e Measurement of te Gravitational Constant wit Kater s Pendulum Abstract A Kater s pendulum is set up to measure te period of oscillation usin a lamppotocell module and a ektronix oscilloscope. Usin repeated measurements of te period and various lent measurements, we determine te value of ravity to witin 0.% accuracy. e addition of a ruler to te apparatus allows us to consider te effects of dampin on te pendulum s oscillation and to test te small anle approximation. We used some variations in te period measurements to increase our precision and furter explore te dynamics of te pendulum motion. Submitted February 7, 005 Daniel Cooper Carolyne Pickler Mein Fon William Paul
I. INRODUCION Kater s pendulum was desined in 87 by Captain enry Kater, an Enlis pysicist, to be used to determine te acceleration of free fall (ravity witout knowlede of te radius of yration. It consists of a riid metal bar wit two knife edes at eiter end. Between te knife edes are two masses tat can be adjusted until te period of te pendulum, wen un and allowed to swin freely under te influence of ravity, will be te same for eiter knife ede. is new pendulum proved to ave a distinct advantae over te simple pendulum for accurately measurin te value of. e measurement of by tis metod is one of te most precise measurements in classical pysics. e oal of our experiment is to obtain to an accuracy of 0.% wit respect to te accepted value for te underraduate pysics lab. II. EORY A simple pendulum can also be used to measure ravity. owever, tere are a few problems wit te simple pendulum tat affect te accuracy of te measurement. ese issues include te stretcin of te strin, te inomoenous nature of te masses, air buoyancy and te nature of te support, wic can eiter be simple, but prone to dampin, or solid, but more difficult to analytically describe. Kater s pendulum, a compound pendulum (see fi., overcomes tese difficulties by enablin te comparison of two very precise measurements, te periods of oscillation on bot ends of te pendulum. e error due to te amount of air bein
displaced by te pendulum is corrected by usin two lare and two small masses of different densities, one of wood and one of metal, at eac end of te pendulum. ese lare and small masses at bot ends ave identical pysical dimensions but different densities. Wen arraned symmetrically around te knife edes, te effect of air sould be eliminated. Fiure. Diaram of Kater's pendulum. (from reference 3 Given a compound pendulum wit pivot axis O and centre of ravity G, its moment of inertia I about an axis trou G parallel to O is I Mk ( were M is te mass of te pendulum and its radius of yration. By te parallel-axis teorem, te moment of inertia I t about E is I t I M ( e period of oscillation about pivot E is iven by
is can be simplified to 4 ( I M (3 M 4 ( k (4 us, for Kater s pendulum, periods and are iven by Combinin te two equations and factorin out 4 ²/ results in: 4 ( k (5 4 ( k (6 4 ( ( (7 If and are exactly equal, equation (7 simplifies as 4 (8 III. APPARAUS AND PROCEDURE able. Symbols and definitions wit units were applicable. S Knife Ede S Knife Ede Period wit S as pivot (s Period wit S as pivot (s M Slidin Mass M Slidin Mass G Center of Gravity of pendulum Distance from S to G (m Distance from S to G (m Defined as (m x Distance of M and M at wic (m 3
