Oscillations • Universe (Previous | Next)

Youth researches 1983

Subject area physics

A novel method for the determination of the periodic time of the mathematical pendulum with large amplitudes and the discussion for the applicability on other selected oscillations of the plane

Compiled by Boris Haase (18),

Theodor-Heuss-Gymnasium in Göttingen

Abstract of the present paper

The oscillation period of mass points on curved oscillation ways in the vertical plane - like e.g. the mathematical pendulum and the cycloid pendulum -, is to be computed in the present paper in a simpler way than so far.

Most of these oscillation periods are - if the mass point on the oscillation way is far from the rest position, whose lowest point - dependent on this distance, which makes the computation more difficult generally.

In order to be able to compute an of this distance, the amplitude, independent periodic time, one must know except the mass only the direction parameter, the quotient from repelling force and corresponding amplitude respectively direction way.

This knowledge is extended and applied in this paper to the oscillation period dependent on its amplitude.

Here partially differences result to the conventional methods - by the example of the mathematical pendulum up to at the most one per cent with 90° of amplitude - partly do not as by the example of the cycloid pendulum however. To the likewise treated ellipse pendulum no literature data could be found.

In the experimental part the calculated periodic time for the mathematical pendulum in consideration of the sources of error and the measuring could be confirmed.

The author comes to the conclusion that the independently developed procedure could by all means be used in physics.

Table of contents

1. | Introduction |

2. | Main Part |

2.2. | Experiment to the Mathematical Pendulum |

2.3. | Calculations of Errors |

2.4. | Discussion |

3. | Conclusion |

4. | Appendix |

5. | Bibliography |

1. Introduction

1.1. Reason and Basic Idea

The periodic time of the mathematical pendulum with large amplitudes was determined so far by quite difficult procedures of the infinitesimal analysis and series expansion. The in the present paper developed new procedure for the determination of the periodic time for selected oscillations of the plane is to simplify the calculation method on the one hand, in addition be compared regarding his by two examples with the old procedure.

Basic idea is the simultaneous simulation of the respective pendulum oscillation by a straight-lined harmonic oscillation (spring pendulum, oscillator).

1.2. Presuppositions

All selected oscillations emanate from a mass point, which is to swing in the vertical plane at a thread without mass around a fix point.

The pendulum thread may be pulled to the side here - seen from the rest position - at the most by 90°, since the maximally repelling force must lie in the initial height, in which no additional acceleration is to arise.

In order to prevent a free fall, otherwise bars would have to be used instead of threads, which could however not (as necessary) be huddled against evolutes of the oscillation ways. Despite the huddling against for simplification the length of the pendulum in rest position is assumed.

The radius of curvature of the oscillation way must remain either constant from the rest position or be continuously reduced, so that a free fall is likewise prevented.

Friction and damping aspects are to be neglected.

Otherwise further computations are necessary for the periodic time, which would go however beyond the scope of this paper.

2. Main Part

2.1. Theoretical Solution of the Problem

2.1.1. General Solution

The periodic time of a harmonic oscillation is given by:\[(a)\;T = 2\pi \sqrt {{\textstyle{m \over D}}}.\]After the principle of conservation of energy holds:\[(b)\;\tfrac{1}{2} m v^{2} = \tfrac{1}{2} D s^{2} = m g h.\]During an oscillation form the potential energy \(W_{pot}\) and the kinetic energy \(W_{kin}\) the constant sum:\[(c)\;W_{pot} + W_{kin} = W_{pot_{0}}.\]\(W_{pot_{0}}\) is to be expressed now by the tension energy \(W_{Sp}\), in order to determine a direction parameter \(D(\varphi_{0})\), dependent on the amplitude. \(D(\varphi_{0})\) is to be inserted then into the formula (a) for the determination of the periodic time.

