arXiv:1809.03871v1 [physics.pop-ph] 8 Sep 2018 tde ic bu 50 n h emccodwsevi- was cycloid in term Galileo the by and coined dently 1500, about since studied fms c)o h iko radius of disk the center of the (cm) variables of mass the of motion of superposition of translational the constant-velocity terms from the in derived demonstration, paths versus this cycloidal P of such point parametric of for the coordinates derive time we Cartesian A the II cycloid of Sec. a blackboard equations In out the traces blackboard. disk of then the tray wooden chalk on chalk chalk the circular the and of a slipping, along piece of without disk edge a the out the roll put traces near and a can then hole on one source a slipping Alternatively, light of through without the edge cycloid. disk where the a the to [6], roll source surface and light flat disk a circular classroom affix physics a to introductory An is magnetic [5]. demonstration and cycloid electric a crossed is static fields starting uniform particle in charged a rest of from path the example, For lems. [4]. available published still was is paths and cycloidal 1878 various study in and detailed A cycloid so the periodically. loops itself of by crosses rounded, replaced path are become cusps the cycloid the that cycloid the prolate which of a in in bottom of one and the is irrespective at cycloid cycloid, curtate cusps A the the [3]. of position bottom starting of the its amount with reach same contact the to which in takes time surface for of cycloid shape frictionless influence the the the is under which cycloid particle solved, inverted a was an problem that tautochrone showed the work, early oino h etro-as(m ftebd n h ro- the mov- and body body translational rigid the the of a of (cm) on center-of-mass superposition the or of a in motion is P space point through a ing of the demonstrates motion to nicely the addition and that in path, generated cycloid be above-mentioned to paths cycloid prolate oino on ffie oteds hti dis- a is that disk the to affixed P point tance a of motion h ahmtclpoete fteccodhv been have cycloid the of properties mathematical The yodlptsotnocri rcia hsc prob- physics practical in occur often paths Cycoidal r msLbrtr n eateto hsc n srnm,Io Astronomy, and Physics of Department and Laboratory Ames rmtec.Ti rcdr nbe utt and curtate enables procedure This cm. the from etro as(m fteds n h oainlmto fPa P of motion rotational the superpositi and the disk from the P of of (cm) path mass the b of for the center of derived tray are chalk the time along versus slipping without rolling is that netdtruhahl tapitPwt radius with P point a at hole a through inserted R < r pnrsnneaefud hc hwta ylia ah ca paths cycloidal that show which found, the magneti where are the derived, of resonance path are spin the a star for a of time star about versus equations a orbit mod parametric to circular a is respect in with P is orbit of that the for position time the sl versus frictionless of equations with coordinates rolling Cartesian for sinusoidal function cycloidal a still oua lsro eosrto st rwaccodo bl a on cycloid a draw to is demonstration classroom popular A .INTRODUCTION I. craeccod and cycloid) (curtate ∼ 60[,2.A ato this of part As 2]. [1, 1600 R n h rotational the and ylia ah nPhysics in Paths Cycloidal R > r Dtd etme 2 2018) 12, September (Dated: poaeccod.I sfrhrsonta h aho is P of path the that shown further is It cycloid). (prolate ai .Johnston C. David r ≈ atfeuny h ahi eia aho h surface the on path helical a is path the frequency, nant ieaedrvdfrudme S ntegnrlcase general versus For the path in parameters. ESR the relevant undamped for of for equations derived In parametric are knowledge. time the versus our to D path II before the reported Sec. for been not equations have parametric time fig- but several in [10], shown ures and described qualitatively been has ahta sdtrie yboth by determined is that path iclryplrzdmcoaemgei field magnetic microwave circularly-polarized incodntso h oiino P. but of function, position Carte- cycloidal the sinusoidal of a the coordinates still of slipping, sian dependence is frictionless time P modified with point a rolls B with the II disk of Sec. In the path if cm. the the that about show P we point the of motion tational etprui oue rcse ntepeec fan of presence the field nu- in magnetic precesses or static volume) applied (ESR) unit electron per ment atomic magnetization the (NMR) clear how experiments, [10–13] measures (ESR) one resonance spin electron and oscil- distance. small Earth-Sun very the with to Moon cycloid relative curtate the amplitude a of lation is orbit Sun the approxi- the that about that idealization, above orbits for the with parameters mate known system, the from Moon-Earth-Sun planet show the our We and ro- star. planet relative the the about about the moon and of distance moon-planet periods tational planet-star the of the ratio to the distance on solu- depending The paths epi-prolate- star. cycloid or the epi-curtate-cycloid, about epi-cycloid, orbit yields its translational tion in the planet and the planet of of the motion motion to rotational respect the parametric with of of moon the superposition the motion the the B is are II that moon moon principle Sec. the the the In using of again coordinates determined, are Cartesian [7]. the the orbits of star for the orbits equations the where cycloidal about star, shows that moon a available orbits is turn coplanar, in that planet n,teha fthe of head the ing, to perpendicular aligned pig u hr h iedpnec fthe of dependence time the where but ipping, R nbt ula antcrsnne(M)[,9] [8, (NMR) resonance magnetic nuclear both In a orbiting moon a of motion the of simulation nice A rmtecne fawo iko radius of disk wood a of center the from akor.Hr h aaercequations parametric the Here lackboard. aSaeUiest,Ae,Iw 01,USA 50011, Iowa Ames, University, State wa ainvco uigudme electron- undamped during vector zation risaecpaa.Fnly h general the Finally, coplanar. are orbits oni iclrobtaotaplanet a about orbit circular a in moon fid nasmlrwyteparametric the way similar a In ified. cu ne eti conditions. certain under occur n no h rnltoa oino the of motion translational the of on otti mfor cm this bout cbadwt ic fchalk of piece a with ackboard M etrflosatime-dependent a follows vector H nteasneo damp- of absence the In . r M H H = 1 R aeaemgei mo- magnetic (average n continuous-wave a and ≪ H (cycloid), H and n ttereso- the at and H R 1 hspath This . H 1 htis that 2

