ICTCM 28th International Conference on Technology in Collegiate Mathematics

THE BRACHISTOCHRONE ON A ROTATING EARTH

J Villanueva Florida Memorial University 15800 NW 42nd Ave Miami, FL 33054 [email protected]

1. Introduction 1.1 The brachistochrone problem 1.2 The

2. Free fall through the Earth 2.1 Homogeneous stationary Earth: simple harmonic motion 2.2 Linear gravity rotating Earth: trochoid

3. Fall trajectory 3.1 Trochoid: epicycles 3.2 Trochoid: hypocycles

4. Examples 4.1 Free fall in the equatorial plane 4.2 Free fall in the northern hemisphere

References:

1. R Calinger, R, 1995. Classics of Mathematics. Englewood Cliffs, NJ: Prentice-Hall 2. Hawking, S, 1988. A Brief History of Time. NY: Bantam. 3. Maor, E, 1998. Trigonometric Delights. Princeton, NJ: Princeton University Press. 4. Simoson, A, 2010. Voltaire’s Riddle. Washington, DC: Mathematical Association of America. 5. Simoson, A, 2007. Hesiod’s Anvil. Washington, DC: Mathematical Association of America. 6. Stewart, J, 2012. Single Variable Calculus. Belmont, CA: Brooks/Cole. 7. Thomas, G, 1972. Calculus. Reading, MA: Addison-Wesley.

In 1696 Johann Bernoulli challenged the mathematicians of his day to find the of quickest descent for a body to fall through a frictionless homogeneous Earth between two given points in a vertical plane [Calinger, 1995]. The problem languished for months until Newton chanced

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upon the problem, quickly solved it, and published the solution anonymously. But Bernoulli recognized the “lion by its claw” – the path is a cycloid. We have shown the solution of this problem by various authors in previous communcications. We will instead consider an extension of the problem, that is, to fall through the rotating Earth with a variable gravity. A complete solution was first presented by Simoson in 2010, and the path is a trochoid [Simoson, 2010, 2007].

We will trace the development of the solution to this problem, including a historical background, and displaying the equations of motion. Our free falling pebble may be realized by a black hole with the mass of a mountain and an event horizon of an atomic nucleus, from a recent suggestion from Hawking [1998].

We will briefly review the properties of the cycloid and the trochoid as they relate to the free fall through a rotating Earth. Examples will be provided for the case of motions in an equatorial plane, and those initiating in the northern hemisphere. The path for the former is a . For the latter, it turns out that the motion is not planar, but rather is three- dimensional whose projection on the -plane is either a or an depending on the rate of rotation.

The cycloid is the curve traced out by a point on the circumference of a rolling without slipping on a straight line. Its parametric equations are:

where r = the radius of the𝑥𝑥 = circle,𝑟𝑟(𝜃𝜃 − and𝑠𝑠𝑠𝑠𝑠𝑠 𝑠𝑠) is, the rotation𝑦𝑦 = 𝑟𝑟(1 angle− 𝑐𝑐𝑐𝑐 𝑐𝑐𝑐𝑐( ), when the tracing point is at the origin) (Figure 1). 𝜃𝜃 𝜃𝜃 = 0

Figure 1. The cycloidal path

Two noteworthy points about the cycloidal track are that: (1) Of all joining the two points A and in a vertical plane, the cycloid gives the shortest time of descent, a brachistochrone; and (2) The cycloid is also a tautochrone, that is, no matter where the particle is placed on an inverted cycloid, it takes the same time to slide to the bottom.