e apparatus tat we used is accurately described in te Pysics 57/58 Lab Notes, 004, includin a diaram of te modified Kater s pendulum (fiure 5, pae 44. e addition of a meter stick eld in place by beaker stands and clamps, and kept level (see fi. allows us to measure te anle of te oscillation usin te trionometry of a rit trianle. e procedure is identical to tat in te Lab Notes for te initial calibration of te pendulum (usin setup 3 on te ektronix DS 0 oscilloscope. Our calibration procedure, performed before any period measurements, is as follows: te slidin masses M and M are set at a specific distance x (sown in fiure 5 of Lab Notes usin a diital caliper. en te pendulum is un, and te position of te upper knife ede is verified (te position itself is not important, as lon as it is on te quartz block on te support, and consistent between trials. is knife ede as a tendency to slide wen te pendulum is swun, so it must be frequently verified. In te rest position of te pendulum, te meter stick can be rouly centered. Set te pendulum in motion and usin te meter stick, verify tat te amplitude of te swin is equal to eiter side, by ceckin one side ten te oter. Once te meter stick is centered, we take readins of te period as described in te lab manual, wit a few extra considerations. We release te pendulum at a position on te meter stick approximately ¾ to cm beyond an 8 cm amplitude (so 8 ¾ to 9 cm. is allows te pendulum s swin to stabilize, and any visible vibrations perpendicular to its orizontal swin trajectory will damp out in tis time. 4
Fiure. Scematic diaram of experimental setup. We use a time base of M ms on te ektronix DS 0 oscilloscope, wic is smaller tan tat indicated in te lab manual, resultin in a period readin of reater precision. Measurements of te time taken for, 4 and 8 periods were made by manually startin te oscilloscope as wen te amplitude of te period reaced 8 cm, as read on te meter stick. e period is ten recorded to 5 decimal places by settin cursor to exactly 8.0000 seconds and usin te delta function on te oscilloscope to read te location of te cursor wic is placed on te rise of te step function. 5
e measurement of te knife to knife distance is done as per te lab manual, but for te sake of accuracy, te diital caliper is used to find te distance from te knife ede to te balance point. en measurements are taken from te first knife ede to te balance point. ese measurements are subtracted from te total knife to knife distance to find a value for te distance between te second knife ede and te balance point. is procedure is as suc because of te inadequate lent of te caliper. IV. DAA AND ANALYSIS An initial estimate for te correct distance of te calibration masses was found from Mein Fon s lab report usin pendulum number 4. Wit te knowlede tat te intersection was around x 6 cm, 5 readins were taken for bot and in 0. cm increments from 5.3 cm to 6.3cm. Since we are dealin wit a small part of te and curves, a linear fit was used (wit error as weit to find a ood estimate of te intersection (see fi.3. See table for period measurements, and tables 3 and 4 for, and measurements. 6
Fiure 3. Calibration plot of versus x. e data points for and are fit wit linear least squares fit, and te intersection calculated from te lines. Equation of te fit for data: y -0.0093x 8.0549 Equation of te fit for data: y -0.0045x 8.059 e calculated value of x for wic is ten: x 5.73 cm 7
able. Complete period measurement data and results usin x 5.73 cm. rial No. Data for (4 periods (in s Data for (4 periods (in s (wit 5 anle compensation (in s (wit 5 anle compensation (in s 8.0004 8.003.999300336.9993703 8.00 8.00.9999034.9993403 3 8.00084 8.00.9995036.9993403 4 8.0009 8.00.9997035.9993403 5 8.00 8.00.99934037.9993033 6 8.0009 8.003.9997035.9993703 7 8.004 8.006.9993503.9993303 8 8.0008 8.006.9993033.9993303 9 8.00 8.00.9993036.9993403 0 8.003 8.008.99937030.9993603 8.004 8.00.9993503.9993403 8.006 8.003.9993303.9993703 3 8.00 8.006.9993036.9993303 4 8.004 8.004.9993503.9993503 5 8.008 8.0008.999360307.9993033 6 8.0008 8.006.9993033.9993303 7 8.008 8.0036.999360307.9993803 8 8.0048 8.0036.9994083.9993803 9 8.0048 8.00.9994083.9993403 0 8.004 8.008.9993503.9993603 8.0044 8.0008.99940088.9993033 8.0036 8.004.99938097.9993503 3 8.0036 8.003.99938097.9993703 4 8.0044 8.008.99940088.9993603 5 8.0036 8.00.99938097.9993403 6 8.00 8.00.99934037.9993403 7 8.003 8.004.99937030.9993503 8 8.0008 8.003.9993033.9993703 9 8.006 8.006.9993303.9993303 30 8.008 8.004.999360307.9993503.99934649.99934698 0.000048074 0.0000969 0.0000076396 0.0000034998 Assumin, we find:.99934484 0.00003389 0.00000590059 8
