Chapter 1 Units and Vectors: Tools for Physics

1 Chapter 1 Units and Vectors: Tools for The Important The SI SystemPhysics is based on measurement. Measurements are made by comparisons to well definedstandardswhich define theunitsfor our system(popularly known as themetric system) is the one used in Physics . Itsunit of length is the meter, its unit of time is the second and its unit of mass is the quantities in Physics are derived from these. For example the unit of energy is thejoule, defined by 1 J = 1kg a convenience in using the SI system we can associate prefixes with the basic Units torepresent powers of 10. The most commonly used prefixes are given here:FactorPrefixSymbol10 12pico-p10 9nano-n10 6micro- 10 3milli-m10 2centi-c103kilo-k106mega-M109giga-GOther basic Units commonly used in Physics are:Time: 1 minute = 60 s1 hour = 60 : 1 atomic mass unit = 1 u = 10 27kg12 Chapter 1. Units AND VECTORS: Tools FOR Changing UnitsIn all of our mathematical operations we mustalwayswrite down the Units and wealwaystreat the unit symbols as multiplicative factors .

2 For example, if me multiply kg by get( kg) ( ) = msWe use the same idea in changing the Units in which some physical quantity is can multiply the original quantity by aconversion factor, a ratio of values forwhich the numerator is the same thing as the denominator. Theconversion factor is thenequal to 1, and so wedo not changethe original quantity when we multiply by the of conversion factors are:(1 min60 s) (100 cm1 m)(1 day)(1 ft) DensityA quantity which will be encountered in your study of liquidsand solids is thedensityof asample. It is usually denoted by and is defined as the ratio of mass to volume: =mV( )The SI Units of density arekgm3but you often see it expressed Dimensional AnalysisEvery equation that we use in Physics must havethe same type of unitson both sides of theequals sign. Our basic unit types (dimensions) are length (L), time (T) and mass (M).When we dodimensional analysiswe focus on the Units of a Physics equation withoutworrying about the numerical Vectors; Vector AdditionMany of the quantities we encounter in Physics have bothmagnitude( how much ) anddirection.

3 These can represent vectors graphically as arrows and then the sum of two vectors is found(graphically) by joining the head of one to the tail of the other and then connecting head totail for the combination, as shown in Fig.. The sum of two (or more) vectors is oftencalled can add vectors in any order we want:A+B=B+A. We say that vector additionis commutative .We express vectors incomponent formusing theunit vectors i,jandk, which eachhave magnitude 1 and point along thex,yandzaxes of the coordinate system, THE IMPORTANT STUFF3 ABABA+B(a)(b)Figure :Vector addition. (a) shows the vectorsAandBto be summed. (b) shows how to perform thesum :Addition of vectors by components (in two dimensions).Any vector can be expressed as a sum of multiples of these basic vectors; for example,for the vectorAwe would write:A=Axi+Ayj+ we would say thatAxis thexcomponent of the vectorA; likewise Fig. we illustrate how we get the components for a vector which is thesumof twoother vectors.

4 IfA=Axi+Ayj+AzkandB=Bxi+Byj+BzkthenA+B= (Ax+Bx)i+ (Ay+By)j+ (Az+Bz)k( )Once we have found the (Cartesian) component of two vectors,addition is simple; just addthecorresponding componentsof the two vectors to get the components of the we multiply a vector by a scalar, the scalar multiplies each component; IfAis avector andnis a scalar, thencA=cAxi+cAyj+cAzk( )4 Chapter 1. Units AND VECTORS: Tools FOR PHYSICSIn terms of its components, the magnitude ( length ) of a vectorA(which we write asA) is given by:A= A2x+A2y+A2z( )Many of our Physics problems will be in two dimensions (xandy) and then we can alsorepresent it inpolarform. IfAis a two dimensional vector and as the angle thatAmakes with the +xaxismeasured counter-clockwisethen we can express this vector in termsof componentsAxandAyor in terms of its magnitudeAand the angle . These descriptionsare related by:Ax=Acos Ay=Asin ( )A= A2x+A2ytan =AyAx( )When we use Eq.

5 To find fromAxandAywe need to be careful because the inversetangent operation (as done on a calculator) might give an angle in the wrong quadrant; onemust think about the signs Multiplying VectorsThere are two ways to multiply two vectors product(ordot product) of the vectorsaandbis given bya b=abcos ( )whereais the magnitude ofa,bis the magnitude ofband is the angle scalar product is commutative:a b=b a. One can show thata bis related tothe components ofaandbby:a b=axbx+ayby+azbz( )If two vectors are perpendicular then their scalar product product(orcross product) of vectorsaandbis a vectorcwhose mag-nitude is given byc=absin ( )where is thesmallestangle betweenaandb. The direction ofcis perpendicular to theplane containingaandbwith its orientation given by theright hand rule. One wayof using the right hand rule is to let the fingers of the right hand bend (in their naturaldirection!)

