# Lab #3: Pendulum Period and Proportionalities

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1 Physics 144 Chwdary Hw Things Wrk Spring 2006 Name: Partners Name(s): Intrductin Lab #3: Pendulum Perid and Prprtinalities Smetimes, it is useful t knw the dependence f ne quantity n anther, like hw the acceleratin f an bject affects the distance traveled. Des dubling the acceleratin duble the distance traveled? des dubling the acceleratin f an bject triple the distance traveled? etc. Smetimes, ne quantity des nt depend n anther. An example f this is hw the mass f an bject affects its acceleratin near the surface f the earth. We knw that if we neglect the effects f air resistance, the mass f an bject desn t affect its acceleratin due t gravity near the surface f the earth. In tday s explratin, we will investigate the pendulum. We already lked at the pendulum as an example f cnservatin f mechanical energy. The pendulum is an imprtant cmpnent in many instruments requiring regular timing, such as clcks and ther peridic machinery. Althugh the pendulum has been replaced by electrnic devices and atmic clcks in many applicatins, it is still used in precisin machinery (the mst cmmn examples are wristwatches and grandfather clcks). The purpse f tday s lab is t investigate the factrs that affect the perid f a pendulum. The perid f the pendulum is the amunt f time it takes fr the pendulum t cmplete ne full cycle f its mtin. We ll first identify several factrs that culd affect the perid f the pendulum, and then see whether r nt they d actually affect the perid. Then, we ll chse ne factr that des affect the perid, and see if we can determine the relatinship between that factr and the perid (i.e. is the relatinship linear, s that if yu duble ne, yu duble the ther? Is it quadratic, s that if yu duble ne, yu quadruple the ther? etc.) In making these decisins, we will rely heavily n the tls f statistical analysis we develped during the first lab. We ll use a cmputer t help us with these repetitive calculatins. There are several gals f this lab: in particular, after finishing this lab, yu shuld a) gained further understanding f the ideas f true mean, experimental mean,, and f the mean, alng with hw t present an experimental result including uncertainty; b) get sme practice using a cmputer spreadsheet prgram t help yu with repetitive calculatins; and c) understand sme f the factrs which d and d nt affect the perid f a pendulum. Specifically, we ll explre the fllwing questin: Equipment What factrs affect the perid f a pendulum? The equipment fr this lab is very simple: a lng lightweight string cnnected t a spherical bb made f metal. There are als several smaller spheres alng with magnets; yu ll use these t increase the mass f the bb. Yu ll als need a 2-meter meter stick and a stp-watch, as well as sme masking tape, a marker, and a prtractr. Yu ll need t get up and dwn frm the table fairly frequently, s make sure there is a lt f clear space. Als, yu ll use the balance up frnt t btain the mass f yur pendulum bb alng with the added masses. Since the masses are spherical, they tend t rll; place the spheres n the brass weight prvided at the frnt fr stability.

2 Getting Started (1) Make sure yur pendulum cnsists nly f the lng string and the single bb. Make sure n magnets are stuck t the bb. Nw, pull up the bb, keeping the string taut. Release the bb frm rest, and bserve its subsequent mtin. Write dwn sme bservatins. Make sure that yu discuss the mtin f the bb, whether it seems t reach the same height n either side, whether it reaches the same height each time, whether it seems t take the same amunt f time t make ne full back-and-frth cycle each time, etc. (2) Discuss with yur partners sme f the factrs that yu think might affect the perid f the pendulum, and sme factrs which yu dn t think will affect the perid f the pendulum. Write dwn sme ntes abut yur discussin. (3) We ll discuss yu ideas as a class. While yu re waiting, yu can (carefully!) climb up n t yur table, pen up the clip hlding yur pendulum string, and take yur bb t be weighed. Use the flat brass weight that is there t help hld the bb statinary. mass f brass weight: g mass f brass weight + pendulum bb: g mass f pendulum bb: g = kg

