Conical Pendulum: Part 2 A Detailed Theoretical and Computational Analysis of the Period, Tension and Centripetal Forces

Transcription

1 European J of Physics Education Volume 8 Issue Dean onical Pendulum: Part A Detailed heoretical and omputational Analysis of the Period, ension and entripetal orces Kevin Dean Physics Department, Khalifa University of Science and echnoloy, Petroleum Institute, PO Box 533, Abu Dhabi United Arab Emirates kdean@pi.ac.ae (Received: , Accepted: ) Abstract his paper represents a continuation of the theoretical and computational work from an earlier publication, with the present calculations usin exactly the same physical values for the lenths (0.435 m.130 m) for the conical pendulum, mass m k, and with the local value of the acceleration due to ravity ms -. Equations for the followin principal physical parameters were derived and calculated: period, anular frequency w, orbital radius R, apex anle f, tension force and centripetal force (additional functions were calculated when required). alculations were performed over a wide rane of values of the apex anle (0 f 85 ), correspondin to a calculated tension force rane of approximately (m 1 N) or alternatively (m 11 m) for the strin. A technique is demonstrated to determine an accurate value of an unknown pendulum mass, by usin a raphical analysis. Intercepts and asymptotic lines with respect to both the horizontal and vertical axes are described and fully explained. he main emphasis for this paper is to present hihly detailed raphical charts for the calculated theoretical functions and appropriate physical parameters. heoretical analysis is presented in comprehensive detail, showin full mathematical derivations and alternative equations when this approach is considered to be advantaeous for both understandin and computational presentation. Keywords: onical pendulum, theoretical analysis, tension force, centripetal force, period, anular frequency, hih precision, computational analysis. INRODUION All that follows is based on and makes reference to the fiure below, which was first used in the previous publication (refer to Dean & Mathew, 017). he fiure defines the basic conical pendulum parameters for the subsequent theoretical analysis and clearly illustrates the overall eometry of a typical underraduate experimental system and the vector nature of the tension and centripetal forces. Basic trionometric functions are used to express the adjacent side of the Pythaorean trianle in terms of the orbital radius R and lenth of the conical pendulum. he mathematical analysis contains all relevant sequential derivation steps and clearly shows how the final equation (that is enerally numbered in sequence) is obtained. When it is necessary to make specific reference to a previously derived equation, the relevant equation is often reproduced for convenience. 11

2 European J of Physics Education Volume 8 Issue Dean iure 1. onical Pendulum Schematic he in-depth physics of a conical pendulum can frequently appear to be mathematically demandin for freshman university students (zudková & Musilová, 000). Standard conical pendulum lenths that have been investiated experimentally (and subsequently published) usually rane from approximately 0.0 m (onaonkar & Khadse, 011) up to approximately 3.0 m (Mazza et al, 007). In the typical rane of underraduate laboratory experiments, the conical pendulum is frequently included alon with the relevant theory. he conical pendulum can also be used to explain concepts such as: enery and anular momentum conservation (Bambill, Benoto & Garda, 004), mechanical potential enery (Dupré & Janssen, 1999) as well as more complex rotational dynamic interactions of mechanical systems (acunza, 015). In a typical experiment, the conical motion is confined to the horizontal plane, however, it is possible to extend the mathematical analysis to three dimensions (Barenboim & Oteo, 013). In the usual case of horizontal planar motion, this can frequently be observed as elliptical rather than bein truly circular. his has been studied in detail in order to determine the physical nature of orbital precession (Deakin, 01). he strin tension force has been measured as a function of the rotational period (or anular frequency) and has been documented (Moses & Adolphi, 1998). or the analysis and discussion that follows, the five specific parameters are considered to be: the local value of the acceleration due to ravity ms - (Ali, M.Y.et al., 014), the mass m k of the conical pendulum, the pendulum lenth (measured to the centre of the spherical mass), the orbital period and the tension force (it is important to note that the anular frequency w is calculated directly from the orbital period, so it is therefore rearded as bein the sixth fundamental parameter). or the purposes of calculation, the theoretically ideal strin is considered to be essentially mass-less (when compared to the conical pendulum mass) and physically inextensible for the present analysis. Reardin the mathematical representation, when theoretical derivations are presented, all of the analytical steps are provided, in a manner that is considered to be self-explanatory. he arrow 1

