aa r X i v : . [ phy s i c s . pop - p h ] S e p Cycloidal Paths in Physics
David C. Johnston
Ames Laboratory and Department of Physics and Astronomy, Iowa State University, Ames, Iowa 50011, USA (Dated: September 12, 2018)A popular classroom demonstration is to draw a cycloid on a blackboard with a piece of chalkinserted through a hole at a point P with radius r ≈ R from the center of a wood disk of radius R that is rolling without slipping along the chalk tray of the blackboard. Here the parametric equationsversus time are derived for the path of P from the superposition of the translational motion of thecenter of mass (cm) of the disk and the rotational motion of P about this cm for r = R (cycloid), r < R (curtate cycloid) and r > R (prolate cycloid). It is further shown that the path of P isstill a cycloidal function for rolling with frictionless slipping, but where the time dependence of thesinusoidal Cartesian coordinates of the position of P is modified. In a similar way the parametricequations versus time for the orbit with respect to a star of a moon in a circular orbit about a planetthat is in a circular orbit about a star are derived, where the orbits are coplanar. Finally, the generalparametric equations versus time for the path of the magnetization vector during undamped electron-spin resonance are found, which show that cycloidal paths can occur under certain conditions. I. INTRODUCTION
The mathematical properties of the cycloid have beenstudied since about 1500, and the term cycloid was evi-dently coined by Galileo in ∼ R and the rotationalmotion of a point P affixed to the disk that is a dis-tance r from the cm. This procedure enables curtate andprolate cycloid paths to be generated in addition to theabove-mentioned cycloid path, and nicely demonstratesthat the motion of a point P in or on a rigid body mov-ing through space is a superposition of the translationalmotion of the center-of-mass (cm) of the body and the ro- tational motion of the point P about the cm. In Sec. II Bwe show that if the disk rolls with frictionless slipping,the path of the point P is still a cycloidal function, butwith a modified time dependence of the sinusoidal Carte-sian coordinates of the position of P.A nice simulation of the motion of a moon orbiting aplanet that in turn orbits a star, where the orbits arecoplanar, is available that shows cycloidal orbits of themoon about the star [7]. In Sec. II B the parametricequations for the Cartesian coordinates of the moon aredetermined, again using the principle that the motion ofthe moon is the superposition of the rotational motion ofthe moon with respect to the planet and the translationalmotion of the planet in its orbit about the star. The solu-tion yields epi-cycloid, epi-curtate-cycloid, or epi-prolate-cycloid paths depending on the ratio of the moon-planetdistance to the planet-star distance and the relative ro-tational periods of moon about the planet and the planetabout the star. We show from the known parameters forour Moon-Earth-Sun system, with orbits that approxi-mate the above idealization, that the orbit of the Moonabout the Sun is a curtate cycloid with very small oscil-lation amplitude relative to the Earth-Sun distance.In both nuclear magnetic resonance (NMR) [8, 9]and electron spin resonance (ESR) [10–13] experiments,one measures how the atomic electron (ESR) or nu-clear (NMR) magnetization M (average magnetic mo-ment per unit volume) precesses in the presence of anapplied static magnetic field H and a continuous-wavecircularly-polarized microwave magnetic field H that isaligned perpendicular to H . In the absence of damp-ing, the head of the M vector follows a time-dependentpath that is determined by both H and H . This pathhas been qualitatively described and shown in several fig-ures [10], but parametric equations for the path versustime have not been reported before to our knowledge. InSec. II D the parametric equations for the path versustime are derived for undamped ESR in the general caseof relevant parameters. For H ≪ H and at the reso-nant frequency, the path is a helical path on the surface