4 Newton Dan Energi

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Lecture IVLecture IV

General Physics (PHY 2130)

• Applications of Newton’s Laws (cont.)

inclined surfaces, connected objects• Energy

Work

Kinetic and potential energy

Conservative and non-conservative

forces

Other forms of energy

http://www.physics.wayne.edu/~apetrov/PHY2130/



Lightning ReviewLightning Review

Last lecture:

1. Motion in two dimensions (projectile motion):

easier to solve in components vertical motion: a=const, horizontal motion: a=0!

2. Laws of motion (Newton’s laws)

forces, three Newton’s laws

free body diagrams and applications of NL

Review Problem: If an astronaut has a mass of 90 kg on Earth, what is theastronaut’s mass on the Moon? The acceleration of gravity on the Moon

is 1/6 of that on Earth.

(1) 15 kg

(2) 90 kg

(3) 147 kg

(4) 540 kg



Applying Newton’s Laws Applying Newton’s Laws

►►Make a sketch of the situation described in theMake a sketch of the situation described in theproblem,problem, introduce a coordinate frameintroduce a coordinate frame

►►Draw aDraw a free body diagramfree body diagram for the isolated objectfor the isolated object

under consideration and label all the forces actingunder consideration and label all the forces actingon iton it

►►Resolve the forces into xResolve the forces into x-- and yand y--componentscomponents,,using a convenient coordinate systemusing a convenient coordinate system

►► Apply equations Apply equations, keeping track of signs, keeping track of signs►► SolveSolve the resulting equationsthe resulting equations



Example1: Inclined PlanesExample1: Inclined Planes

Problem:Problem:

A child holds a sled at rest on A child holds a sled at rest on

frictionless, snowfrictionless, snow--covered hill,covered hill,as shown in figure. If the sledas shown in figure. If the sledweights 77.0 N, find the forceweights 77.0 N, find the force TT

exerted by the rope on the sledexerted by the rope on the sledand the forceand the force nn exerted by theexerted by thehill on the sled.hill on the sled.



Example: Inclined PlanesExample: Inclined Planes

►► Choose the coordinate systemChoose the coordinate systemwith x along the incline and ywith x along the incline and yperpendicular to the inclineperpendicular to the incline

►► Replace the force of gravityReplace the force of gravitywith its componentswith its components

Given:

angle: =30°

weight: w=77.0 N

Find:

Tension T=?

Normal n=?

1. Introduce coordinate frame:

Oy: y is directed perp. to incline

Ox: x is directed right, along incline

N N mgT

mgT F Ox x

5.38)30(sin0.77)30(sin

,0sin:

0: F Note

N N mgn

mgnF Oy y

7.66)30(cos0.77)30(cos

,0cos:



Example 2: Connected ObjectsExample 2: Connected Objects

►► Apply Newton’s LawsApply Newton’s Laws separatelyseparatelyto each objectto each object

►► TheThe accelerationacceleration of both objectsof both objectswill bewill be the samethe same

►► TheThe tension istension is the samethe same in eachin eachdiagramdiagram

►► Solve the simultaneous equationsSolve the simultaneous equations

Problem:

Two objects mTwo objects m11=4.00 kg and=4.00 kg andmm22=7.00 kg are connected by a=7.00 kg are connected by alight string that passes over alight string that passes over africtionless pulley. Thefrictionless pulley. Thecoefficient of sliding frictioncoefficient of sliding frictionbetween the 4.00 kg object anbetween the 4.00 kg object anthe surface is 0.300. Find thethe surface is 0.300. Find theacceleration of the two objectsacceleration of the two objectsand the tension of the string.and the tension of the string.



Example: Connected ObjectsExample: Connected Objects

Given:

mass1: m1=4.00 kg

mass2: m2=7.00 kg

friction: =0.300

Find:

Tensions T=?

Acceleration a=?1. Introduce two coordinate frames:

Oy: y’s are directed up

Ox: x’s are directed right

: ,k

Note F ma and f n

.0:

,::1

11

11

gmnF Oy

am f T F Ox Mass

y

k x

.::2 222 amT gmF Oy Mass y

Solving those equations:

a = 5.16 m/s2

T = 32.4 N



EnergyEnergy



IntroductionIntroduction

►► Forms of energy:Forms of energy: MechanicalMechanical

►► focus for nowfocus for now

chemicalchemical

electromagneticelectromagnetic nuclearnuclear

►► Energy can be transformed from one form to anotherEnergy can be transformed from one form to another Essential to the study of physics, chemistry, biology, geology, astronomyEssential to the study of physics, chemistry, biology, geology, astronomy

►► Can be used in place of Newton’s laws to solve certainCan be used in place of Newton’s laws to solve certainproblems more simplyproblems more simply



Work Work ►► Provides a link between force and energyProvides a link between force and energy

