European J of Physics Education Volume 7 Issue 3 309-70 Dean et al. Conical Pendulum inearization Analyses Kevin Dean Jyothi Mathew Physics Department he Petroleum Institute Abu Dhabi, PO Box 533 United Arab Emirates kdean@pi.ac.ae jmathew@pi.ac.ae (Received: 08.0.07, Accepted: 3.0.07) Abstract A theoretical analysis is presented, showin the derivations of seven different linearization equations for the conical pendulum period, as a function of radial and anular parameters. Experimental data obtained over a lare rane of fixed conical pendulum lenths (0.35 m.30 m) are plotted with the theoretical lines and demonstrate excellent areement. wo of the seven derived linearization equations were considered to be especially useful in terms of student understandin and relative mathematical simplicity. hese linear analysis methods consistently ave an areement of approximately.5% between the theoretical and experimental values for, the acceleration due to ravity. An equation is derived theoretically (from two different startin equations), showin that the conical pendulum lenth appropriate for a second pendulum can only occur within a defined limit: [ / ( )]. It is therefore possible to calculate the appropriate circular radius R or apex anle (0 / ) for any lenth in the calculated limit, so that the conical pendulum will have a one second period. A eneral equation is also derived for the period, for periods other than one second. Keywords: Conical pendulum, theoretical linearization, experimental results INRODUCION he physics of an oscillatin pendulum (simple and conical) can often be considered as somewhat challenin for pre-university students (Czudková and Musilová, 000), but should not pose a sinificant problem for typical underraduates. Conical pendulum lenths investiated experimentally (and published) usually rane from approximately 0 cm (onaonkar and Khadse, 0) up to reater than 3 m (Mazza, Metcalf, Cinson and ynch, 007). he typical conical pendulum is a comparatively standard experiment that is frequently included in underraduate laboratory syllabi. In some cases, the aim of the experiment is for underraduate students to have the opportunity to desin and then perform an experiment (either individually, or in a small team of up to three students). It is usual for students to record 38
European J of Physics Education Volume 7 Issue 3 309-70 Dean et al. appropriate data in tabular form usin suitable readily available software (such as Excel). he conical pendulum is also used to teach enery and anular momentum conservation (Bambill, Benoto and Garda, 00) and potential enery (Dupré and Janssen, 999) as well as the rotational dynamic interactions of classical mechanical systems (acunza, 05). Althouh the present work restricts the conical motion to bein in a horizontal plane, it is possible to extend the mathematical analysis to three dimensions (Barenboim and Oteo, 03). In the case of horizontal planar motion, this is frequently observed to be elliptical rather than truly circular, which has been studied in detail to determine the nature of orbital precession (Deakin, 0). Strin tension measured as a function of the period (or rotational speed) has been documented, and is easily understood by pre-university (or first year university) students (Moses and Adolphi, 998). Based upon several years of experience acquired at he Petroleum Institute, the standard conical pendulum experiment is relatively straihtforward for students to understand, then set up and obtain acceptable data. he principal aim of the conical pendulum as a desin experiment is for students to investiate how the conical pendulum period could depend on the radius R of the conical pendulum path, or on a measured anle as the physical variable. he experimental data analysis can be used to determine a value for the acceleration due to ravity, (if the lenth of the conical pendulum is known), or the lenth can be calculated assumin a value for the acceleration due to ravity. Fiure. Conical Pendulum Schematic he pendulum mass (frequently referred to as a bob ) performs uniform horizontal circular motion over a rane of circular radii such that 0 < R < (where represents the lenth of the 39
European J of Physics Education Volume 7 Issue 3 309-70 Dean et al. pendulum strin). It should be noted that the strin is considered to be both mass-less and inextensible as well as bein effectively devoid of air resistance (as is the pendulum mass itself). he present purpose is to extend the effectiveness of this experiment by enablin students to produce suitable linear data, usin one of the linearization methods that are derived in the subsequent theoretical section. Excel can then be used to prepare a straiht-line chart, includin a linear reression analysis, from which the slope (and) or the intercept can be used to obtain an unknown physical parameter ( or ). For the analysis that follows, it is not necessary to restrict the orbital period to approximate isochronous behavior, which would be the standard approach for analyzin the planar oscillations of a simple pendulum. he analysis presented below applies to the full rane of radii (subject to the physical restrictions that R and ( / ), even thouh mathematically, the analysis allows for these unphysical limits). he physical implications of R = and = ( / ) will be considered at the appropriate point(s) later in the analysis. HEOREICA ANAYSIS he seven theoretical linearization analyses presented in this section are all based on the conical pendulum fiure that is shown in the section above, which defines the appropriate physical parameters that will be used for each of the analyses: DERIVAION OF HE CONICA PENDUUM PERIOD From the fiure, the period can be readily derived for a conical