v cm cm xyz v cm v cm = v cm i ^ R P r v cm = R FIG. 1: Experimental configuration of a disk of radius R rolling towards the right on a horizontal surface without slip-ping. The speed of the center of mass (cm) of the disk is v cm = ωR and hence the velocity of the cm is v cm = ωR ˆ i .The point P is a distance r from the cm at a negative instan-taneous polar angle φ measured clockwise from the positive x axis. of a sphere as shown in Sec. II D. However, when H is a significant fraction of H , a cycloidal path is found,the nature of which depends on the parameters of theelectron-spin resonance. A summary is given in Sec. III. II. RESULTS: CYCLOIDAL PATHSA. Rolling without Slipping
The experimental configuration is shown in Fig. 1. Thedisk is rolling without slipping towards the right. Theazimuthal angle φ of the point P is measured with respectto the positive x axis and is negative because P is rotatingclockwise instead of counterclockwise for which φ wouldbe positive. Because the disk is rolling without slipping,the constant speed of the center of mass (cm) with respectto the stationary surface on which the disk rolls is v cm = ωR, (1)where ω = | dφ/dt | is the angular speed of P with respectto the cm. The position r cm of the cm is r cm = ωRt ˆ i , (2a)where t is the time. The rotational motion of P withrespect to the cm is described by r P , rot = r h cos( ω z t )ˆ i + sin( ω z t )ˆ j i . (2b)Since ω z = − ω , one obtains r rot = r h cos( ωt )ˆ i − sin( ωt )ˆ j i . (2c)The position r of point P is the superposition of thecenter of mass position and the rotational position inEqs. (2a) and (2c), respectively. The Cartesian compo-nents of r are therefore x = ωRt + r cos( ωt ) , (3a) y = − r sin( ωt ) . (3b) Dividing both sides of each of these equations by R givesdimensionless components xR = ωt + rR cos( ωt ) , (4a) yR = − rR sin( ωt ) . (4b)These are the parametric equations for the reducedCartesian 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 .The parametric equations for the cycloid are conven-tionally written [2] xR = θ − sin θ, (5a) yR = 1 − cos θ. (5b)One can obtain these equations from Eqs. (4) by the sub-stitutions ωt → θ + π/ r/R →
1, and a y -axis offset y/R → y/R + 1.Shown in Fig. 2(a) are plots of x/R and y/R versus ωt for a curtate cycloid with r/R = 0 . y/R versus x/R with ωt as the implicit parameter. A curtate cycloid canbe traced on the blackboard by drilling a hole in thewooden disk at radius r < R in which to insert a pieceof chalk and then rolling the disk along the chalk tray asin the above procedure to generate a chalk trace of a cy-cloid. Corresponding plots for the cycloid with r/R = 1are shown in Fig. 3. The derivative dy/dx of the cy-cloid curve in Fig. 3(b) is discontinuous at the minima,whereas for the curtate cycloid in Fig. 2(b) the minimaare rounded. Because Figs. 2(a) and 3(a) are so sim-ilar, one might not anticipate the significant differencebetween Figs. 2(b) and 3(b). At sufficiently small valuesof r/R the y versus x curves become nearly sinusoidal.Figure 4 shows corresonding plots for a prolate cycloidwith r/R = 1 .
5. Here loops appear in the y/R versus x/R plot in Fig. 4(b).
B. Rolling with Frictionless Slipping
Here we consider a disk of radius R that is rotating atangular speed ω ′ = v cm /R and thus rolling with slippingwithout friction on a surface. Here we ask the same ques-tion as in this last section: what is the path through spaceof a point P that is fixed on the disk a distance r from itscenter? We initially assume that the angular velocity ofthe disk is in the same direction as in the previous sectionfor rolling without slipping, but the following results areeasily generalized to the case where the angular velocityof the disk with slipping is in the opposite direction of thecase without slipping by simply changing the sign of theparameter α introduced below from positive to negative.Referring again to Fig. 1, here we write v cm = ωR where ω is the angular speed of the disk if it were rolling FIG. 2: (a) Cartesian components x/R and y/R of point Pversus reduced time ωt in radians for r/R = 0 .
