SERIES ⎜ ARTICLE Snippets of Physics 8. Foucault Meets Thomas T Padmanabhan T he Foucaultpendulum isan elegantdevice that dem onstrates the rotation of the Earth. A fter describingit,w e w illelaborateon an interesting geom etrical relationship betw een the dynam ics of the Foucaultpendulum and T hom as preces- T Padmanabhan works at sion d iscu ssed in th e last installm en t1 . T h is w ill IUCAA, Pune and is interested in all areas helpus to understand both phenom ena better. of theoretical physics, T h e ¯ rst titular p residen t of th e F ren ch rep u b lic, L ou is- especially those which have something to do with N ap oleon B on ap arte,p erm itted F ou cau lt to u se th e P an - gravity. th eon in P aris to give a d em on stration of h is p en d u lum (with a 67 m eterw ireand a 28 kg pendulum bob)on 31 M arch 1851. In th is im p ressive ex p erim en t, on e cou ld see the plane of oscillation of the pendulum rotating in a clockw ise direction (w hen view ed from the top) w ith afrequency! = − cosμ,where − isthe angular fre- 1 Thomas Precession, Reso- nance, Vol.13, No.7, pp.610– quen cy of E arth 's rotation and μ is th e co-latitud e of 618, July 2008. P aris. (T h at is, μ isthe standard polarangle in spher- ical p olar co ord inates w ith th e z-axisbeingtheaxisof rotation of E arth . S o ¼=2 μ is th e g e o g ra p h ic a l la ti- tude).Foucaultclaimed,quitecorrectly,that¡ thise®ect arises d u e to th e rotation of th e E arth an d th u s sh ow ed th at on e can d em on strate th e rotation of th e E arth b y an in situ ex p erim ent w ithou t look ing at celestial ob jects. T h is resu lt is q u ite easy to u n d erstan d if th e exp erim en t w as p erform ed at th e p oles or eq u ator (instead of P aris!). T h e situation at th e N orth P ole is as sh ow n in Figure1. H ere w e see th e E arth as rotating (from w est to east, in th e cou n ter-clock w ise d irection w h en v iew ed from th e Keywords top) underneath the pendulum, m aking one fullturn Spin, Thomas precession, Earth's rotation. in 24 hours. It seem s reasonableto deduce from this 706 RESONANCE ⎜ August 2008 SERIES ⎜ ARTICLE Figure 1. th at, as v iew ed from E arth , th e p lan e of oscillation of the pendulum w illm ake one fullrotation in 24 hours; so the angular frequency ! ofthe rotation of the plane of the Foucaultpendulum isjust ! = − . (Through- ou t th e d iscu ssion w e are con cern ed w ith th e rotation ofthe plane ofoscillation ofthe pendulum;not the pe- riod ofthependulum 2¼=º,which{ofcourse{isgiven b y th e stan d ard form u la involving th e len gth of th e su s- pension w ire,etc.).A t the equator,on the other hand, th e p lan e of oscillation d o es n ot rotate. S o th e form u la, Jean Bernard Lèon ! = − cosμ, cap tu res b oth th e resu lts correctly. Foucault (1819 –1868) was a French It is trivial to w rite d ow n th e equ ation s of m otion for th e physicist, famous for pendulum bob in the rotating fram e of the Earth and the demonstration of solve them to obtainthisresult[1,2]at thelinear order Earth's rotation with in − . Essentially,the Foucaultpendulum e®ect arises his pendulum. due to the C oriolis force in the rotating fram e of the Although Earth's Earthwhichleadstoanacceleration2v − ,where v , rotation was not thevelocity ofthependulum bob,isdirected£ tangential unknown then, but this to th e E arth 's su rface to a goo d ap p rox im ation . If w e easy-to-see ch oose a local coord inate system w ith th e Z -axispoint- experiment caught ing n orm al to th e su rface of th e E arth an d th e X;Y everyone's coordinates in the tangent plane at the location, then it is easy to sh ow th at th e eq u ation s of m otion for th e imagination. RESONANCE ⎜ August 2008 707 SERIES ⎜ ARTICLE pendulum bob arew ellapproximated by XÄ + º 2 X =2− Y_ ; YÄ + º 2 Y = 2− X;_ (1) z ¡ z whereº istheperiod ofoscillation ofthependulum and − z = − cosμ is th e n orm al com p on en t of th e E arth 's angular velocity. In arriving at these equations w e have ignored the term s quadraticin − and the vertical dis- placem ent ofthependulum. T he solution to thisequa- tion is ob tained easily b y intro d u cing th e variab le q(t) ´ X (t)+ iY (t): T h is satis¯es th e eq u ation 2 qÄ +2i− z q_+ º q =0: (2) T h e solution , to th e ord er of accu racy w e are w ork ing with,isgivenby q = X (t)+ iY (t)= (X (t)+ iY (t)) ex p ( i− t); (3) 0 0 ¡ z whereX 0 (t);Y 0 (t) isthe trajectory ofthe pendulum in th e ab sen ce of E arth 's rotation . It is clear th at th e n et e® ect of rotation is to cau se a sh ift in th e p lan e of rota- tion at the rate − z = − cosμ. B ased on thisknow ledge and the resultsfor the poleand the equator one can give a `plainEnglish'derivation oftheresultforinterm ediate la titu d e s b y sa y in g so m e th in g lik e : \ O b v io u sly , it is th e com ponent of − n orm al to th e E arth at th e location of thependulum w hich m attersand hence ! = − cosμ." T h e ¯ rst-p rinciple ap p roach , b ased on (1), of cou rse h as the advantage ofbeing rigorous and algorithm ic;forex- am ple, ifyou w ant to take into account the e®ects of ellipticity of E arth , you can d o th at if you w ork w ith theequationsofm otion. B utitdoesnotgive you an in- tuitive understan d ing of w h at is going on , an d m u ch less a uni¯ed view of allrelated problem s havingthe sam e stru ctu re. W e sh all n ow d escrib e an ap p roach to th is problem w hich has the advantage of providing a clear geom etrical p ictu re an d con n ecting it u p { som ew h at q u ite su rp risingly { w ith T h om as p recession d iscu ssed in th e last installm en t. 708 RESONANCE ⎜ August 2008 SERIES ⎜ ARTICLE O ne point w hich causes som e confusion as regards the Foucaultpendulum isthe follow ing. W hileanalyzing the behavior ofthe pendulum at the pole,one assum es th at th e p lan e of rotation rem ains ¯ x ed w h ile th e E arth rotates underneath it.Ifw e m ake thesam e claim for a pendulum experiment done at an interm ediatelatitude, { i.e ., if w e sa y th a t th e p la n e o f o sc illa tio n re m a in s invariant w ith resp ect to, say, th e \¯ x ed stars" an d th e A Earth rotatesunderneath it{ itseem s naturalthat the period ofrotation ofthependulum plane shouldalways b e 24 h ou rs irresp ective of th e location ! T h is, of cou rse, is n o t tru e a n d it is a lso in tu itiv e ly o b v io u s th a t n o th in g h ap p en s to th e p lan e of rotation at th e eq u ator. In th is w ay ofapproachingtheproblem ,itisnotvery clearhow exactlytheEarth'srotation in°uencesthem otion ofthe pendulum. Figure 2. T o p rov ide a geom etrical ap p roach to th is p rob lem , w e w ill re p h ra se it a s fo llo w s [3 , 4 ]. T h e p la n e o f o sc illa - tion of the pendulum can be characterized by a vector norm alto itor equivalently by a vector w hich islying in th e p la n e and tan gen tial to th e E arth 's su rface. L et us now introduce a cone w hich iscoaxialw ith the axis of rotation of th e E arth an d w ith its su rface tan gen tial to the Earth at the latitude of the pendulum (see Fig- ure 2). T he base radiusof such a cone w illbe R sin μ , whereR is th e rad ius of th e E arth an d th e slant h eigh t of the cone w illbe R tan μ.