FZX Physics Lecture Notes reserved FZX: Personal Lecture...

FZX ‐‐ Physics Lecture Notes Copyright 1995, 2011, D. W. Koon. All Rights reserved

FZX: Personal Lecture Notes from Daniel W. Koon

St. Lawrence University Physics Department

CHAPTER 6

Please report any glitches, bugs or errors to the author: dkoon at stlawu.edu.

6. Work, Energy, and Power Work Kinetic Energy Work-Energy Theorem Conservation of Energy Examples of Potential Energy Power

page 1 http://it.stlawu.edu/~koon/classes/103.104/FZX103-LectureNotes2011.pdf


FZX, Chapter 6: WORK, ENERGY, and POWER Newton’s laws form a very useful approach for analyzing the question of dynamics -- why something moves as it does, or how it will move in the future. However, we have already seen some of the limitations of this approach. First of all, although we could conceivably analyze motion when the net force is changing, we do not know how to deal with the kinematics (the ‘how’ of motion) of nonuniform acceleration. The other problem with Newton’s approach -- let’s face it -- is that it often requires us to deal with resolving vectors, which is hardly ever any fun. Fortunately, there is an alternate approach one can use with some dynamics problems. And since both approaches should give the same results, you should only ever use that approach which gives those results with the least amount of sweat. Your job in this chapter will be to learn that approach -- the energy approach -- and be able to figure out when you can get away with using it, since it is usually easier to apply than the ‘force approach’. WORK: We begin by defining work. Rather than try to fit our definition to any preconceived notion of what work is, let’s just define it and alter our own concept of work to fit the definition. To simplify things a little, let’s start in just a single dimension, x. If a force,

r, acts on some object which is traveling some distance ∆x, then the work is defined as: F

xFW xΔ= , [ Definition of work ] where Fx is the component of in the x-direction. If

rF

rF is pointing in some other direction, it might be more convenient

to use the following, equivalent form of this definition: θcosxFW Δ= . [ ReDefinition of work ] It is important to realize what all the terms stand for, so here goes... = what we call ‘work’ W = magnitude of force acting on an object F = total distance traveled in one dimension by the object while work is done xΔ θ = angle between and the direction of motion

rF

It is important to see that work can be positive or negative. Positive work means that the force is exerted in the direction of motion, which means it will tend to speed up the object or accelerate it. It is HELPING the object to get where it is going. Negative work means that the force is exerted in a direction opposing the motion, which means that it tends to slow down the object or decelerate it. It is IMPEDING the object from getting where it is going. We redefined ‘work’ once, but we will need to do it again. What happens if the force is not uniform? We should probably fill in something for the average value of Fx. A more general solution would be to plot Fx vs x on a graph, and find the area under the curve between the initial and final values of x. W = area under Fx vs x curve [ ReReDefinition of Work ] Let’s illustrate the kind of work that a force can perform on an object. If the force acts in the same direction as the motion, it does positive work and speeds up the object. This happens when gravity acts on a falling object. The object speeds up because the net work done on it is positive: W = (-mg) ∆y > 0 [Work done by gravity on falling object] (Notice that ∆y is negative, because it is falling.) If an object is ascending, however, the force (weight) acting on it acts in a direction opposite to the motion. Thus negative work is done, and the object slows down: W = (-mg) ∆y < 0 [ Work done by gravity on climbing object] (Here ∆y is positive, because it is ascending.) Now consider an object -- the Moon for example -- travelling in a circle. The only force acting is gravity, and it is perpendicular to the motion, so there is no component of force in the direction of motion. The work is zero, and the force neither speeds up nor slows down the motion: W = 0 ∆x = 0 [ Work done by centripetal force ] page 2 http://it.stlawu.edu/~koon/classes/103.104/FZX103-LectureNotes2011.pdf


KINETIC ENERGY: The connection between work and force is clear. It comes from the definition of work. We will now introduce the quantity of energy, which arises for the notion of work, and so is distantly related to force, but which will provide an independent and easier approach for tackling some problems. First consider what we know from kinematics. From our ‘fourth law’ of kinematics, namely , [ ‘4th Law’ of kinematics ] xavv Δ+= 22

