential energy. Potential Energy is the potential an object has to do work.
Gravitational Potential Energy:
 The most common example of potential energy is the gravitational potential energy.
Suppose I raise a ball to a height h = yf - yi. How much work has been done on the ball by me, by gravity, and in total:
WME =Fdcosq = mg(yf - yi) = mgh
The change in kinetic energy is 0, so the "net work" must also be 0. Therefore, we know that the work done by gravity is equal,
but opposite in sign.
WG = mg(yi - yf) = mgyi - mgyf = PEi - PEf where PE is called the
potential energy. (The book uses PE for potential energy. You will sometimes see U for potential energy).
Note that this is the initial minus final energy, so we say,
WG= mg(yi - yf)
WG= -DPE. (since DPE = PEf - PEi)
Look at the definition of gravitational potential energy PE = mgy.
That means that the larger y is (the higher something is above the ground), the larger the potential energy is. The book uses the symbol h to mean the object's height and then defines potential energy as:
The gravitational potential energy is the potential to do work due to
gravity. If I raise this book it has the potential to do work. If I drop it, I could drive a nail. I could attach it to a car and accelerate the car. The
object has the potential to do work due to gravity. There is no absolute scale for potential energy. We always measure the height with respect
to some reference point. For instance, if I define the gravitational potential energy to be 0 on the table, then I raise the book - it has a potential energy of mgh relative to the table. Something that is higher has a higher potential energy than something that is lower. It has
more ability to do work.
Suppose I take two different paths to raise the book 1 meter. One is
straight up and the other is very circuitous. Compare the ability to do work with this book from the two different methods. It is exactly the
same. This illustrates one of the most important principles regarding gravitational potential energy. PEGravitational
is always the same at the same height regardless of the path taken to reach that height.
Springs:
Another form of potential energy is
elastic potential energy. The potential energy stored in a spring. If I compress a spring, it has the
potential to do work. Old wind up watches used this principle. The amount of energy stored in a spring depends on how much it is
compressed. In many cases the force necessary to compress a spring is proportional to the distance it is compressed (x). The force of the spring is then given by:
The minus sign means that the spring wants to push in the opposite
direction from the direction it was compressed. Now, suppose, I compress the spring a distance x. When I first start to compress it takes almost
no force. When it is very compressed it takes a force of kx to compress it. The average force it takes is (1/2)(0 + kx) = (1/2)kx.
So the work it takes to compress the spring is:
where x is the distance the spring has been compressed or stretched.
This is the energy stored in a spring. I can use this energy to do work. So we have talked about two kinds of potential energy, Gravitational = mgh, and Elastic =(1/2)kx2.
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Conservative & Non-Conservative Forces
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As we said, the work done to move an object in a gravitational field
does not depend on the path taken. Such a field is called a conservative field and the force, here the force of gravity, is called a conservative force. If work done by a force in moving an object between two positions is independent of the path of motion, the force is called a conservative force. Friction is not such a conservative force because W
= Fd. If I take a longer distance the work increases because "d" is always opposite of "F." (For gravity "d" can either be in the opposite
direction of "F" or in the same direction, and this contributes to the force being conservative). Because potential energy is the energy associated with an object's location, potential energy can only be defined for a conservative field. We can extend the work energy theorem to include
non-conservative forces as well. In that case:
 (recall that the net work done is the change in KE)
For gravity, we saw that
 , so
Dropping something that smashes when it hits the ground: Initially a lot
of potential energy, then a lot of kinetic energy -- finally, the energy changes to non-conservative forms: heat and deformation.
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Conservation of Mechanical Energy
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This equation can be rewritten as;
So the work done by non-conservative forces is equal to the change in
mechanical energy. If there are no non-conservative forces, (no friction, heat, etc), then WNC = 0, and we find that mechanical energy is conserved. That is:
If only conservative forces are acting, the total mechanical energy of a system neither increases nor decreases in any process. It stays constant. It is conserved.
This principle can be used to solve many problems in physics where there are no non-conservative forces.
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 Problem: A child and sled with mass of
50 kg slide down a frictionless hill. If the sled starts from rest and has a speed of 3.00 m/s at the bottom, how high is the hill?
We can't do this with kinematic equations unless we know the angle of the hill, or at
least some more information. However, we can do it with conservation of mechanical energy. Let's say that at the
bottom of the hill, potential energy is 0J.
