EXPERIMENT 4 DETERMINATION OF ACCELERATION DUE TO GRAVITY AND NEWTON S SECOND LAW

Transcription

1 EXPERIMENT 4 DETERMINATION OF ACCELERATION DUE TO GRAVITY AND NEWTON S SECOND LAW I. Introduction. Thr ar two objctivs in this laboratory xrcis. Th first objctiv, (A), is th study of th bhavior of a body in th gravitational fild. Th scond objctiv, (B), is to vrify Nwton s scond law. II. Thory A. Dtrmination of th valu of th acclration arising from th gravitational fild. Th forc btwn two bodis is attractiv and dpnds on th masss of th bodis and th distanc btwn thm. Th magnitud of th forc on a body of mass m, at a distanc R from th cntr of th arth and at, or abov, th surfac of th Earth, is givn by, F g mm G R 2 (1) whr G is th univrsal gravitational constant, M is th mass of th arth and R is th distanc from th cntr of th arth. If th body is on th arth, th radius of th arth. Th forc F g is calld th gravitational forc. Nwton s scond law givs us th acclration, a, of a body of mass, undr a nt xtrnal forc F as: F a (2) m Thus, if a body has no othr forc on it than th gravitational forc, th magnitud of th acclration of a body of mass m in fr-fall at th surfac of th arth is givn by: GM g a R 2 (3) that is, th acclration of a body at th surfac of th Earth dpnds only on th mass of th arth, M and it radius R - it is indpndnt of th mass of th body. This acclration is trmd "th acclration of gravity", dnotd by g. It is takn as a constant but actually dpnds slightly on th location (bcaus of inhomognity in th arth's crust, such as dnsr rock formations and th diffrnt lvations of th surfac, tc.).

2 III. Equipmnt and Procdur (acclration du to gravity) Equipmnt Photogat, pickt fnc, PASCO 750 Intrfac and computr. Procdur Find th valu of th acclration of gravity, g, at Athns, OH 1. Connct th photogat to th first digital input in th intrfac and turn on th Intrfac. Slct Data Studio from th dsktop. Follow th "Computr Instructions" to st up th photogat to collct position and vlocity data. A photogat is st up at th nd of th laboratory tabl so that a pickt fnc may b droppd through it. 2. On studnt should drop th pickt fnc through th photogat and on should start th collction of data bfor th pickd fnc starts to cross th photogat. Mak sur th pickt fnc dos not hit th photogat. 3. Entr th vlocity data into your notbook. Fit th position data with a quadratic and th vlocity (rally spd) data with a linar fit as indicatd in th computr instructions. Th cofficint of t 2 yilds on-half th valu of th acclration in th gravitational fild. Th slop of th fit of th vlocity data may b takn to b th avrag acclration and th associatd standard dviation in th slop to b th standard dviation in th acclration. dv a a slop (6) dt In this cas ths ar, rspctivly, th avrag valu of th acclration du to gravity and its standard dviation. Entr th avrag valu of th acclration and th standard dviation of th man of th acclration in your notbook and print th graphs showing th curv fits to th position and vlocity data. III. Data Print out and ntr into your notbook th position and vlocity data for th pickt fnc droppd vrtically through th photogat and analyz it to obtain th valu of th acclration of gravity it according to th procdurs outlind in sction II and th Computr Instructions. IV. Rsults Dtrmin th valu of g from fitting to th position and vlocity data. For th linar fits, dtrmin th standard dviation in th man of from th standard dviation in th man of and writ th rsult as g g g. Show your calculation for th standard dviations in th man of on your summary sht. Compar ach valu of g to th accptd valu in Athns, OH, 9.800±0.003ms -2 (from th National Godtic Survy ) and dtrmin th prcnt rror in your masurmnt with

