chapter 1 fundamental of physics pham hong quang

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Fundamental of Physics

PETROVIETNAM UNIVERSITY

FUNDAMENTAL SCIENCES DEPARTMENT

Hanoi, August 2012

Pham Hong QuangE-mail: [email protected]



References

Pham Hong Quang Fundamental Science Department 2

[1] Cơ sở Vật lý, Tập I, II, III: Cơ học, nhiệt học, David

Halliday, Robert Resnik, Jearl Walker, bản dịch tiếng

Vit, !"# $i%o d&c '1((()*

[+] Fundamentals o ph!sics, "th ed#, $%tended,

David Halliday, Robert Resnick and Jearl Walker, John

Wiley -ons '+../)*

[0] Vt l2 34i c56ng, t7 1, 869!hit, :56ng D;y<n

#=nh ch> bi<n



Chapter 1


1.1 Measurement, Standards of

Length, Mass, and Time

1.2 Dimensional Analysis

1.3 on!ersion of "nits

1.# oordinate Systems

1.$ %e&tors and s&alars

1.' The Dis(la&ement, %elo&ity,

and A&&eleration %e&tors

1.) "niform ir&ular Motion

1.* Angular %elo&ity and

A&&eleration %e&tor

1.* Pro+e&tile Motion

inemati&s of a (oint-lie

o/+e&t



1.1 Standards of Length, Mass, and Time


Length@nits

-A B Ceter, C

Dened in terCs oE a CeterB the distance traveled bylight in a vac;;C d;ring agiven tiCe

-ee Fable 1*1 Eor soCeeGaC7les oE lengths



Length, Cont.






Mass@nits

-A B kilograC, kgDened in terCs oE akilograC, based on as7ecic cylinder ke7tat the Anternational

#;rea; oE -tandards-ee Fable 1*+ EorCasses oE vario;sobects



Mass,Cont.






Time@nits

seconds, sDened in terCs oE theoscillation oE radiation EroCa cesi;C atoC

-ee Fable 1*0 Eor soCea77roGiCate tiCe intervals



Time,Cont.






System ofMeasurements

@- 8;stoCaryeveryday ;nits:ength is Ceas;red inEeet

FiCe is Ceas;red in

secondsIass is Ceas;red insl;gs

oEten ;ses eight, in

7o;nds, instead oECass as a





Pre0esLreGes corres7ondto 7oers oE 1.Mach 7reG has as7ecic naCe

Mach 7reG has as7ecic abbreviation

Fhe 7reGes can be;sed ith any base;nits

Fhey are C;lti7liersoE the base ;nitMGaC7lesN

1 CC O 1.90 C1 Cg O 1.90 g



1.2 Dimensional Analysis


•DiCension has as7ecic Ceaning B it

denotes the 7hysicalnat;re oE a K;antity•DiCensions aredenoted ith sK;arebrackets

:ength [:]Iass [I]

FiCe [F]

12D lA l C t



1.2 Dmensonal Analyss, Cont.


• FechniK;e to check the correctness oE an eK;ation orto assist in deriving an eK;ation

•DiCensions 'length, Cass, tiCe, coCbinations) can betreated as algebraic K;antities

• add, s;btract, C;lti7ly, divide•#oth sides oE eK;ation C;st have the saCediCensions•DiCensions oE soCe coCCon K;antities are givenbelo



14C d t S t



1.4 Coordnate Systems


•@sed to describe the 7osition oE a 7oint in

s7ace

•

8oordinate systeC consists oE P Ged reEerence 7oint called the origin

-7ecic aGes ith scales and labels

Anstr;ctions on ho to label a 7ointrelative to the origin and the aGes

e



1.4 Coordinate Systems

Pham Hong Quang Fundamental Science Departmentdinate 17

ye

x y zr OM xe ye ze.= = + +

;;;;rr r r r

Des&artesoordinate System

Loints are labeled

' % , !,& )

θ



1.4 Coordinate Systems


θ

S(heri&al oordinateSystem

G O r sinTcosUyO rsinTsinU

O rcosTPndr O

cosT O

U O arctg

222 z y x ++

222 z y x

z

++

x

y



1.5 Vectors and Scalars


•P scalar quantity is coC7letely

s7ecied by a single val;e ith an

a77ro7riate ;nit and has no direction*

•P vector quantity is coC7letely

described by a n;Cber and

a77ro7riate ;nits 7l;s a direction*





%e&tor am(le•

P 7article travels EroC Pto # along the 7athshon by the dotted redline Fhis is the distance

traveled and is ascalar

• Fhe displacement isthe solid line EroC P to #

Fhe dis7laceCent isinde7endent oE the7ath taken beteenthe to 7ointsDis7laceCent is a

vector





%e&tor otation• FeGt ;ses bold ith arro to denote a vectorN

•Plso ;sed Eor 7rinting is siC7le bold 7rintN A

•When dealing ith ;st the Cagnit;de oE a vector,

an italic letter ill be ;sedN ' or

Fhe Cagnit;de oE the vector has 7hysical ;nits

Fhe Cagnit;de oE a vector is alays a 7ositive

n;Cber

•When handritten, ;se an arroN

A

A

A





Adding %e&tors 4ra(hi&ally

•8hoose a scale

•Dra the rst vector, , ith the a77ro7riate length

and in the direction s7ecied, ith res7ect to a

coordinate systeC

•Dra the neGt vector ith the a77ro7riate length

and in the direction s7ecied, ith res7ect to a

coordinate systeC hose origin is the end oE vector

and 7arallel to the coordinate systeC ;sed Eor

A

AA





Adding %e&tors

4ra(hi&ally, &ont.•8ontin;e draing the

vectors Xti79to9tailY

• Fhe res;ltant is dran

EroC the origin oE to

the end oE the last

vector

•Ieas;re the length oEand its angle

@se the scale Eactor

to convert length to

act;al Cagnit;de

A

R





om(onent Method of Adding%e&tors

•$ra7hical addition is not recoCCended hen

High acc;racy is reK;ired

AE yo; have a three9diCensional 7robleC

•8oC7onent Cethod is an alternative Cethod

At ;ses 7roections oE vectors along

coordinate aGes





om(onents of a

%e&tor, 5ntrodu&tion•P &om(onent is a 7roection

oE a vector along an aGis

Pny vector can be

coC7letely described by its

coC7onents

•At is ;seE;l to ;se

re&tangular &om(onents

Fhese are the 7roections oE

the vector along the G9 and





om(onents of a%e&tor

•Pss;Ce yo; are given a

vector•At can be eG7ressed in

terCs oE to other

vectors, and

• Fhese three vectors

EorC a right triangle

•

A

x A y A

x y = +A A A





om(onents of a

%e&tor, 0nal

• Fhe coC7onents can be

7ositive or negative and

ill have the saCe ;nits

as the original vector

• Fhe signs oE the

coC7onents ill de7end

on the angle





"nit %e&tors

•P unit vector is a vector ith a

Cagnit;de oE eGactly 1*•@nit vectors are ;sed to s7eciEy a

direction and have no other 7hysical

signicance





"nit %e&tors, &ont.• Fhe syCbols

re7resent ;nit vectors• Fhey EorC a set oE

C;t;ally 7er7endic;lar

vectors in a right9handed

coordinate systeC

•ReCeCber,

kZand, HZ,iZ

= = =ˆˆˆ 1i j k

5 d l





Adding %e&tors "sing "nit

%e&tors

@sing

Fhen

and so ( % O ' % ) % and ( ! O ' ! ) !

= +R A B

( ) ( )

( ) ( )

= + + +

= + + += +

r

r

r

ˆˆˆˆ

ˆˆ

ˆˆ

x y x y

x x y y

x y

A A B B

A B A BR R

R i j i j

R i j

R i j

2 2 1tan

y

x y

x

R R R R

R θ

−= + =

5 d l





S&alar (rodu&t of t6o !e&tors

Fhe dot product oE to vectors is the s;C oE the7rod;cts oE their corres7onding coC7onents* AE

aO\a1, a+ and /O\b1, b+, then a^/O a1b1a+b+ *

MG* AE aO\1, and /O\0,/, then

a./O00+O0Q

AE T is the angle beteen vectors a and /, then

!oteN these are ;st to ays oE eG7ressing the dot

7rod;ct

!ote that the dot 7rod;ct oE to vectors 7rod;ces a

θ cosbaba =⋅

15V dS l





%e&tor (rodu&t of t6o !e&tors

Fhe cross 7rod;ct a G / oE to

vectors a and /, ;nlike the dot 7rod;ct,

is a vector*

_or this reason, it is also called the vector

7rod;ct*

!ote that a G / is dened only hen a and /

are three9diCensional '09D) vectors*AE a O ‹a1, a+, a0› and / O ‹*1, *+, *0›, then

the cross 7rod;ct oE a and / is the vector

a G / O ‹a+*0 9 a0*+, a0*1 9 a1*0, a1*+ 9 a+*1›

15V t dS l





%e&tor (rodu&t of t6o !e&tors, &ont.

An order to Cake Denition 1 easier

to reCeCber, e ;se the notation oE deterCinants*

P deterCinant oE order 0 can be dened in terCs oE

second9order deterCinants as EollosN

1 2 3

2 3 1 3 1 2

1 2 3 1 2 32 3 1 3 1 2

1 2 3

a a ab b b b b b

b b b a a ac c c c c cc c c

= − +

15V t dS l





%e&tor (rodu&t of t6o !e&tors, &ont.

We see that the cross 7rod;ct oE the vectorsa O a1i a+ + a0 and / O *1i *+ + *0

isN

2 3 1 3 1 2

2 3 1 3 1 2

a a a a a ab b b b b b

× = − +a b i !

1 2 3

1 2 3

a a a

b b b

× =

i ! a b

15V t dS l





%e&tor (rodu&t of

t6o !e&tors, &ont.Dire&tion7AE the ngers oE yo;r righthand c;rl in the directionoE a rotation 'thro;gh anangle less than 1/.`)EroC a to /, then yo;rth;Cb 7oints in thedirection oE a G /*

Length7AE + is the angle beteena and / 'so . + ), then

a G / O a/ sin +

. , ,A l ti V t



. , , Acceleration Vectors


Position and

Dis(la&ement

• Fhe 7osition oE an obect

is described by its

7osition vector,

• Fhe dis(la&ement oE

the obect is dened as

the change in its

position

∆ ≡ −f i r r r

1.6TheDsplacement,Velocty,and



1.6 The Dsplacement, Velocty, and Acceleration Vectors


A!erage !elo&ity

• Fhe average velocity is the ratio oE the dis7laceCent to

the tiCe interval Eor the dis7laceCent

• Fhe direction oE the average velocity is the direction oE

the dis7laceCent vector

∆≡ ∆

ravg

t r v






5nstantaneous

!elo&ity• Fhe instantaneo;s velocityis the liCit oE the averagevelocity as Δt a77roaches

ero

Ps the tiCe interval

becoCes sCaller, thedirection oE thedis7laceCenta77roaches that oE the

line tangent to the

∆ →

∆≡ =

∆

r

0

limt

d

t dt

r r v






5nstantaneous %elo&ity, &ont.• Fhe direction oE the instantaneo;s

velocity vector at any 7oint in a

7articles 7ath is along a line tangentto the 7ath at that 7oint and in the

direction oE Cotion

• Fhe Cagnit;de oE the instantaneo;svelocity vector is the s7eed

Fhe s7eed is a scalar K;antity






− ∆≡ =− ∆

r f i avg

f i t t t v v va

A!erage a&&eleration

• Fhe average acceleration oE a

7article as it Coves is dened as

the change in the instantaneo;s

velocity vector divided by the

tiCe interval d;ring hich that

change occ;rs*






A!erage

A&&eleration, &ont•Ps a 7article Coves, the

direction oE the change in

velocity is Eo;nd by

vector s;btraction

• Fhe average acceleration

is a vector K;antity

directed along

∆ = −f i v v v

∆v






5nstantaneous a&&eleration

• Fhe instantaneo;s acceleration is the liCiting

val;e oE the ratio as Δt a77roaches ero

Fhe instantaneo;s eK;als the derivative oE

the velocity vector ith res7ect to tiCe

∆ →

∆≡ =

∆r

0

limt

d t dt v va

∆ ∆t v






Produ&ing An A&&elerationVario;s changes in a 7articles Cotion Cay

7rod;ce an acceleration

•

Fhe Cagnit;de oE the velocity vector Caychange

• Fhe direction oE the velocity vector Cay

changeMven iE the Cagnit;de reCains constant

• #oth Cay change siC;ltaneo;sly

1.7Unform CrcularMoton



1.7 Unform Crcular Moton


•Uniform circular motion occ;rs hen an

obect Coves in a circ;lar 7ath ith a

constant s7eed

• Fhe associated analysis Cotion is a particle

in uniorm circular motion

•Pn acceleration eGists since the direction oE

the Cotion is changing

• Fhis change in velocity is related to an

acceleration

• Fhe velocity vector is alays tangent to the

7ath oE the obect

V




. U o C cua oto


hanging %elo&ity in

"niform ir&ular

Motion

• Fhe change in the velocityvector is d;e to the

change in direction

• Fhe vector diagraC shosf i

= + ∆v v v





entri(etal A&&eleration

• Fhe acceleration is alays

7er7endic;lar to the 7ath oE the

Cotion

• Fhe acceleration alays 7oints

toard the center oE the circle oE

Cotion• Fhis acceleration is called the

centripetal acceleration





entri(etal A&&eleration, &ont.• Fhe Cagnit;de oE the centri7etal acceleration

vector is given by

• Fhe direction oE the centri7etal acceleration

vector is alays changing, to stay directedtoard the center oE the circle oE Cotion

2

C var =





Period

• Fhe period , T , is the tiCe

reK;ired Eor one coC7lete

revol;tion

• Fhe s7eed oE the 7article o;ld

be the circ;CEerence oE the

circle oE Cotion divided by the7eriod

• FhereEore, the 7eriod is dened

as

2 r T

v

π ≡

1.7 Tangental and Radal Acceleraton



g


• Fhe Cagnit;de oE the velocity co;ld also be

changing•An this case, there o;ld be a tangential

acceleration

• Fhe Cotion o;ld be ;nder the in;ence oE

both tangential and centri7etal accelerations

• !ote the changing acceleration vectors

17TangentialandRadialAcceleration



1.7 Tangential and Radial Acceleration


Total A&&eleration

• Fhe tangential acceleration ca;ses the

change in the s7eed oE the 7article

• Fhe radial acceleration coCes EroC a

change in the direction oE the velocity

vector

17TangentialandRadialAcceleration



1.7 Tangential and Radial Acceleration


Total A&&eleration, e8uations

• Fhe tangential accelerationN

• Fhe radial accelerationN

• Fhe total accelerationN

Iagnit;de

Direction-aCe as velocity vector iE - is increasing,o77osite iE - is decreasing

t

dv a

dt =

2

r C

v a a

r = − = −

2 2

r t a a a= +

18AngularVelocityandAccelerationVector



1.8 Angular Velocity and Acceleration Vector


z ze eω = θ = ω

r r rγ = θ = γ z ze e

r r r

v R= ω ∧;rrr

1.9ProjectileMotion



1.9 Projectile Motion


•Pn obect Cay Cove in

both the % and ! directions

siC;ltaneo;sly

• Fhe EorC oE to9

diCensional Cotion e ill

deal ith is called

(ro+e&tile motion



1.9ProjectileMotion





Pro+e&tile Motion Diagram



1.9ProjectileMotion





;ange and Maimum

Height of a Pro+e&tile

•When analying 7roectile

Cotion, to characteristics

are oE s7ecial interest

• Fhe range, (, is the

horiontal distance oE the

7roectile

• Fhe CaGiC;C height the

7roectile reaches is h

1.9ProjectileMotion





Height of a Pro+e&tile,

e8uation

• Fhe CaGiC;C height oE the 7roectile

can be Eo;nd in terCs oE the initial

velocity vectorN

• Fhis eK;ation is valid only Eor

syCCetric Cotion

2 2sin

2

i i v h

g

θ =

1.9ProjectileMotion





More A/out the ;ange of a

Pro+e&tile




.9 j


;ange of a Pro+e&tile, 0nal

• Fhe CaGiC;C range occ;rs at θ i O

Qo

•

8oC7leCentary angles ill 7rod;cethe saCe range

Fhe CaGiC;C height ill be

dierent Eor the to angles Fhe tiCes oE the ight ill be

dierent Eor the to angles



Thank you!

chapter 1 fundamental of physics pham hong quang

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