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.

2 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. 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.

3 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.

4 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. 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.

5 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.

6 We illustrate how we get the components for a vector which is thesumof twoother vectors. 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.

7 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. 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.

8 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!) fromatob; the direction of the thumb is the direction ofc=a b.

9 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.

10 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. 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!