Dividing both sides of each of these equations by R gives y dimensionless components x r cm v = ωt + cos(ωt), (4a) R cm x R R z y r = − sin(ωt). (4b) r P  ^ R R vcm = vcm i vcm = R These are the parametric equations for the reduced Cartesian components in terms of the implicit param- eter ωt. The amplitudes of the sinusoidal components of x/R and y/R have the same value r/R. FIG. 1: Experimental configuration of a disk of radius R The parametric equations for the cycloid are conven- rolling towards the right on a horizontal surface without slip- ping. The speed of the center of mass (cm) of the disk is tionally written [2] vcm = ωR and hence the velocity of the cm is vcm = ωRˆi. x = θ − sin θ, (5a) The point P is a distance r from the cm at a negative instan- R taneous polar angle φ measured clockwise from the positive y x axis. = 1 − cos θ. (5b) R One can obtain these equations from Eqs. (4) by the sub- of a sphere as shown in Sec. IID. However, when H1 stitutions ωt → θ + π/2, r/R → 1, and a y-axis offset is a significant fraction of H, a cycloidal path is found, y/R → y/R + 1. the nature of which depends on the parameters of the Shown in Fig. 2(a) are plots of x/R and y/R versus electron-spin resonance. A summary is given in Sec. III. ωt for a curtate cycloid with r/R = 0.5 using Eqs. (4). Figure 2(b) shows the parametric plot of y/R versus x/R II. RESULTS: CYCLOIDAL PATHS with ωt as the implicit parameter. A curtate cycloid can be traced on the blackboard by drilling a hole in the A. Rolling without Slipping wooden disk at radius r < R in which to insert a piece of chalk and then rolling the disk along the chalk tray as in the above procedure to generate a chalk trace of a cy- The experimental configuration is shown in Fig. 1. The cloid. Corresponding plots for the cycloid with r/R =1 disk is rolling without slipping towards the right. The are shown in Fig. 3. The derivative dy/dx of the cy- azimuthal angle φ of the point P is measured with respect cloid in Fig. 3(b) is discontinuous at the minima, to the positive x axis and is negative because P is rotating whereas for the curtate cycloid in Fig. 2(b) the minima clockwise instead of counterclockwise for which φ would are rounded. Because Figs. 2(a) and 3(a) are so sim- be positive. Because the disk is rolling without slipping, ilar, one might not anticipate the significant difference the constant speed of the center of mass (cm) with respect between Figs. 2(b) and 3(b). At sufficiently small values to the stationary surface on which the disk rolls is of r/R the y versus x become nearly sinusoidal. vcm = ωR, (1) Figure 4 shows corresonding plots for a prolate cycloid with r/R = 1.5. Here loops appear in the y/R versus where ω = |dφ/dt| is the angular speed of P with respect x/R plot in Fig. 4(b). to the cm. The position rcm of the cm is

rcm ˆi = ωRt , (2a) B. Rolling with Frictionless Slipping where t is the time. The rotational motion of P with respect to the cm is described by Here we consider a disk of radius R that is rotating at ′ angular speed ω 6= vcm/R and thus rolling with slipping ˆ ˆ rP, rot = r cos(ωzt) i + sin(ωzt) j . (2b) without friction on a surface. Here we ask the same ques- tion as in this last section: what is the path through space Since ω = −ω, one obtainsh i z of a point P that is fixed on the disk a distance r from its center? We initially assume that the angular velocity of rrot = r cos(ωt)ˆi − sin(ωt)ˆj . (2c) the disk is in the same direction as in the previous section The position r of pointh P is the superpositioni of the for rolling without slipping, but the following results are center of mass position and the rotational position in easily generalized to the case where the angular velocity Eqs. (2a) and (2c), respectively. The Cartesian compo- of the disk with slipping is in the opposite direction of the nents of r are therefore case without slipping by simply changing the sign of the parameter α introduced below from positive to negative. x = ωRt + r cos(ωt), (3a) Referring again to Fig. 1, here we write vcm = ωR y = −r sin(ωt). (3b) where ω is the angular speed of the disk if it were rolling 3

FIG. 2: (a) Cartesian components x/R and y/R of point P FIG. 3: (a) Cartesian components x/R and y/R of point P versus reduced time ωt in radians for r/R = 0.5. (b) y/R versus reduced time ωt in radians for r/R = 1. (b) y/R versus versus x/R with ωt as an implicit parameter where this path x/R with ωt as an implicit parameter where this path of point of point P is a curtate cycloid. These plots were obtained P is a cycloid. These plots were obtained using Eqs. (4). using Eqs. (4).

′ without slipping. Therefore one again has We write the relationship between ω and ω as ′ rcm = ωRtˆi (6a) ω = αω, (8) However, the rotational motion of P with respect to the where α is a dimensionless constant. Then Eqs. (7) be- cm is now described by come x r rrot = r cos(ω t)ˆi + sin(ω t)ˆj (6b) = ωt + cos(αωt), (9) z z R R h ′ ′ i y r = r cos(ω t)ˆi − sin(ω t)ˆj , (6c) = − sin(αωt). R R h i where ω′ is the angular speed of the disk which satisfies Thus the motion of point P when the disk is rolling ω′ 6= ω for rolling with slipping. The reduced x and y at constant angular speed with frictionless slipping is a components of r are now cycloidal function, but where the time dependence of the

x r ′ sinusoidal parts of x and y are changed in the same way = ωt + cos(ω t), (7a) compared with the case of rolling without slipping. In R R y r ′ particular, the path is a curtate cycloid if |α|r/R < 1, a = − sin(ω t). (7b) R R cycloid if |α|r/R = 1, and a prolate cycloid if |α|r/R > 1. 4

FIG. 4: (a) Cartesian components x/R and y/R of point P versus reduced time ωt in radians for r/R = 1.5. (b) y/R versus x/R with ωt as an implicit parameter where this path of point P is a prolate cycloid. These plots were obtained using Eqs. (4).

For the case where the disk is rotating in the opposite direction while slipping compared to the case of rolling without slipping, one replaces the positive α in Eqs. (9) by −α and the path becomes inverted. Example plots of Eqs. (9) are shown in Fig. 5 for α = 1/2 and r/R = 3 (prolate cycloid), α =1/2 and r/R = 1 (curtate cycloid), and α = −1/2 and r/R = 2 (inverted cycloid).

C. Path of a Moon with respect to a Star while Orbiting a Planet that is Orbiting the Star

Here we consider the coplanar orbits of a moon orbiting FIG. 5: Cartesian component y/R versus x/R of the path of point P with T = αωt as an implicit parameter for (a) α = 1/2 a planet while the planet orbits a star that is stationary and r/R = 3 (prolate cycloid), (b) α = 1/2 and r/R = 1 with respect to the distant stars, where the moon, planet (curtate cycloid), and (c) α = −1/2 and r/R = 2 (inverted and star are spherically symmetric, the two orbits are cycloid) calculated using Eqs. (9). circular and lie in the xy plane, and the moon and planet 5 are both moving counterclockwise in their orbits when viewed from the positive z axis. The Moon orbiting the Earth that orbits the Sun approximately satisfies these conditions and this case will be discussed below. We first define the following abbreviations for this sec- tion: r = moon to planet distance (center to center) R = planet to star distance (center to center) TM = period of the moon’s orbit about the planet TP = period of the planet’s orbit about the star ωM =2π/TM = angular speed of the moon with respect to the planet ωP =2π/TP = angular speed of the planet with respect to the star (x, y) = Cartesian coordinates of the moon’s center with respect to the star (xP,yP) = Cartesian coordinates of the planet’s center with respect to the star One expects r ≪ R, TM ≪ TP, ωM ≫ ωP. Thus we have

xP = R cos(ωPt), (10a) yP = R sin(ωPt), (10b) x = r cos(ωMt)+ R cos(ωPt), (10c) y = r sin(ωMt)+ R sin(ωPt). (10d)

We define the dimensionless parameter α ≡ ωP/ωM ≪ 1 and from Eqs. (10c) and (10d) obtain

x = r cos(ωMt)+ R cos(αωMt), (11a) y = r sin(ωMt)+ R sin(αωMt). (11b)

Using the generic expression ω =2π/T one obtains

x = r cos(2πt/TM)+ R cos(2παt/TM), (12a) y = r sin(2πt/TM)+ R sin(2παt/TM). (12b)

Finally, introducing the dimensionless reduced time

T ≡ t/TM (13) and dividing both sides Eqs. (12) by R, one obtains the dimensionless parametric equations for the path of the moon with respect to the star versus reduced time T as x r = cos(2πT ) + cos(2παT ), (14a) R R y r = sin(2πT ) + sin(2παT ). (14b) R R Parametric plots of y/R versus x/R using Eqs. (14) are shown in Fig. 6 for α = 0.1 and r/R = 0.05, 0.1, and 0.15. The latter three parameters are unrealistically large in order to clearly show the structure of the paths. The paths are analogous to those in Figs. 2–4, except that the x axes in those figures are bent here into . FIG. 6: Parametric plots of y/R versus x/R for cycloidal Thus for r/R < α one obtains an epi-curtate cycloid, for paths of a moon orbiting a planet that orbits a star with r/R = α an epi-cycloid, and for r/R > α an epi-prolate α = 0.1 and (a) r/R = 0.05 (epi-curtate cycloid), (b) r/R = cycloid, where here the prefix epi refers to linear cycloidal 0.10 (epi-cycloid), and (c) r/R = 0.15 (epi-prolate cycloid), calculated using Eqs. (14). motion bent into a . 6

The orbit of the Moon about the Earth and the Earth z, H about the Sun are approximately coplanar. When viewed from the North, the Earth rotates counter-clockwise about the Sun at a distance R = 1.50 × 1011 m with rotation period TP = 365.4 d and angular speed ωP = 2π/TP. The Moon rotates counter-clockwise around the 8 H Earth at a distance r = 3.84 × 10 m with rotation eff M period TM = 27.3 d and angular speed ωM = 2π/TM. Thus for the Moon orbiting the Earth, the parameter  α = 27.3/365.3 ≈ 0.075, roughly the same as the value α = 0.1 used to construct Fig. 6. However, the ratio  r/R ≈ 0.0026 is much smaller than the values of 0.05 to y 0.15 in Fig. 6. Thus r/R ≪ α and hence the Moon has an epi-curtate cycloidal path around the Sun corresponding  to the linear rolling with slipping path in Fig. 5(b), but with a very small amplitude of oscillation (not shown) that is barely visible on the scale of the plot in Fig. 6(a). x

D. Paths of the Magnetization Vector in FIG. 7: of the precession of the magnetization M Undamped Electron-Spin Resonance in the presence of an applied static field H and a microwave magnetic field H1. M precesses clockwise about the effective field Heff , forming the surface of a cone with cone half-angle The Bloch equations are often the starting point for ∆θ, while Heff precesses counter-clockwise about the applied analyzing experimental electron-spin resonance (ESR) field H with azimuthal angle φ = ωt on the surface of a cone data. In the absence of damping, the Bloch equations with cone angle θ0 given by Eq. (20f). After Ref. [15]. give the Cartesian components of the magnetization M (average magnetic moment per unit volume) that is pre- cessing around the applied uniform, static magnetic field For ESR experiments, an additional circularly- polarized microwave magnetic field H1 with angular fre- H = H0kˆ, (15) quency ω is applied that rotates in the xy plane about the z axis in the same direction that M is precessing in as [14] the absence of H1, given by

dMx ˆ ˆ = −γ(M × H) , (16a) H1 = −H1[cos(ωt)i + sin(ωt)j], (19) dt x dMy where the negative-sign prefactor is due to the negative = −γ(M × H)y, (16b) charge on the electron that applies to ESR as in Eqs. (16). dt Thus H1 is always antiparallel to the projection of M dMz = −γ(M × H)z, (16c) onto the xy plane. dt Qualitatively, M precesses around an effective mag- where the negative sign prefactors arise from the negative netic field Heff at angular frequency ωeff while Heff pre- charge on the electron appropriate for ESR, and γ is the cesses around the applied field H at the angular fre- gyromagnetic ratio (γ = gµB/~ for Heisenberg spins, g quency ω as shown in Fig. 7, where [15] is the spectroscopic splitting factor, µB is the Bohr mag- ~ ω neton, and is Planck’s constant divided by 2π). The Heff = −H1[cos(ωt)ˆi + sin(ωt)ˆj]+ H0 − kˆ, (20a) γ Gaussian cgs system of units is used in this section.   In the absence of damping and additional magnetic 2 2 ω fields, the Bloch equations yield a magnetization that Heff = |Heff | = H1 + H0 − , (20b) s γ precesses around H at angular frequency   2 2 ω0 = γH0 (17) ωeff = γHeff = ω1 + (ω0 − ω) (20c) ω1 ≡ γH1, q (20d) according to φ = ωt, (20e) Mx = M0 sin θ cos(ω0t), (18a) ω0 − ω θ0 = arctan (0 ≤ θ0 ≤ π/2), (20f) My = M0 sin θ sin(ω0t), (18b) ω1   Mz = M0 cos θ, (18c) θ = θ0 + ∆θ cos(ωeff t) , (20g) π where M0 = |M| and θ is the constant angle that M and the cone half-angle ∆θ with 0 < ∆θ ≤ 2 − θ0 is an makes with the z axis during the precession. adjustable parameter. Here we derive an expression for 7

as H ¯ B z, Mx = cos∆θ, ¯ B My = sin ∆θ sin(¯ωeff T ), (24) ¯ B Mz = sin ∆θ cos(¯ωeff T ).

Next we rotate MB clockwise about the y axis by a neg- π ative angle − 2 − θ0 so that Heff is at an angle of θ0 y with respect to the z axis according to Fig. 7, yielding M   ¯ C Mx = − sin ∆θ cos θ0 cos(¯ωeff T ), ¯ C M = 1 My = sin ∆θ sin(¯ωeff T ), (25) ¯ C r = sin( ) Mz = cos∆θ cos θ0 + sin ∆θ sin θ0 cos(¯ωeff T ), x Finally, rotating MC about the z axis by a positive angleωT ¯ to obtain the precessing magnetization M(T ) FIG. 8: First step of generating the path of the precessing in Fig. 7 gives magnetization M in Fig. 7. M¯ x = [cos ∆θ sin θ0 − sin ∆θ cos θ0 cos(¯ωeff T )]cos(¯ωT ) − sin ∆θ sin(¯ωT ) sin(¯ωeff T ), M the path that the head of the magnetization vector M¯ y = [cos ∆θ sin θ0 − sin ∆θ cos θ0 cos(¯ωeff T )] sin(¯ωT ) follows when both H and H1 are present. + sin ∆θ cos(¯ωT ) sin(¯ωeff T ), For subsequent calculations and plots, we normalize M¯ = cos∆θ cos θ0 + sin ∆θ sin θ0 cos(¯ωeff T ). (26) all angular frequencies by ω0, the time by 1/ω0, and the z magnetization magnitude M by M0, yielding the dimen- There are many combinations ofω ¯1,ω ¯, and ∆θ that sionless reduced parameters can be considered. Here we discuss a few representative ¯ cases. In actual ESR experiments one usually hasω ¯1 ≪ 1 M = 1, (21a) (unsaturated condition), but we need not be restricted to T = ω0t, (21b) this inequality here. Shown in Fig. 9 are plots forω ¯1 ≪ 1 ω¯0 = 1, (21c) at resonance (whereω ¯ = 1), withω ¯1 = 0.025 and ∆θ = θ0 = π/2. Figure 9(a) shows M¯ versus reduced time T = ω¯ = ω/ω0, (21d) x ω0t for one-half period of the effective frequencyω ¯eff , and ω¯1 = ω1/ω0, (21e) a 3D plot of M¯ z versus M¯ x and M¯ y with T as the implicit ω¯eff = ωeff /ω0. (21f) parameter is shown for the same time period in Fig. 9(b). The values of θ0 and ∆θ were chosen to give the initial From Eqs. (20) one then obtains conditions Mx = My = 0 and M¯ z = 1 at T = 0, so one can follow the path of M¯ versus time starting from T =0 2 2 at the top to T = π at the bottom of Fig. 9(b). During ω¯eff = ω¯1 + (1 − ω¯ ), (22a) the second half of the period the path rotates with the θ = qθ0 + ∆θ cos(¯ωeff T ), (22b) same chirality upward on the spherical surface until the φ =ωT, ¯ (22c) initial position is reached. Of more interest to the present paper is the behavior of ω¯1 θ0 = arctan . (22d) the path whenω ¯1 becomes appreciable compared to the 1 − ω¯   resonant frequencyω ¯ = 1, which is termed the condition for “saturation” in the field of ESR, a condition that To generate the path of the head of M versus time is usually avoided in practice. A commensurate value according to Fig. 7, we first consider the configuration in ofω ¯eff occurs whenω ¯eff /ω¯ is an integer n ≥ 1. This is A ¯ Fig. 8 where the initial position M of M at time t =0 termed commensurate because a 3D plot of M¯ z versus M¯ x is and M¯ y for such values ofω ¯eff andω ¯ overlaps after each period of reduced time T = 2π/ω¯eff . Using Eq. (22a), ¯ A Mx = cos∆θ, the equalityω ¯eff /ω¯ = n for any value of n> 1 yields ¯ A My = 0, (23) 2 ω¯1 = ω¯ [(n − 1)¯ω + 2] − 1. (27) ¯ A Mz = sin ∆θ. p Shown in Fig. 10 are 3D parametric plots of M¯ z versus A Rotating M clockwise about the x axis by the negative M¯ x and M¯ y with T as the implicit parameter according angle −ω¯eff T gives the precession of M about the x axis to Eqs. (26) and (27) for n = 6 and other parameters 8