In addition an ersatz movement of the mass \(m\) is necessary by force \(F\) on the direction way \(s\). It applies:\[(d)\;D = F/s \text{ (Hooke's law) and}\]\[{\rm{(e)}}\;{\rm{F}}\, = \,{\rm{m}}\,{\rm{a}} = \,{\rm{m}}\,{\rm{l}}\,\ddot \varphi \,\,\left( {{\text{Newton}}} \right).\]In the further one always harks back to the direction parameter \(D\) of the plane thread pendulum with minimum amplitude. Here applies to the repelling and to the oscillation way tangential force \(F_{R}\), since the normal force is compensated by the tension force of the thread (1, 2):\[{\rm{(f)}}\;{{\rm{F}}_{\rm{R}}} = \,{\rm{m}}\,{\rm{g}}\,{\rm{sin}}\,\varphi = \, - m\,l\,\ddot \varphi .\]From this follows:\[(g)\;\ddot \varphi \, = \,\,{\rm{sin}}\,\varphi \,{\textstyle{g \over l}}.\]For smallest amplitude angles \(\varphi \; sin(\varphi)\) can be replaced by \(\varphi\), and the conditions apply to the harmonic oscillation. Therefore is:\[(h)\;\ddot \varphi \, = {\omega ^2}\,\varphi \]and from (g) follows:\[(i)\;\omega = \sqrt {{\textstyle{g \over l}}.} \]Thus applies:\[(j)\;D = m\,{\omega ^2} = m\,{\textstyle{g \over l}}.\]For larger amplitudes the direction parameter \(D\) must consist of the product of the direction parameter for the harmonic oscillation with a function value \(f(\varphi_{0})\) of the amplitude angle \(\varphi\) determining the inharmonic oscillation.

One writes:\[(k)\;D({\varphi _0}) = \left( {m{\textstyle{g \over l}}} \right)\,f({\varphi _0}).\]The function value \(f(\varphi_{0})\) is not however defined for the harmonic condition of the energy theorem, so that it must cancel itself during an introduction. With \(s = x l\) applies:\[(l)\;\frac{1}{2}\frac{{\frac{{f({\varphi _0})mxg}}{{xl}}{{(xl)}^2}}}{{f({\varphi _0})}} = m\,g\,h.\]Hereby the not changing direction way \(s\) can be computed by the energy theorem, if one assigns the function value \(f(\varphi_{0})\) to the repelling force \(F_{R}\). With\[{\textstyle{1 \over 2}}\left( {m{\textstyle{g \over l}}} \right){s^2} = m\,g\,h\]arises for the direction way:\[(m)\;s = \sqrt {2lh} .\]The repelling force \(F_{R}\) must be developed thus independently of the energy theorem. It is rather fixed by the initial oscillation as the in the initial height of the oscillation way tangential attacking and thus maximally repelling force. From\[(n)\;F_{R} = (-) m a\]one obtains the periodic time with (a) and (d):\[(o)\;T = 2\pi \sqrt {\frac{{\sqrt {2lh} }}{a}} .\]Concretely the direction way \(s\) represents with (m) the chord of a circle with the radius l. Since this circle is circle of curvature for the rest position of the oscillation way, this chord must lead from one point of the circular path in the initial height \(h\) to the rest position.

The simplest form of an oscillation way which aligns with the circle of curvature and possesses a punctiform evolute is those of the mathematical pendulum (fig. 1). This is to be first treated therefore.

2.1.2. The Mathematical Pendulum

From fig. 1 one obtains for the direction way \(s\):\[s = 2\sin \left( {{\textstyle{{{\varphi _0}} \over 2}}} \right)l.\]With (f) applies to the maximally repelling force \(F_{R}\):\[F_{R} = (-) m g\, \text{sin}(\varphi_{0}).\]The periodic time is with (a) and (d) therefore:\[T = 2\pi \sqrt {\frac{l}{{\cos \left( {{\textstyle{{{\varphi _0}} \over 2}}} \right)g}}.} \]

2.1.3. The Cycloid Pendulum

To the common cycloid (fig. 2) the parameter equation (2) applies:\[x = a (\Phi - \text{sin}(\Phi)) \text{ and } y = a (\text{cos}(\Phi) - 1).\]For the radius of curvature \(\rho\) one obtains:\[\rho = 4a\sin \left( {{\textstyle{\phi \over 2}}} \right).\]Since the angles \(\Phi\) increase from the initial height \(h\) to the rest position with \(\Phi = \pi\) respectively the sine decreases from \(\Phi/2\) after exceeding of the rest position to the reversal point again, the conditions for the computation of the periodic time are fulfilled.

The initial height is:\[h = 2 a + y = a (\text{cos}(\Phi_{0}) + 1).\]If one inserts \(h\) in (m) ein, applies with \(l = 4 a\):\[s = l\sqrt {{\textstyle{{\cos \,{\phi _0} + 1} \over 2}}} = \cos \left( {{\textstyle{{{\phi _0}} \over 2}}} \right)l.\]Since the tangential acceleration \(b_{t}\) works always vertically to the radius of curvature \(\rho\), one obtains for it:\[{b_t} = \alpha \cos \left( {{\textstyle{{{\phi _0}} \over 2}}} \right)g.\]If \(\Phi_{0} = 0\) holds, then \(b_{t} = g\) holds, if \(\Phi_{0} = \pi\) holds, then \(b_{t} = 0\) holds. Therefore must hold the multiplier \(\alpha = 1\) and apply:\[{b_t} = \cos \left( {{\textstyle{{{\phi _0}} \over 2}}} \right)g.\]With (o) arises finally for the periodic time:\[T = 2\pi \sqrt {{\textstyle{l \over g}}} .\]This was already proven before by Huygens centuries ago.