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igre hortest descent tie

ycloid 𝑎𝑎 𝑎𝑎 = 15.0 𝑓𝑓𝑓𝑓 𝑇𝑇0 = 𝜋𝜋√𝑔𝑔 = 2.144 𝑠𝑠 traight line arabola 𝑇𝑇 = 1.185𝑇𝑇0 = 2.541 𝑠𝑠 ircle 𝑇𝑇 = 1.160𝑇𝑇0 = 2.487 𝑠𝑠 𝑇𝑇 = 1.149𝑇𝑇0 = 2.463 𝑠𝑠

igre ae tie ro A or to inish

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n etension o the roble is to consider alling throgh the whole arth irst when the arth is taen stationary and net to allow it to rotate e hae always been ascinated by a all throgh a hole in the arth

 n the ree oet esiod claied that an anil wold all ro the heaens to the arth in days and throgh the arth again in days beore reaching hell  lice in throgh ewis arroll ell down a rabbit hole or eer so long that she grew sleepy and thought she must have fallen through the Earth’s center.  enri an tten in arged that a illstone droed ro a hole wold reach the Earth’s center in 2.5 days and “hang” in the air.  ltarch arond ointed ot that in a sherical arth bolders wold ass through the Earth’s center and beyond and again oscillating in a perpetual motion.  alileo in analyzed the sile haronic otion o a cannonball droed ro a hole in the arth  awing in dros a blac hole with ass o a ontain and radis that o a ncles o an atom, from the Earth’s surface. his entry is iortant to s becase it is the closest thing we hae or now o a article alling throgh the arth by graity alone nieded by riction

hrogh a stationary hoogeneos arth the otion is shown to be sile haronic with a nown eriod o oscillation igre

𝐹𝐹 = 𝑚𝑚𝑚𝑚 𝑚𝑚𝑚𝑚′ 2 𝑚𝑚𝑟𝑟̈ = −𝐺𝐺 𝑟𝑟

𝐺𝐺𝐺𝐺 4 3 2 = − 𝑟𝑟 ∙ 3 𝜋𝜋𝑟𝑟 𝜌𝜌

4 = −𝑘𝑘𝑘𝑘, 𝑘𝑘 = 3 𝜋𝜋𝜋𝜋𝜋𝜋𝜋𝜋 ie with eriod 𝑘𝑘 𝑚𝑚 𝑟𝑟̈ = − 𝑚𝑚 𝑟𝑟 ⇒ 𝑆𝑆𝑆𝑆𝑆𝑆 𝑇𝑇 = 2𝜋𝜋√ 𝑘𝑘 = 84.4 𝑚𝑚𝑚𝑚𝑚𝑚. e note that this transit tie o intes oneway is the sae ale between any two antiodes on the arth t trns ot that the hole throgh the arth need not ass throgh the center any chordal tnnel in the arth gies rise to a sile haronic otion with the sae eriod as the diaetrical ath igre

′ 𝑚𝑚𝑀𝑀 𝑥𝑥 2 𝑚𝑚𝑥𝑥̈ = −𝐺𝐺 𝑟𝑟 𝑠𝑠𝑠𝑠𝑛𝑛𝜃𝜃, 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 = 𝑟𝑟

𝐺𝐺𝐺𝐺 4 3 𝑥𝑥 2 = − 𝑟𝑟 ∙ 3 𝜋𝜋𝑟𝑟 𝜌𝜌 ∙ 𝑟𝑟

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4 = −𝑘𝑘𝑘𝑘, 𝑘𝑘 = 3 𝜋𝜋𝜋𝜋𝜋𝜋𝜋𝜋 i.e., with period 𝑘𝑘 𝑚𝑚 𝑥𝑥̈ = − 𝑚𝑚 𝑥𝑥 ⇒ 𝑆𝑆𝑆𝑆𝑆𝑆 𝑇𝑇 = 2𝜋𝜋√ 𝑘𝑘 = 84.4 𝑚𝑚𝑚𝑚𝑚𝑚.

igure . ole along a diameter igurer 5. ole along a chord

llowing for a rotating Earth, the motion now adds a new dimension. his was first completely described by imoson in 2. rom a frame of reference on a fied star, the simple harmonic motion through a tunnel in the Earth appears as an ellipse that is precessing in a clocwise sense. rom the point of view of an observer on the Earth, the toandfro motion along the tunnel appears as a trochoidal motion. his is reminiscent of the motion of a oucault pendulum, first demonstrated by oucault in 5. anging a 2lb bob by a string of length 22 ft at the antheon in aris, latitude the pendulum appeared to precess around every hour clocwise, for a total period of0 2.′ hours. 0 48 51 , 11

igure . oucault pendulum antheon igure . oucault pendulum, model

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The equations of motion for motion on a plane (the equatorial plane) is given by Newton’s laws:

