The Moving Center of Mass of a Leaking Bob
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1 The Moving Center of Mss of Leking Bob rxiv: v1 [physics.pop-ph] 21 Feb 2010 P. Arun Deprtment of Electronics, S.G.T.B. Khls College University of Delhi, Delhi , Indi. Februry 2, 2010 Abstrct The evlution of vrition in oscilltion time period of simple pendulum s its mss vries proves rich source of discussion in physics clss-room, overcoming erroneous notions crried forwrd by students s to wht constitutes pendulum s length due to picking up only the results of pproximtions nd ignoring the rigorous definition. The discussion lso presents exercise for evluting center of mss of geometricl shpes nd system of bodies. In ll, the pedgogicl vlue of the problem is worth both theoreticl nd experimentl efforts. This rticle discusses the theoreticl considertions. 1 Introduction Wht hppens to simple pendulum s oscilltion time period with vrying mss? This question is of pedgogicl interest. An rticle in Americn Journl of Physics [1] ddresses this issue from n experimentl point of view, explining vrition in time period with oscilltion of burrette whose liquid content drips. From n introductory clss point of view, the experiment misses mny importnt issues by using rigid pendulum insted of simple pendulum. In this rticle, using theoreticl considertions bsic ides of defining length of the pendulum (simple s it my look, the fine print is mostly overlooked, clcultion of position of center of mss of body nd there-fter system of two bodies etc. The oscilltion time period of simple pendulum is given s L T o = 2π g (1 e-mil:runp92@physics.du.c.in 1
2 where L is the length of the pendulum nd g, the ccelertion due to grvity. The bsence of mss term in the bove expression would either imply the incompleteness of the expression vi pproximtions in derivtion or some oversight. A brief review of the derivtion shows F = mgsinθ where the generl expression of force is F = m ( dv (2 giving m ( dv = mgsinθ The liner velocity v, cn be converted to ngulr velocity with useful pproximtion (the smll ngle pproximtion, i.e. sinθ θ using the reltion Thus, ( d 2 θ L 2 ( d 2 θ 2 v = L dθ = gθ = g L = ω2 o θ ( It is from ω o = g, tht we obtin eqn(1. As per the eqution, the undmped motion of L the simple pendulum is indeed mss independent. But is the bove derivtion rigorous nd exhustive? 2 Time period of Leking Pendulum The bove derivtion innocuously drops n importnt definition of force, defined s rte of chnge of momentum, i.e. F = dp = d(mv. Eqn(2 follows only if mss is constnt, which is not the cse for leking pendulum. Thus, the derivtion would require modifictions. d(mv [ d m d(lθ ] = mgsinθ = mgθ 2
3 L d [ m dθ ] d 2 ( θ 1 + dm dθ m = mgθ ( = ω 2 oθ (5 Eqn(5 is typiclly tht of dmped pendulum nd the time period would be given s T = 1 L g T o ( 1 m (6 2 dm The expression typiclly shows how the time period would vry with vrition in mss. However, from the observtions of the burette experiment s lso from our observtions in cse of pendulum mde with hollow bob filled with wter, the time period initilly increses nd then strts flling. Eqn(6 cn not explin this observtion considering the rte of chnge of mss of leking pendulum will lwys hve vlues greter thn or equl to zero. To investigte further into the eqution, we consider the length of the pendulum lso to be chnging with time. Thus, the bove derivtion chnges from the point of eqn(. Tht is, d 2 ( θ 2 + dl 2 L + 1 m ( dm dθ 1 + d 2 L L ml giving n expression for time period s 2π = g ( 1 T L + Ld 2 L ml dm d(mv [ d m d(lθ ] dmdl = mgsinθ = mgθ θ = ω 2 o θ dl 1 ( 2 dl L + 1 m 2 dm (7 For those who missed