// Program to find the time spent falling through a hole in the earth assuming // a non-uniform earth. The density of the earth is modeled using several // layers which model closely the known density of the earth. // This program is written in Frink: // https://frinklang.org/ // // Alan Eliasen, eliasen@mindspring.com // The mass of a whole shell with inner radius d0 and outer radius d1 // and inner density r0 and outer density r1 wholeShellMass[d0 is length, d1 is length, r0 is mass_density, r1 is mass_density] := { ((d1 - d0) pi (2 d0 d1 (r0 + r1) + d0^2 (3 r0 + r1) + d1^2 (r0 + 3 r1))) / 3 } // The mass of the partial shell with inner radius d0 and outer radius d1 // and inner density r0 and outer density r1 which is below an object // at radius d. partShellMass[d is length, d0 is length, d1 is length, r0 is mass_density, r1 is mass_density] := { (pi (3 d^4 (r0 - r1) + d^3 (-4 d1 r0 + 4 d0 r1) - d0^3 (-4 d1 r0 + d0 (3 r0 + r1)))) / (3 (d0 - d1)) } // The mass of a whole or partial shell with inner radius d0 and outer radius // d1 and inner density r0 and outer density r1 which is below an object at // radius d. shellMass[d is length, d0 is length, d1 is length, r0 is mass_density, r1 is mass_density] := { if (d < d0) 0 kg // inside inner surface of shell else if d > d1 wholeShellMass[d0, d1, r0, r1] // completely outside shell else partShellMass[d, d0, d1, r0, r1] // partway through shell } // The different layers of density in the Earth shells = [[ 0 km, 1221 km, 13.1 g/cm^3, 12.8 g/cm^3], [1221 km, 3480 km, 12.2 g/cm^3, 9.9 g/cm^3], [3480 km, 5651 km, 5.6 g/cm^3, 4.4 g/cm^3], [5651 km, 6341 km, 4.4 g/cm^3, 3.4 g/cm^3], [6341 km, earthradius, 2.9 g/cm^3, 2.2 g/cm^3]] // This finds the mass that's still below you at a given distance from the // earth's center. mass[d is length, shells] := { var m is mass = 0 kg for [d0, d1, r0, r1] shells m = m + shellMass[d, d0, d1, r0, r1] // Fudge factor to make integral come out // with the known mass of the earth. return 0.995775976555 m } // Find the acceleration at a given distance from the core. a[dist is length, shells] := G mass[dist, shells]/dist^2 var v is velocity = 0 m/s // Velocity at end of timestep var stepsize is time = 1/100 s var d is length = earthradius var t is time = 0 s var a is acceleration = 0 gravity while (d > 0 m) { t = t + stepsize a = a[d, shells] v = v + a stepsize d = d - v stepsize // Print results every second if (t mod sec == 0 s) println[(t -> sec) + "\t" + (d->km) + "\t" + (v->mph) + "\t" + (a->m/s^2)] } // Print final times println["Time to core : " + (1. t -> ["min", "sec"]) + "\t, " + (v->"mph")] println["Time through earth: " + (2. t -> ["min", "sec"])]