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Exact Solution for the Nonlinear Pendulum (Solu¸C˜Aoexata Do Pˆendulon˜Aolinear)

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Exact Solution for the Nonlinear Pendulum (Solu¸C˜Aoexata Do Pˆendulon˜Aolinear) Revista Brasileira de Ensino de F¶³sica, v. 29, n. 4, p. 645-648, (2007) www.sb¯sica.org.br Notas e Discuss~oes Exact solution for the nonlinear pendulum (Solu»c~aoexata do p^endulon~aolinear) A. Bel¶endez1, C. Pascual, D.I. M¶endez,T. Bel¶endezand C. Neipp Departamento de F¶³sica, Ingenier¶³ade Sistemas y Teor¶³ade la Se~nal,Universidad de Alicante, Alicante, Spain Recebido em 30/7/2007; Aceito em 28/8/2007 This paper deals with the nonlinear oscillation of a simple pendulum and presents not only the exact formula for the period but also the exact expression of the angular displacement as a function of the time, the amplitude of oscillations and the angular frequency for small oscillations. This angular displacement is written in terms of the Jacobi elliptic function sn(u;m) using the following initial conditions: the initial angular displacement is di®erent from zero while the initial angular velocity is zero. The angular displacements are plotted using Mathematica, an available symbolic computer program that allows us to plot easily the function obtained. As we will see, even for amplitudes as high as 0.75¼ (135±) it is possible to use the expression for the angular displacement, but considering the exact expression for the angular frequency ! in terms of the complete elliptic integral of the ¯rst kind. We can conclude that for amplitudes lower than 135o the periodic motion exhibited by a simple pendulum is practically harmonic but its oscillations are not isochronous (the period is a function of the initial amplitude). We believe that present study may be a suitable and fruitful exercise for teaching and better understanding the behavior of the nonlinear pendulum in advanced undergraduate courses on classical mechanics. Keywords: simple pendulum, large-angle period, angular displacement. Este artigo aborda a oscila»c~aon~ao-linearde um p^endulosimples e apresenta n~aoapenas a f¶ormula exata do per¶³odo mas tamb¶ema dependencia temporal do deslocamento angular para amplitudes das oscila»c~oese a freqÄu^enciaangular para pequenas oscila»c~oes. O deslocamento angular ¶eescrito em termos da fun»c~aoel¶³ptica de Jacobi sn(u;m) usando as seguintes condi»c~oesiniciais: o deslocamento angular inicial ¶ediferente de zero enquanto que a velocidade angular inicial ¶ezero. Os deslocamentos angulares s~aoplotados usando Mathematica, um dispon¶³vel programa simb¶olicode computador que nos permite plotar facilmente a fun»c~aoobtida. Como ver- emos, mesmo para amplitudes t~aograndes quanto 0,75¼ (135o) ¶eposs¶³vel usar a express~aopara o deslocamento angular mas considerando a express~aoexata para a freqÄu^enciaangular w em termos da integral el¶³pticacompleta de primeira esp¶ecie.Conclu¶³mosque, para amplitudes menores que 135o, o movimento peri¶odicoexibido por um p^endulosimples ¶epraticamente harm^onico,mas suas oscila»c~oesn~aos~aois¶ocronas(o per¶³odo ¶euma fun»c~aoda amplitude inicial). Acreditamos que o presente estudo possa tornar-se um exerc¶³cioconveniente e frut¶³feropara o ensino e para uma melhor compreens~aodo p^endulon~ao-linearem cursos avan»cadosde mec^anicacl¶assicana gradua»c~ao. Palavras-chave: p^endulosimples, per¶³odo a grandes ^angulos,deslocamento angular. Perhaps one of the nonlinear systems most studied ferential equation with a nonlinear term (the sine of an and analyzed is the simple pendulum [1-12], which is angle). It is possible to ¯nd the integral expression for the most popular textbook example of a nonlinear sys- the period of the pendulum and to express it in terms of tem and is studied not only in advanced but also in elliptic functions. Although it is possible in many cases introductory university courses of classical mechanics. to replace the nonlinear di®erential equation by a corre- The periodic motion exhibited by a simple pendulum sponding linear di®erential equation that approximates is harmonic only for small angle oscillations [1]. Be- the original equation, such linearization is not always yond this limit, the equation of motion is nonlinear: feasible. In such cases, the actual nonlinear di®erential the simple harmonic motion is unsatisfactory to model equation must be directly dealt with. the oscillation motion for large amplitudes and in such In this note we obtain analytical exact formulas for cases the period depends on amplitude. Application of the period of a simple pendulum and we present some Newton's second law to this physical system gives a dif- ¯gures in which the angular displacement is plotted as 1E-mail: [email protected]. Copyright by the Sociedade Brasileira de F¶³sica.Printed in Brazil. 