Classical Mechanics Lecture 7

Today’sConcepts:Work&Kine6cEnergy

MechanicsLecture7,Slide1

KarateWillnotdoSession3ofUnit8.

ItisaKaratething.

WewillonlymarkSession2ofunit8.

Youcandoitasa“project”inUnit14insteadoftheassignedprojectonthependulum.

Which Job?Allthreejobspaythesameamountofmoney.Whichonewouldyouchooseforyourcrew?

A.Job1

B.Job2

C.Job3

SESSION ONE: PHYSICAL WORK AND POWER30 min

The Concept of Physical Work

Suppose you are president of the Load 'n' Go Co. A local

college has three jobs available and will allow you to

choose which one you want before offering the other two

jobs to rival companies. All three jobs pay the same

amount of money. Which one would you choose for your

crew?

Figure 10-1: A description of jobs to bid on

!Activity 10-1: Choosing Your Job

Examine the descriptions of the jobs shown in Figure 10-1.

Which one would you be most likely to choose? Least likely to

choose? Explain the reasons for your answer.

Workshop Physics: Unit 10 – Work and Energy Page 10-3

Authors: Priscilla Laws, Robert Boyle, and John Luetzelschwab

© 1990-94 Dept. of Physics and Astronomy, Dickinson College Supported by FIPSE (U.S.

Dept. of Ed.) and NSF. Modified for SFU by N. Alberding, 2005, 2008.

A B

B

C

Joke of the day

Don'tdrinkandDERIVE!

B

Stuff you asked about:WHATISGOINGONWITHALLTHEINTEGRALS??????HELP:(Ihatesprings,butweshouldtalkaboutthem.sta6cfric6onEverythingisfineexcepttheworkdonebyGravityfarfromearthTheslideshadalotofinforma6onthatwasconfusinganditflewby.I

didn'thave6metounderstandwhatwasgoingon.ThedotproductisocnfusingPredymucheverything.Thisprelecturesortofwentovermyhead.I'll

definitelyhavetowatchitagain.Theconceptofposi6veandnega6veworkisveryconfusing.please

explain

MechanicsLecture7,Slide2

Your comments and questions

Will we have to use integrals to calculate work? clarification on under what circumstances the dot product

should be used would be helpful.Are you going to test us on the dot product or is it just a concept

you want us to know?

The Dot Product

MechanicsLecture7,Slide3

~A =A

x

ı + A

y

| + A

z

k

~B =B

x

ı + B

y

| + B

z

k

ı · ı =1| · | =1

k · k =1

ı · | =0

| · k =0

k · | =0

Dot Product using Unit Vectors

etc.

~A · ~B =(A

x

ı + A

y

| + A

z

k) · (Bx

ı + B

y

| + B

z

k)

=(Ax

B

x

ı · ı + A

x

B

y

ı · | + · · · + A

z

B

y

k · j + A

z

B

z

k · k)=(A

x

B

x

⇥ 1 + A

x

B

y

⇥ 0 + · · · + A

z

B

y

⇥ 0 + A

z

B

z

⇥ 1)~A · ~B =(A

x

B

x

+ A

y

B

y

+ A

z

B

z

)

Dot Product in Rectangular Coordinates

~A · ~B =(A

x

B

x

+ A

y

B

y

+ A

z

B

z

)

=|~A||~B| cos ✓

✓ = arccos

0BBBB@~A · ~B|~A||~B|

1CCCCA

Finding the angle between vectorsIfweknowthexyzcomponentsoftwovectorswecanfindtheanglebetweenthem:

therefore

|⇥A| =|⇥B| = 5

⇥A · ⇥B =3 · 4 + 4 · 3 = 24

� = arccos

0BBBB@⇥A · ⇥B|⇥A||⇥B|

1CCCCA

� = arccos

24

25

!= arccos 0.96

=0.28 rad = 16

�

~A = 3ı + 4 |~B = 4ı + 3 |

Example

Work-Kinetic Energy TheoremTheworkdonebythenetforceFasitactsonanobjectthatmovesbetweenposi6onsr1and r2 isequaltothechangeintheobject’skine6cenergy:

MechanicsLecture7,Slide4

W =

~r2Z

~r1

~F · d~l

W =

~r2Z

~r1

~F · d~l = ~F · ~d0BBBBBBBBB@

~r2Z

~r1

d~l = ~d

1CCCCCCCCCA

Work-Kinetic Energy Theorem: 1-D Example

Iftheforceisconstantandthedirec6onsaren’tchangingthenthisisverysimpletoevaluate:

MechanicsLecture7,Slide5

Inthiscase sincecos(0)=1

ThisisprobablywhatyourememberfromHighSchool.(Thenota6onmaybeconfusingthough.)