2.1.4. The Ellipse Pendulum

To the ellipse (fig. 3) the parameter equation (3) applies:\[\frac{{{x^2}}}{{{a^2}}} + \frac{{{y^2}}}{{{b^2}}} = 1.\]For the radius of curvature \(\rho\) results after differentiation (see appendix):\[\rho = \frac{{\left({{{a^4} - {a^2}{x^2} + {b^2}{x^2}}}\right)}^{\tfrac{3}{2}}}{{{a^4}b}}.\]From the rest position with \(x = 0\) to the reversal points \(x\) increases, and since \(a^{2} x^{2} > b^{2} x^{2}\) holds, the radius of curvature is continuously reduced. In the rest position \(\rho\) is therefore:\[\rho = l = \frac{{{a^2}}}{b}.\]With\[y = - b\sqrt {1 - \frac{{{x^2}}}{{{a^2}}}} \]the initial height \(h\) is:\[h = b + y = b\left( {1 - \sqrt {1 - \frac{{{x^2}}}{{{a^2}}}} } \right).\]With (m) arises therefore for the direction way \(s\):\[s = \sqrt {2a\left( {a - \sqrt {{a^2} - {x^2}} } \right)} .\]Thus the direction way is independent of the semi-minor axis of the ellipse.

The tangential acceleration \(b_{t}\) is (after fig. 3):\[{b_t} = \cos \,\beta \,g = \frac{{bxg}}{{\sqrt {{a^4} - {a^2}{x^2} + {b^2}{x^2}} }}.\]With (o) one obtains for the periodic time:\[T = 2\pi \sqrt {\frac{{\sqrt {{a^4} - {a^2}{x^2} + {b^2}{x^2}} \sqrt {2a\left( {a - \sqrt {{a^2} - {x^2}} } \right)} }}{{bxg}}} .\]For \(x = 0\) the periodic time is not defined, for very small \(x\) applies however:\[T = 2\pi \sqrt {\frac{{{a^2}}}{{bg}}} = 2\pi \sqrt {\frac{l}{g}} .\]For \(x = a\) the proposition results:

The periodic time on an ellipse with maximum amplitude is independent of the semi-minor axis.

2.2. Experiment to the Mathematical Pendulum

For temporal reasons the experimental part was limited to the mathematical pendulum. Not only making of evolutes would be on the other hand very complex, but during the cycloid respectively ellipse oscillation also larger sources of error would be opened. To this the difficulty belongs to let the pendulum swing exactly to the evolute, since already after few oscillations the pendulum leaves the vertical plane. In addition impulse respectively friction forces arises with the beginning of the swinging and in the support, which with large amplitudes the respective pendulum let become the mathematical pendulum.

The latter can be regarded therefore as part for the whole.

2.2.1. Experimental Setup

Two bars of a meter are connected with a third and four sleeves and pushed symmetrically through two ceiling hooks. In the centre additionally a protractor and the support for the pendulum thread of 1.60 m length are fastened.

An air cushion lane is placed in such a way parallel to it on the bench that its centre is vertically under the point of the protractor. An on the centre placed lane carriage of 0.2 kg mass is clamped in such a way by two springs that between the clamping at the end of the bench and the carriage a direction parameter of 2 N/m prevails.

A stop watch is set up ready to hand by the initial height of the pendulum and the carriage. An illustration to the experiment is on the title page. In the appendix also all used devices are specified.

2.2.2. Test Procedure and Results

First was convinced by the independence of the periodic time of the mass of the pendulum, as once one approx. 5 kg heavy ball, another time the lane carriage weighted on 1 kg of mass were fastened to the pendulum thread. The difference of the measured periodic time amounted to hundredths a second on each of the ten periods.

During the simulation of the mathematical pendulum by the harmonic oscillator the time difference per oscillation amounted to 0.6 s (back and forth) with ten oscillations and 60° of maximum amplitude of the pendulum.

For the computation of the mass of the harmonic oscillator the formula was used:\[m = \frac{{2lD}}{{\cos \left( {{\textstyle{{{\varphi _0}} \over 2}}} \right)g}}.\]The latter can be determined directly from the periodic times for the two systems.

Here the direction parameter \(D_{0}\) for the oscillator is the double of the spring constants equal in size of the used tension springs. The periodic time of the oscillator is therefore:\[T = 2\pi \sqrt {\frac{m}{{2D}}} \,(4).\]The periodic time of the mathematical pendulum calculated above is:\[T = 2\pi \sqrt {\frac{l}{{\cos \left( {{\textstyle{{{\varphi _0}} \over 2}}} \right)g}}} .\]The damping of the pendulum during a amplitude of 60° amounted to 1°, those of the oscillator 0.01 m during a amplitude of 0.9 m.

The measured values for amplitudes of the mathematical pendulum from 0° to 90° are specified in tab. 1 in the appendix.

2.3. Computations of errors

The periodic time of the mathematical pendulum became so far specified over the elliptical integral\[\int\limits_0^{{\textstyle{\pi \over 2}}} {\frac{{d\phi }}{{\sqrt {1 - {k^2}{{\sin }^2}\phi } }}} \,\]with\[\,k = \sin \left( {{\textstyle{{{\varphi _0}} \over 2}}} \right)\,\](2):\[T = 2\pi \sqrt {{\textstyle{l \over g}}\left( {1 + {{\left( {{\textstyle{1 \over 2}}} \right)}^2}{k^2} + {{\left( {{\textstyle{{1 \cdot 3} \over {2 \cdot 4}}}} \right)}^2}{k^4} + {{\left( {{\textstyle{{1 \cdot 3 \cdot 5} \over {2 \cdot 4 \cdot 6}}}} \right)}^2}{k^2} + ...} \right)} .\]In order to be able to compare directly with the new formula for the periodic time:\[T = 2\pi \sqrt {\frac{l}{{\cos \left( {{\textstyle{{{\varphi _0}} \over 2}}} \right)g}}} .\]the function value\[f({\varphi _0}) = \sqrt {\frac{1}{{\cos \left( {{\textstyle{{{\varphi _0}} \over 2}}} \right)}}} \]is converted with the help of the series with \(|x| < 1\):\[{(1 + x)^p} = 1 + px + \frac{{p(p - 1){x^2}}}{{1 \cdot 2}} + \frac{{p(p - 1)(p - 2){x^3}}}{{1 \cdot 2 \cdot 3}} + ...\,\,(2)\]and the relationship:\[f({\varphi _0}) = \frac{1}{{\sqrt[4]{{1 - {{\sin }^2}\left( {{\textstyle{{{\varphi _0}} \over 2}}} \right)}}}}\]with\[x = - {\sin ^2}\left( {{\textstyle{{{\varphi _0}} \over 2}}} \right) = - {k^2}\]into the comparison series:\[1 + {\left( {{\textstyle{k \over 2}}} \right)^2} + {\left( {{\textstyle{{1 \cdot 5} \over {1 \cdot 2}}}} \right)^2}{\left( {{\textstyle{k \over 2}}} \right)^4} + {\left( {{\textstyle{{1 \cdot 5 \cdot 9} \over {1 \cdot 2 \cdot 3}}}} \right)^2}{\left( {{\textstyle{k \over 2}}} \right)^6} + ...\;.\]From this results that the two series correspond up to the first square member.

If one divides the old by the new series and subtracts 1, then results the after the amount smaller series of the relative error: \[{\textstyle{1 \over {64}}}{k^4} + {\textstyle{1 \over {64}}}{k^6} + {\textstyle{{231} \over {16384}}}{k^8} + ...\,\,.\]If one divides in reverse with same subtraction, then the relative error is larger:\[{\textstyle{1 \over {64}}}{k^4} + {\textstyle{1 \over {64}}}{k^6} + {\textstyle{{235} \over {16384}}}{k^8} + ...\,\,.\]The number of the absolute error arises as a result of subtraction of the old from the new series:\[{\textstyle{1 \over {64}}}{k^4} + {\textstyle{5 \over {256}}}{k^6} + {\textstyle{{335} \over {16384}}}{k^8} + ...\,\,.\]The values for selected angles from 0° to 90° of all series specified here are held in the appendix in tables.

2.4. Discussion

Here is once to think about the proportionalities arising in the formula:\[\,T = 2\pi \sqrt {\frac{{\sqrt {2lh} }}{a}}.\]First it is to be noted that the direction way \(s\) is not proportional to the acceleration \(a\) of the repelling force \(F_{R}\), but the function value \(f(\varphi_{0})\), contained in \(a\), must be considered.

The length of the oscillation way is adapted over the direction way:

If the initial height \(h\) is very small, becomes the distance to cover on the oscillation way in \(y\)-direction and concomitantly the periodic time smaller.

If the oscillation way is very flat in addition, the small initial height confronts a large pendulum length, so that one obtains a mean for the length of the direction way, as it appears extra clearly with maximum amplitudes of the ellipse pendulum.

With highly curved oscillation ways and large initial heights as with the mathematical pendulum the periodic time is relatively larger.

In the direct proximity of the rest position the numerical values of direction way and acceleration cancel out, so that the direction way becomes the length \(l\) of the pendulum and the acceleration becomes the standard acceleration due to gravity \(g\).

In the rest position itself the periodic time is strictly taken not defined, because the initial height \(h\) and the acceleration \(a\) become there zero.

The larger the acceleration \(a\) is, the smaller is the periodic time, otherwise reverse conditions are present.

These realisations could be relative well confirmed in the experiment to the mathematical pendulum.

3. Conclusion

In the review can be said that the real approach agrees not completely with in this for clarity reasons stated.

Thus only the problem of the mathematical pendulum was solved and then a general problem solution was established. All periodic times were computed without knowledge of the periodic times indicated in the corresponding literature, their insight like the experiment took place only weeks later.

The calculation method could be indeed very simplified and regarding the to the old procedure the margin of error of 0.01 respectively one per cent did not need to be exceeded.

Here the ellipse pendulum could not be incorporated, since about its periodic time no literature data could be found.

Most likely the margin of error will shift however upward, since the mathematical pendulum as special case of the ellipse pendulum, whose periodic time can be transferred with equal long semi axes into those of the mathematical pendulum, possesses already a deviation from scarcely one per cent with maximum amplitude in respect of its periodic time.

The uncontradictedness of the periodic time of the cycloid pendulum to the literature data is however given.

3.1. Critical Appreciation of the Paper

Without doubt the evolutes problem is not yet completely solved. If one does not presuppose anymore the pendulum length in rest position, one must perhaps nevertheless hark back to procedures of the infinitesimal analysis and series expansion, in order to obtain a larger precision.

Thereto however is contradictory on the one hand the mathematical pendulum, on the other hand the Cycloid pendulum.

With the first a distinct evolute is missing despite deviation, with the latter exists a deviation despite the presence of an evolute.

Thus the new procedure must be considered as independent.

The measurements to further periodic times than that of the mathematical pendulum unfortunately missing in this paper would afford a more exact clarifying of circumstances. Here however very high demands are to be made on the measuring and the run of the experiments.

Altogether the procedure developed in this paper requires a certain adjustment in the way of thinking, yet it should be able to be applied due to the obtained good results in physics.

4. Appendix

Figures

Fig. 1

Fig. 2

Fig. 3

All constructions took place by (9).

Auxiliary Calculations

The general formula (3) applies for the radius of curvature:\[\rho = \frac{{{{\left( {{{\dot x}^2} + {{\dot y}^2}} \right)}^{{\textstyle{3 \over 2}}}}}}{{\dot x\ddot y - \dot y\ddot x}}.\]The common cycloid has the parameter equation (2):\[x = a (\Phi - \text{sin}\,(\Phi)) \text{ and } y = a (\text{cos}(\Phi) - 1).\]For\[{\left( {{{\dot x}^2} + {{\dot y}^2}} \right)^{{\textstyle{3 \over 2}}}}\]one can write therefore\[{\left( {{a^2} - 2\cos \,\phi \,{a^2} + {{\cos }^2}\phi \,{a^2} + {{\sin }^2}\phi \,{a^2}} \right)^{{\textstyle{3 \over 2}}}} = {\left( {2{a^2} - 2\cos \phi \,{a^2}} \right)^{{\textstyle{3 \over 2}}}}.\]With\[\dot x\ddot y - \dot y\ddot x = (a - \cos \,\phi \,a)( - \cos \,\phi \,a) - ( - \sin \,\phi \,a)\sin \,\phi \,a = {a^2} - \cos \,\phi \,{a^2}\]applies thus to the radius of curvature\[\rho = 2a\sqrt {2(1 - \cos \,\phi )} = 4\,a\,\sin \left( {{\textstyle{\phi \over 2}}} \right).\]If one writes with the ellipse temporarily \(x = \text{sin}\,(\varphi) a\) and\[y = - b\sqrt {1 - {\textstyle{{{x^2}} \over {{a^2}}}}} = - \cos \,\varphi \,\,b,\]then arises:\[\dot x\ddot y - \dot y\ddot x = \cos \,\varphi \,\,a(\cos \,\varphi \,\,b) - \sin \,\varphi \,\,b( - \sin \,\varphi \,\,a) = ab.\]Thereby is the radius of curvature:\[\rho = \frac{{{{\left( {{{\cos }^2}\varphi \,\,{a^2} + {{\sin }^2}\varphi \,\,{b^2}} \right)}^{{\textstyle{3 \over 2}}}}}}{{ab}}.\]If one revokes the transformation, one obtains after extension of the fraction with \(a^{3}\) for \(\rho\):\[\rho = \frac{{{{\left( {{a^4} - {a^2}{x^2} + {b^2}{x^2}} \right)}^{{\textstyle{3 \over 2}}}}}}{{{a^4}b}}.\]The evolute of the common cycloid is likewise a cycloid with same \(a\) (5).

To the evolute for the ellipse applies however (6):\[\xi = \frac{{({a^2} - {b^2}){{\cos }^3}t}}{a}\]and\[\,\eta = \frac{{({a^2} - {b^2}){{\sin }^3}t}}{b}\]with \(x = a \text{ cos}(t)\) and \(y = b \text{ sin}\,(t)\).

In order to be able to compute the tangential acceleration \(b_{t}\) of the ellipse, the following initial considerations are necessary:\[\tan \gamma = \frac{{\sqrt {{a^2} - {x^2}} }}{x} = \frac{q}{{\sqrt {{a^2} - {x^2}} }}.\]From this follows directly:\[q = \frac{{{a^2} - {x^2}}}{x}.\]For\[\cos \,\beta = \frac{y}{{\sqrt {{q^2} + {y^2}} }} = \frac{y}{{{{\sqrt {{{\left( {\frac{{{a^2} - {x^2}}}{x}} \right)}^2} + {y^2}} }^{}}}}\]one can write after extending with\[\frac{{ax}}{{\sqrt {{a^2} - {x^2}} }}\]\[\cos \,\beta = \frac{{bx}}{{\sqrt {{a^4} - {a^2}{x^2} + {b^2}{x^2}} }}.\]To convert the formula for the periodic time of the ellipse pendulum:\[T = 2\pi \sqrt {\frac{{\sqrt {{a^4} - {a^2}{x^2} + {b^2}{x^2}} \sqrt {2a\left( {a - \sqrt {{a^2} - {x^2}} } \right)} }}{{bxg}}} \]into those of the mathematical pendulum, let \(x = \text{sin}\,(\varphi_{0}) a\) and \(a = b = 1\). Thus applies:\[T = 2\pi \sqrt {\frac{{{a^2}\sqrt {2\left( {1 - \sqrt {1 - \frac{{{x^2}}}{{{a^2}}}} } \right)} }}{{xg}}} = 2\pi \sqrt {\frac{{l\sqrt {2(1 - \cos \,{\varphi _0})} }}{{\sin \,{\varphi _0}\,\,g}}} .\]Therefore finally is:\[T = 2\pi \sqrt {\frac{l}{{\cos \left( {{\textstyle{{{\varphi _0}} \over 2}}} \right)g}}} .\]

Test devices

2 | table clamps for the bench with two holders for the springs |

2 | 1 m long tension springs with a spring constant of approx. 2 N/m and a diameter of approx. 0.01 m |

1 | 2 m long air cushion lane with air regulator |

2 | lane carriages with blind and external hook of each 0.2 kg of mass |

1 | sphere mass of approx. 5 kg of mass |

1 | pendulum thread of at the most 1.60 m length |

1 | stop watch |

2 | ceiling hooks |

1 | up to 2 m extendable support with holder |

7 | sleeves |

7 | bars with twice 1 m, once 0.6 m, twice 0.5 m, once 0.1 m and once 0.2 m length as angle bars |

1 | clamp for protractor with small piece of wood |

1 | protractor |

1 | adhesive tape |

as well as various burden weights

Tables

Tab. 1

\(\varphi_{0}\)/degree | \(T_{exp}\)/s | \(T_{thn}\)/s | \(T_{tha}\)/s | \(\Delta T_{an}\)/% | \(\Delta T_{aa}\)/% |

90 | 2.95 | 3.02 | 3.00 | 2.26 | 1.52 |

80 | 2.87 | 2.90 | 2.89 | 1.02 | 0.58 |

70 | 2.78 | 2.80 | 2.80 | 0.86 | 0.61 |

60 | 2.72 | 2.73 | 2.72 | 0.26 | 0.13 |

50 | 2.65 | 2.67 | 2.66 | 0.60 | 0.54 |

40 | 2.61 | 2.62 | 2.62 | 0.31 | 0.29 |

30 | 2.58 | 2.58 | 2.58 | 0.09 | 0.08 |

20 | 2.55 | 2.56 | 2.56 | 0.29 | 0.29 |

10 | 2.52 | 2.54 | 2.54 | 0.90 | 0.90 |

5 | 2.51 | 2.54 | 2.54 | 1.15 | 1.15 |

\(T_{exp} =\) experimentally determined periodic time (average values)

