ser way chapter 5

8/13/2019 Ser Way Chapter 5

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Chapter 5

Energy


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Some Energy

ConsiderationsEnergy can be transformed from oneform to another

The total amount of energy in the Universenever changesEssential to the study of physics, chemistry,biology, geology, astronomy

Can be used in place of Newton’s lawsto solve certain problems more simply


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WorkProvides a link between force andenergyThe work, W , done by a constantforce on an object is defined as theproduct of the component of theforce along the direction ofdisplacement and the magnitudeof the displacement


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WorkW = F x

This equation

applies when theforce is in thesame direction asthe displacement

are inthe same directionand F x


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Work GeneralW = (F cos ) x

F is the magnitude

of the forceΔ x is themagnitude of theobject’s

displacement is the anglebetween

and F x


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Work, cont.This gives no information about

The time it took for the displacementto occurThe velocity or acceleration of theobject

Work is a scalar quantity


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Units of WorkSI

Newton • meter = Joule N • m = J J = kg • m 2 / s 2

US Customary

foot • pound ft • lb

no special name


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More About WorkThe work done by a force is zerowhen the force is perpendicular tothe displacement

cos 90° = 0

If there are multiple forces actingon an object, the total work doneis the algebraic sum of the amountof work done by each force


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More About Work, cont.Work can be positive or negative

Positive if the force and thedisplacement are in the samedirectionNegative if the force and the

displacement are in the oppositedirection


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When Work is ZeroDisplacement ishorizontal

Force is verticalcos 90° = 0


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Work Can Be Positive or

NegativeWork is positivewhen lifting theboxWork would benegative iflowering the box

The force wouldstill be upward,but thedisplacementwould be

downward


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Work, FinalWork doesn’t happen by itself Work is done by something in theenvironment, on the object ofinterestThe forces are constant

Varying force will be discussed later


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Work and Dissipative

ForcesWork can be done by frictionThe energy lost to friction by an object

goes into heating both the object andits environmentSome energy may be converted into sound

For now, the phrase “Work done by

friction” will denote the effect of thefriction processes on mechanical energyalone


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Kinetic EnergyEnergy associated with the motionof an object Scalar quantity with the sameunits as workWork is related to kinetic energy

2mv21KE


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Work-Kinetic Energy

TheoremWhen work is done by a net force on anobject and the only change in the object

is its speed, the work done is equal tothe change in the object’s kineticenergy

Speed will increase if work is positiveSpeed will decrease if work is negative

net f i W KE KE KE


17/69Copyright © by Holt, Rinehart and Winston. All rights reserved.

ResourcesChapter menu

Section 2 Energy Chapter 5

Sample Problem

Work-Kinetic Energy Theorem On a frozen pond, a person kicks a 10.0 kg sled,giving it an initial speed of 2.2 m/s. How far does the

sled move if the coefficient of kinetic friction betweenthe sled and the ice is 0.10?





Sample Problem, continued

Work-Kinetic Energy Theorem1. Define

Given: m = 10.0 kgv i = 2.2 m/sv f = 0 m/s

µk = 0.10Unknown: d = ?






Work-Kinetic Energy Theorem2. Plan

Choose an equation or situation: This problem can besolved using the definition of work and the work-kineticenergy theorem.

W net = F net d cos The net work done on the sled is provided by the forceof kinetic friction.

W net = F k d cos = µk mgd cos

S i 2 E






Work-Kinetic Energy Theorem2. Plan, continued

The force of kinetic friction is in the direction opposite d,

= 180°. Because the sled comes to rest, the finalkinetic energy is zero.W net = ∆ KE = KE f - KE i = – (1/2) mv i 2

Use the work-kinetic energy theorem, and solve for d .

