Physics for Scientists and Engineers

Transcription

1 Phsics 111 Sections 003 and 005 Instucto: Pof. Haimin Wang Phone: Office: 460 Tienan Hall Homepage: Office Hou: 2:30 to 3:50 Monda 1:00 to 2:20 Fida Homewok Registation: Tetbook: Sewa and Jewett, Phsics fo Scientists and Enginees, 7 th Edition iclicke equied Phsics fo Scientists and Enginees Week 1 Chapte 1: Measuements and Chapte 3: Vectos Intoduction to Clickes 1. What is ou discipline? A. Engineeing B. Science C. Math D. Humanities E. Othe Intoduction to Clickes 2. How man eas have ou been at NJIT? A. Less than 1 ea B. 1 ea C. 2 eas D. 3 eas E. Moe than 3 eas Phsics Fundamental Science Concened with the fundamental pinciples of the Univese Foundation of othe phsical sciences Has simplicit of fundamental concepts Divided into si majo aeas Classical Mechanics Relativit Themodnamics Electomagnetism Optics Quantum Mechanics Classical Phsics Mechanics and electomagnetism ae basic to all othe banches of classical and moden phsics Classical phsics Developed befoe 1900 Ou stud will stat with Classical Mechanics Also called Newtonian Mechanics o Mechanics Moden phsics Fom about 1900 to the pesent 1

2 Measuements Used to descibe natual phenomena Needs defined standads Chaacteistics of standads fo measuements Readil accessible Possess some popet that can be measued eliabl Must ield the same esults when used b anone anwhee Cannot change with time Standads of Fundamental Quantities Standadized sstems Ageed upon b some authoit, usuall a govenmental bod SI Sstéme Intenational Ageed to in 1960 b an intenational committee Main sstem used in this tet Fundamental Quantities and Thei Units Quantities Used in Mechanics Quantit Length Mass Time Tempeatue Electic Cuent Luminous Intensit Amount of Substance SI Unit mete kilogam second Kelvin Ampee Candela mole In mechanics, thee basic quantities ae used Length Mass Time Will also use deived quantities These ae othe quantities that can be epessed in tems of the basic quantities Eample: Aea is the poduct of two lengths Aea is a deived quantit Length is the fundamental quantit Length Mass Length is the distance between two points in space Units SI mete, m Defined in tems of a mete the distance taveled b light in a vacuum duing a given time See Table 1.1 fo some eamples of lengths Units SI kilogam, kg Defined in tems of a kilogam, based on a specific clinde kept at the Intenational Bueau of Standads See Table 1.2 fo masses of vaious objects 2

3 Standad Kilogam Time Units seconds, s Defined in tems of the oscillation of adiation fom a cesium atom See Table 1.3 fo some appoimate time intevals US Customa Sstem Still used in the US, but tet will use SI Quantit Unit Pefies Pefies coespond to powes of 10 Each pefi has a specific name Each pefi has a specific abbeviation Length foot Mass slug Time second Pefies, cont. The pefies can be used with an basic units The ae multiplies of the basic unit Eamples: 1 mm = 10-3 m 1 mg = 10-3 g Basic Quantities and Thei Dimension Dimension has a specific meaning it denotes the phsical natue of a quantit Dimensions ae denoted with squae backets Length [L] Mass [M] Time [T] 3

4 Dimensions and Units Each dimension can have man actual units Table 1.5 fo the dimensions and units of some deived quantities Dimensional Analsis, eample Given the equation: = ½ at 2 Check dimensions on each side: L 2 L = T = L 2 T The T 2 s cancel, leaving L fo the dimensions of each side The equation is dimensionall coect Thee ae no dimensions fo the constant Dimensional Analsis to Detemine a Powe Law Convesion Detemine powes in a popotionalit Eample: find the eponents in the epession You must have lengths on both sides Acceleation has dimensions of L/T 2 Time has dimensions of T Analsis gives at 2 a t m n Alwas include units fo eve quantit, ou can ca the units though the entie calculation Multipl oiginal value b a atio equal to one Eample 15.0 in =? cm 2.54cm 15.0in = 38.1cm 1in Note the value inside the paentheses is equal to 1 since 1 in. is defined as 2.54 cm Significant Figues Significant Figues, eamples A significant figue is one that is eliabl known Zeos ma o ma not be significant Those used to position the decimal point ae not significant To emove ambiguit, use scientific notation In a measuement, the significant figues include the fist estimated digit m has 2 significant figues The leading zeos ae placeholdes onl Can wite in scientific notation to show moe cleal: m fo 2 significant figues 10.0 m has 3 significant figues The decimal point gives infomation about the eliabilit of the measuement 1500 m is ambiguous Use m fo 2 significant figues Use m fo 3 significant figues Use m fo 4 significant figues 4