646 Bel¶endez at al. a function of the time for di®erent initial amplitudes. where we use the trigonometric relation Moreover, we stress that one may show to the stu- µ ¶ θ dents the advantage of using available symbolic com- cos θ = 1 ¡ 2 sin2 : (8) puter programs such as Mathematica. 2 As we can point out previously, one of the simplest Now let nonlinear oscillating systems is the simple pendulum. This system consists of a particle of mass m attached y = sin(θ=2) (9) to the end of a light inextensible rod, with the motion and taking place in a vertical plane. The di®erential equa- tion modelling the free undamped simple pendulum is 2 k = sin (θ0=2): (10) 2 d θ 2 From Eqs. (3), (9) and (10) we have 2 + !0sinθ = 0; (1) dt p where θ is the angular displacement, t is the time and y(0) = k: (11) !0 is de¯ned as It is easy to obtain the value of dθ=dt as a function r of dy=dt as follows. Firstly, from Eq. (10), we have g !0 = : (2) l dy dy dθ 1 dθ = = cos(θ=2) (12) Here l is the length of the pendulum and g is the ac- dt dθ dt 2 dt celeration due to gravity. Because of the presence of and secondly the trigonometric function sinθ, Eq. (1) is a nonlinear di®erential equation. µ ¶ µ ¶ µ ¶ dy 2 1 θ dθ 2 We consider that the oscillations of the pendulum = cos2 = are subjected to the initial conditions dt 4 2 dt µ ¶ µ ¶ µ ¶ 1 £ ¤ dθ 2 1 ¡ ¢ dθ 2 dθ 1 ¡ sin2(θ=2) = 1 ¡ y2 :(13) θ(0) = θ0 = 0 (3) 4 dt 4 dt dt t=0 Then, we have where θ0 is the amplitude of the oscillation. The sys- tem oscillates between symmetric limits [-θ ,+θ ]. The µ ¶2 µ ¶2 0 0 dθ 4 dy periodic solution θ(t) of Eq. (1) and the angular fre- = 2 : (14) quency ! (also with the period T = 2¼/!) depends on dt 1 ¡ y dt the amplitude θ0. Substituting Eqs. (9), (10) and (14) into Eq. (7), one Equation (1), although straightforward in appear- get ance, is in fact rather di±cult to solve because of the µ ¶ 4 dy 2 ¡ ¢ nonlinearity of the term sinθ. In order to obtain the = 4!2 k ¡ y2 ; (15) exact solution of Eq. (1), this equation is multiplied by 1 ¡ y2 dt 0 dθ=dt, so that it becomes which can be rewritten as follows dθ d2θ dθ µ ¶2 µ ¶ + !2sinθ = 0 (4) dy y2 2 0 = !2k(1 ¡ y2) 1 ¡ : (16) dt dt dt dt 0 k which can be written as We do de¯ne new variables ¿ and z as " µ ¶ # d 1 dθ 2 y ¡ !2cosθ = 0 (5) ¿ = !0t and z = p : (17) dt 2 dt 0 k Equation (5), which corresponds to the conservation Then Eq. (16) becomes of the mechanical energy, is immediately integrable, µ ¶ dz 2 taking into account initial conditions in Eq. (3). From = (1 ¡ z2)(1 ¡ kz2); (18) Eq. (5) we can obtain d¿ where 0 < k < 1, and µ ¶2 dθ 2 µ ¶ = 2!0 (cos θ ¡ cos θ0) (6) dz dt z(0) = 1 = 0: (19) d¿ which can be written as follows ¿=0 Solving Eq. (18) for d¿ gives µ ¶ · µ ¶ µ ¶¸ dθ 2 θ θ dz = 4!2 sin2 0 ¡ sin2 ; (7) d¿ = §p : (20) dt 0 2 2 (1 ¡ z2)(1 ¡ kz2) Exact solution for the nonlinear pendulum 647 The time ¿ to go from the point (1,0) to the point ½ · µ ¶ (z,dz/d¿) in the lower half {plane of the graph of dz/d¿ θ θ θ(t) = 2 arcsin sin 0 sn K sin2 0 ¡ as a function of z is 2 2 ¸¾ Z z d³ 2 θ0 ¿ = ¡ p (21) !0t; sin (31) 2 2 2 1 (1 ¡ ³ )(1 ¡ k³ ) In Figs. 1-6 we have plotted the exact angular dis- Equation (21) can be rewritten as follows placement θ as a function of !0t (Eq. (31)) for di®er- ent values of the initial amplitude θ0. Plots have been R 1 d³ obtained using the Mathematica program. In these ¯g- ¿ = 0 p ¡ (1 ¡ ³2)(1 ¡ k³2) ures we have also© included£ ¡ the angular¢ displacement¤ª θ0 2 θ0 2 θ0 θ(t) = 2 arcsin sin sn K sin ¡ !0t ; sin . R z d³ 2 2 2 p ; (22) As we can see, for amplitudes θ < 0:75¼ (135±) it 0 2 2 0 (1 ¡ ³ )(1 ¡ k³ ) would be possible to use the approximate expression which allows us to obtain ¿ as a function of z and k as ½ · µ ¶ θ θ θ(t) = 2 arcsin sin 0 sn K sin2 0 ¡ ¿(z) = K(k) ¡ F (arcsin z; k); (23) 2 2 ¸¾ 2 θ0 where K(m) and F (';m) are the complete and the in- !0t ; sin (32) complete elliptical integral of the ¯rst kind, de¯ned as 2 follows [13] for the angular displacement of the pendulum and Z Eq.
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