Three Spreadsheet Models Of A Simple Pendulum

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1 Spreadsheets in Education (ejsie) Volume 3 Issue 1 Article Three Spreadsheet Models Of A Simple Pendulum Jan Benacka Constantine the Philosopher University, Nitra, Slovakia, jbenacka@ukf.sk Follow this and additional works at: This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 icense. Recommended Citation Benacka, Jan (008) Three Spreadsheet Models Of A Simple Pendulum, Spreadsheets in Education (ejsie): Vol. 3: Iss. 1, Article 5. Available at: This In the Classroom Article is brouht to you by the Bond Business School at epublications@bond. It has been accepted for inclusion in Spreadsheets in Education (ejsie) by an authorized administrator of epublications@bond. For more information, please contact Bond University's Repository Coordinator.

2 Three Spreadsheet Models Of A Simple Pendulum Abstract The paper ives three spreadsheet models of simple pendulum motion. In two of them, the raph of the pendulum anle is drawn upon the exact interal solution usin numeric interation the simpler model only uses spreadsheet functions, the eneral model uses a VBA proram. The period results directly from the calculations. In the third model, the period is calculated first usin the power series formula. The raph of the pendulum anle is drawn afterwards usin Euler s method of solvin differential equations. The error in ravity acceleration if calculated upon the standard cosine approximate formula instead of the exact one is raphed. Keywords simple pendulum, period, spreadsheet model, ravity acceleration Distribution icense This work is licensed under a Creative Commons Attribution-Noncommercial-No Derivative Works 4.0 icense. Cover Pae Footnote The article was created in the frame of the Socrates Comenius.1 scheme under the project No CP MOTIVATE ME in Maths and Science This in the classroom article is available in Spreadsheets in Education (ejsie):

3 Benacka: THREE SPREADSHEET MODES OF SIMPE PENDUUM Three spreadsheet models of a simple pendulum Jan Benacka Constantine the Philosopher University jbenacka@ukf.sk Abstract The paper ives three spreadsheet models of simple pendulum motion. In two of them, the raph of the pendulum anle is drawn upon the exact interal solution usin numeric interation the simpler model only uses spreadsheet functions, the eneral model uses a VBA proram. The period results directly from the calculations. In the third model, the period is calculated first usin the power series formula. The raph of the pendulum anle is drawn afterwards usin Euler s method of solvin differential equations. The error in ravity acceleration if calculated upon the standard cosine approximate formula instead of the exact one is raphed. Keywords: simple pendulum, period, spreadsheet model, ravity acceleration Quantities and symbols α anle ravity acceleration β anle h pendulum altitude ε computational error in time m pendulum mass ϕ pendulum anle v pendulum velocity η relative error in t Time ω pendulum anular velocity F Force Φ pendulum amplitude G ravitational force: G = m Σ summation of the power series H pendulum startin altitude in the period formula pendulum lenth T pendulum period 1. Introduction Measurin the acceleration of ravity by simple pendulum (a bob on a weihtless cord) is a standard physical experiment in upper secondary schools. Students are acquainted with the fact that the function ( t) = Φcos( t ) ϕ, (1) where t is time, ϕ is the anle, 0<Φ π is the amplitude, and is the lenth of the pendulum (see the table of symbols), is just an approximate one and usable at Φ 5 only, which holds as well for the emerin formulas for the period T and the ravity acceleration T= π, = 4π T. (a-b) 008 Bond University. All rihts reserved. Published by epublications@bond,