5 Opeations with Significant Figues Multipling o Dividing When multipling o dividing, the numbe of significant figues in the final answe is the same as the numbe of significant figues in the quantit having the lowest numbe of significant figues. Eample: m 2.45 m = 62.6 m 2 The 2.45 m limits ou esult to 3 significant figues Opeations with Significant Figues Adding o Subtacting When adding o subtacting, the numbe of decimal places in the esult should equal the smallest numbe of decimal places in an tem in the sum. Eample: 135 cm cm = 138 cm The 135 cm limits ou answe to the units decimal value Catesian Coodinate Sstem Chapte 3 Vectos Also called ectangula coodinate sstem - and - aes intesect at the oigin Points ae labeled (,) Pola Coodinate Sstem Oigin and efeence line ae noted Point is distance fom the oigin in the diection of angle θ, ccw fom efeence line Points ae labeled (,θ) Pola to Catesian Coodinates Based on foming a ight tiangle fom and θ = cos θ = sin θ 5

6 Tigonomet Review Given vaious adius vectos, find Length and angle - and -components Tigonometic functions: sin, cos, tan Catesian to Pola Coodinates is the hpotenuse and θ an angle tanθ = = θ must be ccw fom positive ais fo these equations to be valid Eample 3.1 The Catesian coodinates of a point in the plane ae (,) = (-3.50, -2.50) m, as shown in the figue. Find the pola coodinates of this point. Solution: Fom Equation 3.4, = + = + = ( 3.50 m) ( 2.50 m) 4.30 m and fom Equation 3.3, 2.50 m tanθ = = = m θ = 216 (signs give quadant) Eample 3.1, cont. Change the point in the - plane Note its Catesian coodinates Note its pola coodinates Please inset active fig. 3.3 hee Vectos and Scalas A scala quantit is completel specified b a single value with an appopiate unit and has no diection. A vecto quantit is completel descibed b a numbe and appopiate units plus a diection. Vecto Eample A paticle tavels fom A to B along the path shown b the dotted ed line This is the distance taveled and is a scala The displacement is the solid line fom A to B The displacement is independent of the path taken between the two points Displacement is a vecto 6

7 Vecto Notation Tet uses bold with aow to denote a vecto: A Also used fo pinting is simple bold pint: A When dealing with just the magnitude of a vecto in pint, an italic lette will be used: A o A The magnitude of the vecto has phsical units The magnitude of a vecto is alwas a positive numbe When handwitten, use an aow: A Equalit of Two Vectos Two vectos ae equal if the have the same magnitude and the same diection A = B if A = B and the point along paallel lines All of the vectos shown ae equal Adding Vectos When adding vectos, thei diections must be taken into account Units must be the same Gaphical Methods Use scale dawings Algebaic Methods Moe convenient Adding Vectos Gaphicall Choose a scale Daw the fist vecto, A, with the appopiate length and in the diection specified, with espect to a coodinate sstem Daw the net vecto with the appopiate length and in the diection specified, with espect to a coodinate sstem whose oigin is the end of vecto A and paallel to the coodinate sstem used fo A Adding Vectos Gaphicall, cont. Continue dawing the vectos tip-to-tail The esultant is dawn fom the oigin of A to the end of the last vecto Measue the length of R and its angle Use the scale facto to convet length to actual magnitude Adding Vectos Gaphicall, final When ou have man vectos, just keep epeating the pocess until all ae included The esultant is still dawn fom the tail of the fist vecto to the tip of the last vecto 7

8 Adding Vectos, Rules When two vectos ae added, the sum is independent of the ode of the addition. This is the Commutative Law of Addition A+ B= B+ A Adding Vectos, Rules cont. When adding thee o moe vectos, thei sum is independent of the wa in which the individual vectos ae gouped A + B+ C = A+ B + C This is called the Associative Popet of Addition ( ) ( ) Adding Vectos, Rules final When adding vectos, all of the vectos must have the same units All of the vectos must be of the same tpe of quantit Fo eample, ou cannot add a displacement to a velocit Negative of a Vecto The negative of a vecto is defined as the vecto that, when added to the oiginal vecto, gives a esultant of zeo Repesented as A ( ) 0 A+ A = The negative of the vecto will have the same magnitude, but point in the opposite diection Subtacting Vectos Special case of vecto addition If A B, then use A+ B Continue with standad vecto addition pocedue ( ) Subtacting Vectos, Method 2 Anothe wa to look at subtaction is to find the vecto that, added to the second vecto gives ou the fist vecto A + ( B ) = C As shown, the esultant vecto points fom the tip of the second to the tip of the fist 8