5. (b) y/R versus x/R with ωt as an implicit parameter where this pathof point P is a curtate cycloid. These plots were obtainedusing Eqs. (4). without slipping. Therefore one again has r cm = ωRt ˆ i (6a)However, the rotational motion of P with respect to thecm is now described by r rot = r h cos( ω z t )ˆ i + sin( ω z t )ˆ j i (6b)= r h cos( ω ′ t )ˆ i − sin( ω ′ t )ˆ j i , (6c)where ω ′ is the angular speed of the disk which satisfies ω ′ = ω for rolling with slipping. The reduced x and y components of r are now xR = ωt + rR cos( ω ′ t ) , (7a) yR = − rR sin( ω ′ t ) . (7b) FIG. 3: (a) Cartesian components x/R and y/R of point Pversus reduced time ωt in radians for r/R = 1. (b) y/R versus x/R with ωt as an implicit parameter where this path of pointP is a cycloid. These plots were obtained using Eqs. (4). We write the relationship between ω ′ and ω as ω ′ = αω, (8)where α is a dimensionless constant. Then Eqs. (7) be-come xR = ωt + rR cos( αωt ) , (9) yR = − rR sin( αωt ) . Thus the motion of point P when the disk is rollingat constant angular speed with frictionless slipping is acycloidal function, but where the time dependence of thesinusoidal parts of x and y are changed in the same waycompared with the case of rolling without slipping. Inparticular, the path is a curtate cycloid if | α | r/R <
1, acycloid if | α | r/R = 1, and a prolate cycloid if | α | r/R > FIG. 4: (a) Cartesian components x/R and y/R of point Pversus reduced time ωt in radians for r/R = 1 .
5. (b) y/R versus x/R with ωt as an implicit parameter where this pathof point P is a prolate cycloid. These plots were obtainedusing Eqs. (4). For the case where the disk is rotating in the oppositedirection while slipping compared to the case of rollingwithout slipping, one replaces the positive α in Eqs. (9)by − α and the path becomes inverted. Example plots ofEqs. (9) are shown in Fig. 5 for α = 1 / r/R = 3(prolate cycloid), α = 1 / r/R = 1 (curtate cycloid),and α = − / r/R = 2 (inverted cycloid). C. Path of a Moon with respect to a Star whileOrbiting a Planet that is Orbiting the Star
Here we consider the coplanar orbits of a moon orbitinga planet while the planet orbits a star that is stationarywith respect to the distant stars, where the moon, planetand star are spherically symmetric, the two orbits arecircular and lie in the xy plane, and the moon and planet FIG. 5: Cartesian component y/R versus x/R of the path ofpoint P with T = αωt as an implicit parameter for (a) α = 1 / r/R = 3 (prolate cycloid), (b) α = 1 / r/R = 1(curtate cycloid), and (c) α = − / r/R = 2 (invertedcycloid) calculated using Eqs. (9). are both moving counterclockwise in their orbits whenviewed from the positive z axis. The Moon orbiting theEarth that orbits the Sun approximately satisfies theseconditions 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) T M = period of the moon’s orbit about the planet T P = period of the planet’s orbit about the star ω M = 2 π/T M = angular speed of the moon with respectto the planet ω P = 2 π/T P = angular speed of the planet with respectto the star( x, y ) = Cartesian coordinates of the moon’s center withrespect to the star( x P , y P ) = Cartesian coordinates of the planet’s centerwith respect to the starOne expects r ≪ R , T M ≪ T P , ω M ≫ ω P .Thus we have x P = R cos( ω P t ) , (10a) y P = R sin( ω P t ) , (10b) x = r cos( ω M t ) + R cos( ω P t ) , (10c) y = r sin( ω M t ) + R sin( ω P t ) . (10d)We define the dimensionless parameter α ≡ ω P /ω M ≪ x = r cos( ω M t ) + R cos( αω M t ) , (11a) y = r sin( ω M t ) + R sin( αω M t ) . (11b)Using the generic expression ω = 2 π/T one obtains x = r cos(2 πt/T M ) + R cos(2 παt/T M ) , (12a) y = r sin(2 πt/T M ) + R sin(2 παt/T M ) . (12b)Finally, introducing the dimensionless reduced time T ≡ t/T M (13)and dividing both sides Eqs. (12) by R , one obtains thedimensionless parametric equations for the path of themoon with respect to the star versus reduced time T as xR = rR cos(2 πT ) + cos(2 παT ) , (14a) yR = rR sin(2 πT ) + sin(2 παT ) . (14b)Parametric plots of y/R versus x/R using Eqs. (14)are shown in Fig. 6 for α = 0 . r/R = 0 .
05, 0.1,and 0.15. The latter three parameters are unrealisticallylarge in order to clearly show the structure of the paths.The paths are analogous to those in Figs. 2–4, exceptthat the x axes in those figures are bent here into circles.Thus for r/R < α one obtains an epi-curtate cycloid, for r/R = α an epi-cycloid, and for r/R > α an epi-prolatecycloid, where here the prefix epi refers to linear cycloidalmotion bent into a circle. FIG. 6: Parametric plots of y/R versus x/R for cycloidalpaths of a moon orbiting a planet that orbits a star with α = 0 . r/R = 0 .
05 (epi-curtate cycloid), (b) r/R =0 .
10 (epi-cycloid), and (c) r/R = 0 .
15 (epi-prolate cycloid),calculated using Eqs. (14).
The orbit of the Moon about the Earth and the Earthabout the Sun are approximately coplanar. When viewedfrom the North, the Earth rotates counter-clockwiseabout the Sun at a distance R = 1 . × m withrotation period T P = 365 . ω P =2 π/T P . The Moon rotates counter-clockwise around theEarth at a distance r = 3 . × m with rotationperiod T M = 27 . ω M = 2 π/T M .Thus for the Moon orbiting the Earth, the parameter α = 27 . / . ≈ . α = 0 . r/R ≈ . r/R ≪ α and hence the Moon has anepi-curtate cycloidal path around the Sun correspondingto the linear rolling with slipping path in Fig. 5(b), butwith a very small amplitude of oscillation (not shown)that is barely visible on the scale of the plot in Fig. 6(a). D. Paths of the Magnetization Vector inUndamped Electron-Spin Resonance
The Bloch equations are often the starting point foranalyzing experimental electron-spin resonance (ESR)data. In the absence of damping, the Bloch equationsgive the Cartesian components of the magnetization M (average magnetic moment per unit volume) that is pre-cessing around the applied uniform, static magnetic field H = H ˆ k , (15)as [14] dM x dt = − γ ( M × H ) x , (16a) dM y dt = − γ ( M × H ) y , (16b) dM z dt = − γ ( M × H ) z , (16c)where the negative sign prefactors arise from the negativecharge on the electron appropriate for ESR, and γ is thegyromagnetic ratio ( γ = gµ B / ~ for Heisenberg spins, g is the spectroscopic splitting factor, µ B is the Bohr mag-neton, and ~ is Planck’s constant divided by 2 π ). TheGaussian cgs system of units is used in this section.In the absence of damping and additional magneticfields, the Bloch equations yield a magnetization thatprecesses around H at angular frequency ω = γH (17)according to M x = M sin θ cos( ω t ) , (18a) M y = M sin θ sin( ω t ) , (18b) M z = M cos θ, (18c)where M = | M | and θ is the constant angle that M makes with the z axis during the precession. x yz, H M H eff FIG. 7: Geometry of the precession of the magnetization M in the presence of an applied static field H and a microwavemagnetic field H . M precesses clockwise about the effectivefield H eff , forming the surface of a cone with cone half-angle∆ θ , while H eff precesses counter-clockwise about the appliedfield H with azimuthal angle φ = ωt on the surface of a conewith cone angle θ given by Eq. (20f). After Ref. [15]. For ESR experiments, an additional circularly-polarized microwave magnetic field H with angular fre-quency ω is applied that rotates in the xy plane aboutthe z axis in the same direction that M is precessing inthe absence of H , given by H = − H [cos( ωt )ˆ i + sin( ωt )ˆ j ] , (19)where the negative-sign prefactor is due to the negativecharge on the electron that applies to ESR as in Eqs. (16).Thus H is always antiparallel to the projection of M onto the xy plane.Qualitatively, M precesses around an effective mag-netic field H eff at angular frequency ω eff while H eff pre-cesses around the applied field H at the angular fre-quency ω as shown in Fig. 7, where [15] H eff = − H [cos( ωt )ˆ i + sin( ωt )ˆ j ] + (cid:18) H − ωγ (cid:19) ˆ k , (20a) H eff = | H eff | = s H + (cid:18) H − ωγ (cid:19) , (20b) ω eff = γH eff = q ω + ( ω − ω ) (20c) ω ≡ γH , (20d) φ = ωt, (20e) θ = arctan (cid:18) ω − ωω (cid:19) (0 ≤ θ ≤ π/ , (20f) θ = θ + ∆ θ cos ( ω eff t ) , (20g)and the cone half-angle ∆ θ with 0 < ∆ θ ≤ π − θ is anadjustable parameter. Here we derive an expression for x yz, HM M = 1 r = sin( ) FIG. 8: First step of generating the path of the precessingmagnetization M in Fig. 7. the path that the head of the magnetization vector M follows when both H and H are present.For subsequent calculations and plots, we normalizeall angular frequencies by ω , the time by 1 /ω , and themagnetization magnitude M by M , yielding the dimen-sionless reduced parameters¯ M = 1 , (21a) T = ω t, (21b)¯ ω = 1 , (21c)¯ ω = ω/ω , (21d)¯ ω = ω /ω , (21e)¯ ω eff = ω eff /ω . (21f)From Eqs. (20) one then obtains¯ ω eff = q ¯ ω + (1 − ¯ ω ) , (22a) θ = θ + ∆ θ cos(¯ ω eff T ) , (22b) φ = ¯ ωT, (22c) θ = arctan (cid:18) ¯ ω − ¯ ω (cid:19) . (22d)To generate the path of the head of M versus timeaccording to Fig. 7, we first consider the configuration inFig. 8 where the initial position M A of ¯ M at time t = 0is ¯ M A x = cos ∆ θ, ¯ M A y = 0 , (23)¯ M A z = sin ∆ θ. Rotating M A clockwise about the x axis by the negativeangle − ¯ ω eff T gives the precession of M about the x axis as ¯ M B x = cos ∆ θ, ¯ M B y = sin ∆ θ sin(¯ ω eff T ) , (24)¯ M B z = sin ∆ θ cos(¯ ω eff T ) . Next we rotate M B clockwise about the y axis by a neg-ative angle − (cid:0) π − θ (cid:1) so that H eff is at an angle of θ with respect to the z axis according to Fig. 7, yielding¯ M C x = − sin ∆ θ cos θ cos(¯ ω eff T ) , ¯ M C y = sin ∆ θ sin(¯ ω eff T ) , (25)¯ M C z = cos ∆ θ cos θ + sin ∆ θ sin θ cos(¯ ω eff T ) , Finally, rotating M C about the z axis by a positiveangle ¯ ωT to obtain the precessing magnetization M ( T )in Fig. 7 gives¯ M x = [cos ∆ θ sin θ − sin ∆ θ cos θ cos(¯ ω eff T )] cos(¯ ωT ) − sin ∆ θ sin(¯ ωT ) sin(¯ ω eff T ) , ¯ M y = [cos ∆ θ sin θ − sin ∆ θ cos θ cos(¯ ω eff T )] sin(¯ ωT )+ sin ∆ θ cos(¯ ωT ) sin(¯ ω eff T ) , ¯ M z = cos ∆ θ cos θ + sin ∆ θ sin θ cos(¯ ω eff T ) . (26)There are many combinations of ¯ ω , ¯ ω , and ∆ θ thatcan be considered. Here we discuss a few representativecases. In actual ESR experiments one usually has ¯ ω ≪ ω ≪ ω = 1), with ¯ ω = 0 .