the rigorous definition of wht constitutes the pendulum length would ponder how the length of the string used to suspend the bob would vry with time. Tht is, students crry wrong notion tht the length of the pendulum is the length of the string. This, however is only true if the bob is dense nd considered point mss with bob s rdius fr smller thn the length of the string. Prcticlly, this is not the cse nd the length of the pendulum would be length of the string nd the rdius of the bob. The rdius is included since the whole mss of the bob is concentrted t its center, or its center of mss. The length of the pendulum is hence rigorously defined s distnce between point of suspension to the center of mss of the pendulum. In cse of leking bob, the decresing wter would give moving center of mss (see fig 1. The length of the pendulum then cn be written s L = l+r o χ (8
4 where l, r o nd χ is the length of the string, rdius of the bob nd the position of the center of mss written with respect to the bob s center (set s origin respectively. It is cler s wter drips, initilly the center of mss moves down giving +dl/. At some point when wter content is low, the center of mss would tend to move bck to the center, leding to -dl/. This chnge in sign would explin incresing time period followed by decrese in it. The leking pendulum thus not only helps to illustrte the requirement to pprecite the wht is the length of the pendulum, but lso dds the need for n expression of center of mss s function of the wter level or its mss. In the pssges below we proceed to find n expression for the center of mss of our bob consisting shell with wter. y o x Figure 1: Vrition of COM with wter level nd resultnt vrition in time period of oscilltion. In the following section we derive n expression for the vrible mss simple pendulum, eseenetilly considering the bob to be hollow shell filled with wter nd s it leks the wter level vries. Where is the Center of Mss.1 Of Shell The clcultion of the center of mss (COM of body strts with evlution of the body s mss. Since the shell in question hs sphericl symmetry, the integrl to clculte the mss is best done in polr coordintes. Thus, m shell = ρ shell o r= 2π π/2 θ=0 φ= π/2 dr rdθ rcosφdφ
5 = ρ shell o r= 2π π/2 θ=0 φ= π/2 r 2 cosφdrdθdφ = πρ shell( o (9 where o nd re the outer nd inner rdius of the shell whose density is ρ shell. The generl expression for COM is given s χ = Σm ir i M where M is the totl mss of the body nd m i is the mss of smll volume (dx i dy i dz i of the body t r i from the origin. Applying this to the problem of shell (using eqn 9, we hve (10 = ρ o 2π shell m shell 0 = ρ shell( o m shell = ρ shell( o m shell = ρ shell( o m shell = 0î+0ĵ +0ˆk π/2 π/2 2π π/2 0 π/2 2π π/2 0 [ r 2 cosφdrdθdφ(rcosθcosφî+rsinθcosφĵ +rsinφˆk π/2 0î+0ĵ +2π The COM is t the center of the shell. cosφdθdφ(cosθcosφî+sinθcosφĵ +sinφˆk (cosθcos 2 φî+sinθcos 2 φĵ +sinφcosφˆkdθdφ π/2 π/2 (sinφcosφdφˆk ] (11.2 Wter Body The wter mss filled in the shell, when the shell is filled cn be considered to be sphere of rdius, the shell s inner rdius. Now, consider disc of thickness dz is cut from the sphere t distnce z from the center. The disc hs rdius r (see fig 2 nd hence it s re would be The disc volume would be A = πr 2 dv = πr 2 dz The mss ssocited with this disc would be dm = ρ wter dv = ρ wter πr 2 dz (12 5