pendulum of lenth and with apex anle as shown above, where the subscript c indicates the centripetal acceleration and centripetal force actin on the conical pendulum mass: F c m tan m a c m v R v R tan R R A straihtforward re-arranement of the above ives: R R R v R v In the form shown by the equation derived above for the period, the conical pendulum period can be seen to be a definite function of the circular radius R althouh the functional relationship is neither straihtforward nor linear. his functional relationship is illustrated in Chart (Appendix ), for nine different theoretical values of alon with the appropriate experimental data. he lenths of the conical pendulum for the experiments were in the rane 0.35 m.30 m (the theoretical and experimental data for specific lenths are indicated in the chart leend). Data for = 0.99 m is included and referenced (onaonkar and Khadse, 0). 0
European J of Physics Education Volume 7 Issue 3 309-70 Dean et al. For details of all the charts and the chart leend (which applies for all charts) refer to the Appendices. Inspection of Chart (Appendix ) and the equation for the period, shows that the horizontal axis intercept (for = 0) occurs for when R =, whereas the vertical axis intercept occurs when the period is the same as for a simple pendulum of the same lenth (with motion confined to the x y or z y plane for example, see Fiure ). It is possible (and considered to be analytically desirable) to re-express the functional relationship between the conical pendulum period and either the circular radius R, or the apex anle (or alternatively the anle measured with respect to the horizontal) to achieve linearization. Several straihtforward methods of accomplishin the mathematical linearization are presented below and the most appropriate method(s) for ease of student use is specified in a later part of this analysis. Each of the charts that are shown contains both the theoretical calculations and experimental data values. inearization Method he startin point for this analysis is the period equation that was derived above, which is then squared as shown. R R Chart (Appendix ) demonstrates this very straihtforward method of linearization that directly produces a straiht line passin throuh the oriin of coordinates, which then enables the acceleration due to ravity to be easily calculated from the slope. he mathematical analysis is uncomplicated and should be readily understood by typical students. By usin a standard linear reression with the condition that the straiht line should pass throuh the oriin, this linearization method should produce ood results. inearization Method he period equation that was derived earlier can be developed further to ive, 6 6 R c m R Althouh this analysis ives rise to an equation that involves the fourth power of the period as a function of the conical radius squared, the equation is basically nothin more than a straiht line, with a positive intercept that depends on the lenth of the conical pendulum, and a neative slope that is independent of. his can be clearly seen in the Chart 3 (Appendix ), which shows the theoretical parallel straiht lines and the experimental data that are displayed in Chart (Appendix ). It can be noted that as a student exercise, it is possible (in principle) to calculate a value for the lenth of a conical pendulum if this is not known in advance, when a value for the
European J of Physics Education Volume 7 Issue 3 309-70 Dean et al. acceleration due to ravity is provided. his is readily determined by calculatin the numerical ratio of the vertical axis intercept divided by the slope, which ives and therefore. herefore, various values of can be used by different student roups. By referrin to the linearization equation, it can be observed that the point where the line intercepts the horizontal axis corresponds to the situation where R = so that could be obtained directly. Alternatively, if the pendulum lenth is iven, then both the slope and intercept can be used to calculate separate values for the acceleration due to ravity. inearization Method 3 he first analysis can be extended and re-arraned to make use of the fact that the straiht line intercept c = (m ) in the followin way: R R 6 m In this form, the linearization ives rise to a straiht line with positive slope m passin throuh the oriin of coordinates (which corresponds to the mathematical condition that = 0 when R = ). Althouh in practice this situation is clearly physically unattainable since it would require that the pendulum speed v =, it is nevertheless mathematically precise and could be pointed out to students. Inspection of the above linearization equation shows that the sinle straiht line passes throuh the coordinate oriin and has a constant positive slope that is independent of the pendulum lenth. Consequently, therefore, all experimental data points (for all lenths of conical pendulum) will fit on the same straiht line. It is of interest to note that experimentally, havin a lare value for the conical pendulum lenth would be hihly beneficial in terms of obtainin ood results compared to usin a smaller lenth pendulum. his linearization is shown in Chart (Appendix ). By knowin the conical pendulum lenth and usin the fact that theoretically the straiht line must pass throuh the coordinate oriin, it is only necessary for students to use suitable linear reression analysis, with the requirement that the trend-line line passes throuh the oriin. It is then a simple matter for students to calculate a value for (the local acceleration due to ravity) from the slope. inearization Method Startin with the linearization equation derived above, further analysis ives: 6 6 R ln ln ln R his is calculated and shown in Chart 5 (Appendix ).