3 European J of Physics Education Volume 8 Issue Dean symbol that is shown here within quotation marks is used frequently in mathematical derivations and is understood to mean ivin, or will ive, or leads to. he alternative arrow symbol in equations is used to mean approaches, or tends to. When a derivation can be clearly written on a sinle line usin a short-form mathematical layout, this method is used. A preliminary section dealin with the essential physical parameters is intended as a theoretical introduction, and the mathematical complexity has been deliberately minimized, to be readily accessible for readers. If convenient alternative mathematical representations of equations are readily available, these are frequently included in the principal analysis when this is considered to be helpful. HEOREIA RESUS AND ANAYSIS or the theoretical derivation of the conical pendulum period see (Dean & Mathew, 017); the resultin equation itself is presented below and will be developed further, in order to derive suitable equations for calculatin the orbital radius R and the conical pendulum apex anle f. Additional uncomplicated analysis will follow, providin two exceedinly simple and eleant equations for the tension force and centripetal force. p cos f p - R p w 1 w - R 4 w - R R - 4 w æ ö - ç è w ø (1) he final Equation (1) above shows that the orbital radius R can be calculated directly since the required parameters are known (namely, and w). he apex anle f is consequently directly obtained from the followin Equation (): R sin f f sin -1 æ ç è R ö ø () he apex anle f can be calculated from an alternative derivation usin the conical pendulum orbital radius R as the startin point. his alternative approach provides an equation that is only dependent on three of the fundamental conical pendulum physical parameters. In terms of its mathematical usefulness and applicability, the equation that is derived immediately below, is considered to be the most suitable equation for the calculation of the apex anle f. It can be noted that f is readily calculated for any particular pendulum lenth and anular frequency w (calculated from the period ). It is also suested below how an appropriate raphical analysis could be suitably performed, for the accurate determination of the local value of the acceleration due to ravity. 13

4 European J of Physics Education Volume 8 Issue Dean R - 4 w R - 4 w w - R w cos f f cos -1 æ ç è ö w ø (3) If an experiment is suitably desined so the apex anle f can be determined accurately, then the above analysis readily provides a straihtforward raphical method for the determination of the local acceleration due to ravity, from the slope of the raph with the followin axes: æ 1 ö! w ç Slope y axis - è cos %"$"# f ø x-axis (4) Multiple pendulum lenths coverin a suitably wide rane (for example the lenths reported in the present paper) could readily enable an accurate value of to be determined experimentally (see also onaonkar & Khadse, 011). By referrin to iure 1, and with a straihtforward understandin of the basic physics principals involved in the circular motion of a conical pendulum, the tension force can be shown to have a proportionality with respect to the reciprocal of the orbital period squared and consequently the square of the anular frequency: cos f p m cos f ü ý þ 4 p m (5) æ 1 ö ( 4 p m ) ç m w è ø (6) It can be readily seen that Equation (5) is obtained by makin a straihtforward substitution, usin the second equation for the tension force. A direct re-arranement of Equation (5) enables it to be written in a mathematically more eleant way in terms of the anular frequency squared. In this form, a raph of plotted on the y-axis aainst either ( w ) or just w on the x-axis, would enable the pendulum mass m to be calculated from the slopes of each of the resultin straiht lines. By usin experimental data acquired from multiple pendulum lenths, it is suested that 14