2 2 2 𝑑𝑑 𝑟𝑟 ℎ 𝑑𝑑 𝑧𝑧 𝑘𝑘 2 3 2 2 3 with r as the radial𝑑𝑑𝑡𝑡 + distance𝑘𝑘𝑘𝑘 = 𝑟𝑟 ,from the 𝑑𝑑Earth’s𝑡𝑡 + 𝑧𝑧 center= ℎ 𝑧𝑧 to the particle, the radial angle of the position vector, = the angular momentum, and is a proportionality constant to the 𝜃𝜃 gravitation force The solutions are (details from imoson, ): 𝑧𝑧 = 1⁄𝑟𝑟, 𝐹𝐹 = −𝑘𝑘𝑘𝑘𝑘𝑘. and 2 1/2 2 𝜔𝜔 2 𝑟𝑟(𝑡𝑡) = 𝑅𝑅 [𝑐𝑐𝑐𝑐𝑐𝑐 (√𝑘𝑘𝑡𝑡) + 𝑘𝑘 𝑠𝑠𝑠𝑠𝑠𝑠 (√𝑘𝑘𝑡𝑡)] , with −1/2 2 𝑘𝑘 2 2𝜋𝜋 2 𝑟𝑟(𝜃𝜃) = 𝑅𝑅 [𝑐𝑐𝑐𝑐𝑐𝑐 𝜃𝜃 + 𝜔𝜔 𝑠𝑠𝑠𝑠𝑠𝑠 𝜃𝜃] 𝜔𝜔 = 𝑄𝑄 , = Earth’s period, and Earth’s angular frequency. The solution can be written as a matri:

𝜔𝜔 =

𝑥𝑥(𝑡𝑡) 𝑐𝑐𝑐𝑐𝑐𝑐(√𝑘𝑘𝑡𝑡) 𝑃𝑃(𝑡𝑡) = [ ] = 𝑅𝑅 [ 𝜔𝜔 ], 𝑦𝑦(𝑡𝑡) 𝑘𝑘 which is an elliptical path, for an observer√ sin( fied√𝑘𝑘𝑡𝑡) on the stars. or an Earth observer, this path is made to precess by introducing a dynamic rotation matri:

cos (𝛼𝛼𝛼𝛼) −sin (𝛼𝛼𝛼𝛼) 𝑄𝑄(𝛼𝛼, 𝑡𝑡) = [ ]. The resulting path, sin (𝛼𝛼𝛼𝛼) iscos a (hypoc𝛼𝛼𝛼𝛼) ycloid along the equatorial plane of a rotating homogeneous spherical Earth. 𝑄𝑄(−𝜔𝜔, 𝑡𝑡)𝑃𝑃(𝑡𝑡),

igure . ebble motion relative to the stars and the Earth.

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trochoid is generated by a point on a circle rolling, without slipping, on another circle. f the rolling circle is outside the other circle, the curve generated is called an epitrochoid, inside the curve is a hypotrochoid. hen the cusps of the curves are on the stationary circle, the curves are called and (igure ). enerating a trochoid is shown in igure . The eact shape of the curve depends on the ratio of the radii of the two , the ring, r = the wheel. f this ratio is a fraction in lowest terms, the curve will have ′ cusps, 𝑟𝑟′⁄𝑟𝑟, 𝑟𝑟 = and it will be completely traced after moving the wheel n times around the inner rim. f this 𝑚𝑚⁄𝑛𝑛 ratio is irrational, the curve will not close, although going many times around the rim will nearly close it.

