Approximation for the large-angle simple pendulum period

Transcription

1 Approximation for the large-angle simple pendulum period A. Beléndez, J. J. Rodes, T. Beléndez and A. Hernández Departamento de Física, Ingeniería de Sistemas y Teoría de la Señal. Universidad de Alicante. Apartado 99. E Alicante. SPAIN a.belendez@ua.es ABSTRACT An approximation scheme to obtain the period for large amplitude oscillations of a simple pendulum is analysed and discussed. The analytical approximate formula for the period is the same as that suggested by Hite (2005 Phys. Teach ), but it is now obtained analytically by means of a term-by-term comparison of the power-series expansion for the approximate period with the corresponding series for the exact period. 1

2 The simple pendulum is one of the most popular examples analysed in textbooks [1], Physics Education journals [2-16] and undergraduate courses, being also one of the nonlinear systems most studied not only in advanced but also in introductory university courses of classical mechanics. The simple pendulum is usually the first example of a nonlinear problem that many university teachers discuss in their classes and probably it is the first time that students analyze and understand the differences between linear and nonlinear physical problems. If students just can understand the physical and mathematical aspects of this example, the will be able to understand in the future more complicated nonlinear problems in physics, engineering and mathematics and the common methods used to deal with these nonlinear problems such as perturbation theory [17, 18], whereby the solution is analytically expanded in the power series of a small parameter, or harmonic balance method [10]. Several approximation schemes have been developed to investigate the situation for large amplitude oscillations of a simple pendulum, and a variety of approximations have been proposed for calculating its large-angle period with precision [2-16]. Through the use of ingenious strategies, different authors [2-11] have obtained explicit formulae for the period of the pendulum, all of which are deduced through a heuristic derivation and these expressions are accurate for large oscillation amplitudes. In this note we analyze another analytical approximate formula for the period, which it is analogous to that suggested by Hite [8] using a reasoning based on a graphical procedure. However we obtain this expression analytically by means of a term-by-term comparison of the power-series expansion for the approximate period with the corresponding series for the exact period. In 2002, Kidd and Fogg [2] proposed an approximate formula for the large-angle pendulum period. This expression was based on the analogy between the equation of the period of a pendulum in the small-angle regime and the period of a simple harmonic oscillator and it may be expressed as [2] T app1 / T 0 = cos "1/2 (# 0 / 2) = (1" k 2 ) "1/ 4 (1) where k = sin(! 0 / 2) and θ 0 is the amplitude of oscillations. This approximation is good for small values of k. This approximate expression, as others proposed by Ganley [3], Cadwell and Boyco [4], Molina [5] or Parwani [6], depends only on one parameter. We have verified in all these cases that this parameter can be found by comparing term-byterm the power-series expansions of the exact and the approximate expressions for the period. Subsequently, Millet [7] proposed a mathematical justification for the Kidd and Fogg formula by considering trigonometric relations and small-angle approximations for sine and cosine functions, as well as the comparison between the sine function and various of its linear approximations (see Fig. 1 in reference 7). Beléndez et al [19] proposed an approximate formula depending on two parameters and Hite [8] also suggested a new more accurate approximate expression for the period of a simple pendulum. Unlike all expressions previously published, and as in the Beléndez at al formula [19], the Hite formula depends on two parameters instead of one, and he found them by using a graphical analysis. He compared the plot of the neperian logarithm of the quotient between the exact and the approximate periods as a function of k. 2

3 As Millet did with the Kidd and Fogg expression, in this paper we suggest an alternative and easier justification of the approximate expression for the period of a simple pendulum proposed by Hite. Instead of using plots of complicated functions and graphical reasoning, we will see how is possible to obtain the two parameters Hite expression by doing a term-by-term comparison of the series expansion for the approximate period with the corresponding series for the exact period. The period of a simple pendulum T depends on the amplitude θ 0 and its exact expression is [1] T / T 0 = 2K(k ) / " (2) where K(k) is the complete elliptic integral of the first kind, k has been defined previously and T 0 is the period for small-angle oscillations, T 0 = 2! l / g, being l the length of the pendulum and g the acceleration due to gravity. The values of K(k) have been tabulated for various values of k. The approximation for the period T app1 in Eq. (1) proposed by Kidd and Fogg may be generalized by means of the equation [2] T app2 / T 0 = cos "n/2 (# 0 / 2) = (1" k 2 ) "n/ 4 (3) where n is a constant to be determined. Eq. (3) is a one-parameter expression. Hite plotted ln{t 0 / T}/ ln(1" k 2 ) versus k, where T 0 / T = " /[2K(k )] (see Eq. (2)), and he obtained that for small values of k this expressions tends to 1/4. However, as k increases the curve falls rapidly and finally approaches zero for k equal to 1 [8]. By following the Hite reasoning, a better approximation could be obtained by replacing the constant n in Eq. (3) by one that depends on k (or on cos(! 0 / 2) ) by means of a function F(k) and we have to write nf(k) instead of n in Eq. (3) T app3 / T 0 = [cos(" 0 / 2)] # n 2 F(k ) = (1# k 2 ) # n 4 F(k ) (4) Hite [8] also concluded that the function F(k) should be equal to 1 for k = 0 and to zero for k = 1. Taking this into account, we can consider the following functional form for this F(k) F(k ) = (1" k 2 ) m/ 2 = cos m (# 0 / 2) (5) where m is a constant. Then, the improved approximation T app3 for the period may be written as T app3 / T 0 = (1" k 2 ) " n 4 (1"k 2 ) m / 2 = [ cos(# 0 / 2) ] " n [ 2 cos(# 0 /2) ] m (6) which is a two parameters expression, m and n, which are unknown and must be found by using some technique. A simple way to obtain them will be presented next. The 3