5 European J of Physics Education Volume 8 Issue Dean an accurate value of m can be readily calculated. rom the tension force and the ravitational force (m), as well as the trianular eometry; the two equations below are readily derived, where Equation (8) is observed to be independent of the pendulum mass m. m cos f f cos -1 æ ç è m ö ø (7) Pythaoras f cos -1 æ ç - R ç è ö ø (8) As a natural physical extension of the precedin analysis concernin the tension force, the centripetal force can also be considered in terms of the rotational properties of the pendulum. he short (one line) self-explanatory derivation below (usin v R w) addresses this issue: a v R m v R m R w R m a m R w (9) Equation (9) shows that the centripetal force is directly proportional to the anular frequency squared (see also onaonkar & Khadse, 011). Substitution of Equation (1) for the orbital radius R enables the centripetal force equation to be written in the followin way (by aain makin use of the short-form sinle-line mathematical layout): æ ö 4 m R w usin R - ç m w èw ø - (10) he resultin Equation (10) above makes it clear that the centripetal force can therefore be calculated directly without havin to calculate a value for the orbital radius R first. It is observed from the above equation that the conical pendulum mass m is specifically required (Mazza et al, 007). Makin reference to iure 1, and usin a Pythaorean trianle analysis, the followin eleantly simple mathematical relation between the tension force and centripetal force can be immediately obtained (aain usin the short-form mathematical layout): m and from iure 1: m tan f cos f sin f (11) his particular equation is considered to be especially useful due to its simplicity; however a prior calculation of the orbital radius R is required before the apex anle f can be calculated. A substitution for R ives rise to a mathematically more complex equation, which is now seen to be 15

6 European J of Physics Education Volume 8 Issue Dean dependent on four of the primary parameters. It is important to recall that (as stated earlier) the anular frequency w is rearded as bein a known parameter, since it is calculated directly from the conical pendulum period. R sin f æ ö - ç è w ø R ü ý þ æ 1 - ç è w ö ø (1) Despite the increased mathematical complexity, the final Equation (1) for the centripetal force is directly calculated from the available parameters and does not require either the apex anle f or the orbital radius R to be calculated in advance. he calculation of is now mathematically independent of the conical pendulum mass m, which consequently extends the applicability of the above centripetal force equation. he two equations for the tension force and the centripetal force, that are expressed in trionometric terms of the apex anle f can be re-expressed in the followin alebraic way: m cos f m - R (13) m tan f m R - R (14) he and equations can be used to investiate their dependence on the pendulum orbital radius R, as shown in the chart below, which clearly demonstrates the complex manner in which and depend on R, for a wide rane of conical pendulum lenths. It is of considerable interest to make direct reference to the functional dependence of the conical pendulum period on the orbital radius, throuh the equation below, which is referred to as Equation (1) in the paper by (Dean & Mathew, 017) and reproduced below for convenient reference: p - R he two forces and depend on R in a very different way (althouh mathematical similarities exist in the form of the equations), which will be considered below. 16

7 European J of Physics Education Volume 8 Issue Dean hart 1. and dependence on Orbital Radius for nine different values In order to ain a detailed understandin of the physical information displayed in the chart, it is advisable to make reference to the Appendix, which contains the colour-coded chart leend and provides an identification of the lines (solid and dashed) as well as the solid data markers. he shapes of the and lines in hart 1 can be readily explained by considerin the alebraic (rather than the trionometric) form of the appropriate equations, because the orbital radius R is specifically included. Since 0 R, it is of sinificant interest to examine the end-points of the two raphical plots. he two extreme values of R will be considered below, startin with the situation where R 0:! R 0 m - R %"$"# R 0 m (15)! R 0 m R - R %"$"# R 0 0 (16) he nine calculated curves that are shown on the chart above, clearly demonstrate that in the asymptotic limit that arises when the conical pendulum orbital radius decreases proressively to zero, then the tension force becomes equal to m. he physical interpretation of this specific limit is that the conical pendulum strin would be hanin, such that when projected in one of the two vertical planes (x-y or x-z) the mass would appear to be vertically downwards. A zero orbital radius also requires that the centripetal force becomes asymptotically zero, since Equation (9) clearly shows this to be the case. 17

8 European J of Physics Education Volume 8 Issue Dean When the apex anle f is lare and approachin (but physically never reachin) 90 it is appropriate to consider the mathematical upper orbital radius limit R ivin the followin:! R m - R %"$"# R (17)! R m R - R %"$"# R (18) Assumin the conical pendulum mass m remains constant, it can be seen that when the orbital radius R 0, the vertical axis intercepts of hart 1 will all have exactly the same tension force, namely, (m) the ravitational force (hart 3 demonstrates the situation for different masses). he simple analysis above also shows that the centripetal force will always start at the coordinate oriin and initially appear to be independent of the pendulum mass. As the orbital radius R increases from zero, then the apex anle f correspondinly increases, accordin to 0 f 90 and consequently the trionometric functions become:! m sin f cos f %"$"# ( sin 90 ) m cos f %" $" "#! f 90 f 90 f 90 f 90 (19) hart 1 unquestionably shows that the centripetal force becomes proressively closer to as the orbital radius increases, correspondin to the proressive increase in the apex anle f. Both forces are asymptotic (mathematically) to infinity on the vertical force axis; this would correspond to the conical pendulum bein at f 90, therefore horizontal and consequently physically unattainable. However, a straihtforward projection of the respective asymptotic forces downwards onto the horizontal radius axis, occurs at R for all of the conical pendulum lenths that have been investiated computationally. heoretically from Equations (13 & 14), the downwards projection limit R must apply for all possible conical pendulum lenths when both and. An inspection of hart 1 clearly supports this interpretation of the theoretical chart plots. When the tension force and centripetal force are, both plotted aainst the apex anle f, the calculated chart lines for all selected pendulum lenths, falls on only two and lines as shown on hart below. It is important to note that this chart contains all of the theoretical lines for the nine values of that were used computationally. he theoretical lines are superimposed where they overlap when plotted, hence the appearance of only two theoretical lines: 18

9 European J of Physics Education Volume 8 Issue Dean hart. and dependence on Apex anle f for a constant mass his is explained by makin reference to iure 1, where the apex anle f can be seen to be iven by Equation (0) below. When the apex anle f is held constant at any specific value, there are an infinite set of paired values for the orbital radius R and pendulum lenth that will satisfy the arcsine equation for f. onsequently, provided the pendulum mass remains constant, all values of and for any pendulum lenth must be positioned on only two lines (one each for the tension force and the centripetal force ) as shown on hart above. R R 1 1 sin f ü ý sin f þ f sin -1 æ R ç è i i ö ø ( where i inteer ) (0) It is noted that from the earlier analysis (Equation 11), is proportional to as: m tan f sin f As the apex anle f proressively increases towards 90 then the sine term correspondinly increases towards the mathematically limitin value of +1. onsequently the centripetal force proressively approaches (but physically can never actually equal) the tension force, which is observed as the asymptotic behavior on the above chart. he fact that the tension force and the centripetal force are specifically mass-dependent is demonstrated by the theoretically calculated chart shown below. It is essential to note that this particular chart is unique in one respect; the self-explanatory colour-coded lines refer to different masses as indicated in the included chart leend, for the and lines that are plotted. All the 19

10 European J of Physics Education Volume 8 Issue Dean remainin charts that are presented in this paper possess the same colour code, which refers only to the individual pendulum lenths (refer to the Appendix for details). hart 3. and dependence on Apex anle f with four different mass values Attention is directed to the vertical axis intercepts for the tension force solid lines that clearly indicate the initial values of this force are equal to the ravitational force (m), which is expected. he initial anular situation is taken to be f 0, correspondin to the pendulum strin bein vertical. he dashed lines on hart 3, which apply to the centripetal force all start from the coordinate oriin, as expected from the derived Equations (11) presented earlier. herefore, this particular chart helps to confirm the validity of the theoretical analysis, with respect to the functional dependence of the two forces and on the apex anle f. rom an earlier analysis, it was explained that the tension force and centripetal force are related throuh the above Equation (11), where it is clear that depends on as: sin f It is a mathematical point of interest to draw attention to the fact that as the orbital radius R proressively increases, then the apex anle f correspondinly increases towards the maximum possible value of 90 (althouh it is not physically possible for f 90 ). herefore, from the above equation it is readily observed that: hart 4 below shows this relationship theoretically up to approximately 1 N (which corresponds to an apex anle f» 85 for a conical pendulum mass m k). his is rearded as the maximum feasible tension force that is attainable under reasonable conditions and has therefore been selected for computational purposes. 0

