ۥ-!@ <-t3*$*$nnnnnnJXn\ppppp"ppqLqqqqqqqqqqqqqqet4t/qknqq10.1: Analyzing the Motion of a Pendulum PROBLEM: What is the relationship between the length and the frequency of a simple pendulum? DIAGRAM: **See back of title page** APARATUS: SYMBOL 183 \f "Symbol" \s 10 \h apparatus as illustrasted SYMBOL 183 \f "Symbol" \s 10 \h stopwatch SYMBOL 183 \f "Symbol" \s 10 \h protractor METHOD: 1. A simple pendulum was attached to a rigid support, with the centre of the bob about 100 cm below the pivot point. The length of the pendulum from the pivot point to the centre of the bob was measured. 2. The length of the 100cm pendulum was decreased, in steps of approximately 20cm. The frequency of the pendulum was determined for each length. Observations were recorded in a chart, as seen in OBSERVATIONS. 3. A graph was plotted of the pendulums frequency (f) against its length (L). 4. The length of the pendulum was adjusted to the identical length used in the first observation. This time, the bob on the pendulum was released from aproximatley 10o so the pendulum would vibrate with a smaller amplitude. The frequency was compared with that obtained when the pendulum oscillated with a larger amplitude, in step 1. 5. The bob was replaced with one of a larger mass, making sure that the length of the pendulum remained the same. The frequency was determined and compared with that obtained in step 4. OBSERVATIONS: Mass (g) Angle (o) Length (cm) Cycles Time (s)  20g 20o 100cm 10 20.40.5  20g 20o 80cm 10 18.120.5  20g 20o 60cm 10 15.750.5  20g 20o 40cm 10 12.710.5  20g 20o 20cm 10 9.290.5  20g 10o 100cm 10 20.490.5  20g 20o 100cm 10 20.830.5   ANALYSIS: Frequency(Hz) =cycles/time or =1/T Period(s) =time/cycles or =1/f  0.49019 2.04  0.55187 1.812  0.63492 1.575  0.78678 1.271  1.0642 0.929  0.48804 2.049  0.48007 2.083   **See following page** Sources of Error The angle of release was changed slightly after measurement due to the unsteadiness of the hand holding it and the fact that the person letting go pulled it back slightly before release. This change in the angle would change the amplitude but it would not directly affect the results because only the number of cycles and the time were being measured and they depend only on the length of the pendulum. This error could be minimized by using a stationary device to hold the pendulum at the desired angle. The length of the pendulum was not exact due to the gradations on the metre stick and human error. A change in this length would directly affect the results obtained for the length of the time that each cycle took. This would then in turn affect the calculations obtained for frequency and period. This error could be minimized by using a millimetre ruler, and by being more careful. The string could not be exactly fixed to the rubber stopper. The weight of the bob caused the string to slide up and down through the hole in the stopper which changed the length of the pendulum several times during each experiment. A change in this length would directly affect the results obtained for the length of the time that each cycle took. This would then in turn affect the calculations obtained for frequency and period. This error could be minimized by using something other than a rubber stopper to attach the string to. The starting and stopping times were not exactly matched to the real time due to the slow reaction time of the person and the difficulty that exists in determining the exact point in which to start and stop. These discrepancies in the length of time would affect the results that were calculated for frequency and period. This error could be minimized by using a computer with motion detectors to record the exact times at which the pendulum started its first cycle and ended its tenth. In