igure . Trochoid curves

igure . enerating a trochoid

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s an application of the equations of motion, we can calculate the transit times between any two cities along the equator. ne further feature of the equations of the path is that it allows for the variation of the rate of rotation of the Earth. The solutions, for the equatorial plane, may be written:

with the Earth’s𝑟𝑟(𝑡𝑡)[ 𝑐𝑐𝑐𝑐𝑐𝑐angular(𝜃𝜃(𝑡𝑡 frequency,) − 𝜔𝜔𝜔𝜔), 𝑠𝑠 𝑠𝑠𝑠𝑠and(𝜃𝜃 (𝑡𝑡) − 𝜔𝜔𝜔𝜔)],

𝜔𝜔 = 𝜔𝜔 ( −1)𝑠𝑠𝑠𝑠𝑠𝑠√𝑘𝑘𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡√𝑘𝑘𝑡𝑡 −1 √𝑘𝑘 𝜔𝜔 2 𝜃𝜃(𝑡𝑡) = √𝑘𝑘𝑡𝑡 + 𝑡𝑡𝑡𝑡𝑡𝑡 [ 1+(√𝑘𝑘−1)𝑠𝑠𝑠𝑠𝑠𝑠 √𝑘𝑘𝑡𝑡 ] , 𝑡𝑡 > 0. This last equation allows us to calculate the transit time between launch point A and resurface point for various hypothetical rates of rotation of the Earth (igure ), with t = the number, in hours, as argument in the parentheses of . or eample, corresponds to a stationary Earth is for normal rotating Earth at hoursday and for the pebble 𝐵𝐵(∞) immediately veers off into space because the centripetal acceleration eceeds gravitational 𝐵𝐵(24) 𝐵𝐵(1.5), acceleration. The hypocycloid between A and () is the same hypocycloid shown in igure b. n particular, for the hypocycloid between launch point A: uito, Ecuador resurface point : Entebbe, ganda the transit time is . minutes. ompared0 to the transit (78 𝑊𝑊), time for a stationary Earth, .0 minutes, this is slower by . seconds. (32.5 𝐸𝐸),

igure . ypocycloid arches with respect to Earth period

or release points A in higher latitudes, say the Northern emisphere (igure ), the paths through the tunnel get a new shape. rom an observer in the stars, the path is still elliptical, but for an Earth observer, a new factor comes in. Since the pebble’s initial tangential speed at latitude is less by a factor of compared to that of a particle at the equator, the semi minor aial length for the ellipse originating at will be smaller by the same factor to that of the 𝜓𝜓 𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝜓𝜓

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ellipse originating at the equator. esides, the tunnel curve between A and will no longer be planar, but a curve between , resembling the curves on a wicer baset, igure . heir proection onto the equatorial plane is still a hypocycloid. Simoson gives an eample of a ±𝜓𝜓 drop at orino, taly and a resurface point at which is miles southeast off the coast from0 ellington,0 ew ealand. o transit0 time was0 given, however. (45 𝑁𝑁, 7.5 𝐸𝐸) (45 𝑆𝑆, 177 𝐸𝐸),

igure . Ellipse of motion for latitude

𝜓𝜓

igure . ath of fall for a pebble dropped at latitude

Such paths as shown for a free fall at higher latitudes are not unfamiliar. hey are reminiscent𝜓𝜓 of the orbits of lowaltitude polar satellites around the Earth, as shown in igure a, b.

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igure a. olar lowaltitude satellite orbits b.

Summary of esults

 he path of quicest descent in a free fall through a constant gravity field is a cycloid.  n etended fall through a stationary Earth is a simple harmonic motion, with the same period between any two points on the Earth.  Even when the tunnel is a chord not through the center, the motion is still simple harmonic, with the same period as before an isochrone.  hen rotation is allowed, the path of the fall becomes a trochoid along the equator, which is still the least time path. long the equator, the trochoidal motion is planar.  n the orth or South emisphere, the motion is nonplanar, but its proection on the equator is still trochoidal. he motion between release point and resurface point is accomplished in the least time.

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