11 European J of Physics Education Volume 8 Issue Dean hart 4. as a function of up to 1.0 N he appearance of the above chart line requires some explanation; from the basic physics equations for the anular dependence of the tension force, the minimum possible value for has previously been shown as bein equal to the ravitational force (m), which accounts for the startin position of the line alon the horizontal axis. At the point where (m) then 0, (and the conical pendulum strin is hanin vertically down so f 0). It is appropriate at this juncture to once aain refer to Equations (11), which are reproduced below for convenience: m and from iure 1: m tan f cos f sin f rom these and equations, it is apparent that as the apex anle f increases from zero, the centripetal force proressively becomes closer to the tension force (i.e. more linear), althouh always remainin lower in value. he table presented below (which is calculated from the above equations) clearly indicates the very rapid transition of both forces towards linearity, over the scale of their respective hart 4 axes. Since the lowest possible value of is m» 1.1 N, the reion from the coordinate oriin to (m) is mathematically undefined and therefore physically unattainable experimentally. Attention is drawn to the fact that althouh this paper is essentially theoretical in nature; any obvious physical limitations are taken into consideration. he below table indicates that the initial value of the tension force» 1.1 N (which is m) and the centripetal force 0; when the tension has increased to 1.5 N, then 1.0 N which corresponds to the apex anle havin a value of f his relatively lare and rapid anular chane (startin from f 0) occurs within the small curved section of the line shown on hart 4. he initially curved (vertically projectin) line proressively becomes more linear as the two forces continue to increase and become asymptotically equal in manitude. In a mathematical sense, this can be interpreted as the positive first derivative asymptotically approachin the value of (+1). 1

12 European J of Physics Education Volume 8 Issue Dean able 1. orces and with reciprocals and correspondin Apex anle f It is of interest to analytically explore the variation (and physical dependence) of two primary physical parameters that form the basis of this paper. Namely, the conical pendulum tension force and orbital period. he principal equation that will be used for this parametric analysis, namely Equation (5) is reproduced below for convenient reference: 4 p m Inspection of the basic mathematical format of the above equation clearly reveals the familiar symbolic format of an inverse-square law. or this particular case, the parametric dependence of the tension force is the inverse square of the conical pendulum period. his is an extremely familiar raphical line-shape that is tauht in all standard physics courses and appears in many analyses of different types of force laws. he basic theoretical analysis can be visualized by closely examinin the chart below, where the x-axis and y-axis are essentially taken to be the two relevant computationally determined values of (y-axis) and (x-axis) and follow the mathematical dependence implied above (as an inverse square law).

13 European J of Physics Education Volume 8 Issue Dean hart 5. as a function of Period It is observed that the apparent horizontal asymptotes abruptly terminate at well-defined values of the period ; this corresponds to the period that is appropriate for a simple pendulum of the same lenth (this is the lonest possible period and occurs for when R 0). A straihtforward analysis will be helpful with understandin the physical reason for the behavior shown above. In what immediately follows, the subscripts SP and P refer to the simple pendulum and conical pendulum respectively: SP P 4 p 4 p - R ü ý þ P 4 SP -16 p 4 R R 0 P SP P SP (1) he above concise analysis therefore confirms the behavior observed in hart 5, for when the tension force has the minimum possible value ( m). A consideration of the above chart with respect to Equation (5) directly suests an alternative functional scalin for the horizontal axis and is shown in the chart below. p æ 1 ö ( 4 m ) ç ø è 3

14 European J of Physics Education Volume 8 Issue Dean hart 6. as a function of (1/Period ) In this linear form, the slopes of the straiht lines (4 p m ) can provide a straihtforward raphical technique to determine an unknown conical pendulum mass m experimentally. As shown above, the straiht lines cannot physically pass throuh the coordinate oriin because the lonest period must be equal to the period of a simple pendulum of the same lenth (this defines the smallest attainable value for the reciprocal of the conical pendulum period squared). he second reason for the above observation is due to the fact that the tension force must have a minimum non-zero value on the vertical axis ( m). When the two axes are interchaned in hart 5 the followin alternative presentation is obtained: hart 7. Period as a function of 4