►►The work,The work, WW, done by a constant force on an, done by a constant force on anobject is defined as the product of theobject is defined as the product of the

component of the force along the direction of component of the force along the direction of displacementdisplacement andand the magnitude of thethe magnitude of thedisplacementdisplacement

xF W )cos(

(F cos(F cos θ)θ) is the component of is the component of the force in the direction of thethe force in the direction of thedisplacementdisplacement

Δx Δx is the displacementis the displacement



Work Work

►►This gives no information aboutThis gives no information about the time it took for the displacement to occurthe time it took for the displacement to occur

the velocity or acceleration of the objectthe velocity or acceleration of the object

►►Note:Note: work work isis zero whenzero when►►there isthere is no displacementno displacement (holding a bucket)(holding a bucket)

►► force and displacement areforce and displacement are perpendicularperpendicular

to each otherto each other, as, as cos 90cos 90°° = 0= 0 (if we are(if we are

carrying the bucket horizontally, gravitycarrying the bucket horizontally, gravity

does no work)does no work)

(different from everyday “definition” of work)(different from everyday “definition” of work)

xF W )cos(



More About Work More About Work

►► ScalarScalar quantityquantity

►► If there are multiple forces acting on an object, the totalIf there are multiple forces acting on an object, the total

work done is the algebraic sum of the amount of work work done is the algebraic sum of the amount of work done by each forcedone by each force

Units of WorkUnits of Work

SISI joule (J=N m) joule (J=N m)

CGSCGS erg (erg=dyne cm)erg (erg=dyne cm)

US CustomaryUS Customary footfoot--pound (footpound (foot--pound=ft lb)pound=ft lb)



More About Work More About Work

►► Work can beWork can be positivepositive oror negativenegative PositivePositive if the force and the displacement areif the force and the displacement are in thein the

same directionsame direction

NegativeNegative if the force and the displacement areif the force and the displacement are in thein theopposite directionopposite direction

►► Example 1: lifting a cement block…Example 1: lifting a cement block… Work doneWork done by the personby the person::

isis positivepositive when lifting the boxwhen lifting the box

isis negativenegative when lowering the boxwhen lowering the box

Example 2: … then moving it horizontallyExample 2: … then moving it horizontally

Work doneWork done by gravityby gravity:: isis negativenegative when lifting the boxwhen lifting the box

isis positivepositive when lowering the boxwhen lowering the box

isis zerozero when moving it horizontallywhen moving it horizontally

00: 321 mghmghW W W W work Total

lifting lowering moving total



Problem:Problem: cleaning the dorm roomcleaning the dorm room

Eric decided to clean his dorm room

with his vacuum cleaner. While doing

so, he pulls the canister of the vacuum

cleaner with a force of magnitudeF=50.0 N at an angle 30.0°. He moves

the vacuum cleaner a distance of 3.00

meters. Calculate the work done by all

the forces acting on the canister.



Problem:Problem: cleaning the dorm roomcleaning the dorm room

Given:

angle: =30°

force: F=55.0 N

Find:

Work WF=?

Wn=?

Wmg=?

1. Introduce coordinate frame:

Oy: y is directed up

Ox: x is directed right

2. Note: horizontal

displacement only,

Work: W=(F cos ) s

J m N m N sF W F 130130)00.3)(0.30)(cos0.50()cos(

J mnsnW n 0)00.3)(0.90)(cos()cos(

J mnsmgW mg 0)00.3))(0.90)(cos(()cos(

No work as force is

perpendicular to the

displacement



Graphical Representation of Work Graphical Representation of Work

ii xF W )cos(

SplitSplit totaltotal displacement (xdisplacement (xf f --xxii))

intointo many small displacementsmany small displacements xx

For each small displacement:For each small displacement:

Thus, total work is:Thus, total work is:

which iswhich is total area under the F(x) curve!total area under the F(x) curve!

i

i x

i

itot xF W W



Kinetic EnergyKinetic Energy

►► EnergyEnergy associated with theassociated with the motionmotion of an objectof an object

►► Scalar quantityScalar quantity with thewith the same units as work same units as work

►► Work is related to kinetic energyWork is related to kinetic energy►► Let F be aLet F be a constantconstant force:force:

.2

1

2

1

2:

.