3 Setting Up The first thing we ll explre tday is whether the amplitude (in ther wrds, the initial height r better yet angle) f the swing affects the perid. (1) Since we will be explring the effect f initial amplitude (initial angle) n the perid, we ll need t make sure we can reliably reprduce the starting angle. One persn shuld climb back up the table, and reattach the end f the string t the clip. Use the 2-meter meter stick, and make sure that the length f the pendulum (frm the bttm f the clip where the string is attached t the middle f the bb is 145 cm (1.45 m). It s imprtant that everyne start with the same length f pendulum, s take sme care with this step. (2) We need t mark ff sme angles. There may already be sme angles measured ff, but dn t trust them: mark ff yur wn. Place hrizntal strips f masking tape n the wall (cver any tape that is already there). Yu ll need a strip f tape lw n the wall and anther strip f tape high n the wall. The pendulum shuld hang straight dwn; mark that psitin as 0 n bth pieces f tape. (3) Then, smene shuld hld the pendulum bb, keeping the string taut (dn t pull t hard r yu might cause the string t slip at the clip end). A persn at the tp shuld have a prtractr, and tell the persn hlding the bb when they have swung thrugh 5. The 5 psitin shuld be marked n bth pieces f tape. (4) Repeat fr 10 and 15. (5) Befre making any timing measurements, make yur best guess. Yu will be measuring the time fr small swings, fr medium swings, and fr big swings. The medium swings will be twice as big as the small swings, and the big swings will be abut 3 times larger than the small swings. What d yu think the difference in the perid f scillatin will be? This is just a guess. Dn t wrry abut being right r wrng at this pint. Please write dwn yur guess: (6) Shw yur guess t yur instructr. Make sure yu instructr initials that he has seen yur guess: Instructr s initials: Making Measurements There are lts f ways t measure the perid f a pendulum. Sme techniques are much better than thers. Fr example, it is quite unsatisfactry fr ne experimenter t watch the pendulum and call ut "ready, set, g" and fr the ther t start r stp the timer n "g." This wuld be like ne umpire watching the runner's ft tuch first base and anther umpire watching the ball hit the first baseman's glve! Als, it's better t start timing after the pendulum has swung a cuple f times s that timing the start will be exactly like timing the finish. And dn't make the mistake f having ne partner measure all the small swing times and anther all the larger swing times. This culd intrduce a bias due t different techniques r judgments. Finally, start and stp yur measurements at the bttm f a swing, where the pendulum is traveling fastest (it is hard t tell just when the pendulum has reached the tp f its swing, since it is mving s slwly there). Here are my suggestins fr gd ideas in making the mst careful timing measurements yu can. Yu might have imprvements n sme f these ideas; please discuss yur ideas with me.

4 (a) The same persn shuld always release the pendulum frm rest. The ther persn shuld always measure the time. Try t have the persn with the better reflexes be the timer. (b) The persn lifting the pendulum bb shuld cnsult with the persn n the grund t make sure the pendulum is released frm the crrect angle. The best way t d this is fr the persn n the grund t stand directly in frnt f the 0 mark, with the pendulum hanging mtinless in line with the 0 mark. The persn n the grund shuld then tell the lifter when they have gtten the string t line up at the crrect angle. (c) The persn releasing the pendulum bb shuld hld the bb with tw fingers, and make sure t simply release the bb frm rest (dn t push it ff). (d) DON T start timing upn the release f the bb. Wait a few swings, and start timing as the bb passes the bttm (the 0 mark) ging t the right. Start cunting, ut lud with yur partner, at Zer. Then, when the bb makes ne full swing and passes the 0 mark, again ging t the right, cunt ut lud One. Repeat until yu reach Ten. This insures that yu have measured the time fr 10 full swings (ften, peple incrrectly start cunting at One ). Yu want t measure 10 full swings since that will tend t average any starting/stpping errr ver 10 perids instead f just ne perid. (e) DON T make all yur 5 measurements, then yur 10 measurements, then yur 15 measurements. Rather, make ne 5 measurement, then a 10 measurement, then a 15 measurement. This will help t insure that any imprvement yu make in timing as yu get mre practice will be spread ut amng all the measurements. (1) Fllw the suggestins abve. Fill in yur data in the table belw. Yu ll make 10 separate measurements fr each angle. Dn t fill in any f the shaded bxes yet; yu ll have a cmputer d mst f that fr yu. Trial T 5 (s): time fr 10 swings at 5 T 10 (s): time fr 10 swings at 10 T 15 (s): time fr 10 swings at 15 average f the mean