3 rspct to th accptd valu. V. Conclusion and Discussion (acclration du to gravity) Dos th valu of g fall within th standard dviation of th accptd valu? Discuss any actual sourcs of rror. What ar th sourcs of rror (if any) in your procdur? VI. Equipmnt and Procdur (Nwton s Scond Law) Equipmnt Airtrack with glidr, photogat, PASCO 750 Intrfac and computr. Procdur 1. Mak sur th airtrack is lvl. Th instructor will show you how to dtrmin if it is lvl and how to lvl it if ncssary. Th glidr must mov smoothly (it should not appar to slow down or "catch" along th track). If it dosn't mov smoothly ask th instructor to look at it with you. 2. Follow th computr instructions for Nwton s Scond Law at th nd of this lab. 3. Attach a mass holdr to a string about 3.0 m long and pass it ovr th pully, tying th othr nd to th hook on th glidr. Plac th assignd numbr of grams on th glidr and us th scal providd to dtrmin th mass of th assmbly of glidr plus masss, including th holdr. Entr this data into your laboratory notbook. 4. Transfr masss of 2-g ach, on at a tim, from th glidr to th hangr so that th total mass of th systm rmains constant. Bgin with a run with only th holdr to provid th acclrating forc ( =m holdr g ). Thraftr transfr a 2-g mass to th holdr and obtain th avrag acclration and standard dviation again. Rpat th procdur until all th 2-g masss hav bn transfrrd to th holdr. Th magnitud of th nt forc is th wight of th hangr and masss on it so that th acclration of th glidr is givn by dv mholdr g a (9) dt m 5. To hold th glidr at rst, plac a pncil rasr or fingr on th sid of th glidr. Rlas th glidr. Lt th mass mov som distanc and tak position and spd data (which will appar undr Run#1 in th lft window). But stop th glidr so that th masss don t hit th floor. Mak sur th string rmains on th pully. Th computr will provid a st of positions spds for ach mass placd on th glidr from which th acclration is obtaind by taking th slop of th vlocity vrsus tim graph, Eq. (9). total

4 VII. Data Aftr th first run (with all th masss on th glidr), transfr a fixd amount of mass ach tim to th mass holdr and rlas th glidr as indicatd in th procdur. Entr into your laboratory notbook th total mass (includs th holdr), th acclration, and th standard dviation of th acclration for ach additional mass addd to th holdr. Data: Part B Nwton s Scond Law Mass of hangr, m h = kg Addd masss to glidr m a kg Mass of glidr from balanc kg Total Mass of th Systm m s = (holdr + mass on glidr +glidr) kg Tabl 1. Masss, forc and linar acclrations for a mass m s. Run 1 Mass of Holdr + Addd Mass Forc (=(m h +m a )g) Acclration (Fit to Position) Acclration (Fit to Spd). Std. Dviation Mass of th systm from th invrs slop of a vs F, th mass. m s F a, and th standard dviation of Mass of glidr = m glidr = m s - (m h + m a ) VIII. Rsults a vs F Plot, on sparat graphs, i i for th acclration obtaind from fitting th position vrsus tim and th slop of th vlocity vrsus tim data and attach th standard dviation of th man for th ach acclration dtrmind from th slop. Obtain th bst fit to th data kping in mind th standard dviations associatd with th ach acclration. Th invrs slop of th acclration vrsus th forc is th mass of th systm (total mass). Th standard rror in th mass is th dtrmind by standard rror in th fit. Dtrmin th mass of th systm on th scal providd.