\(T_{thn} =\) theoretical periodic time by the new procedure

\(T_{tha} =\) theoretical periodic time by the old procedure

\(\Delta T_{an} =\) absolute difference of \(T_{exp}\) and \(T_{thn}\) with \(T_{thn}\) as reference value

\(\Delta T_{aa} =\) absolute difference of \(T_{exp}\) and \(T_{tha}\) with \(T_{tha}\) as reference value

Here are:

l = 1.60 m

g = 9.80665 m/s\({}^{2}\)

Tab. 2

\(\varphi_{0}\)/degree | \(f(T_{0})_{n}\) | \(f(T_{0})_{a}\) | \(\Delta f(T_{0})_{rkl}\) | \(\Delta f(T_{0})_{rgr}\) | \(\Delta f(T_{0})_{a}\) |

90 | 1.18921 | 1.18034 | 0.74558 | 0.75118 | 0.88665 |

80 | 1.14254 | 1.13749 | 0.44214 | 0.44411 | 0.50517 |

70 | 1.10489 | 1.10214 | 0.24815 | 0.24877 | 0.27418 |

60 | 1.07457 | 1.07318 | 0.12916 | 0.12933 | 0.13879 |

50 | 1.05042 | 1.04978 | 0.06045 | 0.06049 | 0.06350 |

40 | 1.03159 | 1.03134 | 0.02418 | 0.02418 | 0.02494 |

30 | 1.01748 | 1.01741 | 8·10\({}^{-3}\) | 8·10\({}^{-3}\) | 8·10\({}^{-3}\) |

20 | 1.00768 | 1.00766 | 1·10\({}^{-3}\) | 1·10\({}^{-3}\) | 1·10\({}^{-3}\) |

10 | 1.00191 | 1.00191 | 1·10\({}^{-4}\) | 1·10\({}^{-4}\) | 1·10\({}^{-4}\) |

5 | 1.00048 | 1.00048 | 6·10\({}^{-6}\) | 6·10\({}^{-6}\) | 6·10\({}^{-6}\) |

4 | 1.00030 | 1.00030 | 3·10\({}^{-6}\) | 2·10\({}^{-6}\) | 2·10\({}^{-6}\) |

3 | 1.00017 | 1.00017 | 8·10\({}^{-7}\) | 7·10\({}^{-7}\) | 8·10\({}^{-7}\) |

2 | 1.00008 | 1.00008 | 3·10\({}^{-7}\) | 2·10\({}^{-7}\) | 2·10\({}^{-7}\) |

1 | 1.00002 | 1.00002 | 2·10\({}^{-7}\) | 1·10\({}^{-7}\) | 1·10\({}^{-7}\) |

0 | 1.00000 | 1.00000 | 0 | 0 | 0 |

\(f(T_{0})_{n} =\) function value of the periodic time of the mathematical pendulum, with which the periodic time of the Cycloid pendulums must be multiplied, by the new procedure

\(f(T_{0})_{a} =\) the same by the old procedure

\(\Delta f(T_{0})_{rkl} =\) smaller relative difference of the two

\(\Delta f(T_{0})_{rgr} =\) larger relative difference of the two

\(\Delta f(T_{0})_{a} =\) absolute difference of the two

Here are all \(\Delta f(T_{0})_{x}\) specified in per cent.

5. Bibliography

1. | Arnold, Günter Formeln der Mathematik, Hrsg. von H. Netz 3. Auflage, 1977, Carl Hauser, München (9) |

2. | Beyer, William H. (Herausgeber) CRC Handbook of Mathematical Sciences 5. Auflage, 1978, CRC Press Inc., 2255 Palm Beach Lakes Blvd. (8) |

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4. | Dorn, Friedrich / Bader, Franz Physik in einem Band (4) und Physik-Oberstufe, Band O (1) Neubearb. Ausgabe, 1976, Hermann Schroedel, Hannover |

5. | Fichtenholz, G. M. Differential- und Integralrechnungen 1 dt. Orig.ausg., 1964, VEB Deutscher Verlag der Wissenschaften, Berlin (6) |

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7. | Sieber, Helmut Mathematische Begriffe und Formeln 1. Auflage, 1973, Ernst Klett, Stuttgart (7) |

8. | Spiegel, Murray R. Allgemeine Mechanik dt. Orig.ausg., 1976, McGraw-Hill Inc., Düsseldorf (2) |

Addendum:

Today, I am certainly in a position to calculate the correct periodic times and to state the error concerning my approximate formula.