2

2

1 – cos2

–2 cos

i k

i

k

mv mgd

v d

g

S i 2 E






Work-Kinetic Energy Theorem3. Calculate

Substitute values into the equation:

2

2

(–2.2 m/s)2(0.10)(9.81 m/s )(cos180 )

2.5 m

d

d

S i 2 E






Work-Kinetic Energy Theorem4. Evaluate

According to Newton’s second law, the accelerationof the sled is about -1 m/s 2 and the time it takes thesled to stop is about 2 s. Thus, the distance the sledtraveled in the given amount of time should be lessthan the distance it would have traveled in the

absence of friction.2.5 m < (2.2 m/s)(2 s) = 4.4 m


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Work and Kinetic EnergyAn object’s kineticenergy can also bethought of as theamount of work themoving object coulddo in coming to rest

The moving hammer

has kinetic energyand can do work onthe nail


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Types of ForcesThere are two general kinds offorces

ConservativeWork and energy associated with theforce can be recovered

NonconservativeThe forces are generally dissipative andwork done against it cannot easily berecovered


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Conservative ForcesA force is conservative if the work itdoes on an object moving between two

points is independent of the path theobjects take between the pointsThe work depends only upon the initial andfinal positions of the object

Any conservative force can have a potentialenergy function associated with it


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More About Conservative

ForcesExamples of conservative forcesinclude:

GravitySpring forceElectromagnetic forces

Potential energy is another way oflooking at the work done byconservative forces


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Nonconservative ForcesA force is nonconservative if thework it does on an object depends

on the path taken by the objectbetween its final and startingpoints.Examples of nonconservativeforces

Kinetic friction, air drag, propulsiveforces


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Friction Depends on the

PathThe blue path isshorter than thered pathThe work requiredis less on the bluepath than on thered pathFriction dependson the path andso is a non-conservative force


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Potential EnergyPotential energy is associated withthe position of the object withinsome system

Potential energy is a property of thesystem, not the object

A system is a collection of objectsinteracting via forces or processesthat are internal to the system


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Work and Potential EnergyFor every conservative force apotential energy function can befoundEvaluating the difference of thefunction at any two points in an

object’s path gives the negative ofthe work done by the forcebetween those two points


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Gravitational Potential

EnergyGravitational Potential Energy isthe energy associated with therelative position of an object inspace near the Earth’s surface

Objects interact with the earth

through the gravitational forceActually the potential energy is forthe earth-object system


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Work and Gravitational

Potential EnergyPE = mgy

Units of PotentialEnergy are thesame as those ofWork and KineticEnergy

f igravity PEPEW


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Work-Energy Theorem,

ExtendedThe work-energy theorem can beextended to include potential energy:

If other conservative forces are present,potential energy functions can be

developed for them and their change inthat potential energy added to the rightside of the equation

( ) ( )nc f i f i W KE KE PE PE

Section 3 Conservation of


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Copyright © by Holt, Rinehart and Winston. All rights reserved.


Section 3 Conservation of Energy Chapter 5

Sample Problem

Conservation of Mechanical EnergyStarting from rest, a child zooms down a frictionlessslide from an initial height of 3.00 m. What is her

speed at the bottom of the slide? Assume she has amass of 25.0 kg.



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Conservation of Mechanical Energy1. DefineGiven:

h = h i = 3.00 mm = 25.0 kgv i = 0.0 m/sh

f = 0 m

Unknown: v f = ?



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Conservation of Mechanical Energy2. Plan

Choose an equation or situation: The slide isfrictionless, so mechanical energy is conserved.Kinetic energy and gravitational potential energy arethe only forms of energy present.

21 2

KE mv

PE mgh



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Conservation of Mechanical Energy2. Plan, continued

The zero level chosen for gravitational potentialenergy is the bottom of the slide. Because the childends at the zero level, the final gravitational potentialenergy is zero.

PE g,f = 0



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Conservation of Mechanical Energy2. Plan, continued

The initial gravitational potential energy at the top of

the slide isPE g,i = mgh i = mgh

Because the child starts at rest, the initial kineticenergy at the top is zero.