2

,2

:,)(

2

0

2

2

0

2

2

0

22

0

2

mvmvvv

mW Thus

vvsaor savv

but smaFsW

net

net

2

2

1mvKE This quantity is called kinetic energy:



Work Work--Kinetic Energy TheoremKinetic Energy Theorem

►►When work is done by a net force on anWhen work is done by a net force on anobject and the only change in the object is itsobject and the only change in the object is its

speed, the work done is equal to the changespeed, the work done is equal to the changein the object’s kinetic energyin the object’s kinetic energy

►►

Speed will increase if work is positiveSpeed will increase if work is positive

Speed will decrease if work is negativeSpeed will decrease if work is negative

KE KE KE W i f net



Work and Kinetic EnergyWork and Kinetic Energy

►► An object’s kinetic An object’s kineticenergy can also beenergy can also bethought of as thethought of as the

amount of work theamount of work themoving object couldmoving object coulddo in coming to restdo in coming to rest The moving hammerThe moving hammer

has kinetic energy andhas kinetic energy and

can do work on the nailcan do work on the nail



Let’s watch the movie!Let’s watch the movie!



Potential EnergyPotential Energy

►►Potential energyPotential energy is associated with theis associated with thepositionposition of the object within someof the object within some

systemsystem Potential energy is a property of thePotential energy is a property of the

system, not the objectsystem, not the object

A system is a collection of objects or A system is a collection of objects or

particles interacting via forces or processesparticles interacting via forces or processesthat are internal to the systemthat are internal to the system

►►UnitsUnits of of Potential EnergyPotential Energy are the sameare the sameas those of as those of Work Work andand Kinetic EnergyKinetic Energy



Gravitational Potential EnergyGravitational Potential Energy

►►Gravitational Potential Energy is the energyGravitational Potential Energy is the energyassociated with the relative position of anassociated with the relative position of anobject in space near the Earth’s surfaceobject in space near the Earth’s surface

Objects interact with the earth through theObjects interact with the earth through thegravitational forcegravitational force

Actually the potential energy of the earth Actually the potential energy of the earth--objectobject

systemsystem



Potential Energy: examplePotential Energy: example



Work and Gravitational PotentialWork and Gravitational Potential

EnergyEnergy►► Consider block of mass m at initial heightConsider block of mass m at initial height yyii

►► Work done by the gravitational forceWork done by the gravitational force

f igravity PE PE W

.:

,1cos,

:,)cos(cos

f i f igrav

f i

grav

mgymgy y ymgW Thus

y ys

but smgsF W

This quantity is called potential energy:

mgyPE

Note:Note: Important: work is related to the difference in PE’s!



Reference Levels for GravitationalReference Levels for Gravitational

Potential EnergyPotential Energy►► A location where the gravitational potential energy A location where the gravitational potential energy

is zero must be chosen for each problemis zero must be chosen for each problem

The choice is arbitrary sinceThe choice is arbitrary since the changethe change in the potentialin the potential

energy is the important quantityenergy is the important quantity Choose a convenient location for the zero referenceChoose a convenient location for the zero reference

heightheight

►►often the Earth’s surfaceoften the Earth’s surface

►►may be some other point suggested by the problemmay be some other point suggested by the problem



Reference Levels forReference Levels for

Gravitational Potential EnergyGravitational Potential Energy A location where the A location where thegravitational potential energy isgravitational potential energy iszero must be chosen for eachzero must be chosen for each

problemproblem The choice is arbitrary sinceThe choice is arbitrary sincethe changethe change in the potentialin the potentialenergy gives the work doneenergy gives the work done

.

.

,

,

321

3

2

1

33

22

11

gravgravgrav

f igrav

f igrav

f igrav

W W W

mgymgyW

mgymgyW

mgymgyW



ConcepTest 2ConcepTest 2

At the bowling alley, the ball-feeder mechanism must exerta force to push the bowling balls up a 1.0-m long ramp.The ramp leads the balls to a chute 0.5 m above the baseof the ramp. Approximately how much force must beexerted on a 5.0-kg bowling ball?

Please fill your answer as question 3 of

General Purpose Answer Sheet

1. 200 N

2. 50 N

3. 25 N

4. 5.0 N

5. impossible todetermine



Conservative ForcesConservative Forces

►► A force is A force is conservativeconservative if the work it does onif the work it does onan object moving between two points isan object moving between two points isindependent of the pathindependent of the path the objects takethe objects takebetween the pointsbetween the points

The work depends only upon the initial and finalThe work depends only upon the initial and finalpositions of the objectpositions of the object

Any conservative force can have a potential Any conservative force can have a potentialenergy function associated with itenergy function associated with it

Note: a force is conservative if the work it does on an object moving

through any closed path is zero.



Examples of Conservative Forces:Examples of Conservative Forces:

►►Examples of conservative forces include:Examples of conservative forces include:

GravityGravity

Spring forceSpring force

Electromagnetic forcesElectromagnetic forces

►►Since work is independent of the path:Since work is independent of the path:

:: only initial and final pointsonly initial and final points

f ic PE PE W



Nonconservative ForcesNonconservative Forces

►► A force is A force is nonconservativenonconservative if the work itif the work itdoes on an objectdoes on an object depends on the pathdepends on the pathtaken by the object between its final andtaken by the object between its final andstarting points.starting points.