European J of Physics Education Volume 7 Issue 3 309-70 Dean et al. All experimental data points must lie on a sinle straiht line of constant positive slope havin a value of exactly m = ¼ and with a vertical axis intercept that is independent of the conical pendulum lenth. he intercept of the straiht line on the vertical axis can be used to provide a value for the local acceleration due to ravity. It can be seen that where the straiht line crosses the horizontal axis, requires ln() = 0 (which therefore corresponds to the conical pendulum havin a period = second). he physical consequences of this are outlined below. ln 6 ln ln R 0 R 6 sin sin 6 cos 6 cos cos 0 Inspection of the final equations in the above analysis shows that the conical pendulum lenth that is appropriate for a second s pendulum, falls within a precisely defined rane. his rane limit for will re-occur at the end of a subsequent derivation in a later section, and will be considered in context with the simple pendulum. his linearization analysis is considered to be somewhat complicated for use by most students takin a first level Physics course. As a result, the application of loarithmic functions by students to experimental data analysis and interpretation is avoided whenever possible. his is due to an intrinsic lack of in-depth understandin of the loarithmic function and how these should be treated. inearization Methods 5 & 6 By makin reference to the conical pendulum fiure, it can be noted that the followin two trionometric functions can be obtained: sin R R sin and cos R R cos It now becomes a simple matter to substitute for R to ive the followin: 3
European J of Physics Education Volume 7 Issue 3 309-70 Dean et al. R sin sin 6 6 sin his is once aain in the same format as the first linearization analysis where the intercept c equals the slope m and ives rise to the followin two equations: 6 6 sin cos he first linearization equation is shown in Chart 6 (Appendix ), which aain includes the theoretical straiht lines and the experimental data sets that were shown in Chart (Appendix ) earlier: It is interestin to make note of the fact that the horizontal axis intercept occurs at exactly is a direct result of the fact that = 0 when = / rad (or 90). his mathematical result can be thouht of as definin an anchor point on the horizontal axis, for the straiht-line data chart. An alternative substitution for the conical pendulum radius R (usin the second equation for the cosine function in place of the sine) will ive rise to the linearization shown in Chart 7 (Appendix ). For this representation, the oriin of coordinates is obtained as a direct result of the fact that = 0 when = ( / ) rad (or 90). As a result of the straiht line passin throuh the coordinate oriin, this analysis is slihtly more preferable for student use, when compared to the linearization of Chart 6 (Appendix ). When experimental data is plotted as a chart, it can be used to obtain a value for either if is iven, or if is provided. inearization Method 7 It is possible to extend the second version of the two previous linearization equations by takin natural loarithms. 6 cos ln 6 ln ln cos ln 6 ln ln cos ln ln cos 6 Usin the same data sets as for earlier analyses produces Chart 8 (Appendix ).
European J of Physics Education Volume 7 Issue 3 309-70 Dean et al. his linearization ives rise to a series of parallel straiht lines with slope m = ½ and with a vertical axis intercept that will provide a value for either if is iven, or if is known. he analysis below uses an abbreviated equation form. ln ln a ln cos 0 ln a ln cos 0 ln a ln cos 0 ln cos ln a ln a cos a 6 cos Inspection of the final equation above shows (aain) that the conical pendulum lenth that is appropriate for a second s pendulum falls within a well-defined rane. he rane of acceptable values for the cosine function readily ives the simple analysis below. 0 It is therefore possible to determine the appropriate apex anle for any lenth within the calculated rane, such that the conical pendulum will have a one second period. It is of interest to note that in terms of the linearization of a simple pendulum (see later section for full details), the above limit equation can be expressed below, where the simple pendulum slope is iven by m SP (see later). m SP Usin the local value for acceleration due to ravity = 9.79 ms - ( decimal places) ives, > 0.798 m (.798 cm). he standard value for the acceleration due to ravity = 9.8 ms - ( decimal places) ives, > 0.89 m (.89 cm). his is the same as the small-anle approximation period for the simple pendulum underoin simple harmonic motion, as shown below settin = : 5
European J of Physics Education Volume 7 Issue 3 309-70 Dean et al. Reference to Chart (Appendix ) confirms that a horizontal line at = intercepts all the lines for pendulum lenths satisfyin the above analysis. If periods other than one second are to be specifically investiated, then a simple re-arranement of the initial period equation will enable such a study, by usin: R R 6 Additional details reardin the functional dependence of the conical pendulum lenth on rotational anular speed and the derivation of the critical lenth can be obtained from published previously published work (Klosteraard, 976). INEARIZAION OF A SIMPE AND CONICA PENDUUM COMPARED Squarin both sides of the familiar SHM pendulum equation readily ives the familiar straiht line equation, with the square of the pendulum period bein directly proportional to the pendulum lenth. For the purpose of comparison with the present work, the nearest equivalent linearization for the conical pendulum is also shown. In the linearization equations, the subscripts SP and CP refer to the simple pendulum and conical pendulum respectively. It can be observed that these two pendulums exhibit similar physical behavior, which is inextricably linked throuh the mathematical term ( /) that appears as the slope msp of the straiht line for the simple pendulum, and appears squared as the slope mcp of several of the conical pendulum linearization equations. SP Slope m SP SP m SP CP 6 Slope m CP R m R SP Due to the absence of a vertical axis intercept, both equations provide a simple and direct technique of displayin experimental data that yields a straiht line passin throuh the coordinate oriin. It can be seen that the slope for both pendula is the same, bein the square of the slope for a simple pendulum. 6
European J of Physics Education Volume 7 Issue 3 309-70 Dean et al. In terms of experimental usefulness, a sinle calculation of the slope of the best straiht line passin throuh the oriin will ive the local value for the acceleration due to ravity. It is noted that the conical pendulum lenth remains constant, but not for the simple pendulum. CONICA PENDUUM EXPERIMEN AND DAA When a conical pendulum experiment is performed, there are several parameters that can be quantified and measured. It is usual to have some form of timin measurement to determine the orbital period and some way of measurin any one (or several) of the followin: the orbital radius R, the cone apex anle or equivalently, the strin anle measured with respect to the horizontal. able. Experimental Data (*onaonkar and Khadse, 0) (m) =.30 (m) =.003 (m) =.875 R (m) Apex (de) (s) R (m) Apex (de) (s) R (m) Apex (de) (s) 0.00 5.3878.90 0.00 5.7305.85 0.00 6.3.78 0.50 6.70.930 0.300 8.639.837 0.300 9.069.736 0.300 8.0967.95 0.00.59.89 0.00.378.730 0.00 0.80.905 0.500.55.83 0.500 5.660.76 0.500 3.576.890 0.600 6.36.875 (m) =.73 (m) =.85 (m) =.5 R (m) Apex (de) (s) R (m) Apex (de) (s) R (m) Apex (de) (s) 0.00 6.6657.6 0.00 7.70.0 0.00 9.9.6 0.300 0.07.68 0.300.655. 