1 Poblem Pendulum in Obit Kik T. McDonald Joseph Heny Laboatoies, Pinceton Univesity, Pinceton, NJ 08544 (Decembe 1, 2017) Discuss the fequency of small oscillations of a simple pendulum in obit, say, about the Eath, supposing that the point of suppot of the pendulum is much moe massive than the bob of the pendulum, and the suppot point is in a cicula obit. 2 Solution The suppot point is taken to be at adius R fom the cente of the (spheical) Eath whose mass is M. Then, this point moves with angula velocity Ω = GM/R 3 = g/r with espect to an inetial fame which the cente of the Eath is at est, whee g = GM/R 2 is the acceleation due to gavity at adius R. 1 We ae pehaps most inteested in the motion as would be epoted by an obseve at the point of suppot of the pendulum, so we wok in a otating fame, centeed on the Eath, whose angula velocity is Ω, which is pependicula to the plane of the obit of the suppot point. Of couse, the suppot point is at est in this fame. We use a ectangula coodinate system centeed on the suppot point (O in the figue above, fom [1]), with the z-axis along the vecto R fom the cente of the Eath to the 1 Note that the angula velocity of small oscillations of a pendulum of length l is ω = g/l if its suppot point is at est in an inetial fame with acceleation g due to gavity. 1
suppot point, and the x-axis in the plane of the obit of the latte. Then, the angula velocity of the otating fame is Ω =Ωŷ. The (simple) pendulum has length l andabobofmassm, and is at position x =(x, y, z) whee x 2 + y 2 + z 2 = l 2. The distance fom the cente of the Eath to the bob is x E = R + x =(x, y, z + R), x E R + z. (1) The foces in the otating fame on the bob ae that due to gavity, GMm x E x 3 E centifugal foce, Coiolis foce, = m Ω2 R 3 x E m ( Ω2 x E m x 3 Ω2 1 3z ) (x ˆx + y ŷ + R(1 + z/r) ẑ) E (1 + z/r) 3 R m Ω 2 (x ˆx + y ŷ +(R 2z) ẑ), (2) m Ω (Ω x E )=m Ω 2 [x ˆx +(R + z) ẑ], (3) 2m Ω ẋ =2m Ω(ż ˆx ẋ ẑ), (4) and the tension T = T x in the massless od/sting of the pendulum. To avoid need fo knowledge of the constaint foce T, we conside the toque, τ = x F total, and angula momentum, L = x mẋ of the bob, about the suppot point, dl dt = mx ẍ = m[(y z zÿ) ˆx +(zẍ x z) ŷ +(xÿ yẍ) ẑ] = τ = x m[ Ω 2 y ŷ +3Ω 2 z ẑ +2Ω(ż ˆx ẋ ẑ) T ˆx] = m[ω 2 (yz ˆx xy ẑ)+3ω 2 (yz ˆx xz ŷ) 2Ω(yẋ ˆx +(xẋ zż) ŷ + yż ẑ)]. (5) Hence, the equations of motion can be witten as y z zÿ = 4Ω 2 yz 2Ω yẋ, (6) zẍ x z = 3Ω 2 xz 2Ω(xẋ zż), (7) xÿ yẍ = Ω 2 xy 2Ω yż. (8) The conditions fo equilibium, at which all time deivatives vanish, ae xy = yz = xz =0. These ae satisfied at the six locations (±l, 0, 0), (0, ±l, 0) and (0, 0, ±l) of the bob, as shown in the figue on p. 1. 2.1 The Equilibia at (±l,0, 0) ae Unstable Fo motion nea these equilibium points, both ẋ and ẍ ae small. Fo motion in the x-y plane (with z = 0), eq. (8) implies that ÿ = Ω 2 y,sosmall oscillations can exist in y. Howeve, fo motion in the x-z plane (with y = 0), eq. (7) implies that z =4Ω 2 z,soany small petubation in z would gow exponentially with time. Hence, these equilibia ae unstable. 2
2.2 The Equilibia at (0, ±l, 0) ae Unstable Fo motion nea these equilibium points, both ẏ and ÿ ae small. Fo motion in the x-y plane (with z = 0), eq. (8) implies that ẍ =Ω 2 x,soanysmall petubation in x would gow exponentially with time. Fomotioninthey-z plane (with x = 0), eq. (6) implies that z =4Ω 2 z,soanysmall petubation in z would gow exponentially with time. Hence, these equilibia ae unstable. 2.3 The Equilibia at (0, 0, ±l) ae Stable Fo motion nea these equilibium points, both ż and z ae small. Fomotioninthex-z plane (with y = 0), eq. (7) implies that ẍ = 3Ω 2 x, 2 so small oscillations in x can exist with angula fequency ω xz = 3Ω = 3g/R. Fomotioninthey-z plane (with x = 0), eq. (6) implies that ÿ = 4Ω 2 y,sosmall oscillations in y can exist with angula fequency ω yz =2Ω=2 g/r. That the two fequencies ω xz and ω yz ae diffeent is a consequence of the diffeent symmeties of the gavitational and centifugal foces; the fome is spheically symmetic while the latte is axially symmetic (about y). The peiods 2π/ 3Ω and π/ω of these oscillations ae independent of the length l of the pendulum, and ae of the same ode as the peiod 2π/Ω ( 90 min) of the (low-eath-) obital motion. 