KE i = 0 Therefore, the final kinetic energy is as follows:

212f f

KE mv


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Reference Levels for

Gravitational Potential EnergyA location where the gravitationalpotential energy is zero must be chosenfor each problem

The choice is arbitrary since the change inthe potential energy is the importantquantityChoose a convenient location for the zeroreference height

Often the Earth’s surface May be some other point suggested by theproblem

Once the position is chosen, it must remainfixed for the entire problem


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Reference Levels, contAt location A, thedesk may be theconvenient referencelevelAt location B, thefloor could be usedAt location C, theground would be themost logicalreference levelThe choice isarbitrary, though


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Conservation of

Mechanical EnergyConservation in generalTo say a physical quantity is conserved is tosay that the numerical value of the quantity

remains constant throughout any physicalprocess although the quantities may changeform

In Conservation of Energy, the totalmechanical energy remains constant

In any isolated system of objects interactingonly through conservative forces, the totalmechanical energy of the system remainsconstant.


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Conservation of Energy,

cont.Total mechanical energy is thesum of the kinetic and potential

energies in the system

Other types of potential energyfunctions can be added to modify thisequation

f f ii

f i

PEKEPEKE

EE


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Problem Solving with

Conservation of EnergyDefine the system

Verify that only conservative forces are presentSelect the location of zero gravitationalpotential energy, where y = 0

Do not change this location while solving theproblem

Identify two points the object of interestmoves between

One point should be where information is givenThe other point should be where you want to find outsomething


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Problem Solving, contApply the conservation of energyequation to the system

Immediately substitute zero values, then dothe algebra before substituting the othervalues

Solve for the unknown

Typically a speed or a positionSubstitute known valuesCalculate result


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Work-Energy With

Nonconservative ForcesIf nonconservative forces arepresent, then the full Work-Energy

Theorem must be used instead ofthe equation for Conservation ofEnergy

Often techniques from previouschapters will need to be employed


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Potential Energy Stored in

a SpringInvolves the spring constant , kHooke’s Law gives the force

F = - k xF is the restoring forceF is in the opposite direction of x

k depends on how the spring wasformed, the material it is made from,thickness of the wire, etc.


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Potential Energy in a

SpringElastic Potential Energy

Related to the work required to

compress a spring from itsequilibrium position to some final,arbitrary, position x 2

s kx21

PE


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Spring Potential Energy,

ExampleA) The spring is inequilibrium, neitherstretched orcompressedB) The spring iscompressed, storingpotential energyC) The block isreleased and thepotential energy istransformed tokinetic energy of theblock


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Work-Energy Theorem

Including a SpringWnc = (KE f – KE i) + (PE gf – PE gi) +(PE sf – PE si)

PEg is the gravitational potentialenergyPE s is the elastic potential energy

associated with a springPE will now be used to denote thetotal potential energy of the system

Section 2 Energy Ch 5


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Chapter 5

Sample Problem

Potential Energy A 70.0 kg stuntman is attached to a bungee cord withan unstretched length of 15.0 m. He jumps off a

bridge spanning a river from a height of 50.0 m.When he finally stops, the cord has a stretchedlength of 44.0 m. Treat the stuntman as a point mass,and disregard the weight of the bungee cord.

Assuming the spring constant of the bungee cord is71.8 N/m, what is the total potential energy relative tothe water when the man stops falling?

Section 2 Energy Ch 5


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Chapter 5


Potential Energy1. DefineGiven: m = 70.0 kg

k = 71.8 N/mg = 9.81 m/s 2 h = 50.0 m – 44.0 m = 6.0 m

x = 44.0 m – 15.0 m = 29.0 mPE = 0 J at river level

Unknown: PE tot = ?

Section 2 Energy Ch t 5


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Chapter 5


Potential Energy2. PlanChoose an equation or situation: The zero level for

gravitational potential energy is chosen to be at thesurface of the water. The total potential energy is thesum of the gravitational and elastic potential energy.

212

tot g elastic

g

elastic

PE PE PE

PE mgh

PE kx



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Chapter 5


Potential Energy3. CalculateSubstitute the values into the equations and solve:

2 3

2 4

3 4

4

(70.0 kg)(9.81 m/s )(6.0 m) = 4.1 10 J1

(71.8 N/m)(29.0 m) 3.02 10 J2

4.1 10 J + 3.02 10 J

3.43 10 J

g

elastic

tot

tot

PE

PE

PE

PE



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Chapter 5


Potential Energy4. Evaluate

One way to evaluate the answer is to make an

order-of-magnitude estimate. The gravitationalpotential energy is on the order of 10 2 kg 10m/s 2 10 m = 10 4 J. The elastic potential energyis on the order of 1 10 2 N/m 10 2 m 2 = 10 4 J.