►►Examples of nonconservative forcesExamples of nonconservative forces

kinetic friction, air drag, propulsive forceskinetic friction, air drag, propulsive forces



Example: Friction as aExample: Friction as a

Nonconservative ForceNonconservative Force►►The friction force transforms kinetic energyThe friction force transforms kinetic energy

of the object into a type of energyof the object into a type of energyassociated with temperatureassociated with temperature

the objects are warmer than they were beforethe objects are warmer than they were beforethe movementthe movement

Internal EnergyInternal Energy is the term used for the energyis the term used for the energy

associated with an object’s temperatureassociated with an object’s temperature



Friction Depends on the PathFriction Depends on the Path

►►TheThe blueblue path ispath isshortershorter than thethan the redred

pathpath►►The work required isThe work required is

less on the blueless on the bluepath than on the redpath than on the redpathpath

►► FrictionFriction depends ondepends onthe paththe path and so is aand so is anonconservativenonconservativeforceforce



Conservation of MechanicalConservation of Mechanical

EnergyEnergy►►Conservation in generalConservation in general

To say a physical quantity isTo say a physical quantity is conservedconserved is to say thatis to say thatthe numerical value of the quantity remains constantthe numerical value of the quantity remains constant

►► In Conservation of Energy, the total mechanicalIn Conservation of Energy, the total mechanicalenergy remains constantenergy remains constant

In any isolated system of objects that interact onlyIn any isolated system of objects that interact onlythrough conservative forces, the total mechanicalthrough conservative forces, the total mechanical

energy of the system remains constant.energy of the system remains constant.



Conservation of EnergyConservation of Energy

►►Total mechanical energy is the sum of theTotal mechanical energy is the sum of thekinetic and potential energies in the systemkinetic and potential energies in the system

Other types of energy can be added to modifyOther types of energy can be added to modify

this equationthis equation

f f ii

f i

PE KE PE KE E E



Problem Solving withProblem Solving with

Conservation of EnergyConservation of Energy►►Define the systemDefine the system

►► Select the location of zero gravitational potentialSelect the location of zero gravitational potentialenergyenergy

DoDo notnot change this location while solving the problemchange this location while solving the problem►►Determine whether or not nonconservative forcesDetermine whether or not nonconservative forces

are presentare present

►► If only conservative forces are present, applyIf only conservative forces are present, apply

conservation of energy and solve for the unknownconservation of energy and solve for the unknown



Potential Energy Stored in aPotential Energy Stored in a

SpringSpring►► Involves theInvolves the spring constantspring constant (or(or

force constant), k force constant), k

►►Hooke’s Law gives the forceHooke’s Law gives the force

F =F = -- k xk x►► F is the restoring forceF is the restoring force

►► F is in the opposite direction of xF is in the opposite direction of x

►► k depends on how the spring was formed, thek depends on how the spring was formed, thematerial it is made from, thickness of the wire,material it is made from, thickness of the wire,

etc.etc.



Potential Energy in a SpringPotential Energy in a Spring

►► Elastic Potential EnergyElastic Potential Energy

related to the work required to compress a spring fromrelated to the work required to compress a spring fromits equilibrium position to some final, arbitrary, positionits equilibrium position to some final, arbitrary, position

xx

2

2

1kxPE s

.2

1

20:

22

0

2,1cos

:,cos

2

0

xk xkx

W Thus

kxF F F F

but xF W

spr

x x

spr

This is called elastic potential energy:



Power, cont.Power, cont.

►►US Customary units are generally hpUS Customary units are generally hp(horsepower)(horsepower)

need a conversion factorneed a conversion factor

Can define units of work or energy in terms of units of Can define units of work or energy in terms of units of

power:power:►►kilowatt hours (kWh) are often used in electric billskilowatt hours (kWh) are often used in electric bills

W746s

lbft550hp1



Center of MassCenter of Mass

►►The point in the body at which all the massThe point in the body at which all the massmay be considered to be concentratedmay be considered to be concentrated

When using mechanical energy, the change inWhen using mechanical energy, the change inpotential energy is related to the change inpotential energy is related to the change inheight of the center of massheight of the center of mass



Work Done by Varying ForcesWork Done by Varying Forces

►►The work done by aThe work done by avariable force actingvariable force actingon an object thaton an object that

undergoes aundergoes adisplacement is equaldisplacement is equalto the area under theto the area under thegraph of F versus xgraph of F versus x



Spring ExampleSpring Example

►► Spring is slowly stretchedSpring is slowly stretchedfrom 0 to xfrom 0 to xmaxmax

►► FFappliedapplied == --FFrestoringrestoring = kx= kx

►►W = {W = {area under the curvearea under the curve} =} =½(kx) x = ½kx²½(kx) x = ½kx²

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