15 European J of Physics Education Volume 8 Issue Dean his selection of the chart axes implies that the period can be considered to be dependent on the tension force, while the earlier chart presentation implied the reverse. he selection of parametric axis enerally follows the convention where the independent variable is plotted alon the horizontal axis (x-axis) and the dependent variable is plotted vertically (y-axis). A classic example of when this format is usually not adhered to is the well-known equation V I R often referred to as Ohm s aw (V I R y m x with the straiht line of slope R º m passin exactly throuh the coordinate oriin). It is clearly apparent from the laws of Physics that the potential V does not depend on the current I flowin throuh the resistance R, however, the inverse is true (the current I depends on the potential V across the resistance). When to follow the axes convention can frequently be determined from the equation that relates the dependent and independent parameters. In the present case, it is assumed that the rotational period is in fact dependent on the strin tension force even thouh the usual representative convention has not been applied inflexibly. By referrin to Equation (5), it can be readily observed that the two principal theoretical parameters can be re-arraned to yield Equation () below: æ 1 ( ) ö 4 p m ç è ø () Usin this equation form, the chart appears as shown below: hart 8. Period as a function of (1/ ) With respect to this particular chart axes representation, the straiht lines can mathematically pass throuh the coordinate oriin, althouh the situation cannot be physically achieved. Once aain, just as for hart 6, the slopes of the straiht lines (4 p m ) can be used to determine an unknown conical pendulum mass m. It is of interest to note that the riht-side of every line stops 5

16 European J of Physics Education Volume 8 Issue Dean abruptly alon the horizontal axis at a calculated value of (1/ ) , correspondin to the minimum possible value of the tension force ( m), with the mass hanin vertically down. As an extension of the above chart and analysis, it is appropriate to consider the variation of the period squared with respect to the centripetal force that is directly responsible for the horizontal circular motion of the conical pendulum. Equations (9 & ) provide the followin: æ 1 ( ) ö 4 p m R ç è ø (3) he chart below shows the above equation for each of the nine values of the lenth, where it is noted from Equation (3) that due to the non-constancy of the orbital radius R, the plotted lines will not be linear: hart 9. Period as a function of (1/ ) his chart can be explained by notin that the initially curved lines proressively become liner as the reciprocal of the centripetal force (1/ ) increases to the riht (the centripetal force itself increases towards the left), which corresponds to a proressive decrease in itself. he square of the rotational periods (on the y-axis) initially increase as expected, however, they subsequently become asymptotically horizontal with values of the period squared correspondin to the period squared appropriate for a simple pendulum of the same lenth. he rapidity with which the lines become horizontal is due to the non-linear reciprocal scale of the horizontal axis, as shown in the appropriate column of able 1 above. heoretical analysis presented in earlier sections of this paper detailed the derivation of two concise equations for the tension force and centripetal force namely Equations (6 & 9) as reproduced below for convenient reference. m w and m R w 6

17 European J of Physics Education Volume 8 Issue Dean It is of interest to consider the simultaneous variation of these two forces as a function of the square of the anular frequency, as shown in the chart below: hart 10. and as a function of w With reference to this fiure and the Appendix, it is of importance to note that the solid lines and dashed lines represent and respectively. As expected from earlier discussion, the lines all start from the minimum tension force value of (m) on the vertical (y-axis) and with a horizontal displacement correspondin to the minimum value of the square of the anular frequency. his corresponds to the larest possible value of the orbital period squared, which occurs for a simple pendulum (subscript SP) of the same lenth in Equation (4) below: 4 p 4 p 4 w min max w p SP (4) he centripetal force (dashed lines) must also start from the same minimum value of the anular frequency alon the horizontal axis, but from 0 on the vertical (y-axis), meanin the conical pendulum strin would consequently be hanin vertically downwards. It can be observed from hart 10 that the plotted tension force lines are all straiht and would all pass directly throuh the coordinate oriin if this was physically possible, with slopes of value (m ) for each lenth. Reference to Equation (3) provides the followin: -1 æ ö f cos ç 0 è w ø w 1 7