We have, vi = 0m/s, PEf = 0J, and
because we have a frictionless hill then, WNC= 0J. Thus;
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Problem: A motorcycle rider leaps across a river canyon with an initial
speed of 38.0 m/s from a height of 70.0 m. He lands at a height of 35.0 m. What is his final velocity? (Assuming no air resistance, WNC=0)
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 Problem: A 0.5 kg block is used to
compresses a spring with a spring constant of 80.0 N/m a distance of 2.0 cm (.02 m). After the spring is released, what is the final speed of the block?
We have, vi = 0m/s, PEf = 0J (spring un-compresses, or xf = 0m).
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 Problem: A 70 kg person bungee jumps
off of a 50 m bridge with his ankles attached to a 15 m long bungee cord. Assuming he stops just at the edge of the water and he is 2.0 m tall, what is
the spring constant of the bungee cord?
How much does the bungee cord "stretch?"
50m – 15m – 2m = 33m
This is the final amount the bungee cord stretches.
So xi = 0m and xf = 33m
In addition, hf = 0m, vi = vf =0m/s, now we can use conservation of
mechanical energy to find the spring constant.
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Non-Conservative Forces & Conservation
of Energy
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So far in our study of energy, we have neglected friction and other
non-conservative forces. When there are no non-conservative forces present, then the total mechanical energy is conserved, as we have seen. However, there is a more fundamental law of the conservation of
energy that includes non-conservative forces. It states the total energy
in any process is not increased or decreased. Energy can be transformed from one form to another and transferred from one body to another, but the total energy remains constant. This is ALWAYS true! For example, when we compress a spring then use it to accelerate a block, the potential energy of the spring is changed into the
kinetic energy of the block. What about if I slide something across the floor and it comes to rest? It started with a lot of kinetic energy, and at
the end it has no greater potential energy. What happened to it? The kinetic energy changed into thermal energy or heat and the book and
floor heated up. The forms of forces (like friction) that reduce the mechanical energy of a system are called dissipative forces
. When dissipative forces are present, the total energy remains constant, but the mechanical energy decreases.
Energy can be transformed from one form to another. There are many forms of energy: electrical, chemical, nuclear, thermal, gravitational, elastic, kinetic... Although energy can be transformed from one of these forms to another;
it cannot ever be created or destroyed. We use this fact in solving problems where the mechanical energy is not conserved.
Recall, the equation we derived which included non-conservative forces.
If there are no non-conservative forces, then the mechanical is
conserved. If there are non-conservative forces, then the loss in mechanical energy goes into work done due to these forces. For instance, we did a problem with a child on a sled who weighed 50 kg on a
hill that was 0.46 m high, and had a final velocity of 3.0 m/s. Instead suppose the final velocity is 2.6 m/s on the same hill.
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Problem: Children and sled with mass of 50 kg slide down a hill with a height of 0.46
m. If the sled starts from rest and has a speed of 2.6 m/s at the bottom, how much thermal energy is lost due to friction (i.e. what is the work that
friction does)? If the hill has an angle of 20° above the horizontal what was the frictional force.
Since vi = 0, and PEf = 0,
The force done by friction is determined from;
 q=180° (angle between Ffr & displacement directions)
Also, d is the length (hypotenuse) of the hill.
Since sin(20°) = o/h Þ h = o/sin(20°) = (0.46 m)/sin(20°) = 1.34 m
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So, if the mechanical energy is not conserved, this means "work"
must have been done by non-conservative forces. That is, energy was transformed from mechanical energy to other forms of energy. No energy was created or destroyed.
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Power
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 Sometimes it is more important to know not just the work that has been done, rather the rate at which work has been done. For instance, I may have two identical cars, except
for the engines. The two cars my require the same amount of work to go from 0 to 60 mph, but one car can do it in 5 seconds while one car can do it in 12 seconds. The one
that can do it in 5 sec is a more powerful car. This illustrates the concept of power. Power is
defined as the rate at which work is done.
The SI unit of power is "watts" (W).
Sometimes we measure power in horsepower.
One horsepower = 550 ft-lb/s and is equal to 746 W.
Power can also be written as;
Whenever you want to determine power, you must first determine the force and the velocity or the work being done and the time.
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 Problem: A deep-sea observation chamber
is raised from the bottom of the ocean 1700 m below the surface by means of a steel cable. The chamber moves at constant
velocity and takes 5.00 minutes to reach the surface. The cable has a constant tension of 8900 N. How much power is required to pull
thecable in. Give the answer in horsepower.
How about a conversion to horsepower?
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 Problem: A 1000 kg elevator carries an 800 kg load. A
constant frictional force of 4000 N retards its motion. What minimum power must the motor deliver to lift the elevator at a constant rate of 3.00 m/s?
The power of the motor is used to support the tension in the cable.
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where E is mechanical energy.
