Period vs. Length of a Pendulum

Size: px

Start display at page:

Download "Period vs. Length of a Pendulum"

Lambert Newman
5 years ago
Views:

1 Gaphcal Mtho n Phc Gaph Intptaton an Lnazaton Pat 1: Gaphng Tchnqu In Phc w u a vat of tool nclung wo, quaton, an gaph to mak mol of th moton of objct an th ntacton btwn objct n a tm. Gaph a on of th bt wa to ctl vualz th quanttatv latonhp btwn two vaabl n oth wo, whth th vaabl a ctl popotonal, nvl popotonal, not lat at all, o omthng l ntl. Whn w contuct a gaph, w plot th npnnt vaabl th vaabl that th pmnt contol on th -a, an th pnnt vaabl th vaabl that pon whn th npnnt vaabl chang on th -a. Th a alo contol vaabl vaabl that a kpt contant thoughout th pmnt o that th o not nflunc th ata. So, fo ampl, f ou w tng to tmn how th po of a pnulum chang whn th lngth of th pnulum va, th pnnt vaabl woul b th pnulum po, an th npnnt vaabl woul b th pnulum lngth. Contoll vaabl woul nclu th pnulum ma an th angl at whch th pnulum wa launch. An appopat gaph fo th pmnt hown blow. Po v. Lngth of a Pnulum P o Lngth m Notc that th ttl lt th pnnt vaabl, whch plott on th -a, ft, an th npnnt vaabl, whch plott on th -a, con. Th a a coctl labl wth th appopat unt. Th gaph bgn at, wth no jump, an ncmnt a quall pac. In th pmnt, w can clal that a th lngth of th pnulum nca, th po alo nca, but a th vaabl ctl popotonal? In oth wo, can w wt an quaton fo th latonhp n th fom = m + b? Ecl wll aw a tn ln fo a gaph that can hlp u to tmn th.

2 Po v. Lngth of a Pnulum P o = R² = Lngth m Whl th gaph appa to b omwhat lna, w can a fw poblm ft, th majot of th pont o not fall on th ln; con, th ln o not co th -a at zo, an w woul pct t to aft all, a pnulum wth a lngth v clo to zo mt houl hav a po v clo to zo con. To tmn th coct latonhp btwn th vaabl, w wll hav to lnaz th gaph. Pat 2: Lnazaton Lnazng a gaph man mofng th pnnt an/o npnnt vaabl o that whn ou gaph thm, a taght ln appa. Th mplt wa to o th to match th hap of ou gaph to on of val tpcal hap that ou woul pct to n a Phc cla. Appn A pla th common tp of gaph ou wll ncount n th cla. Ou gaph appa to match th lat latonhp, th qua of popotonal to. To lnaz th gaph, follow th cton n th th column n oth wo, plot a gaph n whch th qua of th po on th -a, an th lngth of th pnulum on th -a. Th nw, lnaz gaph hown blow: Po Squa v. Lngth of a Pnulum P o q u a ^ = R² = Lngth m A ou can, th mof gaph pla a taght ln wth a -ntcpt that v clo to zo, a pct. Th pnt a goo lna ft.

3 Pat 3: Ok, now what? Onc w hav ou lna ft, w can tmn th mathmatcal latonhp btwn th vaabl n ou pmnt. In th ca, th gnal quaton of ou ln = Snc w know th actual vaabl, an th unt coponng to thm, that w plott on th an a, w can ubttut thm nto th quaton: Th unt fo th lop an -ntcpt a takn ctl fom th gaph. Th quaton tll u that th qua of th po of a pnulum ctl popotonal to t lngth, wth th lop bng a popotonalt contant, /m. In gnal, th lop ha om ot of phcal manng lat to th vaabl n th pmnt that ou wll b ak to tmn, whch w wll now o. In ou phc ttbook, th quaton that lat th po of a pnulum to t lngth a follow: wh g th acclaton u to gavt at th ufac of Eath. Th quaton can b wttn to mo clol mbl th quaton w tmn fom th gaph: An t can b manpulat vn futh o that th quaton n th fom = m + b: H, w can that th lop of th ln houl b qual to. A tpcal AP Phc 1 Lab Epmnt F Rpon Quton mght ak a tunt to u th lop of a gaph lk th on abov to tmn a contant uch a g. W can now o th b ttng th quaton fo th lop qual to th pmntal valu that w tmn, an thn algbacall olvng fo th unknown contant. Th what th tp woul look lk: Th v clo to th known valu of g, 9.81 m/ 2.

4 Pat 4: You Tun! Th followng gaphcal ntptaton poblm mla to on that appa n po AP Phc B Eam. A tunt wh to tmn th acclaton u to gavt. Sh v an appaatu that tmn th tm t tak fo a mall ball to fall whn opp fom t fom ffnt hght. Th appaatu collct th followng ata: Dtanc m Tm Sh plot th gaph hown blow: D t a n c m Dtanc v. Tm Tm S1 Notc that vn though th tunt chang th hght hlf, tanc plott a th pnnt vaabl bcau w a tng to tmn how th tanc th ball tavl pn on th tm t wa n f-fall. Eplan how ou woul lnaz th ata, an u Ecl Appn B to cat a lnaz gaph. Thn, wt an quaton wth coct vaabl an unt latng tanc to tm fo an objct n f-fall. If th actual quaton tanc = 1/2gt 2, u ou gaph to tmn th valu of g.

Appn A: Gaphcal Mtho-Summa Copght 26, Molng Phc Pojct Th followng gaph hap pnt latonhp btwn vaabl that ou a lkl to ncount n Phc: Gaph hap Wttn latonhp Mofcaton qu to lnaz gaph Algbac pntaton A nca,

5 Appn A: Gaphcal Mtho-Summa Copght 26, Molng Phc Pojct Th followng gaph hap pnt latonhp btwn vaabl that ou a lkl to ncount n Phc: Gaph hap Wttn latonhp Mofcaton qu to lnaz gaph Algbac pntaton A nca, man th am. Th no latonhp btwn th vaabl. Non b, o contant A nca, nca popotonall. Non m b Y ctl popotonal to. A nca, ca. Gaph v 1 m 1 b Y nvl popotonal, o v -1 to. Y popotonal to th qua of. Gaph v 2 m 2 b Th qua of popotonal to. Gaph 2 v 2 m b

6 Appn B: Plottng a Gaph n Ecl 1. Opn Ecl. Un column A, lt ou ata that wll b plac on th a of th gaph. Un column B, lt ou ata that wll b plac on th a of th gaph. 2. Ung ou mou, hghlght th ata. 3. Go to th Int mnu, an lct Scatt. 4. Choo th ft bo no ln. 5. Ung th Chat Tool mnu, ttl ou gaph an labl th an a, wth coct unt. 6. Clck th chat aa. Go to th op-own mnu un Chat Tool. Slct Tnln, an Mo Tnln Opton 7. Choo Lna f ou blv ou gaph lna. Othw, mof th ata a n an cat a nw lna gaph.

Load Equations. So let s look at a single machine connected to an infinite bus, as illustrated in Fig. 1 below.

Load Equations. So let s look at a single machine connected to an infinite bus, as illustrated in Fig. 1 below. oa Euatons Thoughout all of chapt 4, ou focus s on th machn tslf, thfo w wll only pfom a y smpl tatmnt of th ntwok n o to s a complt mol. W o that h, but alz that w wll tun to ths ssu n Chapt 9. So lt