able 3. Data from measurement of usin lab setup for suc purpose. Diital caliper readin (in mm Knife to Knife distance (in m 45.8 0.9998 45.7 0.9997 45.5 0.9995 45. 0.99 45.4 0.9994 45.8 0.9998 45. 0.99 45.8 0.9998 45.8 0.9998 45.8 0.9998 0.99976 0.00009 0.000006 able 4. Data from measurement of distances from knife edes to G. Diital caliper readin (in m (in m 9.36 0.936 0.7006 9.3 0.93 0.70066 9.34 0.934 0.7006 9. 0.9 0.7009 9.5 0.95 0.70043 9.08 0.908 0.7009 9.7 0.97 0.70083 90.97 0.9097 0.700 9.0 0.90 0.70097 90.99 0.9099 0.70099 0.984 0.70079 0.00085 0.00098 0.0000586 0.000067 Usin equation (7: 9.7968 ± 0.000 m/s Usin equation (8: 9.79683 ± 0.00006 m/s 9
e error in all measured quantities (,,,, was found usin te standard deviation of te values obtained. Error calculations can be found in Appendix C. V. DISCUSSION Wit 9.7968 ± 0.000 m/s as our experimental, and 9.80643 m/s as te accepted value, we ave acieved a percent difference of 0.098%. Wit a reater amount of time, an even more accurate value could be obtained wit an extended investiation into te dampin of te period measurements. e accuracy of our results improved wen considerin te small anle approximation of te oscillations, as described in te teory section. Wile measurin te period, we observed tat tere was sinificant dampin as sown in fiure 4. In experiments usin Kater s pendulum, it is usual to measure te time taken for a lare number of oscillations. owever, due to te observed dampin, tis cannot be done wit our pendulum. us, it was essential to take readins at consistent amplitude, assumin tat durin te 4 period readin, te period decay is neliible. Period measurements deacreasin in time 8.0050 8.0045 ime of 4 periods (s 8.0040 8.0035 8.0030 8.005 8.000 0 4 6 8 ime (in min Fiure 4. Quantitative plot of damped oscillations as a function of time. 0
Measurin consistently at amplitude correspondin to a 5 deviation also allowed for adjustment of te period wit respect to te small anle approximation. Usin equation 9 wic describes te variation of te period wit respect to te amplitude of oscillation, te period was corrected. corrected θ 6 (9 Variations in te experimental environment were also considered. e temperature of te lab, and te noise level were recorded to see if tese factors affected results as suested by Prof. Warburton. Variations of 0. C were recorded and assumed to be neliible. A micropone was also attaced to te steel beam supportin te pendulum to quantify te vibrations in te support due to traffic in te alls above te lab. ese vibrations may ave traveled into te pendulum via te support and affected its swin. owever, no sinificant vibrations were recorded. VI. CONCLUSION e objective of tis experiment was to measure te value of te acceleration due to ravity to an accuracy of 0.%. e value obtained was 9.7968 ± 0.000 m/s. is as a per cent difference of 0.098% wen compared to te teoretical value for in te underraduate pysics lab of 9.80643 m/s. is deree of accuracy was accomplised by usin a meter stick to determine te anle at wic te period was measured, usin a time base of ms on te oscilloscope, manually startin te oscilloscope, and measurin te time for 4 periods.