F d

= Fd

car

Clicker Question

Alightercarandaheaviervan,eachini6allyatrest,arepushedwiththesameconstantforceF.Aierbothvehiclestraveladistanced,whichofthefollowingstatementsistrue?(Ignorefric6on)

MechanicsLecture7,Slide6

A)TheywillhavethesamevelocityB)Theywillhavethesamekine6cenergyC)Theywillhavethesamemomentum

F d

F d

car

van

~r2Z

~r1

~F · d~l = �K

Aforcepushingoversomedistancewillchangethekine6cenergy.

MechanicsLecture7,Slide7

Concept–veryimportant

Deriva6on–notsoimportant

θ

Work done by gravity near the Earth’s surface

MechanicsLecture7,Slide8

mgpat

h

mg

dl1dy1

dx1

mg

dl1

dl2

dlN

Work done by gravity near the Earth’s surface

MechanicsLecture7,Slide9

= m~g · d~l1 + m~g · d~l1 + · · · + m~g · d~lN2

mg

dl1

dl2

dlN

Work done by gravity near the Earth’s surface

MechanicsLecture7,Slide10

Δy

= m~g · d~l1 + m~g · d~l1 + · · · + m~g · d~lN

Work-Kinetic Energy Theorem

Ifthereareseveralforcesac6ngthenWistheworkdonebythenet(total)force:

MechanicsLecture7,Slide12

Youcanjustadduptheworkdonebyeachforce

H

FreeFallFric6onlessincline String

CheckPoint

Threeobjectshavingthesamemassbeginatthesameheight,andallmovedownthesamever6caldistanceH.Onefallsstraightdown,oneslidesdownafric6onlessinclinedplane,andoneswingsontheendofastring.

MechanicsLecture7,Slide13

Inwhichcasedoestheobjecthavethebiggestnetworkdoneonitbyallforcesduringitsmo6on?

A)FreeFall B)Incline C)String D)Allthesame

vp

Clicker Question

Threeobjectshavingthesamemassbeginatthesameheight,andallmovedownthesamever6caldistanceH.Onefallsstraightdown,oneslidesdownafric6onlessinclinedplane,andoneswingsontheendofastring.Whatistherela6onshipbetweentheirspeedswhentheyreachthebodom?

H

FreeFallFric6onlessincline Pendulum

A)vf > vi > vp B)vf > vp > vi C)vf = vp = vi

MechanicsLecture7,Slide14

v f vi

CheckPoint / Clicker Question

Onlygravitywilldowork:Wg = Δ K

H

FreeFallFric6onlessincline String

A) vf > vi > vp B) vf > vp > vi C) vf = vp = vi

Wg = mgH

Δ K = 1/2 mv22

MechanicsLecture7,Slide15

CheckPoint

Acardrivesupahillwithconstantspeed.WhichstatementbestdescribesthetotalworkWTOTdoneonthecarbyallforcesasitmovesupthehill?

About50%gotthisright…

A)WTOT = 0

B)WTOT > 0

C)WTOT < 0

v

MechanicsLecture7,Slide16

v

Clicker Question

Acardrivesupahillwithconstantspeed.Howdoesthekine6cenergyofthecarchangeasitmovesupthehill?

A)ItincreasesB)ItstaysthesameC)Itdecreases

MechanicsLecture7,Slide17

v

CheckPoint

Acardrivesupahillwithconstantspeed.WhichstatementbestdescribesthetotalworkWTOTdoneonthecarbyallforcesasitmovesupthehill?

A) WTOT = 0

B) WTOT > 0

C) WTOT < 0

A)Thechangeinkine6cenergyiszerobecausethevelocityisconstant.Bythework-kine6cenergytheorem,weknowthatworkmustalsobezero.

