Preamble: Mind your language

Transcription

1 Peamble: Mind you language The idiom of this Physics couse will be a mixtue of natual language and algebaic fomalism equiing a cetain attention. So, teat you algeba with the same espect that you offe to you eveyday palance. Hee is an indispensable albeit incomplete list of equiements: Use symbols consistent thoughout you solution, and avoid using the same symbol fo diffeent quantities in the same agument Adapt the geneic equations to the language of the poblem and always show symbolic expessions befoe feeding in the numbes Ex: F = ma is a geneic fomula fo foce. If in a poblem two masses m 1,2 ae acted by foces F 1,2, wite distinct expessions: F 1 = m 1 a 1 and F 2 = m 2 a 2 Avoid using numbes in algebaic manipulations. Cay out you agument using symbols and feed the numbes only in the final expession Build you aguments in clea, complete, and meaningful sentences Ex: F/m = a is a fomally coect algebaic statement meaning that the atio between F and m is equal to a. A stay F/m followed by no opeato doesn t state anything! Make sue that the tems on both sides of the = sign ae indeed equal, including all tems in a chain of equalities. Fo instance, make sue that simplifying tems on two sides of one equality in a chain doesn t falsify anothe equality in the chain Ex: This succession of equalities may be tue: but it becomes false if you simplify caelessly: F = ma= mv t F = ma= m v t

2 Chapte 1 & Chapte 3 Intoductoy Fomalism and Vectos What is Physics? Physical quantities Units. Intenational System Dimensions and dimensional analysis Measuement and uncetainty. Significant figues Vectos: Components, unit vectos Vecto addition: gaphical and based on components Vecto poduct: scala and vectoial Modeling Motion Solving poblems Motion diagams Kinematic vectos

3 What is Physics? The atue of Science Science is the activity fo acquiing and oganizing knowledge based on the scientific method developed pimaily duing the last few centuies. It employs systematically: Obsevations: impotant fist step towad scientific theoy; equies imagination to tell what is impotant. Theoies: ceated to explain obsevations and to conceptualize vaious instances of the Natue; will make pedictions; must be falsifiable and always pefectible. Expeiments: Systematic and intelligent obsevations oganized into data which will tell if the pediction is accuate within some limits. Then the cycle goes on. Physics (fom the Geek, φύσις (phúsis), "natue") is the fundamental physical science concened with the stuctued undestanding of the undelying pinciples of the natual wold. Physics deals with the elementay constituents of the Univese, that is, all classes of matte and enegy, and thei inteactions, as well as the analysis of systems which ae best undestood in tems of these fundamental pinciples. Banches of Classical Physics: Mechanics the study of motion of physical bodies in its causal emegence: classical (o Newtonian) mechanics is just the macoscopic limit of quantum mechanics, and the small speed limit of elativistic mechanics Themodynamics the balance of heat, wok and intenal enegy of an object Electicity and Magnetism the study of electic and magnetic fields

4 Physical Quantities Basic quantities Physics is an expeimental science, that is, any of its statement must be veifiable via an oganized test upon natue. Duing an expeiment one measues physical quantities Ex: mass, length, time, tempeatue, cuent, etc. The physical quantities descibe an objective eality Some quantities ae consideed as basic physical quantities: fo instance, in mechanics Length [L] Time [T] Mass [M] ae consideed basic since the othe physical quantities ae deived fom them Ex: velocity, acceleation, enegy, momentum, etc. Consequently, the units fo the deivable quantities can be expessed in units of length, mass and time What ae units?

5 Physical Quantities Standads of units Any measuement makes necessay a standadized system of units. Ex: kilogams, slugs, metes, inches, seconds, hous etc. Defining units allows a consistent way of poviding numeical values fo physical quantities measued in an expeiment The unit standadization is just a convention ageed upon by some authoity. Examples of unit standads: Système Intenational (SI) Gaussian System (cgs) Bitish System We shall be woking in the SI system, whee the basic units ae kilogams [kg], metes [m], and seconds [s]. Quiz 1: Why does the SI povide units only fo mass, length and time?

6 Physical Quantities the Intenational System Système Intenational - SI Quantity Unit Standad Length [L] Mete, m Length of the path taveled by light in 1/299,792,458 second. Time [T] Second, s Time equied fo 9,192,631,770 peiods of adiation emitted by cesium atoms Mass [M] Kilogam, kg Platinum cylinde in Intenational Bueau of Weights and Measues, Pais

7 Physical Quantities Witing and conveting units Othe unit systems can have significant, albeit local, impotance: Ex: cgs units ae [L]: [M]: [T]: centimetes (cm) gams (g) seconds (s) Bitish system has foce (weight) instead of mass as one of its basic quantities: [L]: foce: [T]: feet pounds seconds Since units ae conventional, they can be easily conveted feet Ex: 6.00 metes ~ ( 6.00 metes) 3.28 = 19.7 feet metes Howeve, convesion should be pefomed consistently, such that all quantities involved in an opeation ae specified by numbes with the same units When a cetain quantity is vey lage o vey small in diect SI units, it is customay to use powes and ten and coesponding pefixes Ex: mico 10-6, mili 10-3, kilo 10 3, mega 10 6

8 Physical Quantities Dimensions and Dimensional Analysis The dimension of a quantity is given by the basic quantities that make it up; they ae geneally witten using squae backets. Ex: Speed = distance / time Dimensions of speed: [L]/[T] Quantities that ae being added o subtacted must have the same dimensions. Any physical equation must always be dimensionally consistent (i.e. all tems must have the same dimension). A quantity calculated as the solution to a poblem should have the coect dimensions. This can be used to veify the necessay (but not sufficient) validity of a cetain esult. Poblem: 1. Dimensional analysis: The speed, v, of an object is given by the equation A v = + Bt t whee t is time. What ae the dimensions of the quantities A and B? 2

9 Physical Quantities Measuement and Uncetainty. Significant Figues Uncetainties ae inheent to any measuement Evey measuing tool is associated with an uncetainty which can be used to specify the instument s accuacy Ex: Time = 1.67 ± 0.05 s Length = 1.2 ± 10% m The uncetainty can be indicated using significant figues Ex: Mass = 148 kg Uncetainty ± 1 kg Speed = 2.21 m/s (3 s.f.) Uncetainty ± 0.01 m/s Speed = 0.21 m/s (2 s.f.) Uncetainty ± 0.1 m/s Witing out the numbes in scientific notation clealy delineates the coect numbe of significant figues:

10 Physical Quantities Deived significant figues Results of poducts o divisions etain the uncetainty of the least cetain tem Results of summations o subtactions etain the least numbe of decimal figues Ex: = 640 Uncetainty ± = 12.9 Uncetainty ± 0.01 Numeic intege o factional coefficients in equations have no uncetainty. Caution: calculatos will not give you the ight numbe of significant figues; they usually give too many but sometimes give too few (especially if thee ae tailing zeoes afte a decimal point) Ex: a) the calculato shows the esult of 2.0 / 3.0 b) the calculato shows the esult of Quiz 2: What should be the answes with the coect numbe of significant figues in the two cases?

11 Vectos Definition and epesentation Scalas ae physical quantities completely descibed only by thei magnitude. Ex: time, mass, tempeatue, etc. Vectos descibe physical quantities having both magnitude and diection. Ex: position, displacement, velocity, acceleation, foce, etc. V o V V magnitude y y diection θ diection o θ x x The diection of a vecto depends on the abitay system of coodinates Howeve, the magnitude does not depend on how you choose to span the space

12 Vecto Popeties Vectos can be added o subtacted in any ode, but, if the vectos epesent physical quantities, they must have the same natue Multiplying a vecto by a positive numbe multiplies its magnitude by that numbe (if the numbe is negative the vecto flips in the opposite diection): V 2V 2V V 2V 2V Theefoe, any vecto can be witten as a numbe (its magnitude) times a unit vecto with the diection of the vecto: V V V Vvˆ ˆv ˆ Vv v ˆ = 1 unit vecto

13 Vectos One Dimensional The simplest physical situations that we ae going to encounte will involve vectos along the same staight line, such that they can have only two diections which can be abitaily consideed as negative and positive Moeove, fo simplicity, the aows on top of the symbols can be dopped: v v v 1 + v this sign means equivalent to not equal : neve use an equal sign between a vecto and a numbe In these cases, vectos can be added gaphically and as numbes: Ex: Say that we have 3 aows (vectos) along the same line (1D) with magnitudes povided in abitay units on the diagam, and we want to add them: Gaphically, chain the vectos tail to tip: the esultant connects the tail of the fist on to the tip of the last on in the chain v + 1 v 2 v = R umeically: add togethe the vectos epesented by thei espective magnitude and the sign R= v1 + v2+ v3 + v1+ v2 v3 = = + 4 v 2 v A vecto of magnitude 4 units pointing to the ight

14 Vectos 2D Vecto Gaphical Addition In geneal, even if the vectos ae not along the same axis, they can be added gaphically by using the same tail-to-tip method: The vecto sum can be obtained gaphically by chaining the vectos each with the tail to the tip of the pevious: then the vecto esultant connects the tail of the fist vecto to the tip of the last one. The opeation can be done in any ode. Ex: Say that we have 3 aows (vectos) in a plane (2D) and we want to add them up: v + 1 v 2 v 3 R= v + v + v = The method offes a qualitative idea about the esultant: in ode to obtain the esultant numeically (magnitude and diection), one has to use scaled gid pape which is a athe cumbesome technique R v 1 v 2 Notice that in 2D, the aows above the vecto symbols cannot be skipped since a vecto can have an infinity of diections not only two as in the 1D case: the opeation between the aows cannot be educed to an immediate algebaic addition o subtaction v 3

15 Ex: Physical example: Successive 2D displacements can still be added to obtain the total displacement An application of vecto summation in mechanics is calculating the net displacement of an object taveling fom an initial position to a final one via seveal successive patial displacements final d 3 d 2 If we denote d 1, d 2 and d 3 thee successive displacements the net displacement is d = d + d + d net It is given by the vecto sum (o esultant) of the patial displacements d net initial d 1 Notice that adding the patial displacement follows the logic of tail-to-tip method

16 Vectos Gaphical Subtaction In ode to subtact vectos, we can still use the addition pocedue by adding the negative of the aow being subtacted We define the negative of a vecto to be a vecto with the same magnitude but pointing in the opposite diection. v v Ex: Say that we have 2 aows (vectos) in a plane and we want to subtact them: v 1 _ R= v v = v + v ( ) = + = R v 2 v 1 v 2 v 1 v 2

17 Ex: Physical example: linea displacement is defined as the final position minus the initial position If we denote 1 and 2 two positions successively occupied by a moving objects, the displacement is = 2 1 = + ( ) efeence 2 final 1 initial

18 Vectos Components Note that, in ode to obtain magnitudes and diections, the gaphical methods should be used on gid pape. A moe computational way to get magnitudes and diections is by using vecto components in abitay systems of coodinates: V y y otation: V ( V, ) x Vy θ V V x y 2 2 V= Vx + Vy x V= V + V V V θ x x y = V cosθ = V sinθ V 1 y = tan Vx Components fom diection and magnitude Diection and magnitude fom components Caution: The components ae not ae not vectos o vecto magnitudes. They can be negative if the coesponding vecto components point in the negative diection of the espective axis.

19 Vectos Axial unit vectos Fo any system of coodinates (1-D, 2-D o 3-D), one can use unit vectos to define positive diections pointing along the axes. Popula notations: ˆ, ˆ, ˆ, xˆ, yˆ, zˆ 2-D case: V y vecto component ( ) i j k ( ) component = V ˆj y ĵ y V î V ˆ x = Vxi V = V + V = V iˆ + V ˆ j V, V ( ) x y x y x y x magnitude V = V + V diection 2 2 x y θ = 1 tan Vy Vx

20 Vectos Components and unit vectos in 3D V z = V kˆ z z V V x = V iˆ x î ˆk ĵ V y = V ˆj y y x V = V + V + V = V iˆ + V ˆ j+ V kˆ V, V, V x y z ( ) x y z x y z x y z V = V + V + V

21 Vectos Addition and subtaction using vecto components The addition and subtaction of vectos can be educed to addition and subtactions of components Recall that the components depend on the system of coodinates, so the opeation fist demands picking a SC. Howeve, the esultant will be the same in any SC. Given n coplana vectos, the addition can be solved in 2D as following: R= V + V V = V i+ V j + V i+ V j V i+ V j ( ˆ ˆ ) ( ˆ ˆ ) ( ˆ ˆ ) 1 2 n 1x 1y 2x 2 y nx ny (.. ) (.. ) R i ˆ + R ˆ j= V + V + + V i ˆ + V + V + + V ˆ j x y 1x 2x nx 1y 2 y ny magnitude: angle with espect to positive x: θ = R= R + R 2 2 x y ( R R ) 1 tan y x Ex: The pocedue can be visualized gaphically: the components (R x, R y ) of the esultant R ae aligned with the components of the vectos involved so they can be added as numbes R= A+ B

22 Poblem 2. Opeating with vectos: Given the two vectos in the figue, find the following vecto esultants R1 = A+ B A B R2 = 2A B whee and ae vectos with magnitudes 4 and 5 units espectively, by using a) Gaphical method b) Vecto components 4 A θ = 30 5 B

23 Vectos Types of poducts Thee ae two types of poduct of vectos, each with a specific applicability: 1. Scala poduct esults in a numbe 2. Vecto poduct esults in a vecto Unlike in the case of vecto addition, vectos with diffeent physical natue can be multiplied via eithe of the two poducts. The esult will be a diffeent physical quantity. Ex: a) The enegy exchanged by the action of a foce is called wok and is given by a scala poduct W = F b) The otational effect of applying a foce is detemined by toque a vecto poduct: τ = F

24 Results in a scala otation: Definitions: A B Vectos Dot poduct dot poduct A B= AB cosθ A B= A B + A B + A B x x y y z z Given vectos: A= A A A (,, ) x y z θ B= B B B ( x, y, z) Intepetation: the scala poduct between two vectos is the poduct between any of the vectos and the pojection (component) of the othe vecto along the fist one: A A B= BAcosθ A B= AB cosθ A B cosθ θ Acosθ B o θ B

25 Application: the scala poduct can be used to find the angle between two vectos with given components: 3D: Ax Bx+ Ay By+ Az B z θ = accos Ax Ay Az Bx By B A B z cosθ = AB 2D: Ax Bx+ Ay B y θ = accos Ax + Ay Bx + B y Poblem: 3. Application of dot poduct: Conside the following pai of vectos: A= ( 2.00,6.00) a) Sketch the vectos in the system of coodinates coesponding to the given components b) Calculate the angle between the two vectos B= ( 2.00, 3.00)

26 Vectos Coss poduct Results in a vecto otation: Definitions: A B o, in deteminant fom, coss poduct A B = AB sinθ A B= A B A B i+ ( ) ˆ y z z y ( A B A B ) ˆ + z x x z j ( A ) ˆ xby Ay Bx k Given vectos: A= A A A (,, ) x y z θ B= B B B ( x, y, z) A A A A B= B B B x y z x y z iˆ ˆj kˆ (No poblem if don t know how to handle this; use the pevious fom!)

27 Vectos Moe about Vecto Poduct A B = BAsinθ Intepetation fo the magnitude: the magnitude of the coss poduct between two vectos is the poduct between one of the vectos and the component of the othe vecto pependicula on the fist one: A Asinθ Diection: given by a ight-hand ule: Align you finges along the fist vecto such that you can cul them towad the second vecto. The vecto poduct is pependicula onto the plane of the vectos in the diection indicated by the thumb θ A A B A B in B B A out B Comment: Notice that, unlike the dot poduct, the vecto poduct is not commutative A B= B A

28 Poblems: 4. Coss poduct of unit vectos: Fo a concete feel of what unit vectos and the coss poduct ae, coss-multiply the unit vectos (i, j, k) between them. 5. Poduct vecto opeations: Conside again the vectos a) Calculate the magnitude of the vecto using b) Calculate A= ( 2.00,6.00) B= ( 2.00, 3.00) A B A B - fist the scala poduct - and then the component definition of the vecto poduct. A A B ( )

29 Modeling Motion Solving physics poblems... means eally undestanding a technique and the theoetical ovelay and being able to handle the necessay instumental mathematics to each a meaningful esult Tentative steps: Identify the elevant concepts and the taget vaiable Set up the poblem by making simplifying assumptions, dawing diagams and gaphical epesentations, and choosing the elevant equations Execute by solving equations. Thee may be new taget vaiables (unknowns which ae not specifically equied in the statement of the poblem) to be identified in ode to identify the final one Evaluate by solving numeically, pefoming a dimensional analysis and tying to see if the esult makes sense physically fo instance by consideing paticula cases o compaing numeical esults to the common sense knowledge