(a)

(b)

(b)

(c)

FIG. 9: These plots are forω ¯ = 1,ω ¯1 = 0.025, and ∆θ = θ0 = π/2 rad. (a) M¯ x ≡ Mx/M0 versus reduced time T = ω0t. (b) Three-dimensional parametric plot of M¯ z versus M¯ x and M¯ y with T as the implicit parameter. The plots in (a) and (b) are for a time T =0 to π/ω¯eff (one-half period ofω ¯eff ).

listed in the figure caption. Since n = 6, one sees a sixfold periodic rotational behavior in each of the three panels. Figure 10(a) shows a case where the path of M¯ is an epi-cycloid, whereas Figs. 10(b) and 10(c) show in- creasingly epi-prolate-cycloid behaviors. With increas- ing values ofω ¯,ω ¯1 increases from 0.296 in panel (a) FIG. 10: 3D parametric plots of M¯ z versus M¯ x and M¯ y with to 6 in panel (c), and θ0 also increases from 0.335 rad T as the implicit parameter according to Eqs. (26) and (27) in panel (a) to π/2 rad in panel (c). These paths and for n = 6 and (a)ω ¯ = 0.15,ω ¯1 = 0.296, and θ0 = 2∆θ = those in Fig. 9(b) are three-dimensional angular varia- 0.335 rad, (b)ω ¯ = 0.3,ω ¯1 = 1.658, and θ0 = 2∆θ = 1.171 rad, tions with time at constant radius M¯ = 1 which are to and (c)ω ¯ = 1,ω ¯1 = 6, and θ0 = 2∆θ = π/2 rad. Panel (a) be contrasted with the cycloidal paths of a moon about shows epi-cycloid behavior, whereas panels (b) and (c) in- creasingly show epi-prolate cycloid behavior. a star in Fig. 6 where the two-dimensional cycloidal vari- ations with time are in the radial distance of the moon from the star. 9

III. SUMMARY a circular orbit about a planet that is in a circular or- bit about a star, where the orbits are coplanar, using the In this paper, the cycloidal paths of a point P in several same approach as for the rolling disk. Here the radial dis- physical situations of practical interest are studied. The tance of the moon from the star is the parameter showing parametric equations for the path of P a distance r from cycloidal paths. Finally, we show that cycloidal paths of the axis of a disk with radius R that is rolling without the magnetization vector versus time can occur during slipping is derived from the superposition of the transla- undamped electron-spin resonance if the amplitude H1 tional motion of the center of mass of the disk and the of the microwave magnetic field is an appreciable frac- rotational motion of P about the center of mass. As pre- tion of the magnitude H of the applied static magnetic viously known, a cycloid path and also curtate and pro- field. late cycloid paths are found for r = R, r < R and r > R, respectively. In these cases the cycloid is described para- metrically in terms of the Cartesian x and y coordinates Acknowledgments of P as a function of time as the implicit parameter. The same forms of cycloidal paths are obtained during rolling This research was supported by the U.S. Department of with frictionless slipping, but where the time dependence Energy, Office of Basic Energy Sciences, Division of Ma- of the sinusoidal Cartesian coordinates of the point P is terials Sciences and Engineering. Ames Laboratory is op- modified. The parametric equations versus time are ob- erated for the U.S. Department of Energy by Iowa State tained for the orbit with respect to a star of a moon in University under Contract No. DE-AC02-07CH11358.

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