5 Analysis (1) Enter the data frm yur table abve int an Excel ntebk. The time data fr T5 shuld g in cells A1 t A10, fr T10 in C1 t C10, and fr T15 in E1 t E10. There is n particular reasn t enter them in these particular clumns ther than fr ease in viewing and s that yu can fllw alng with the instructins in this handut. If yu are already familiar with Excel, yu shuld use yur wn experience. In rder t calculate the average (ie. the experimental mean), g t cell A12 and type =AVERAGE(A1:A10) then press enter. This will take the values in cells A1 t A10 and calculate the average in the familiar way In rder t calculate the, g t cell A13 and type =STDEVP(A1:A10) then press enter. This will take the values in cells A1 t A10 and calculate the like yu did by hand in Lab #1. Excel desn t have a nice frmula fr the f the mean, s yu yu ll need t figure it ut yurself (ask if yu dn t recall the frmula fr f the mean). Yu shuld repeat these Excel calculatins n yur ther tw clumns f data. (2) Write the values fr the average, the, and the f the mean int yur data table (i.e. in the shaded bxes). (3) Yur best estimate fr the true value f T5 will be given by the mean ± the f the mean. Write dwn yur best estimate fr the true value f (use a reasnable number f digits; dn t g verbard!): T 5 = ± s T 10 = ± s T 15 = ± s (4) Based n yur answers abve, including uncertainty, what can yu cnclude abut any dependence f the perid f a pendulum n its initial amplitude? In ther wrds, is the perid fr 5 swings the same r different than the perid fr 10 swings t within experimental uncertainty, r can t yu make a cnclusin? Is the perid fr 10 swings the same r different than the perid fr 15 swings t within experimental uncertainty, r can t yu make a cnclusin? the perid fr 5 swings the same r different than the perid fr 15 swings t within experimental uncertainty, r can t yu make a cnclusin? Discuss yur results belw and with yur instructr.

6 Dependence n Mass Next, yu will repeat yur measurements, this time hlding the initial amplitude t 5, but changing the mass. The easiest way t change the mass is by using the magnets t attach the small spheres t the pendulum bb. Yu need t be careful t attach the small spheres in a symmetric fashin t maintain balance, and als at the center f the bb s t nt shift the center f mass. (1) Cnsult with yur instructr t see hw many small spheres t add. (2) Measure the mass f the small spheres yu are adding. Dn t frget t accunt fr the mass f the brass weight if necessary. Write dwn the new ttal mass f the bb: New ttal mass f yur bb: g (3) Use a 145 cm length pendulum with initial swing angle f 5. Use the same timing techniques as befre. Trial T yur mass (s): time fr 10 swings at 5 fr yur mass average f the mean (4) Use Excel t determine the average and the. Calculate the f the mean. Fill thse in n the table abve. (5) Write dwn yur best estimate fr the true value f the perid f a pendulum with the mass yu used. perid f a g pendulum swinging at 5 : = ± s (6) Write dwn yur answer fr questin (5) n the bard.

7 (7) Cpy the varius class results frm the bard belw. (8) What can yu cnclude abut the dependence f the perid f a pendulum n the mass f the bb? Is there a dependence? Dependence n Length Next, yu will repeat yur measurements, this time hlding the initial amplitude t 5, with the initial mass f the bb, but this time yu will change the length f the string. Yu ll d this fr tw lengths, ne shrt and ne lng. (1) Cnsult with yur instructr t see which lengths t measure. (2) Yu shuld alternate measurements between lng length and shrt length, but that s just t incnvenient fr this part. S, d all yur lng length measurements, then d all yur shrt length measurements. (3) Fill ut the table n the next page.

8 Trial T yur shrt length (s): time fr 10 swings at 5 T yur lng length (s): time fr 10 swings at 5 average f the mean (4) Reprt yur results fr yur shrt length and yur lng length perids (including uncertainty!) belw and n the bard. Write dwn the class results belw as well. (5) What can yu cnclude abut the dependence f the perid f a pendulum n the length f the pendulum? Is there a dependence?

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