5 IX. Conclusion (Nwton s Scond Law) Dos th data vrify a linar rlationship btwn th forc and th acclration of th glidr. Dos th mass dtrmind from th graph of acclration vs forc fall within th standard dviation of th mass masurd on th scal? What ar th possibl sourcs of rror? N.B. Th mass as masurd by th acclrations is calld th inrtial mass bcaus it is a masur of th rsistanc to chang of motion whil that masurd on th scal is calld th gravitational mass which is a masur of th strngth of a body as a sourc of gravitational forc. In principl th proprtis of ths two masss ar diffrnt but vry carful xprimnts yild a ratio of 1 to about fourtn dcimal placs for th two masss. Computr Instructions Acclration du to Gravity Physically plug th photogat into digital input port #1 on th 750 intrfac. 1. In th Exprimnt Stup window (click Stup on th toolbar if it isn t shown), you ll s an imag of th front of th Pasco 750 Intrfac. Click input port #1 on that imag, and slct th photogat snsor from th list. 2. Choos th position and vlocity data sourcs. In th Constants tab, ntr th bar spacing for th pickt fnc so that th intrfac will rport distancs corrctly. For th larg pickt fnc it is 0.05 m and for th smallr pickt fnc it is 0.03 m. You may clos th Exprimnt Stup window. 3. Now to collct som data and display it on a graph: Lft Click on Run#, wait a short whil and lft click again until a rctangl appars. Typ your initial and thos of your partnr into th box. Thn answr ys whn Rnam appars in th main window. This will plac th initials on th graph. Drag and drop th Position data sourc (from th top lft window) onto th graph icon in th Displays window (bottom lft). A graph will appar in th Main window. You'll want to show Position and Vlocity on th sam graph, so slct Vlocity from th data sourc window and drag & drop it onto th Graph that was just cratd. If you drag & drop it onto th Graph itm in th Display window, it will opn a nw graph in th main window. This will also work, but you'll hav to print two pags instad of on. 4. To start th run, slct on studnt to drop th pickt fnc and on studnt to start th data acquisition. It is important that th pickt fnc not hit th sids of th photogat as it travls through it. Th data run is startd with th Start Button on th toolbar. Onc data acquisition is in progrss, th Start button bcoms a Stop button. 5. Your first data st will b namd Run #1. Thr's an autoscal button on th top lft of th graph display. You may hav to furthr corrct th x and y axs by clicking and dragging thm.

6 6. To analyz your data, a) slct Run #1 on th position graph, and thn go to th G button and slct Quadratic fit from th Fit pulldown mnu. Th cofficint of t 2 will yild half th valu of g. b) Slct th vlocity graph, and do a linar fit. Th program will rturn a slop which is th avrag valu of g and a quantity aftr th sign which th is th standard dviation in th valu of g. Nwton s Scond Law 1. Click on th first digital input, and slct th snsor labld Motion Snsor from th mnu that appars. 2. An icon showing th Motion Snsor will appar attachd to th intrfac. Blow th intrfac imag ar th snsor configuration options. Configur th snsor in th following way: in th Masurmnt tab, slct position and vlocity and d-slct acclration. Click on th Motion Snsor tab s if th snsor is working, it should at this point rad th distanc of th glidr on th track, if not s th TA, st th sampling rat to 20 Hz. You ar now rady to mak th masurmnts. You can minimiz th xprimnt stup scrn. 3. Drag th Position, Ch 1 data st from th Data panl (top lft) and drop it onto th Graph icon in th Displays panl (bottom lft). Rpat for vlocity. 4. You'll want to show Position and Vlocity graphs. St up th display as bfor. 5. To start th run, slct on studnt to hold th air car in plac and rlas it whn a scond studnt clicks th start button. You'll want to stop th run bfor th air car hits th nd of th track and bfor th masss hit th ground. 6. Again, rscal th graphs so that your data is most clarly displayd. 7. Analyz your data as bfor, and print a sampl of th position and vlocity graphs. 8. To fit a curv: In th graph window, click th Fit button and slct th fit typ from th pull-down mnu. In calculating g from th position vs tim data choos quadratic. This will fit th data with a quadratic and a box will appar with th valu of th cofficints for ach powr. A is th cofficint for th quadratic trm. Procd similarly for th vlocity but call for a linar fit. Th program will rturn th slop and standard dviation (th quantity following th slop) for th avrag acclration and an RME (Root Man Squar Error). Disrgard th RME. Th slop can b takn to b th avrag valu of th acclration to b usd in Eq. (1). 9. Rcord th valu of th slop and th associatd standard dviation (which oftn is takn as th standard rror). 10. If you ar intrstd in th drag of th systm, curv fit th first 6 points on th position graph to a quadratic and th last 6 points in th position graph to masur th chang in

7 fit to th graph.