4 power-series expansion of the expression for the exact period of the pendulum given in Eq. (2) is [1] T / T 0 = " " " (7) We can now expand in power-series expansion the approximate expression of the period given in Eq. (6). To do it, series can be easily generated with a MATHEMATICA software package. The result obtained is T app3 / T 0 =1+ n 16 " n(4 #12m + 3n) " (8) 3072 By compare term-by-term Eqs. (7) and (8) it can be concluded that values of the parameters must be n = 1 and m = 1/8. By using them, Eq. (6) can be rewritten as T app3 / T 0 = (1" k 2 ) " 1 4 (1"k 2 ) 1/16 = [ cos(# 0 / 2) ] "1 [ 2 cos(# 0 /2) ] 1/ 8 (9) This is the expression obtained by Hite by means of a different procedure. The quality of this approximation can be determined by comparing Eq. (7) with the power-series expansion of Eq. (9), which can be written as follows T app3 / T 0 = " " " (10) From this comparison, we can see that the first three terms of these two series are identical and the fourth term of Eq. (10) approaches that of the exact solution. The difference between the coefficients for the corresponding fourth terms (power of " 0 6 ) is less than 3.3%. If we take into account the power-series expansion of the modified Kidd and Fogg approximation (Eq. (3)) and we compare term-by-term this series with Eq. (7), we can find the value of the unknown parameter n, which must be n = 1. Then, the power-series expansion for the Kidd and Fogg approximation is [2] T app1 / T 0 = " " " (11) The first two terms of series in Eqs. (7) and (11) are identical and the difference 4 between the coefficients of third terms (power of! 0 ) is around 27%. This is a considerable large relative error. By comparing Eqs. (10) and (11) it can be concluded that the Kidd and Fogg approximate formula is not as good as the Hite expression given in Eq. (9), as Hite pointed out in his paper by means of a different type of comparison between them. Finally, we think that the method followed in this paper for justifying approximate large-angle period formulas is easier to understand than the graphic method used previously in Hite s paper. Finally, this method could be possibly used to find 4

5 better and easier approximations using a two parameters scheme and, perhaps, three parameters could even be considered. Acknowledgments This work has supported by the Generalitat Valenciana of Spain (Grant. No. GVPRE/2008/182). References [1] Baker G L and Blackburn J A 2005 The Pendulum: A Case Study in Physics (Oxford University Press: Oxford) [2] Kidd R B and Fogg S L 2002 A simple formula for the large-angle pendulum period Phys. Teach. 40 (2) 81-3 [3] Ganley W P 1985 Simple pendulum approximation Am. J. Phys [4] Cadwell L H and Boyco E R 1991 Linearization of the simple pendulum Am. J. Phys [5] Molina M I 1997 Simple linearizations of the simple pendulum for any amplitude Phys. Teach. 35 (8), [6] Parwani R R 2004 An approximate expression for the large angle period of a simple pendulum Eur. J. Phys. 25, [7] Millet L E 2003 The large-angle pendulum period Phys. Teach. 41 (3) [8] Hite G E 2005 Approximations for the period of a simple pendulum Phys. Teach. 43 (5) [9] Carvalhaes C G and Suppes P 2008 Approximations for the period of the simple pendulum based on the arithmetic-geometric mean Am. J. Phys [10] Beléndez A, Hernández A, Márquez A, Beléndez T and Neipp C 2006 Analytical approximations for the period of a simple pendulum Eur. J. Phys [11] Lima F M S and Arun P 2006 An accurate formula for the period of a simple pendulum oscillating beyond the small angle regime Am. J. Phys [12] Gil S, Legarreta A E and Di Gregorio D E 2008 Measuring anharmonicity in a large amplitude pendulum Am. J. Phys [13] Beléndez A, Hernández A, Beléndez T, Neipp C and Márquez A 2007 Application of the homotopy perturbation method to the nonlinear pendulum Eur. J. Phys [14] Amore P, Valdovinos M C, Orneles G and Barajas S Z 2007 The nonlinear pendulum: formulae for the large amplitude period Rev. Mex. Fis. E

6 [15] Lima F M S 2008 Simple log formulae for pendulum motion valid for any amplitude Eur. J. Phys [16] Beléndez A, Alvarez M L, Fernández E and Pascual I 2009 Linearization of conservative nonlinear oscillators Eur. J. Phys [17] Amore P and Aranda A 2005 Improved Lindstedt-Poincaré method for the solution of nonlinear problems J. Sound Vib [18] He J H 2006 Some asymptotic methods for strongly nonlinear equations Int. J. Mod. Phys. B [19] Beléndez A, Hernández A, Beléndez T, Márquez A and Neipp C 2007 An improved heuristic approximation for the period of a nonlinear pendulum: linear analysis of a nonlinear problem Int. J. Non-linear Sci. Numer. Simulat