B)Theforceoffric6onbetweenthecar'swheelsandtherampexertaforceuptherampoverapar6culardistance.W=Fd,sotheworkisposi6ve.C)gravityisdownwardandcarmovesupthehill.

MechanicsLecture7,Slide18

CheckPoint

Aboxsitsonthehorizontalbedofamovingtruck.Sta6cfric6onbetweentheboxandthetruckkeepstheboxfromslidingaroundasthetruckdrives.

µS

a

Theworkdoneontheboxbythesta6cfric6onalforceasthetruckmovesadistanceDis:

A)Posi6veB)Nega6veC)Zero

About60%gotthisright…

MechanicsLecture7,Slide21

Physics211Lecture7,Slide22

From Before

Aboxsitsonthehorizontalbedofamovingtruck.Sta6cfric6onbetweentheboxandthetruckkeepstheboxfromslidingaroundasthetruckdrives.

Ifthetruckmoveswithconstantaccelera6ngtotheleiasshown,whichofthefollowingdiagramsbestdescribesthesta6cfric6onalforceac6ngonthebox:

µS

a

A B C

CheckPoint

MechanicsLecture7,Slide23

Theworkdoneontheboxbythesta6cfric6onalforceasthetruckmovesadistanceDis:

A)Posi6veB)Nega6veC)Zero

D

A)becausethedirec6onofthesta6cfric6onalforceandthedirec6ontheboxtravelsareallsame.

B)Sincethefric6onforcealwayshasoppositedirec6onfromthemovementdirec6on,theworkdoneisnega6ve

C)Fric6onkeepstheboxfromsliding,thusD=0,andworkdoneis0.

µS

a

F

MechanicsLecture7,Slide24

Work done by a Spring

MechanicsLecture7,Slide25

Iamconfusedabouttheposi6veworkandnega6veworkandalsotheposi6veandnega6veforcesforthespringproblems.

Inthisexamplethespringdoes–veworksince F andΔxareinoppositedirec6on.Theaxesdon’tmader.

Usetheformulatogetthemagnitudeofthework

Useapicturetogetthesign(lookatdirec6ons)

Aboxadachedatresttoaspringatitsequilibriumlength.YounowpushtheboxwithyourhandsothatthespringiscompressedadistanceD,andyouholdtheboxatrestinthisnewloca6on.

Duringthismo6on,thespringdoes:

A)Posi6veWorkB)Nega6veWorkC)Zerowork

MechanicsLecture7,Slide26

Clicker Question

D

Aboxadachedatresttoaspringatitsequilibriumlength.YounowpushtheboxwithyourhandsothatthespringiscompressedadistanceD,andyouholdtheboxatrestinthisnewloca6on.

Duringthismo6on,yourhanddoes:

A)Posi6veWorkB)Nega6veWorkC)Zerowork

Clicker Question

MechanicsLecture7,Slide27

D

Aboxadachedatresttoaspringatitsequilibriumlength.YounowpushtheboxwithyourhandsothatthespringiscompressedadistanceD,andyouholdtheboxatrestinthisnewloca6on.

Duringthismo6on,thetotalworkdoneontheboxis:

A)Posi6veB)Nega6veC)Zero

Clicker Question

MechanicsLecture7,Slide28

D

= GMem

1r2� 1

r1

!

MechanicsLecture7,Slide29

W =

~r2Z

~r1

~F(~r) · d~r = �r2Z

r1

GMemr2 dr

Gravity

W = GMem

1r2� 1

r1

!A)Case1B)Case2C)Theyarethesame

InCase1wesendanobjectfromthesurfaceoftheearthtoaheightabovetheearthsurfaceequaltooneearthradius.

InCase2westartthesameobjectaheightofoneearthradiusabovethesurfaceoftheearthandwesenditinfinitelyfaraway.

InwhichcaseisthemagnitudeoftheworkdonebytheEarth’sgravityontheobjectbiggest?

Clicker Question

Case1 Case 2

MechanicsLecture7,Slide30

W = GMem

12RE� 1

RE

!= �GMem

2RE

W = GMem

11 �

12RE

!= �GMem

2RE

Case1 Case2

Case1:

2RE

RE

Same!

Clicker Question Solution

MechanicsLecture7,Slide31

Case2: