Review Topics Lawrence B. Rees You may make a single copy of this document for personal use without written permission.

Transcription

1 Review Topics Lawence. Rees You ma make a single cop of this document fo pesonal use without witten pemission. R.1 Vectos I assume that ou have alead studied vectos in pevious phsics couses. If ou have not been intoduced to vectos et, ou should look ove the section on vectos in an standad calculus-based phsics tet.. Geometic Model of Vectos In ou pevious phsics couses, ou have had to deal with a numbe of vecto quantities including displacement, velocit, acceleation, and foce. 1 We usuall define a vecto as a quantit that has both magnitude and diection. This is in contast with a scala which has magnitude but no diection. When ou think of a vecto, ou pobabl think of an aow. n aow is a good geometical wa to model a vecto as the magnitude can coespond to the length of an aow and the diection to the diection the aow points. Of couse, a vecto is not eall an aow; but an aow is a good wa of visualiing a vecto. Let s conside how vectos and vecto opeations ae defined when we think of vectos as aows. Magnitude and diection. s an aow in space, a vecto can eist without defining an coodinate sstem. We meel have to somehow specif the length and diection of the aow. Fo eample, we could sa that a vecto is 2.54 cm long and points in the diection of Polais. s long as we ignoe the fact that the diection of Polais diffes slightl fom location to location on the eath, this desciption chaacteies the vecto. Howeve, we could have one vecto on one side of ou oom of length 2.54 cm pointing towad Polais and anothe, diffeent vecto, on the othe side of the oom also 2.54 cm long and pointing towad Polais. The two vectos ae not eall the same vecto; howeve, we define the vectos to be equal as long as thei magnitudes and diections ae equal. If we wish to intoduce a coodinate sstem, we can also define the diection of the vecto with espect to that coodinate sstem. In two dimensions we usuall set up an ais and a ais pependicula to the ais. We specif the vecto b two independent numbes: the vectos length and its angle. We define the angle to be the diection of the vecto measued with espect to the diection. Note that the vecto need not be located at the oigin of the coodinate sstem fo to be defined. In thee dimensions, we need thee independent numbes to specif the diection of a vecto. These numbes ae the length of the vecto and two angles. In spheical coodinates, we usuall label these angles as and φ. In this book, as in most phsics tets, the angle is defined as the angle of the vecto with espect to the ais. We eall won t wo about N. (Note that math tets usuall define and N oppositel to phsics tets.) 1

2 Notation. In this book, we denote vectos with an aow ove the top (e.g., ) and the magnitude of vectos in italic tpe (e.g., ). ddition. To geometicall add two aows, we place them head to tail and daw the sum as the aow which etends fom the head of the fist vecto to the tail of the second. Note that it doesn t matte if move a vecto aound to a new, convenient location as long we don t otate it in the pocess. Figue R.1. dding two vectos geometicall. Dot Poduct. Thee ae two methods of multipling vectos. The fist is called the "dot poduct" because it epesented b the smbol. It is also called the scala poduct because the esult is a scala athe than anothe vecto. The dot poduct of two vectos and is the poduct of (the magnitude) and the component of which is paallel to. That is, (R.1 Dot poduct) cos whee is the angle between the vectos when placed tail to tail. cos Figue R.2. The dot poduct of two vectos. You can convince ouself that the ode in which two vectos appea in a dot poduct is unimpotant. That is: 2

3 (R.2 Dot poduct commutes) When the ode of a poduct does not affect the outcome, we sa the poduct commutes. Coss Poduct. The second method of multipling two vectos is called the coss poduct, epesented b. It is also called the vecto poduct, as the esult is a vecto. The magnitude of the coss poduct of two vectos and is the poduct of and the component of that is pependicula to, that is (R.3 Coss poduct) sin. Note that this is just the fomula fo the aea of the paallelogam fomed b two vectos and, as ou can see fom Fig. R.3. sin Figue R.3. The coss poduct of two vectos. What we want to do now is define the diection of the coss poduct. If ou think about it, it s tue that when we place an two vectos tail to tail, the must lie in the same plane. Let s take the sceen (o page) as the plane in which and lie. Now let s give ou an impotant ule. Fist Rule of Coss Poduct Diection The coss poduct of and is pependicula to the plane fomed when we put the tails of and togethe. This means that the coss poduct is pependicula to both and. Theefoe, in Fig. R.3 the coss poduct points eithe into the sceen o out of the sceen, as these ae the onl diections pependicula to the plane of the sceen. What emains is fo us to detemine in which of these two diections the coss poduct points. In ode to emove this ambiguit, a ight-hand ule is used. Hold ou hand out flat with ou finges pointing in the diection of. Tun ou hand so that ou finges otate towad when ou close ou hand. 3

4 The coss poduct is in the diection of ou thumb. Remembe that the ight-hand ule will onl tell ou which of the two possible diections the coss poduct is in. using the ight hand ule, ou can easil see that if then is into the sceen. Hence: (R.4 Coss poduct anticommutes) is out of the sceen as in Fig. R.3, When evesing the ode of a poduct onl changes the sign of the poduct, we sa the poduct anticommutes.. lgebaic o Component Model of Vectos Unit vectos. unit vecto is a vecto that has a magnitude of one unit. In Catesian coodinates, we will denote the unit vectos along the,, and aes as, ŷ, and ẑ espectivel. In pola coodinates, we wite a unit vecto diected outwad fom the oigin (note that the angle is not specified, it can be in an diection diectl outwad fom the oigin) as. The unit vecto pependicula to in a counteclockwise sense is. Similal, in spheical coodinates the unit vecto diected awa fom the oigin is also called. It is impotant to emembe that the magnitude of a unit vecto is alwas one. Components. In Catesian coodinates, an vecto can be epessed as the sum of vectos in the, ŷ, and ẑ diections. Fig. R.4. The components of a vecto. The vectos that lie in the and diections ae the components of the vecto. We could wite, but the usual notation uses the and unit vectos eplicitl. (R.5 Components of a vecto) 4

5 5 In two dimensions we have some simple elations between the components of a vecto and its magnitude and diection: (R.6 Component elationships) tan sin cos 2 2 ddition. To add vectos, we just add components: (R.7 Vecto addition) ( ) ( ) ( ) Dot poduct. (R.8 Dot poduct) Coss poduct. (R.9 Coss poduct) ( ) ( ) ( ) Note that if we eplace with, with, and with in the fist tem, then we get the second tem. Doing this to the second tem gives us the thid tem. If ou ae acquainted with deteminants, an eas wa to emembe this fom is (R.10 Coss poduct) 1 You also wee intoduced to a few aial vectos o pseudovectos such as toque and angula momentum. These ae quantities that ae simila to vectos in the wa the tansfom unde otations, but diffe in the wa the tansfom unde eflections. In fact, vectos ae

6 technicall defined b how the behave unde tansfomations such as otations and eflections. Since we eall don t cae about eflection popeties at this point, we don t make a distinction between vectos and aial vectos. 6