4 Spreadsheets in Education (ejsie), Vol. 3, Iss. 1 [008], Art. 5 SIMPE PENDUUM Fiure 1: Simple pendulum There are three topic questions from the students: If 5 < Φ 90 : what is the raph of the exact solution like; what are the correct alternatives to formulae (a, b), and how bi is the error if one uses formula (b)? The answers follow from the exact solution to the pendulum problem. That is iven by an analytically incalculable definite interal. Gettin a quarter-period of the raph (which enables us to construct the entire period) requires evaluatin the interal at each ϕ from Φ to 0 by a step Φ 100 e..; the interal is evaluated a hundred times at each chane of Φ, then. Generally, that is impossible without prorammin. However, the monotonicity of the interand enables us to pre-calculate the number of subintervals of the main interval for ettin a iven accuracy. et ε be the computational error in time. If 0 < Φ 90, 10 m, ms -, and ε s, then the number of the subintervals comes out less than 170, which allows us to use standard spreadsheet skills i.e. avoid prorammin. That is the first model. It is simple, and fully interactive. The period is calculated from the quarter-period atϕ = 0. The second model is a eneral variant of the first one (a bob on a weihtless rod) and uses a VBA proram. The model works at any physically correct inputs, i.e., 0 < Φ < 180, > 0 m, > 0 ms -, ε > 0 s, and is only limited by the accuracy of the spreadsheet proram. The calculations take some time, thus the model reacts with a delay. The third model is based upon a numeric solution of the overnin differential equation. The period is calculated first usin the power series formula. The raph of the pendulum anle is drawn afterwards usin Euler s method of solvin differential equations that is simple to understand. Nevertheless, the solution is impossible without usin some tricks that makes it interestin. The model is valid at 0 < Φ 90, 100 m, ms -, ε s, and it is quick and fully interactive. In each model, the exact formula for follows from the period formula. The error in if calculated upon the cosine approximate formula (b) instead of the exact one is computed and raphed at Φ from 0 to 90. ejsie 3(1) In The Classroom 60

5 Benacka: THREE SPREADSHEET MODES OF SIMPE PENDUUM JAN BENACKA. Exact solution to simple pendulum and the exact formula for computin Remark: The theory presented in this section can be found e.. in [1 6]. The enery of the pendulum is iven by the law of enery conservation (Fi. 1) mh = mh+ mv, (3) where m is the mass, v is the velocity, h is the altitude, and H is the startin altitude. Then ( H h) v=. (4) As H = cosφ, h= cosϕ, v= ω= dϕ (ω is the anular velocity), then dϕ = ϕ ( cos cosφ), Φ ϕ Φ, ( 0) = Φ ϕ. (5) This results in the fundamental formula that ives the inverse function to ϕ ( t) t= Φ ϕ dα ( cosα cosφ), Φ ϕ Φ, (6) where α is anle between Φ and ϕ. The interal is incalculable analytically. Moreover, it is improper as lim ( cosα cosφ) = 0 α Φ, and this excludes numeric interation. Applyin formula cosα = 1 sin ( α ) and substitution of sin( ) sin( α ) sin( Φ ) where β is anle, ives π ( Φ ) β =, dβ t =, (7) 1 sin sin β a where a= arcsin[ sin( ϕ ) sin( Φ ) ]. This interal is proper as sin ( ) < 1 If ϕ = 0, then t= T 4, so π ( ) Φ at any Φ. dβ T = 4. (8) 1 sin sin β 0 Φ The interal is the complete elliptic interal of the first kind. The power expansion is 1 T π ( Φ ) = 1+ sin + sin ( Φ ) +... = π Σ, (9).4 Published by epublications@bond,

6 Spreadsheets in Education (ejsie), Vol. 3, Iss. 1 [008], Art. 5 SIMPE PENDUUM where Σ is for the summation in the brackets. The correct formula for computin is then = 4π Σ T. (10) et 0 be the value computed upon Eq. (b). The relative error is then 0 = 1 1 η = 1 Σ. (11) We can see that η depend on amplitude Φ but not on lenth. We note that Eq. (5) is obtainable by solvin the motion equation of pendulum (Fi. 1) dv m + msinϕ= 0, ϕ ( 0) = Φ, ( 0 ) = 0 v. (1) Substitution v= ω= dϕ yields the famous pendulum equation d ϕ + sinϕ = 0 dϕ ϕ, = 0. (13), ( 0) = Φ t= 0 Usin d ϕ dω dω dϕ dω = = = ω we et dϕ d t dω ω + sinϕ= 0, ω ( Φ) = 0 (14) with solution dϕ ω = = ϕ ( cos cosφ), ( 0) = Φ ϕ. (15) 3. Models with usin numeric interation The raph consists of two parts. The first part comprises 108 points ( t,ϕ), where ϕ oes from Φ to 0 (the first quarter-period) by steps ϕ =Φ 100 mostly; it is ϕ 4 over the first four points, and it is ϕ over the next eiht points to make the model more precise if ϕ Φ. The interal is computed at each ϕ usin the trapezoid rule. The minimum number of subintervals for ettin a iven (small) error ε in t is calculable as follows: et a arcsin[ sin( ϕ ) sin( Φ ) ] =, b= π, n be the number of subintervals of ejsie 3(1) In The Classroom 6 4

7 Benacka: THREE SPREADSHEET MODES OF SIMPE PENDUUM JAN BENACKA interval k=, f( β) 1 1 k β a, b, sinφ ( ) = sin be the interand. Then Δ β = ( b a)n, βi = βi 1 + Δβ, β 0 = a, β n = b, and the trapezoid rule ives ( a) + f( b) n 1 f = + t f( βi) Δβ. (16) i= 1 Fiure : The trapezoid rule and the maximum error As f ( β) is ascendin (see sheet Interand and Fi. ), the maximum absolute error ε in time is ε = f ( β ) f( β ) f( β ) f( β ) f( β ) f( β ) f( β ) f( β ) n 1 n n n Δβ,(17) then ( b) f( a) f b a ε =, (18) n which yields n= f ( b) f( a) b a. (19) ε If 0 < Φ 90, 10 m, ms -, and ε s, then n < 170, which allows us to use standard spreadsheet skills only. That is the idea that the first model is based on. The application is in sheet Model1_InteralExcel. The main calculations are in the hidden columns N GH. Function IF is used in columns T H to match the actual value Published by epublications@bond,

Spreadsheets in Education (ejsie), Vol. 3, Iss. 1 [008], Art. 5 SIMPE PENDUUM of n. The inputs stated above are the limitin ones. The raph of the approximate cosine solution (Eq.

8 Spreadsheets in Education (ejsie), Vol. 3, Iss. 1 [008], Art. 5 SIMPE PENDUUM of n. The inputs stated above are the limitin ones. The raph of the approximate cosine solution (Eq. 1) is added to make a comparison. The model is in Fi. 3. Fiure 3: Model 1 The second model is a eneral variant of the first one and uses a VBA proram. The proram is a method of a hidden textbox, and it starts at the chane of the textbox. The textbox is linked with a hidden cell (columns N, O) that contains the sum of all inputs. Hence, if any input is chaned, which is acknowleded by the Enter key, the textbox chanes and the proram starts. The model works at any physically correct inputs, i.e. 0 < Φ < 180, > 0 m, > 0 ms -, ε > 0 s, and is only limited by the accuracy of the spreadsheet proram. The calculations take some time, thus the model reacts with a delay; that is why no scrollbar is used. The runnin calculation is indicated by a label WAIT. If Φ =90, = 1 m, =9. 81 ms -, and ε = s, then the delay is about 3 s; if ε = s, then it is 1 s (at Celeron 1.5 GHz, 51 MB RAM). If Φ >90, then the delay increases huely, thus ε s should be used. If Φ = 179, 99, = 1000 m, =9.81 ms -, and ε = s, then the delay is about 55 s. The model with these inputs is in Fi. 4 (the cosine raph is cleared away because there is no point in drawin that at such amplitude as almost seven rouhly drawn cosine periods come out within the exact one). The application is in sheet Model_InteralVBA. The proram (with comments) is accessible by double-click on the textbox. ejsie 3(1) In The Classroom

Benacka: THREE SPREADSHEET MODES OF SIMPE PENDUUM JAN BENACKA Fiure 4: Model The second part of the raphs in both models comprises 31 points ( t,ϕ), where ϕ oes from 0 to Φ (the second

9 Benacka: THREE SPREADSHEET MODES OF SIMPE PENDUUM JAN BENACKA Fiure 4: Model The second part of the raphs in both models comprises 31 points ( t,ϕ), where ϕ oes from 0 to Φ (the second quarter-period), then ϕ oes back from Φ to Φ (the second half-period), and time t is computed upon the symmetry of the raph usin functions INDIRECT and ADRESS. The sheets are locked with no password. The white cells are accessible only. The formulae are hidden under a white rectanle below the raph. 4. The model with usin Euler s method of solvin differential equations Substitutin k= sinφ ( ) into Eq. (9) ives T = T0 1+ k + k + k (0) The series T major = T0 1+ k + k + k +.. = T0 1 k (1) is a majorant one to Eq. (0) i.e. each term is bier than the correspondin term in Eq. (0). If we take the first n terms, then the remainder (i.e. the error) is Published by epublications@bond,

10 Spreadsheets in Education (ejsie), Vol. 3, Iss. 1 [008], Art. 5 SIMPE PENDUUM ε = T0 1 1 k 1+ k + k + k +.. () This is sure to be bier then the correspondin remainder of period T, i.e. the error that we et if we take the first n terms in Eq. (0) is sure to be less than ε. The spreadsheet application that enables us to find the minimum n is in sheet Minimum N for T. It results that n = 3 is enouh if 0 < Φ 90, 100 m, ms -, and ε s. The redundant terms of the iven n = 5 ones are in red numbers. The model uses Euler s method of solvin differential equations for drawin the raph at 0 t T. Replacin the differentials with differences in Eq. (5) ives ϕ = ( cosϕ cosφ) t, ϕ (0) = Φ. (3) If ϕ oes from Φ to Φ, then ϕ i = ϕ i 1 ϕ, ϕ 0 = Φ, i = 1,, k, where k is the number of subintervals Δ t of period T, i.e. Δ t=t k. If ϕ oes back from Φ to Φ, then ϕ i = ϕ i 1 + ϕ. The application is in sheet Model3_Euler. The chane from to + in the formula for ϕ i is realized by the IF function nested twice (hidden column Q). The problem with the zero derivative of ϕ at t = 0 ( ϕ = Φ then) that causes ϕ = 0 at any i is settled with a trick: if (the IF function) the square root in Eq. (3) ives zero, then we put i ϕ =10 15 (the lowest possible nonzero value in a spreadsheet cell) or else ϕ is iven by Eq. (3) (column S). The exact period is calculated in rane N1:O39. Despite the simplicity of the method, k = 000 is enouh for ettin ϕ = at t= T if the inputs are Φ = 90, 100 m, ms -, and ε s, which are the limitin inputs for the model. It is locked without a password. The formulae are hidden under a white rectanle below the raph. The model is in Fi. 5. ejsie 3(1) In The Classroom

The exact period is computed upon Eq. (0) usin the first n = 5 terms. If Φ =( ) 5, 16, 36, 5, 90, then η = (%) 0.

11 Benacka: THREE SPREADSHEET MODES OF SIMPE PENDUUM JAN BENACKA Fiure 5: Model 3 The application that computes the error η in (Eq. 11) at various Φ and draws the η (Φ) raph is in sheet Error in. The exact period is computed upon Eq. (0) usin the first n = 5 terms. If Φ =( ) 5, 16, 36, 5, 90, then η = (%) (<0.1), 0.97 (<1), 4.86 (<5), 9.99 (<10), 8., respectively. The η (Φ) raph is in Fi. 6. Fiure 6: Error in Published by epublications@bond,

12 Spreadsheets in Education (ejsie), Vol. 3, Iss. 1 [008], Art. 5 SIMPE PENDUUM 5. Conclusions The simple pendulum is involved in the physics curriculum in hiher secondary school (ae 15 19) usually, where measurements of upon formula (b) are carried out. One way of usin the models is just to show them to the students and study the behaviour at various inputs, especially to show that the cosine and exact raphs mere into one at small amplitudes, and that formula (b) is absolutely unusable at Φ >16 as the error is bier than 1 %. Instead, formula (10) has to be used, where the period T is measured by a stopwatch, and the sum is computed (e.. like in sheet Error in ). Another way is to build up a model with students interested in physics and mathematics who have ood spreadsheet skills. The model with the VBA proram is an example of prorammin mathematical tasks, and it is meant for students who are interested in prorammin. The proram is easily transferable to other developin environments (the author has developed a Borland Delphi version, e..). However, the two other pure spreadsheet models are quick and precise enouh, and the input ranes exceed by far the needs of school practice. Quantity Symbol (Unit) Model 1 Model Model 3 Amplitude Φ ( ) enth (m) > Gravity acceleration (ms - ) > Computational error in time ε (s) > Table 1: Summarization of the models ejsie 3(1) In The Classroom

13 Benacka: THREE SPREADSHEET MODES OF SIMPE PENDUUM JAN BENACKA References [1] Fulcher,.P. and Davis, B.F., 1976, Theoretical and experimental study of the motion of the simple pendulum. Am. J. Phys. 44, [] Barer, V. and Olsson, M., 1995, Classical mechanics: a modern perspective, nd ed. McGraw-Hill, New York, pp [3] Kidd, R.B. and Fo, S.., 00, A simple formula for the lare-anle pendulum period. Phys. Teach [4] Fay, T.H, 00, The pendulum equation. Int. J. Math. Educ. Sci. Techn., 33 (4), [5] ima, F.M.S. and Arun P., 006, An accurate formula for the period of a simple pendulum oscillatin beyond the small-anle reime. Amer. J. Phys. 74 (10), [6] Pendulum (mathematics). In Wikipedia, the free encyclopedia, retrieved 1:30, Jun 5, 008, from Published by epublications@bond,

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