addition to swinging from side to side, the mass of the pendulum was also spinning round and round. This spinning would slow the mass down which in turn would affect the time of each cycle, which would then change the calculations obtained for frequency and period. This error could be minimized by using a stationary mass and a better set-up. Due to the amount of people in the room, air currents were created that may have slightly changed the direction of the swinging of the pendulum causing it to swing in more than one plane. This error could be minimized by performing the experiment in a vacuum. The table that the pendulum was set up on was completely stable and due to the members of group sitting near it was moved slightly during each experiment. This would move the pendulum causing it to swing in more than one plane. This error could be minimized by setting up the experiment on a table secured to the floor that is not being used at the time. The pendulum was not swinging is just one plane. There was lateral movement that would change the results. The time of each cycle would be increased slowly which would then change the calculations obtained for frequency and period. This error could be minimized by correcting all of the errors mentioned above. All of these errors are minimal and can be ignored for one cycle, but they grow worse as time continues. Because the time of one cycle was measured by finding the average of 10 cycles, the actual time the first cycle took is less than the average time because these errors when compounded increase the time of each cycle. This experiment could be improved by performing the experiment in a vacuum, on a secured table, with a proper set up and accurate instruments. Questions 1. What effect does a change in the length have on the frequency and period of an oscillating pendulum? As the length decreases, the frequency of an oscillating pendulum increases and the period decreases. As the length increases, the frequency of an oscillating pendulum decreases and the period increases. 2. For a pendulum with a fixed length, what is the effect on the frequency and period of an oscillating pendulum of: a) a change in amplitude? A change in amplitude has no effect on the frequency and period of an oscillating pendulum. b) a change in the mass of the bob? A change in the mass of the bob only affects the amplitude. Therefore a change in the mass will have no effect on the frequency and period of an oscillating pendulum. CONCLUSIONS The frequency and period of a pendulum are directly affected by the length. As the length decreases, the frequency increases and the period decreases. As the length increases, the frequency decreases and the period increases. For a pendulum of fixed length, a change in amplitude and a change in the mass of the bob have no effect on the frequency and period. The results of the experiment were valid as the procedure was followed exactly and the sources of error were mentioned and accounted for. APPARATUSillustratedcentercenterapproximatelymillimeter EMBED MSGraph \s \* mergeformat  Pendulum Frequency vs Pendulum Length EMBED MSGraph \s \* mergeformat Frequency Length What is the relationship between the length and the frequency of a simple6ABabc~13:;RT[\su|},vx $        ( Pgl  !')*>cefl$*MjUj[jpjjjjjkkù̳       $* @R    @ @@<   1 46A~-/'138=DHR.l?y9 y! h hh!h!h!"h!h!!"!%RTY^dhsuz !,;IMS^lpvͤ h!"h!h(l?y9 y! h(l?y9 y! 1vx  $ & " $  "ܮ{u{!!!!!!h!h!hltT! ltT! hltT!  .   gi  13#%'),.02468:<>e常ܦ!.h!"h!h!"hh!!"!!h!h!!!;F  : ^cC"T55&xx   ' R5 ' % `"ArialO-"System-'- 5%-'- 4%---'-- {2a$W\ v2\v2@\@v2\v2G\Gv2\v2\v2O\Ov2 \ v2V \V v2 \ v2 \ v2 \ v2 \ v2e\ev2\v2l\lv2\v2\v2s\sv2\v2{\{v2(\(v2\v2/\/v2\v2\v27\7v2\v2>\>v2\v2\v2\v2\v2M\Mv2\v2T!\T!v2"\"v2"\"v2[#\[#v2--'-- 4%---'-- {2a$W\ v2\v2] \] v2\v2!