Chapter 1. Introduction and Mathematical Concell!..

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1 Chapter 1 Introduction and Mathematical Concell! Units The metric system is also known as the "8 I" system ofunits. (8 I == 8ysteme International). A. Lengfu The unit of length in the metric system is the meter. B. Mass The unit ofmass in the metric system is the kilogram. c. Time The unit oftime in the metric system is the second. The metric system is also known as the "mks" system of units. Mass is a quantity used to measure the resistance of an object to changes in its motion. It is a measure ofthe 1

2 inertia ofthe object. Galileo Galilei was the first person. to introduce the concept of inertia. Density and Atomic Mass A. The mass density p ofany object is defined as the mass per unit volume, i.e., M p= V for liquid water at standard temperature and pressure, STP, the density is p == 1.0 g/cm 3 == 1000 kg/m 3 B. The atomic mass ofan element is equal to the mass of a single atom ofthat element. Atomic masses are measured in amu (atomic mass units) where 11 amu == 1.66x10-27 kgl 2

3 1.3 Conversion ofunits Units can be treated as algebraic quantities that can cancel each other. Go over example 2. ~nificant figures A significant figure is a reliably known digit... A. Adding or Subtracting Numbers: The number of decimal places in the result should equal the smallest number ofdecimal places ofany term in the sum (or subtraction). That is (1) = => (2) = => 21 B. Multiplying or Dividing Numbers: The number of significant figures in the final product is the same as the number of significant figures in the 3

4 factor having the lowest number ofsignificant figures. That is, T~ltlp.._ 1.._? Standard Prefixes Used to Denote Multiples of Ten Prefix Symbol Factor" (1) x 7.89 = => 975 tera T giga b G 10 9 mega M 10 6 (2) 564/ = => 1.7xl0 5 kilo k 10 3 hecto h Trieonometry A. Cartesian (Rectangular) Coordinate System y cd.j:l.j.pc L J I) deka da 10 1 deci d 10-1 centi c 10-2 milli m micro JL nano n 10-9 pico p femto f x (-'J--z.) (x,y) are the coordinates of a point P in the xy-plane. h a = length of side adjacent to the angle B Figure 1.5 A right triangle. h a = length of side opposite the angle B ~(fj):: ~/k 4,fo ktj W.se. U2-(9) = tt&t/~ h'1 F0t-e rn.ts--e ~(e) =- ~ ~dj~ 4

5 B. Polar Coordinates p or, e:=.o (r,8) are the coordinates of a point P in the r8-plane. Note that x = rcos(8) y = rsin(8) I 2 2 r = \/X + y 8 = tan-l(~) 1.5 Scalars and Vectors A. A Scalar quantity is a physical quantity that has magnitude only. Examples include: (a) mass m (in kilograms, kg) (b) time t (in seconds, s or sec) (c) temperature T (in Kelvin, K) (d) volume V (in cubic meters, m 3 ) 5

6 (e) density p (in kglm 3 ) (f) energy E (in Joules, J) (g) distance d (in meters, m) (h) speed v (in meters per second, m/s) B. A vector quantity is a physical quantity that has both magnitude and direction. Examples include: (a) displacement Ai ( in meters, m) (b) velocity v(in meters per second, m/s) (c) acceleration a(in meters per second square, m/s 2 ) -+ (d) force F (in Newtons, N) (e) linear momentum 15 (in kg m/s) (f) angular momentum L (in kg m 2 I sec) -+ (g) electric field E (in Volts per meter, Vim) -+ (h) magnetic field B (in Teslas, T) (i) torque i (in Newton meter, N m) 6

7 The displacement vector AT The displacement vector ~r ofan object is defined as the vector whose magnitude is the shortest distance between the initial and final positions ofthe object, and whose direction points from the initial position to the final position. ~~",i$l \ AY:-\ =. mt\~n~~j.e. 1> Ar -:. S M e- = s '3" ) So fe ""\ +; e s d\ rc2 c.,,4 {b., 4m ~ 5 ~ ~"e <.,-1.., '(" A'r -:::. A~~- - s+cvx 3vn d "\ s +C!tI'n c.jl = 1- tv\. ~ SC-~Al\ Components of a Vector and Unit Vectors A. The components of a vector are the projections ofthe vector along the axes ofa rectangular coordinate system.... Ax == the x-component ofvector A.... A y == the y-component ofvector A. 7

8 +y AI or A denote the magnitude ofvector A. A 1 Figure 1.16 An arbitr~ vecto.:.-aand... its vector components Ax and A,. 8 denotes the direction ofvector A. +Y y +x -- """ A, A'(, e 1 A) '> X "..l-- +x Figure 1.17 This alternative way of drawing the vector A and its vector components is completely eq~_iyalent to )t that shown in Figure Any vector alone gives you the same physics as its two rectangular components together. Thus, \ Ay..::::' A ~[e) %_ - A~ ~lay-=-a~lt}) ) A- ::~ A~ t A1~ A e = ~-I ( *J B. A unit vector is a vector ofmagnitude one. x == a vector ofmagnitude one and in the direction ofthe positive x-axis. y == a vector ofmagnitude one and in the direction ofthe positive y-axis. Z== a vector ofmagnitude one and in the direction ofthe positive z-axis. lc I ~=-..L_~:-- I _! '2... In general, any vector Acan be written in terms of its rectangular components and unit vectors. That is, IA = Axx + ~yy + AzZI ~ ~ " r,x\ 2.. :3 '+ 'X, 8

9 so that its magnitude may also be written as IIAI~ A; + A; + A; 1.6,1.8 Some Mathematical Manipulations of Vectors A. Multiplication (or Division) of Vectors by a Scalar Do some examples... B. Addition ofvectors A and B (i) A and B in the same direction... (ii) A and B in opposite directions... (iii) A and B in 1- directions... (iv) A and B in arbitrary directions Graphical method +Y r ) """'" i' +x 35.0m 2. Algebraic method <'" Figure 1.24 The displacement vectors A, D, and Cadd together to give the 3. "Head-to-Tail" method) resultant displacement R, which lies along the x axis. Note that the vector D is parallel to the x axis. C. Subtraction ofvectors A and B Think of A - B simply as A + (-B) and proceed as in vector addition... 9