6 Z xis + r dz 0 z Figure 2: For deriving the center of mss of wter body, we cn consider the body to be sum of pile of disc of thickness dz t distnce z from the center. The disc rdius r would be function of the distnce of the disc from the center. The net mss of the wter sphere hence would be M wter = ρ wter π Butrdiusofdiscwoulddependonhowfrwyfromthecenter isthedisccut, hencer r(z, which is obtined from simple rule r 2 = 2 z 2. Hence, M wter = ρ wter π ( 2 z 2 dz ( = ρ wter π 2 z z [( ( ] = ρ wter π + ( π = ρ wter In cse the sphere is not completely filled then the clcultions remin the sme, however, the limits chnge. Sy the wter level is t h, the limits chnge nd clcultions proceed s h M wter = ρ wter π ( 2 z 2 dz 6 r 2 dz
7 ( = ρ wter π 2 z z h [( ( ] = ρ wter π 2 h h + ( 2 = ρ wter π +2 h h (1 The position of the center of mss then is clculted by solving the following integrl χ wter = ρ wterπ M wter = ρ wterπ M wter = ρ wterπ M wter = ρ wterπ M wter h ( 2 z 2 2 [( 2 h 2 ( z( 2 z 2 dz 2 z h h + 2 h 2 h 2 ( ] 2 (1. Of Leking Bob The COM of the pendulum s bob mde by thin shell filled with wter cn be evluted using stndrd formul χ bob = m shellχ shell +M wter χ wter m shell +M wter (15 As evluted in eqn(11, the shell s COM will lwys be t it s center, which we tke s the origin. Hence, eqn(15 reduces to Using eqn(9, eqn(1 nd eqn(1 we hve χ bob = = = χ bob = M wterχ wter m shell +M wter (16 ρ wter π ( + 2 h 2 h 2 πρ shell( o +ρ wter π ( 2 +2 h h ( ρ wter + 2 h 2 h 2 ρ ( shell( o +ρ 2 wter +2 h h ρ wter ( +2 2 h 2 h 16ρ shell ( o +ρ wter (2 + 2 h h 7
8 The bove eqution will give the vrition of bob s COM s wter leks. The density of wter is unity, hence χ bob = ( +2 2 h 2 h 16ρ shell ( o +(2 + 2 h h (17 Forthepurposeofplottingitwould bebetter todefinenormlized vrible, (h/ndrewrite the bove eqution. We hve [ 1+2 ( 2 ( ] h h χ bob = = [ (o ] 16ρ shell 1 + [2+ ( h ( h ( 1+2x 2 x 16ρ shell [ (o 1 ]+(2+x x where o would hve vlue greter thn unity. The first term in the denomintor would depend on the shell s thickness nd it s density, hence we write ] (18 χ bob = ( 1+2x2 x K +(2+x x (19 Center of Mss K=0.01 K=0.05 K=0.10 K=0.0 K=0.60 K=1.00 T (in sec K=0.01 K=0.05 K=0.10 K=0.0 K=0.60 K= x/ x/ Figure : Vrition of COM with wter level nd resultnt vrition in time period of oscilltion. The bove mthemtics imply tht the vrying mss of the pendulum (s wter flows from the bob results in chnge in center of mss nd in turn the length of the pendulum (bsiclly from results flowing from eqn 8, eqn 1 nd eqn 19. In other words the length of vrying msspendulumiscloselyrelteothemss. Figureshowsthechnging positionofthebob s 8
9 center of mss s the wter level in it chnges long with resulting oscilltion time period. The curves were generted using eqn(19, eqn(8 nd eqn(1. Though not n exhustive clcultion it esily shows the increse followed by decrese in time period of oscilltion s the pendulum s mss vries. The plots re fmily of curves, generted for vrious shell prmeters (K, which depends of the shell mteril s density nd it s thickness. It is cler from these curves tht to get good resolvble experimentl results, it is best to use shell of very smll thickness nd moderte density ( K smll. This ensures the center of mss of the shell-wter body system is strongly controlled by rhe wter body. Bsed on these ides experimentl results re being gthered nd would be reported in future. Conclusion The rticle discusses the problem of vrible mss pendulum. The incresing followed by decresing time period of oscilltion is novel feture. Though simply explined vi vrition in the pendulum length, the discussion of this problem in clssroom is of rich pedgogicl vlue. Experimentl verifiction of these ides would further enrich the experience of pplying concepts such s center of mss which is usully studied with theoreticl emphsis. References [1] Rymond W. Mires nd Rndll D. Peters, Am. J. Phys. 62 (!
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