\!v2\v2\v2E\Ev2 \ v2 $\ $v2--'-- 4%---'-- 32$9_  $_9f $f9l  $l 9s, $s,--'-- 4%---'-- 32$9c  $c9i $i9p& $p&9v2 $v2--'-- 4%-9\  $\99] ] !!EE   $ $9\9v2\\cciip&p&v2v2--'--  2$W--'-- 2k$- $\  Ecip&v2 $\)- $]#$]?$%$]#Ec$c Ec}+Ec i$iKi1iKp&$o&&o&7&o&v2$u22u27 =2u2---'---  2$W---'--- 4%---'--- 5%---'--- 5%---'--- 5%  2 0Z 2 x10ZZ 2 ^ x20ZZ 2 x30ZZ 2 !x40ZZ 2 x50ZZ 2 x60ZZ 2 Fx70ZZ 2 x80ZZ 2 !x90ZZ 2 k$100ZZZ---'--- 5%---'--- 5% 2 60.49019Z0ZZZZZ2 = 0.55187Z0ZZZZZ2 C0.63492Z0ZZZZZ2 J%0.78678Z0ZZZZZ2 P11.07642Z0ZZZZZ---'--- 5%--'MSGraph` 60B&'()@3BZ31Arial Rounded MT Bold2 "General00.00#,##0 #,##0.00"$"#,##0\ ;\("$"#,##0\)"$"#,##0\ ;[Red]\("$"#,##0\)"$"#,##0.00\ ;\("$"#,##0.00\)#""$"#,##0.00\ ;[Red]\("$"#,##0.00\)0%0.00% 0.00E+00 m\/d\/yy d\-mmm\-yyd\-mmmmmm\-yy h\:mm\ AM/PMh\:mm\:ss\ AM/PMh\:mm h\:mm\:ssm\/d\/yy\ h\:mm@@@@Frequency (Hz)E_?4F?7qrCQ?>$@M-?E*-9? Period (s)RQ@ˡE?333333?tV?|?5^?U@UZZZ,2= c>@Citizen 200GXCIT9USLPT1:DEVBITBLTOUTPUTQUERYDEVICENAMES\PIXEL SCANLR COLORINFO ENUMDFONTS DEVINSTALL^ DEVEXTTEXTOUTADVANCEDSETUPDIALOG]DEVICEBITMAPBITS401 Arial/$33  Period (s)4/$3" 1D+ 3 ! ! 4ND+ 3! ! 44%-6'3&'44 = 0&_URA  METAFILEPICT_U(_URA5   ' f0 ' $ `"ArialF-"System-'- #0$-'- /$---'-- -?#KN -N-N-lNl- N -Y NY - N - N -N-N-2N2-N-N-mNm-N- N -N-N-G NG -!N!---'-- /$---'-- -?#KN -ENE-N-ZNZ-"N"---'-- /$---'-- )-"1 "1"1Y"Y1*("*(--'-- /$---'-- )-"1  " 1"1"""1-"---'-- /$-1N "N11EEZZ""1N1-NN  ""----'-- -"K--'-- -I#-N   "X-N)- $QQQ $ W  !  $+"$"""""X-$-!-Y-]-Y-!---'--- -"K---'--- /$---'--- #0$---'--- #0$---'--- )0$  2 0Z 2  B0.5Z0Z 2 1Z 2 2B1.5Z0Z 2 2Z 2 G#B2.5Z0Z---'--- #0$---'--- )0$ 2 (0.49019Z0ZZZZZ2 0.55187Z0ZZZZZ2 0.63492Z0ZZZZZ2 !0.78678Z0ZZZZZ2 l,1.07642Z0ZZZZZ---'--- #0$--'  5MSGraph 6NB&'()@3BZ31 Arial2 "General00.00#,##0 #,##0.00"$"#,##0\ ;\("$"#,##0\)"$"#,##0\ ;[Red]\("$"#,##0\)"$"#,##0.00\ ;\("$"#,##0.00\)#""$"#,##0.00\ ;[Red]\("$"#,##0.00\)0%0.00% 0.00E+00 m\/d\/yy d\-mmm\-yyd\-mmmmmm\-yy h\:mm\ AM/PMh\:mm\:ss\ AM/PMh\:mm h\:mm\:ssm\/d\/yy\ h\:mm@@@@ FrequencyE_?4F?7qrCQ?>$@M-?E*-9?LengthY@T@N@D@4@U@UZZZ,2= '>@Citizen 200GXCIT9USLPT1:DEVBITBLTOUTPUTQUERYDEVICENAMES\PIXEL SCANLR COLORINFO ENUMDFONTS DEVINSTALL^ DEVEXTTEXTOUTADVANCEDSETUPDIALOG]DEVICEBITMAPBITS401 Arial4%33  Length44%3" 90!3 ! ! 4\0!3! ! 444 = 5'^cC  METAFILEPICT ^ ^cC#   ' R5 ' % `"ArialO-"System-'- 5%-'- 4%---'-- {2a$W\ v2\v2@\@v2\v2G\Gv2\v2\v2O\Ov2 \ v2V \V v2 \ v2 \ v2 \ v2 \ v2e\ev2\v2l\lv2\v2\v2s\sv2\v2{\{v2(\(v2\v2/\/v2\v2\v27\7v2\v2>\>v2\v2\v2\v2\v2M\Mv2\v2T!\T!v2"\"v2"\"v2[#\[#v2--'-- 4%---'-- {2a$W\ v2\v2] \] v2\v2!\!v2\v2\v2E\Ev2 \ v2 $\ $v2--'-- 4%---'-- 32$9_  $_9f $f9l  $l 9s, $s,--'-- 4%---'-- 32$9c  $c9i $i9p& $p&9v2 $v2--'-- 4%-9\  $\99] ] !!EE   $ $9\9v2\\cciip&p&v2v2--'--  2$W--'-- 2k$- $\  Ecip&v2 $\)- $]#$]?$%$]#Ec$c Ec}+Ec i$iKi1iKp&$o&&o&7&o&v2$u22u27 =2u2---'---  2$W---'--- 4%---'--- 5%---'--- 5%---'--- 5%  2 0Z 2 x10ZZ 2 ^ x20ZZ 2 x30ZZ 2 !x40ZZ 2 x50ZZ 2 x60ZZ 2 Fx70ZZ 2 x80ZZ 2 !x90ZZ 2 k$100ZZZ---'--- 5%---'--- 5% 2 60.49019Z0ZZZZZ2 = 0.55187Z0ZZZZZ2 C0.63492Z0ZZZZZ2 J%0.78678Z0ZZZZZ2 P11.07642Z0ZZZZZ---'--- 5%--' Present at the Lab: Andrew Likakis, Michael Hancock, Eva Chen Absent from the Lab: Christina Mitchel Title Page: Andrew Likakis Problem: Andrew Likakis Diagram:Andrew Likakis Aparatus: Andrew Likakis Method: Andrew Likakis Observations: Andrew Likakis Analysis: Mike Hancock, Eva Chen Conclusions: Christina Mitchel OjQjSjUjWjYj[jpjjjjjjjk-kEkckkk!!!!h!3o 3' d[qsu-/13579K X'=PRn13!"!!!!!!"!"!"!"!!!!!!!!!!!!!!!!"!.!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!kRvk6Times New Roman Symbol&ArialArial MT Black&Arial Rounded MT Bold Mead BoldFGradl5Courier New VPlaybill FVivaldi Wide Latin VStencil+K 3999::hou  +9 '=NR13>|) !) *>cefl    MjUj[jpjjjjjk" h,E-E/Andrew LikakisAndrew Likakis