Thus, the total potential energy should be on theorder of 2 10 4 J. This number is close to theactual answer.

Section 3 Conservation of Ch t 5


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Energy Chapter 5

Conservation of Mechanical Energy3. Calculate

Substitute values into the equations:

PE g,i = (25.0 kg)(9.81 m/s2)(3.00 m) = 736 J

KE f = (1/2)(25.0 kg) v f 2 Now use the calculated quantities to evaluate thefinal velocity.

ME i = ME f PE i + KE i = PE f + KE f 736 J + 0 J = 0 J + (0.500)(25.0 kg) v f 2 v f = 7.67 m/s


Section 3 Conservation of EChapter 5


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Energy Chapter 5


Conservation of Mechanical Energy4. Evaluate

The expression for the square of the final speed can

be written as follows:

Notice that the masses cancel, so the final speeddoes not depend on the mass of the child. This

result makes sense because the acceleration of anobject due to gravity does not depend on the massof the object.

v f 2

2 mghm

2 gh


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Conservation of EnergyIncluding a Spring

The PE of the spring is added toboth sides of the conservation of

energy equation The same problem-solvingstrategies apply

f sgisg )PEPEKE()PEPEKE(


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Nonconservative Forceswith Energy Considerations

When nonconservative forces arepresent, the total mechanical energy of

the system is not constantThe work done by all nonconservativeforces acting on parts of a systemequals the change in the mechanical

energy of the system

nc W Energy


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Nonconservative Forcesand Energy

In equation form:

The energy can either cross a boundary

or the energy is transformed into aform of non-mechanical energy such asthermal energy

( )( ) ( )

nc f i i f

nc f f i i

W KE KE PE PE or

W KE PE KE PE


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Transferring EnergyBy Work

By applying a forceProduces a displacement of the system

HeatThe process of transferring heat bycollisions between atoms or moleculesFor example, when a spoon rests in a cup ofcoffee, the spoon becomes hot becausesome of the KE of the molecules in thecoffee is transferred to the molecules of thespoon as internal energy


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Transferring EnergyMechanical Waves

A disturbance propagates through amediumExamples include sound, water, seismic

Electrical transmissionTransfer by means of electrical current

This is how energy enters any electricaldevice


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Transferring EnergyElectromagnetic radiation

Any form of electromagnetic waves

Light, microwaves, radio wavesFor example

Cooking something in your microwaveovenLight energy traveling from the Sun tothe Earth


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Notes About Conservationof Energy

We can neither create nor destroyenergy

Another way of saying energy isconservedIf the total energy of the system doesnot remain constant, the energy must

have crossed the boundary by somemechanismApplies to areas other than physics


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PowerOften also interested in the rate atwhich the energy transfer takes placePower is defined as this rate of energytransfer

SI units are Watts (W)

W Fv

t

2

2

J kg mW

s s


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Power, cont.US Customary units are generally hp

Need a conversion factor

Can define units of work or energy in terms

of units of power:kilowatt hours (kWh) are often used in electricbillsThis is a unit of energy, not power

W746slbft550hp1


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Center of MassThe point in the body at which allthe mass may be considered to be

concentratedWhen using mechanical energy, thechange in potential energy is relatedto the change in height of the centerof mass


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Work Done by VaryingForces

The work done bya variable force

acting on anobject thatundergoes adisplacement is

equal to the areaunder the graphof F x versus x


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Spring ExampleSpring is slowlystretched from 0

to xmax W = ½kx²

applied s= - = kxF F


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Spring Example, cont.The work is alsoequal to the areaunder the curveIn this case, the

“curve” is atriangle

A = ½ B h givesW = ½ k(x max ) 2 and W = PE

ser way chapter 5

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