02

we can rewrite the total work done on an object in terms of initial and final speed. We can write the total work done on the object as

( )

202

1221 mvmv

xmaxFxFW xx

−=

Δ=

Δ=Δ= ∑∑∑

The last line is a result of the ‘4th Law’ stated above. We can define Energy to be the capacity to do work, and so at any time, we can say that this energy equals ½ mv2, and that when we do positive work on an object, we raise its energy. Well, the picture isn’t quite this neat, because there is another form of energy that doesn’t manifest itself in the movement ‘v’ of the object, but which is stored with the potential to do work some other time. We call these two forms of energy ‘kinetic energy’ and ‘potential energy’.



KE = ‘K’ = ‘T’ = ½ mv2 [ Translational kinetic energy ] PE = ‘P’ = ‘U’ = no formula yet [ Potential energy ] Two things to note here: First, I include a couple of different symbols that are used for each quantity. The book you are using might use one or the other. There are some symbols that are not completely standardized in the physics community. You should recognize all these symbols for kinetic energy. Secondly, I called the quantity KE up above the ‘translational’ kinetic energy. This simply means kinetic energy due to motion in a more-or-less straight line. If a wheel is spinning in place, so that its axle does not move, but the parts of the wheel are all moving with different velocities, there is some kinetic motion there. This equation doesn’t help to quantify this case, because which ‘v’ would you use? We will return to circular motion later, and introduce a ‘rotational kinetic energy’. Just leave a space in your brain for it. Till then, we can use ‘kinetic energy’ and ‘translational kinetic energy’ interchangeably. The units of energy are Joules, defined as J = 1 N.m [Units of work, energy] WORK-ENERGY THEOREM: We’ve already done the leg work above for the Work-Energy Theorem, when we arrived at an expression for �W. All that remains is to apply our definition of KE to put it in more concise language: [ Work-Energy Theorem ] KEW Δ=∑ All that this says is that the net work done on an object changes its kinetic energy. If, for instance, we lift a heavy box off the ground, we do work in lifting it, but if we lift it up at a constant speed, the net force is zero, so that the net work is zero. It doesn’t pick up speed. It is vital in considering Work-Energy problems to consider ALL forces acting on an object -- weight and the force WE exert, in this case -- so we should still draw a free body diagram. If we lift the box with more force than the weight, it will acclerate, picking up speed and kinetic energy. We can see from this definition the connection between work and energy. We already saw the connection between force and work. All that remains is to demonstrate the usefulness of energy in analyzing motion. That’s coming up.



CONSERVATION OF ENERGY: Consider a simple motion that does not involve constant acceleration: the motion of a mass tied to the end of a string -- the pendulum. The pendulum executes its simple motion over and over, and would never stop if it weren’t for some fiendish killjoy force like friction or air resistance. The acceleration of a pendulum is nonuniform, both in magnitude and direction. Still, it is a fairly simple motion. It shouldn’t be too difficult to describe. We can try to understand it in terms of force, but as I pointed out, the force -- and thus the acceleration -- is nonuniform, so we can’t expect to write out ‘position’ as a function of time using the kinematics equations. We can, however, use the concept of ‘work’. When the pendulum speeds up, positive work is being done, and when it slows down, negative work is done. The only forces acting on the mass are weight and the tension in the string. Draw a diagram and convince yourself that the tension can do no work. Why? It is perpendicular to the direction of motion. So all the work is being done by gravity. The nice thing about the work done by gravity is that it is ‘bankable’: it is being stored. If negative work is done, this decreases the kinetic energy, but we can imagine a separate ‘banking account’ that increases by the same amount as the kinetic account decreases, so that, on balance, the total energy remains the same. The pendulum spends its entire life



transferring ‘funds’ between the ‘kinetic’ fund and the ‘potential’ fund, just as you can transfer funds between separate banking accounts without decreasing the total amount of money you have. There are a number of forces for which you can do this. They are called ‘conservative forces’, which simply means that you can define a potential energy for them, such that the total energy is conserved: E = E′ [ Conservation of energy ] PE + KE = PE′ + KE′ (only works if no non-conservative forces are present) In the equations above, unprimed quantities refer to the initial conditions, and primed quantities refer to final conditions. In analyzing one of these problems, you want to set it up this way: First, draw your diagram, drawing both a ‘before’ and an ‘after’ picture. Label both figures with all the information you know about the situation both before and after. Second, write down the two equations I just wrote down above. For the particular system of interest, substitute for PE, KE, PE′, and KE′ all the terms that apply. It is possible for there to be more than one type of potential energy in a particular problem, for instance. The rest, hopefully, is algebra.