9 Multipling o Dividing a Vecto b a Scala The esult of the multiplication o division of a vecto b a scala is a vecto The magnitude of the vecto is multiplied o divided b the scala If the scala is positive, the diection of the esult is the same as of the oiginal vecto If the scala is negative, the diection of the esult is opposite that of the oiginal vecto Component Method of Adding Vectos Gaphical addition is not ecommended when High accuac is equied If ou have a thee-dimensional poblem Component method is an altenative method It uses pojections of vectos along coodinate aes Components of a Vecto, Intoduction A component is a pojection of a vecto along an ais An vecto can be completel descibed b its components It is useful to use ectangula components These ae the pojections of the vecto along the - and -aes Vecto Component Teminolog A and A ae the component vectos of A The ae vectos and follow all the ules fo vectos A and A ae scalas, and will be efeed to as the components of A Components of a Vecto Assume ou ae given a vecto A It can be epessed in tems of two othe vectos, A and A These thee vectos fom a ight tiangle A = A + A Components of a Vecto, 2 The -component is moved to the end of the -component This is due to the fact that an vecto can be moved paallel to itself without being affected This completes the tiangle 9

10 Components of a Vecto, 3 The -component of a vecto is the pojection along the -ais A = Acosθ The -component of a vecto is the pojection along the -ais A = Asinθ This assumes the angle θ is measued with espect to the -ais If not, do not use these equations, use the sides of the tiangle diectl Components of a Vecto, 4 The components ae the legs of the ight tiangle whose hpotenuse is the length of A A A= A + A and θ = tan A Ma still have to find θ with espect to the positive -ais Components of a Vecto, final The components can be positive o negative and will have the same units as the oiginal vecto The signs of the components will depend on the angle Unit Vectos A unit vecto is a dimensionless vecto with a magnitude of eactl 1. Unit vectos ae used to specif a diection and have no othe phsical significance Unit Vectos, cont. The smbols î, ĵ, and kˆ epesent unit vectos The fom a set of mutuall pependicula vectos in a ighthanded coodinate sstem Remembe, 垐 i = j = k = 1 Viewing a Vecto and Its Pojections Rotate the aes fo vaious views Stud the pojection of a vecto on vaious planes,, z, z 10

11 Unit Vectos in Vecto Notation A is the same as A î and A is the same as A ĵ etc. The complete vecto can be epessed as A= A 垐 i+ A j Adding Vectos Using Unit Vectos Using Then R = A+ B R = i+ j + i+ j R = ( A ) 垐 + B i+ ( A + B) j R = R 垐 i+ R j ( A 垐 ) ( 垐 A B B ) and so R = A + B and R = A + B R R = R + R θ = tan R Adding Vectos with Unit Vectos Note the elationships among the components of the esultant and the components of the oiginal vectos R = A + B R = A + B Thee-Dimensional Etension Using R = A+ B Then R = ( A 垐垐 ) ( 垐 i+ Aj+ Azk + Bi+ Bj+ Bzk) R = ( A ) 垐 ( ) ( ) + B i+ A + B j+ Az + Bz k R = R 垐 i+ R j+ R k z and so R = A +B, R = A +B, and R z =A +B z R R = R + R + Rz θ = cos, etc. R Eample 3.5 Taking a Hike Eample 3.5 A hike begins a tip b fist walking 25.0 km southeast fom he ca. She stops and sets up he tent fo the night. On the second da, she walks 40.0 km in a diection 60.0 noth of east, at which point she discoves a foest ange s towe. (A) Detemine the components of the hike s displacement fo each da. A Solution: We conceptualize the poblem b dawing a sketch as in the figue above. If we denote the displacement vectos on the fist and second das b and B A espectivel, and use the ca as the oigin of coodinates, we obtain the vectos shown in the figue. Dawing the esultant R, we can now categoize this poblem as an addition of two vectos. 11

12 Eample 3.5 We will analze this poblem b using ou new knowledge of vecto components. Displacement A has a magnitude of 25.0 km and is diected 45.0 below the positive ais. Fom Equations 3.8 and 3.9, its components ae: A = Acos( 45.0 ) = (25.0 km)(0.707) = 17.7 km A = Asin( 45.0 ) = (25.0 km)( 0.707) = 17.7 km The negative value of A indicates that the hike walks in the negative diection on the fist da. The signs of A and A also ae evident fom the figue above. Eample 3.5 The second displacement B has a magnitude of 40.0 km and is 60.0 noth of east. Its components ae: B = Bcos60.0 = (40.0 km)(0.500) = 20.0 km B = Bsin 60.0 = (40.0 km)(0.866) = 34.6 km Eample 3.5 (B) Detemine the components of the hike s esultant displacement R R fo the tip. Find an epession R fo in tems of unit vectos. Solution: The esultant displacement fo the tip R = A+ B has components given b Equation 3.15: R = A + B = 17.7 km km = 37.7 km R = A + B = km km = 16.9 km In unit-vecto fom, we can wite the total displacement as R = (37.7 垐 i j) km Eample 3.5 Using Equations 3.16 and 3.17, we find that the esultant vecto has a magnitude of 41.3 km and is diected 24.1 noth of east. Let us finalize. The units of R ae km, which is easonable fo a displacement. Looking at the gaphical epesentation in the figue above, we estimate that the final position of the hike is at about (38 km, 17 km) which is consistent with the components of R in ou final esult. Also, both components of R ae positive, putting the final position in the fist quadant of the coodinate sstem, which is also consistent with the figue. R 12