Furter study of Kater s pendulum mit include an extensive look at te dampin of oscillations wic became apparent in our experiment. A study of te dampin could enable readins to be taken over a loner period of time by makin compensations for te dampin effects. A dense array of potosensors (peraps several undred per inc could also be used to track te pendulum s decreasin amplitude as a function of time. is experiment successfully demonstrated tat te use of Kater s pendulum to measure is one of te most accurate measurements in pysics. ACKNOWLEDGEMENS Mrs. Edit Enelber Mr. Mario Della Neve Prof. Andreas Warburton REFERENCES. PYS 57/58 Lab Notes. Kater,. An Account of Experiments for Determinin te Variation in te Lent of te Pendulum Vibratin Seconds, at te Principal Stations of te rionometrical Survey of Great Britain. Pilosopical ransactions. 89; 09: 337-508. 3. Experiment 0. ttp://polaris.pys.ualberta.ca/users/austen/pys9x/manual/0katerpendulum99.pdf
Appendix A Our Experimental Setup A potorap of our experimental setup. 3
Appendix B Complete Data ables Periods 8 Periods x 57.3 mm x 57.6 mm rial rial 4.00088 4.00084 6.00684 6.0067 4.00056 4.0000 6.00684 6.0069 3 4.00064 4.00084 3 6.007 6.00684 4 4.00076 4.00084 4 6.00700 6.0067 5 4.0007 -- 5 6.00684 6.00696 Averae 4.0007 4.000880 Averae 6.00693 6.00683 Period.000356.000440 Period.000866.000854 8 Periods 8 Periods x 57.8 mm x 57.8 mm rial rial 6.00744 6.00600 6.00680 6.0064 6.0078 6.00656 6.00660 6.00588 3 6.00700 6.00664 3 6.00640 6.0060 4 6.0069 6.00696 4 6.00676 6.0066 5 6.00676 6.00676 5 -- 6.0065 Averae 6.00708 6.00658 Averae 6.00664 6.0060 Period.000885.00083 Period.000830.000775 8 Periods 8 Periods x 57.3 mm x 57.6 mm rial rial 6.00568 6.00680 6.00680 6.00568 6.00568 6.00680 6.0077 6.00580 3 6.00568 6.00640 3 6.0076 6.00548 4 6.00584 6.00644 4 6.0067 6.00508 5 6.0059 6.00660 5 6.00680 6.006 6 6.00584 6.00664 6 6.00688 6.00636 7 6.00604 6.00660 7 6.00704 6.006 8 6.00588 6.00644 8 6.0069 6.00664 9 6.00576 6.00648 9 6.00664 6.00608 0 6.00564 6.0063 0 6.00744 6.0065 6.00588 6.00636 Averae 6.0070 6.00599 6.00560 6.00656 Period.000877.000749 3 6.0057 6.00644 4 6.00600 6.0064 5 6.00576 6.0068 4
6 6.00560 6.0068 7 6.00588 6.0066 8 6.00544 6.00648 9 6.00560 6.00660 0 6.0060 6.00660 6.00584 6.00668 6.00576 6.00696 3 6.00588 6.00676 4 6.00644 6.00696 5 6.00648 6.0064 6 6.00604 6.0068 7 6.00600 6.00596 8 6.00588 6.00604 9 6.00596 6.006 Averae 6.00586 6.00646 Period.00073.000808 Appendix C Sample Calculations Calculatin from data for 4 periods & applyin small anle approximation for 5 (/ 0.00048 x 4.00048 8.0004 4.00048.999300336 s Calculatin a mean 30 i 30 i 30 (.0006....0003.00030 s Calculatin knife to knife distance, (in m, from caliper readin, x (in mm ( x 946.8 ( 45.8 946.8 000 000 0.9998 m Calculatin from and readins 0.99980-0.936 0.7006 m 5
Calculatin for (usin first equation 9.796776987 m/s 0.70079 (0.984 (.99934698 (.99934649 (0.99976 (.99934698 (.99934649 4 ( ( 4 Calculatin for (usin second equation ( 4 9.79688 0.99976.999345 4 ( 4 m/s Error Calculations: e error in wen usin equation (7 is iven by: First, te partial derivatives are calculated: 6.9666556 0.70079 (0.984.99934698.99934649 (0.99976.99934698.99934649 0.70079 0.984.99934649 0.99976.99934649 4* ( 4* 6
-6.7665775 0.70079 (0.984.99934698.99934649 (0.99976.99934698.99934649 0.70079 0.984.99934698 0.99976.99934698 4* ( 4* ( ( 9.875970376 (0.99976 0.70079 (0.984.99934698.99934649 (0.99976.99934698.99934649.99934698.99934649 * ( * ( ( ( ( -0.0005479 0.70079 (0.984 0.70079 (0.984.99934698.99934649 (0.99976.99934698.99934649.99934698.99934649 8* ( ( 8* 7
( ( ( ( 0.0005479 0.70079 (0.984 0.70079 (0.984.99934698.99934649 (0.99976.99934698.99934649.99934698.99934649 8* ( ( 8* Insertin tese into te error equation, we et ( ( ( ( ( (0.00006748 0.0005479 (0.000058674 0.00055 - (0.00000600 9.8759704 (0.00000350-6.766577 (0.00000763 6.9666556 0.0000989m/s e error in wen usin equation (8 is iven by: 3 3 0.0000607 m/s (0.0000086593 (.999345 4 (0.00000590058 (.999345 (0.99976 8 4 8 8