6 Fromatob; the direction of the thumb is the direction ofc=a b. This isillustrated in Fig. vector product isanti commutative:a b= b among the unit vectors for vector products are:i j=k j k=i k i=j( ) WORKED EXAMPLES5 ABCABC(a)(b)fFigure :(a) Finding the direction ofA B. Fingers of the right hand sweep fromAtoBin theshortest and least painful way. The extended thumb points inthe direction ofC. (b) VectorsA, magnitude ofCisC=ABsin .The vector product ofaandbcan be computed from the components of these vectorsby:a b= (aybz azby)i+ (azbx axbz)j+ (axby aybx)k( )which can be abbreviated by the notation of the determinant:a b= i j kaxayazbxbybz ( ) Worked Changing Units1. The Empire State Building is1472 fthigh. Express this height in both metersand centimeters.[FGT 1-4]To do the first unit conversion (feet to meters), we can use therelation (see the ConversionFactors in the back of this book):1 m = ftWe set up the conversion factor so that ft cancels and leaves meters:1472 ft = (1472 ft)(1 ft)= the height can be expressed as m.

7 To convert this to centimeters, use:1 m = 100 cm6 Chapter 1. Units AND VECTORS: Tools FOR PHYSICSand m = ( m)(100 cm1 m)= 104cmThe Empire State Building is 104cm high!2. A rectangular building lot ft. Determine the area of this lotinm2.[Ser4 1-19]The area of a rectangle is just the product of its length and width so the area of the lotisA= ( ft)( ft) = 104ft2To convert this to Units of m2we can use the relation1 m = ftbut the conversion factor needs to be appliedtwiceso as to cancel ft2 and get m2 . 104ft2= ( 104ft2) (1 ft)2= 103m2 The area of the lot is The Earth is approximately a sphere of 106m. (a) What is itscircumference in kilometers? (b) What is its surface area insquare kilometers?(c) What is its volume in cubic kilometers?[HRW5 1-6](a)The circumference of the sphere of radiusR, the distance around any great circle isC= 2 R. Using the given value ofRwe find:C= 2 R= 2 ( 106m) = convert this to kilometers, use the relation 1 km = 103m in a conversion factor:C= 107m = ( 107m) (1 km103m)= 104kmThe circumference of the Earth is 104km.

8 (b)The surface area of a sphere of radiusRisA= 4 R2. So we getA= 4 R2= 4 ( 106m)2= 1014m2 Again, use 1 km = 103m but to cancel out the Units m2 and replace them with km2 itmust be appliedtwice:A= 1014m2= ( 1014m2) (1 km103m)2= WORKED EXAMPLES7 The surface area of the Earth is 108km2.(c)The volume of a sphere of radiusRisV=43 R3. So we getV=43 R3=43 ( 106m)3= 1021m3 Again, use 1 km = 103m but to cancel out the Units m3 and replace them with km3 itmust be appliedthree times:V= 1021m3= ( 1021m3) (1 km103m)3= 1012km3 The volume of the Earth is Calculate the number of kilometers miusing only the following conver-sion factors :1 mi = 5280 ft,1 ft = 12 in,1 in = cm,1 m = 100 cm,1 km = 1000 m.[HRW5 1-7]Set up the factors of 1 as mi = ( mi) (5280 ft1 mi) (12 in1 ft) ( cm1 in) (1 m100 cm) (1 km1000 m)= kmSetting up the factors of 1 in this way, all of the unit symbols cancel except for km(kilometers) which we keep as the Units of the One gallon of paint (volume= 10 3m3) covers an area m3.

9 Whatis the thickness of the paint on the wall?[Ser4 1-31]We will assume that the volume which the paint occupies whileit s covering the wall isthesameas it has when it is in the can. (There are reasons why this may not be true, butlet s just do this and proceed.)The paint on the wall covers an areaAand has a thickness ; the volume occupied is thearea time the thickness:V=A .We haveVandA; we just need to solve for : =VA= 10 m2= 10 thickness is 10 4m. This quantity can also be expressed as A certain brand of house paint claims a coverage of460ft2gal. (a) Express thisquantity in square meters per liter. (b) Express this quantity in SI base Units . (c)8 Chapter 1. Units AND VECTORS: Tools FOR PHYSICSWhat is the inverse of the original quantity, and what is its physical significance?[HRW5 1-15](a)Use the following relations in forming the conversion factors : 1 m = ft and 1000 liter =264 gal.

10 To get proper cancellation of the Units we set it up as:460ft2gal= (460ft2gal) (1 ft)2 (264 gal1000 L)= (b)Even though the Units of the answer to part (a) are based on themetric system, theyare not made from thebaseunits of the SI system, which are m, s, and kg. To make thecomplete conversion to SI Units we need to use the relation 1 m3= 1000 L. Then we ( ) (1000 L1 m3)= 104m 1So the coverage can also be expressed (not so meaningfully, perhaps) as 104m 1.(c)The inverse (reciprocal) of the quantity as it wasoriginallyexpressed is(460ft2gal) 1= 10 course when we take the reciprocal theunitsin the numerator and denominator alsoswitch places!Now, the first expression of the quantity tells us that 460 ft2are associated with everygallon, that is, each gallon will provide 460 ft2of coverage. The new expression tells us 10 3gal are associated with every ft2, that is, to cover one square foot of surface withpaint, one needs 10 3gallons of Express the speed of light, 108msin (a) feet per nanosecond and (b)millimeters per picosecond.