0.300 3.8639.35 0.00 3.39.67 0.00 5.66.3 0.00 8.630.03 0.500 6.869.603 0.500 9.6760.393 0.500 3.538.76 (m) =.09 (m) = 0.85 (m) = 0.35 R (m) Apex (de) (s) R (m) Apex (de) (s) R (m) Apex (de) (s) 0.00.075.08 0.00.097.806 0.050 6.6003.37 0.300 6.9505.00 0.300.337.798 0.00 3.903.3 0.00.8753.986 0.00 9.005.73 0.00 7.373.86 0.500 9.070.933 *EJPE Data (m) = 0.99 R (m) Apex (de) (s) 0.068 9.9990 0.850 0.97 36.9658 0.775 0.735 60.667 0.65 0.9 75.505 0.60 0.99 78.03 0.00 7
European J of Physics Education Volume 7 Issue 3 309-70 Dean et al. he experimental set-up required two slihtly different methods of riid support dependin on the pendulum lenths to be investiated. Initially, a liht-weiht inextensible strin of lenth.500 m is held between a pair of parallel-sided wooden blocks, which are tihtened toether usin clamps. A concentrically scaled pendulum orbit sheet is placed directly below the clamp. hen a spherical steel bob is then connected so that it is just above the center of the orbit sheet when it is at a vertical stationary equilibrium. he lenth of the pendulum is measured from the bottom ede of the clampin blocks to the center of the bob. A force is applied on the bob at a desired radius R directly alon its tanent. For a sinle measurement at radius R, a diital stopwatch is used to record the time for 5 rotational periods (referred to as orbits in the experiment). o ensure consistent accurate measurements are obtained, this procedure is repeated at least 5 times. he radius of the conical pendulum orbits is then chaned in order to perform the experiment as a function of radius. his procedure is then repeated for different radii and lenths. When sufficient ood quality experimental data is obtained, the periods and other parameters are calculated and the experimental data is plotted on the same chart as the theoretical lines. he two methods of pendulum support (for both a lon and short conical pendulum) are shown photoraphically below the data table. he table below shows typical experimental data for conical pendulum lenths in the rane 0.35 m.30 m that were obtained by the authors. Data previously obtained by other authors (onaonkar and Khadse, 0) and published is also indicated in the table below ((EJPE 0.99 m). For pendulum strin lenths, more than.500 m, the strin support clamp is attached directly to a riidly fixed ceilin rod, which maintains a constant (but adjustable) conical pendulum lenth. Fiure. Short pendulum Fiure 3. on pendulum 8
European J of Physics Education Volume 7 Issue 3 309-70 Dean et al. DISCUSSION OF RESUS AND CONCUSIONS Seven theoretical methods of linearization have been derived for the conical pendulum period and all have been plotted (usin Excel) on appropriate charts for the lenth rane 0.35 m.30 m (with additional calculations for = 0.99 m). he charts demonstrate excellent areement between the theoretical analysis and experimental data over the lenth rane considered. heoretical calculations used the local value of the ravitational acceleration = 9.79 m s -. Althouh all the seven linearization equations are equally capable of providin very closely similar values for the local ravitational acceleration, not all the derived linearization equations are equally simple to apply. As a result, the equations are not necessarily all suitable for use by students. Inspection of the seven theoretically derived linearization equations as well as their appropriate charts, suests that those charts havin a straiht line passin throuh the coordinate oriin would be most appropriate for student use. he table below summarizes the experimental results obtained from the three linearization charts (Charts, and 7) that were considered to be most suitable for underraduate student experimental classes (Appendix ). In the table of results shown below, the followin is defined: = (ocal value) (Experimental value). able : Summary of Results (*onaonkar and Khadse, 0) heory (m) Chart Chart Chart 7 & Expt (m) Slope m Calculated Slope m Calculated Slope m Calculated.30.05 9.76 6. 