30 Modeling Motion Simplifying assumptions Mechanics is the science of motion: this semeste we ll focus on two types of motion: tanslational and otational Most of the time, the motion of a cetain object is bette analyzed by splitting a moe complex motion into simple ones, and by making simplifying assumptions Ex: Pojectile motion is a 2D tanslational motion Ex: A spinning top pefoms otational motion Two simplifying assumptions that we ll consistently make ae: 1. When analyzing tanslational motion, moving objects will be assumed point-like: objects with mass concentated in thei centes of mass, that is, they ae paticles 2. When analyzing otational motion, moving objects will be assumed as being made of igidly bounded paticles, that is, they ae igid bodies Othe simplifying assumptions will be moe paticula: gavity in poximity of the Eath s suface will be consideed constant, sometimes sufaces will be assumed vey smooth in ode to neglect fiction, in some intoductoy systems pulleys will be consideed as vey light, spings will be also light and pefectly elastic, etc. These assumptions can be successfully emoved when building moe complex models

31 Modeling Motion Motion diagams Motion can be made easie to visualize and solved by epesenting it schematically The paticle model in combination with the vecto fomalism ae the backbone of motion diagams: 1. The motion is epesented by dots at the paticle location at equal time intevals 2. Then the motion is chaacteized using motion vectos in evey location: position, displacement, velocity, acceleation Ex: Say that a ca moving in a staight line is filmed using a camea taking snapshots at equal time intevals Each fame is going to show the ca at a diffeent location When combined, the fames mege into a motion diagam In case we want to use the diagam to descibe the tanslation of the ca, it can be simplified by eplacing the ca with a paticle and the gound with an axis x

32 Modeling Motion Position and Displacement How do we chaacteize the motion using kinematic vectos? Kinematic vectos: 1. Position, : a vecto connecting a point of efeence with the location of the paticle Ex: In the case of the ca moving in a staight line: efeence point 2. Displacement, : vecto diffeence between two positions position vectos shifted vetically fo visibility = final initial Displacement can be epesented as a vecto connecting an initial to a final location The dots on a motion diagam can be connected by successive vecto displacements, with the net displacement being the vecto sum of patial displacements Ex: In the case of the ca moving in a staight line, the successive displacements ae equal = = = = Ex: Pojectile motion

33 Modeling Motion Velocity 3. Aveage Velocity, v avg : a vecto given by the displacement divided by the elapsed time v avg = t The aveage velocity between two locations on the tajectoy has the same diection as the associated displacement, so the successive velocities can be epesented using a simila succession of vectos If the displacement changes fom segment to segment, the aveage velocity will change accodingly Ex: In the case of the ca moving in a staight line, the aveage velocities between any two successive locations is the same, equal to the aveage velocity in the whole inteval v v v avg _1 avg _ 2 avg _ 3 v = = v = v = v t avg avg _1 avg _ 2 avg _ 3 v avg _1 v avg _ 2 Ex: Say that an object slides down an incline. Since it speeds up, it coves lage and lage displacements in equal intevals of time v avg _ 3 v < v < v < v avg _1 avg _ 2 avg _ 3 avg _ 4 v avg _ 4

34 Modeling Motion Acceleation 3. Aveage Acceleation, a avg : a vecto given by the change in velocity divided by the elapsed time So, the acceleation is a measue of the change in velocity Since velocity is a vecto, it can change in magnitude, in diection, o in both of them The diection of the acceleation is given by the diection of the change in velocity: 1. If the paticle slows down in a line, the acceleation opposes the velocity 2. If the object speeds up in a line, the acceleation has the same diection as velocity 3. If an object acceleates in plane, the acceleation is inside the cuvatue of the tajectoy On the motion diagams, the diection of the acceleation can be estimated by building gaphically the vecto diffeence between two successive velocities Ex: Object sliding down an incline. The acceleation is a vecto down the incline v 1 v v = v 2 v v v acceleation has the diection of v Ex: Pojectile motion. The acceleation is a vecto vetically downwad v = v v v 2 v v 2 v 1 a avg v = t

35 Poblem: 6. Motion diagam: A skie slides down the constant slope of a hill, then moves on a flat suface, then up the constant slope of anothe hill. Sketch qualitatively the motion diagam of the skie s tanslational motion using aveage velocity and acceleation vectos.