EXAMPLES OF POTENTIAL ENERGY: Okay, but you need to know what the potential energy PE is, right? First of all, it is only defined for a conservative force. If we move some object from one place to another without allowing it to speed up, then the change in potential energy due to that force will be equal to the negative of the work done by that force. What you will notice from the energy conservation equation above is that it is the CHANGE in the potential energy that is important, not its exact magnitude at any time. That means that the ‘zero-point’ of our potential energy has no real physical significance. We can call the gravitational potential energy zero for your pencil sitting on your desk, or for the same pencil after it has fallen to the floor, or your pencil sitting at sea level, or whatever. All that matters is that you consistently use the same zero-point for PE in a given problem. Gravity is the most important source of potential energy. In raising an object, gravity does negative work, so the gravitational potential energy (the negative of the gravitational work done) is negative and just equal to the force -- the weight -- times the distance we have moved it. Choose a zero point for gravity and measure a height, ‘h’, above it, and the gravitational potential energy there is . [ Gravitational P.E. ] mghGPE =



Springs exert forces when they are pulled or squashed. The force, for most simple applications, is proportional to the amount by which you change their length, and the force points in the direction that would return it to its usual length. Using the results of a previous exercise, we can show that the potential energy stored in a spring is proportional to the SQUARE of the change in its length: 2

21 kxSPE = [Spring P.E.]

where ‘k’ is something called the spring constant, which we will describe further in another section, and which tells us how ‘stiff’ the spring is, and ‘x’ is the CHANGE in the length of the spring from its unstretched, equilibrium length. In a problem involving a mass tied to the end of a vertically hanging spring, there is both ‘gravitational’ and ‘spring’ potential energy. In doing conservation of energy problems involving such a system, be sure to include both terms when you plug in for PE and PE′ in the conservation equation. (May you be so lucky that your instructor doesn’t assign any such problems!) What about other forces we have encountered? Do they give rise to potential energies -- or equivalently, do they conserve energy? We can show that for a force to be conservative, the work it does as an object travels over some course that ends where it started must be zero. How about tension? If we pull a box behind us as we walk in a circle, the total work we do on the box as we complete one circle is positive. So tension is not a conservative force. If the box drags along the ground, friction will do consistently negative work. So friction is also nonconservative. What about the normal force? The normal force is directed perpendicular to the surface of interest, and so it does not do any work, positive or negative, so we don’t have to worry about whether it’s conservative or not. (Sorry for evading the issue.)



POWER: An Alfa Romeo and a Hyundai will both do the same net amount of work on you taking you from your home to the mall, so why does one have such a more powerful engine? If you want to get to the mall in a hurry, you want to do that work faster. If you want to climb hills and pass other cars without worry about pushing the limits of your car’s ability, you want a car with a lot of ‘power’. (There is of course, much more to life than ‘power’ and having a muscle car, so if you’re like me, you shouldn’t sweat saving for the Alfa.) Power is just the amount of work done per unit time: [Definition of power] tWtWP // =ΔΔ= The units of power are Watts, W = J/s [Units of power] horsepower = hp = 746W = 550 ft.lb/s [Non-metric units of power] Why the two different expressions for ‘P’ above? Different textbooks use different formulae. The one with the ∆’s in it should remind you that you want to take the limit as ∆t goes to zero to calculate the power at some ‘instant’. The other equation is there to remind you that, if the power is constant, you can just divide the work done by the time. One last equation that may be useful. Since Work is related to force, we can derive the following for some force acting on a moving object: θcosvFP = , [Power exerted by force on moving object ]




where ‘θ’ is the angle between the force and the velocity.

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