9.75 7.6 9.75.003.08 9.666 6.68 9.666 66.9 9.666.875.06 9.7 6.9 9.7 57.97 9.7.73.09 9.66 6.69 9.665 9.5 9.665.85.09 9.66 6.69 9.665 36.80 9.665.5.0 9.68 6.8 9.69 6.35 9.69.09. 9.60 6.88 9.608 7.88 9.608 0.85. 9.57 7.5 9.53.67 9.53 0.35.3 9.558 7.00 9.575 3. 9.575 : Averae = 9.6 Averae = 9.65 Averae = 9.65 : = 0.8 = 0.5 = 0.5 : =.5% =.5% =.5% *0.99 3.87 0.3.77 0.7 0.59 0.7 9
European J of Physics Education Volume 7 Issue 3 309-70 Dean et al. In each of the above three charts, the calculated straiht lines were constrained to pass throuh the oriin of coordinates (which theory shows to be a necessary physical condition). It can be seen that the theoretical derivations are completely self-consistent and ive rise to uniform values for the ravitational acceleration, usin the experimental data that is presented. his analysis method consistently provided an areement of approximately.5% between theoretical and experimental values for, the acceleration due to ravity (the local value bein = 9.79 m/s ). Previously published results (Mazza, Metcalf, Cinson and ynch, 007) for conical pendulum experiments havin a lenth rane that extended up to approximately 3 m (actual rane:.9 m 3. m) were reported to be usually better than %. In conclusion, it can therefore be stated that the theoretical derivations have been thorouhly tested and confirmed as correct for the rane of conical pendulum lenths investiated, includin previously published results (onaonkar and Khadse, 0) for small and (Mazza, Metcalf, Cinson and ynch, 007) for lare. ACKNOWEDGEMEN he authors would like to express their ratitude to the editor EJPE for rantin permission to include previously published experimental data from onaonkar and Khadse (0), for a conical pendulum of lenth 0.99 m, which is referred to in the text as (EJPE 0.99 m). APPENDIX. CHARS FOR HEOREICA & EXPERIMENA RESUS Chart Equation Chart Equation Chart 3 Equation 3 Chart Equation 50
European J of Physics Education Volume 7 Issue 3 309-70 Dean et al. Chart 5 Equation 5 Chart 6 Equation 6 Chart 7 Equation 7 Chart 8 Equation 8 APPENDIX CHAR EGEND FOR HEOREICA & EXPERIMENA RESUS heoretical and experimental results are indicated from larest conical pendulum lenth ( =.30 m) to the shortest ( = 0.35 m). Calculations for = 0.99 m are also included, with experimental data that was obtained by other authors (onaonkar and Khadse, 0) and has been previously published (referred to as EJPE 0.99 m). Fiure - Chart eend 5
European J of Physics Education Volume 7 Issue 3 309-70 Dean et al. APPENDIX 3 EQUAIONS USED FOR HEOREICA CACUAIONS R Equation Chart R Equation Chart 6 6 Equation 3 Chart 3 R 6 R Equation Chart ln 6 ln ln R Equation 5 Chart 5 6 cos Equation 7 Chart 7 6 6 sin Equation 6 Chart 6 6 ln ln ln cos Equation 8 Chart 8 REFERENCES Bambill, H.R., Benoto, M.R. and Garda, G.R., (00), European Journal of Physics, Vol 5, p3-35 Barenboim, G. and Oteo, J.A., (03), European Journal of Physics, Vol 3, p09-065 Czudková,. and Musilová, J., (000), Physics Education, Vol. 35, Number 6, p8-35 Deakin, M.A.B., (0), International Journal of Mathematical Education in Science and echnoloy, Vol., Issue 5, p75-75, 03 Dupré, A. and Janssen, P., (999), American Association of Physics eachers, Vol. 68, p70-7 Klosteraard, H., (976), American Association of Physics eachers, Vol., p68-69 acunza, J.C., (05), Journal of Applied Mathematics and Physics, Vol. 3, p86-98 Mazza, A.P., Metcalf, W.E., Cinson, A.D. and ynch, J.J., (007), Physics Education, Vol., Number, p6-67 Moses,. and Adolphi, N.., (998), American Association of Physics eachers, Vol. 36, p37-373 onaonkar, S.S. and Khadse, V.R. (0), European J of Physics Education, Vol., Issue, 0 5