3 Hence, astonauts in a space station would tend to say that a pendulum does not oscillate (accoding to thei expectations of peiod 2π l/g = l/r 2π/Ω 1s fom expeience on Eath). 4 The equations of motion fo oscillations in the x-z o y-z planes have no tems (at fist ode in ẋ o ẏ) associated with the Coiolis foce, so the small oscillations of a pendulum in obit do not exhibit the pecession fist discussed by Foucault [3]. Howeve, since the fequencies of oscillation in x-z and y-z planes ae incommensuate, the geneal motion of the pendulum ove long times would be consideed as chaotic by an astonaut, even fo small oscillations (unless the oscillation wee puely in the x-z o in the y-z planes). A Appendix: Shote Deivation of Motion in the Plane of the Obit of the Suppot Point A somewhat biefe deivation was given in Appendix 17 of [4], using consevation of enegy, E = T + V, to deduce the motion in the plane of the obit of the suppot point fom the time deivative Ė = T + V =0. 2 Thee is also a second-ode, Coiolis tem 2Ω(ż xẋ/l) that we neglect fo small oscillations in the x-z plane about (0, 0, ±l). 3 These esults agee with those found in [1], whee a cleve vaiant of Lagange s method was employed using a Lagange multiplie, consideing the elation x 2 +y 2 +z 2 = l 2 to be a constaint. Lagange s method was also used in [2], fo spheical coodinates. 4 See, fo example, https://www.quoa.com/would-a-pendulum-swing-in-obit. 3
The potential enegy V of the bob of mass m at distance fom the cente of the Eath is, in the otating fame whee the centifugal can be elated to the centifugal potential, 5 V = GMm m(ω )2 2 ( ) ( mω 2 2 2 + R3, V mω 2 ṙ R3 2 ), (9) whee the appoximation holds fo a pendulum of length l R. Anticipating that the pendulum might oscillate about the vetical fom the cente of the Eath to the suppot point, we use a spheical coodinate system (ρ, θ, φ) (in the otating fame) whose oigin is at the suppot point, whose z-axis points away fom the cente of the Eath, and with φ =0andπ in the plane of the obit of the suppot point. Then, the kinetic enegy of the bob of the pendulum of length l is T = ml2 2 ( θ 2 + φ 2 sin 2 θ), T = ml 2 ( θ θ + φ φ sin 2 θ + φ 2 θ sin 2θ). (10) Futhemoe, to a vey good appoximation, = R + l cos θ, soṙ = l θ sin θ, and { } V mω 2 Rl θ 1 sin θ 1+(l/R)cosθ + 3mΩ 2 l 2 sin θ cos θ. (11) [1 + (l/r)cosθ] 2 Then, fo motion in the plane of the obit of the suppot point, φ =0, Fo θ = ɛ o π + ɛ and small ɛ, wehavethat 0= T + V = ml 2 θ( θ +3Ω 2 sin θ cos θ). (12) 0 ɛ +3Ω 2 ɛ. (13) The the angula velocity of small oscillations of the pendulum in the plane of the obit of the suppot point, about eithe θ =0oπ, is 3Ω = 3g/R, as found peviously in sec. 2.3. Note how eq. (12) also shows that θ = π and 3π/2 coespond to the unstable equilibia of sec. 2.1. Refeences [1] J.L. Synge, On the Behaviou, Accoding to Newtonian Theoy, of a Plumb Line o Pendulum Attached to an Atificial Satellite, Poc. Roy. Iish Acad. A 60, 1 (1959), http://physics.pinceton.edu/~mcdonald/examples/mechanics/synge_piaa_60_1_59.pdf [2] L. Blitze, Equilibium and stability of a pendulum in an obiting spaceship, Am.J. Phys. 47, 241 (1979), http://physics.pinceton.edu/~mcdonald/examples/mechanics/blitze_ajp_47_241_79.pdf 5 Both the Coiolis foce and the tension in the od/sting of the pendulum do no wok, and so do not contibute to the potential enegy in the otating fame. 4
[3] L. Foucault, Démonstation physique du mouvement de otation de la tee au moyen du pendule, Comptes Rendus Acad. Sci. 32, 135 (1851), http://physics.pinceton.edu/~mcdonald/examples/mechanics/foucault_cas_32_135_51.pdf [4] E.J. van de Heide and M. Kuijff, StaTack: A swinging tethe assisted e-enty fo the Intenational Space Station, Eu. Space Agency, EWP 1883 (1996), http://physics.pinceton.edu/~mcdonald/examples/mechanics/vandeheide_ewp_96.pdf 5