It applies:\[mg{h_0} = mgh + {\textstyle{1 \over 2}}m\frac{{d{s^2}}}{{d{t^2}}}.\]With \(x = x(t)\) and \(y = y(t)\) applies:\[T = \sqrt {\frac{8}{g}} \int\limits_0^{{s_0}} {\sqrt {\frac{1}{{{h_0} - h}}} ds = \sqrt {\frac{8}{g}} \int\limits_0^{{x_0}} {\sqrt {\frac{{1 + {{y'}^2}}}{{{y_0} - y}}} } dx} = \sqrt {\frac{8}{g}} \int\limits_0^{{t_0}} {\sqrt {\frac{{{{\dot x}^2} + {{\dot y}^2}}}{{{y_0} - y}}} } dt.\]The approximate formula reads:\[T=2\pi \sqrt{\frac{\sqrt{2lh}}{a}}=2\pi \sqrt{\frac{\sqrt{2{{\left( 1+{y}'{{(0)}^{2}} \right)}^{\tfrac{3}{2}}}{{y}_{0}}\left( {1+{y}'}_{0}^{2} \right)}}{g{{{{y}'}}_{0}}\sqrt{{y}''(0)}}}=2\pi \sqrt{\frac{\sqrt{2{{\left( \dot{x}{{(0)}^{2}}+\dot{y}{{(0)}^{2}} \right)}^{\tfrac{3}{2}}}{{y}_{0}}\left( \dot{x}_{0}^{2}+\dot{y}_{0}^{2} \right)}}{g{{{\dot{y}}}_{0}}\sqrt{\dot{x}(0)\ddot{y}(0)-\dot{y}(0)\ddot{x}(0)}}}.\]With \(x = a \text{ sin} \,\varphi, y = b - b \text{ cos} \,\varphi\),\[ds = \sqrt {d{x^2} + d{y^2}} = \sqrt {{a^2}{{\cos }^2}\varphi + {b^2}{{\sin }^2}\varphi } \,d\varphi ,\,\,k = \sin \,\left( {{\textstyle{{{\varphi _0}} \over 2}}} \right)\]and\[{\varepsilon ^2} = \frac{{{a^2} - {b^2}}}{{{a^2}}}\]exactly applies for the ellipse pendulum:\[T = \sqrt {\frac{{8{a^2}}}{{bg}}} \int\limits_0^{{\varphi _0}} {\sqrt {\frac{{1 - {\varepsilon ^2}{{\sin }^2}\varphi }}{{\cos \,\varphi - \cos \,{\varphi _0}}}} } d\varphi .\]With \(\varepsilon^{2} < 1\) the calculation of the integral yields the series expansion:\[T = \frac{{2\pi a}}{{\sqrt {bg} }}\left\{ {1 + {\textstyle{1 \over 4}}{k^2} - {\varepsilon ^2}{k^2} + {\textstyle{9 \over {64}}}{k^4} + {\textstyle{3 \over 8}}{\varepsilon ^2}{k^4} - {\textstyle{3 \over 4}}{\varepsilon ^4}{k^4} + \mathcal{O}({k^6})} \right\}.\]The series expansion of the approximate formula\[T = \frac{{2\pi a\sqrt[4]{{1 - {\varepsilon ^2}{{\sin }^2}{\varphi _0}}}}}{{\sqrt {bg\cos \left( {{\textstyle{{{\varphi _0}} \over 2}}} \right)} }}\]reads:\[T = \frac{{2\pi a}}{{\sqrt {bg} }}\left\{ {1 + {\textstyle{1 \over 4}}{k^2} - {\varepsilon ^2}{k^2} + {\textstyle{5 \over {32}}}{k^4} + {\textstyle{3 \over 4}}{\varepsilon ^2}{k^4} - {\textstyle{3 \over 2}}{\varepsilon ^4}{k^4} + \mathcal{O}({k^6})} \right\}.\]Both formulas are identical up to the quadratic term as for the mathematical pendulum and confirm the quality of the approximation procedure. To obtain a better approximate formula, \(y(x)\) is to develop into a Taylor series around \(x = 0\). For \(|y'| < 1\), then numerator and denominator of the integral can be developed into series and easily be integrated after multiplying them out. Otherwise, the integral can be computed by numerical quadrature or by development of a Taylor series of the integrand with arbitrary precision. The latter is always possible by suitable substitution of \(x\), and sufficient differentiability of the integrand. Interval arithmetic secures the results.

My actual aim was less the calculation of concrete periodic times, but the mathematical reduction of the difficult integration to a relatively easy differentiation (with simple integration). Precisely this yields the just mentioned development of a Taylor series, which is possibly to repeat for separate interval limits.

© 1983-2011 by Boris Haase

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