025 and ∆ θ = θ = π/
2. Figure 9(a) shows ¯ M x versus reduced time T = ω t for one-half period of the effective frequency ¯ ω eff , anda 3D plot of ¯ M z versus ¯ M x and ¯ M y with T as the implicitparameter is shown for the same time period in Fig. 9(b).The values of θ and ∆ θ were chosen to give the initialconditions M x = M y = 0 and ¯ M z = 1 at T = 0, so onecan follow the path of ¯ M versus time starting from T = 0at the top to T = π at the bottom of Fig. 9(b). Duringthe second half of the period the path rotates with thesame chirality upward on the spherical surface until theinitial position is reached.Of more interest to the present paper is the behavior ofthe path when ¯ ω becomes appreciable compared to theresonant frequency ¯ ω = 1, which is termed the conditionfor “saturation” in the field of ESR, a condition thatis usually avoided in practice. A commensurate valueof ¯ ω eff occurs when ¯ ω eff / ¯ ω is an integer n ≥
1. This istermed commensurate because a 3D plot of ¯ M z versus ¯ M x and ¯ M y for such values of ¯ ω eff and ¯ ω overlaps after eachperiod of reduced time T = 2 π/ ¯ ω eff . Using Eq. (22a),the equality ¯ ω eff / ¯ ω = n for any value of n > ω = p ¯ ω [( n − ω + 2] − . (27)Shown in Fig. 10 are 3D parametric plots of ¯ M z versus¯ M x and ¯ M y with T as the implicit parameter accordingto Eqs. (26) and (27) for n = 6 and other parameters (b) FIG. 9: These plots are for ¯ ω = 1, ¯ ω = 0 . θ = θ = π/ M x ≡ M x /M versus reduced time T = ω t .(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 of ¯ ω eff ). listed in the figure caption. Since n = 6, one sees asixfold periodic rotational behavior in each of the threepanels. 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 of ¯ ω , ¯ ω increases from 0.296 in panel (a)to 6 in panel (c), and θ also increases from 0.335 radin panel (a) to π/ M = 1 which are tobe contrasted with the cycloidal paths of a moon abouta star in Fig. 6 where the two-dimensional cycloidal vari-ations with time are in the radial distance of the moonfrom the star. (a) (b)(c) FIG. 10: 3D parametric plots of ¯ M z versus ¯ M x and ¯ M y with T as the implicit parameter according to Eqs. (26) and (27)for n = 6 and (a) ¯ ω = 0 .
15, ¯ ω = 0 . θ = 2∆ θ =0 .
335 rad, (b) ¯ ω = 0 .
3, ¯ ω = 1 . θ = 2∆ θ = 1 .
171 rad,and (c) ¯ ω = 1, ¯ ω = 6, and θ = 2∆ θ = π/ III. SUMMARY
In this paper, the cycloidal paths of a point P in severalphysical situations of practical interest are studied. Theparametric equations for the path of P a distance r fromthe axis of a disk with radius R that is rolling withoutslipping is derived from the superposition of the transla-tional motion of the center of mass of the disk and therotational motion of P about the center of mass. As pre-viously known, a cycloid path and also curtate and pro-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 coordinatesof P as a function of time as the implicit parameter. Thesame forms of cycloidal paths are obtained during rollingwith frictionless slipping, but where the time dependenceof the sinusoidal Cartesian coordinates of the point P ismodified. The parametric equations versus time are ob-tained for the orbit with respect to a star of a moon in a circular orbit about a planet that is in a circular or-bit about a star, where the orbits are coplanar, using thesame approach as for the rolling disk. Here the radial dis-tance of the moon from the star is the parameter showingcycloidal paths. Finally, we show that cycloidal paths ofthe magnetization vector versus time can occur duringundamped electron-spin resonance if the amplitude H of the microwave magnetic field is an appreciable frac-tion of the magnitude H of the applied static magneticfield. Acknowledgments
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