18 European J of Physics Education Volume 8 Issue Dean here are clearly two extreme mathematical limits to be considered as shown below: w ì 0 í 1 î f 90 w (5) he second limit (for which the Apex anle f 0) must correspond to the minimum possible value for the square of the anular frequency, which therefore immediately provides the equation shown below. rom Equations (4 & 5) it is clear that there must be a minimum threshold (or critical) anular velocity before the centripetal force becomes present and the conical pendulum can start to perform uniform planar circular motion (see also Klosteraard, 1976). 4 p w min max 4 p SP DISUSSION O RESUS AND ONUSIONS It can be readily seen from the charts presented in this paper, that the theoretical analysis is wideranin and continuous, over the complete rane of conical pendulum period, tension force, centripetal force, orbital radius and apex anle values that were computationally studied. With reference to the charts and the accompanyin theoretical equations, the overall shapes of plotted lines, their initial points, end-points and any unmistakably observable axial asymptotes (x-axis and y-axis) were clearly explained. It is proved theoretically that when the tension force and centripetal force are both plotted on the vertical y-axis aainst the apex anle alon the horizontal x-axis, there will be only two lines (despite the two forces bein determined for nine different pendulum lenths). he asymptotic nature of these two forces becomin proressively more equal as the apex anle increases is also explained; as is the effect of usin different masses for the conical pendulum. When the tension force is plotted aainst the period, the abrupt termination of the horizontal asymptotes at well defined values of the period is explained in terms of the period of a simple pendulum havin the same strin lenth as the conical pendulum. When the square of the conical pendulum period is plotted as a function of the reciprocal of the tension force, the abrupt termination of the calculated chart lines alon the horizontal axis is explained as bein unequivocally due to the minimum value of the tension force. he effect that usin different mass values has on the computational results were also demonstrated. APPENDIX - HAR EGEND OR HEORY INES he chart leend used to identify the theoretical lines is presented below (adapted from Dean & Mathew, 017), from the larest pendulum lenth (.130 m) to the shortest ( m). 8

19 European J of Physics Education Volume 8 Issue Dean Solid lines are used for the tension force and dashed lines represent the centripetal force, when these are required to illustrate the theoretical calculations. iure. hart eend Note: if there is either a sinle solid dark-blue line, or a solid dark-blue line and a dashed line of the same colour, plotted on any particular chart, the accompanyin section of text will make it clear that the lines are from a theoretical calculation. It will be clear from the theoretical analysis that lines on the chart are independent of the pendulum lenth, by bein superimposed. REERENES Ali, M.Y., Watts, A.B. and arid, A., (014). Gravity anomalies of the United Arab Emirates: Implications for basement structures and infra-ambrian salt distribution, GeoArabia, volume 19, no. 1, p Bambill, H.R., Benoto, M.R. and Garda, G.R., (004). Investiation of conservation laws usin a conical pendulum, European J of Physics, V. 5, Barenboim, G. and Oteo, J.A., (013). One pendulum to run them all, European Journal of Physics, Vol 34, p zudková,. and Musilová, J., (000). he pendulum: a stumblin block to secondary school mechanics, Physics Education, Vol. 35, Number 6, p Deakin, M.A.B., (01). he ellipsin pendulum, International Journal of Mathematical Education in Science and echnoloy, Vol. 44, Issue 5, p745-75, 013 Dean, K. and Mathew, J., (017). onical Pendulum inearization Analyses, European J of Physics Education, Vol. 7, Issue 3, p38-5 9

20 European J of Physics Education Volume 8 Issue Dean Dupré, A. and Janssen, P., (1999). An accurate determination of the acceleration of ravity in the underraduate laboratory, American Association of Physics eachers, Vol. 68, p Klosteraard, H., (1976). Determination of ravitational acceleration usin a uniform circular motion, American Association of Physics eachers, Vol. 44, p68-69 acunza, J.., (015). he Pendulum of Dynamic Interactions, Journal of Applied Mathematics and Physics, Vol. 3, p Mazza, A.P., Metcalf, W.E., inson, A.D. and ynch, J.J., (007). he conical pendulum: the tethered aeroplane, Physics Education, Vol. 4, Number 1, p6-67 Moses,. and Adolphi, N.., (1998). A New wist for the onical Pendulum, Am. Association of Physics eachers, Vol. 36, onaonkar, S.S. and Khadse, V.R. (011). Experiment with onical Pendulum, European J of Physics Education, Vol. (1),