This notebook contains additional material to the Mathematica GuideBooks  (Springer-Verlag, New York, 2004) http://www.MathematicaGuideBooks.org . It uses the stylesheet of the Mathematica GuideBooks, which is embedded.

Symbolics Volume, Chapter 3

The equations of motion for a flat spinning disk sliding on a plane surface including friction can be written, after an appropriate scaling of variables, in the following form [✶], [✶]:

(Overscript[𝓍, ..])(τ)={({{((4 )/(3 π ε(τ)))((((ε(τ))^2)+1) E(((ε(τ))^2))+((ε^2)-1) K(((ε(τ))^2))), if |ε(τ)|≤1}, {((4 sgn(ε(τ)))/(3 π))  ((((ε(τ))^2)+1) E(((ε(τ))^(-2)))-(((ε(τ))^2)-1) K(((ε(τ))^(-2)))), if |ε(τ)|≥1}})

(Overscript[φ, ..])(τ)={({{(4/(9 π)) ((4-2 ((ε(τ))^2)) E(((ε(τ))^2))+(((ε(τ))^2)-1) K(((ε(τ))^2))), if |ε(τ)|≤1}, {((4 sgn(ε(τ)))/(9π ε(τ))) ((4 ((ε(τ))^2)-2 ((ε(τ))^4)) E(((ε(τ))^(-2)))+(2 ((ε(τ))^4)-5 ((ε(τ))^2)+3) K(((ε(τ))^(-2)))), if |ε(τ)|≥1}})

Here 𝓍(τ) is the position of the disk along its direction of motion, and φ(τ) is the angle of the disk with respect to its center. The friction induces a complicated coupling between the translational and the rotational movement characterized by the quantity ε(τ)=(OverDot[𝓍])(τ)/(OverDot[φ])(τ). E(z) and K(z) are the complete elliptic integrals. For a fixed initial velocity 𝓋(0)=(OverDot[x])(0) and various initial angular velocities ω(0)=(OverDot[φ])(0), solve the equations of motion numerically. Does a disk that initially rotates move faster than one that does not rotate initially?

ℱ[ε_] = 
Which[ε == 1, 8/(3 Pi), (* no rotation *) ε === ComplexInfinity, 1, 
Abs[ε] < 1,  
4/3 ((ε^2 + 1) EllipticE[ε^2] + (ε^2 - 1) EllipticK[ε^2])/(Pi ε),
Abs[ε] > 1,
4/3 Sign[ε] ((ε^2 + 1) EllipticE[1/ε^2] - (ε^2 - 1) EllipticK[1/ε^2])/(Pi)];

𝒯[ε_] = 
Which[ε == 1, 8/(9 Pi), (* no rotation *) ε === ComplexInfinity, 0, 
Abs[ε] < 1, 
4/9 ((4 - 2 ε^2) EllipticE[ε^2] + (ε^2 - 1) EllipticK[ε^2])/Pi,
Abs[ε] > 1,
4/9 Sign[ε] ((4 ε^2 - 2 ε^4) EllipticE[1/ε^2] + 
     (2ε^4 - 5ε^2 + 3) EllipticK[1/ε^2])/(Pi ε)];

Here are the force and the torque as a function of ε. For increasing angular velocity, the force is decreasing and the torque is increasing.

Show[GraphicsArray[
Plot[#[ε], {ε, 0, 4}, DisplayFunction -> Identity,
     Frame -> True, Axes -> False, PlotRange -> All]& /@ {ℱ, 𝒯}]];

Solving the equations of motions and visualizing the results is straightforward. We use 𝓋(0)=2 and integrate the equations as long as (OverDot[x])(τ) is positive. The left graphic shows the positions and the angles (as dashed lines) for four initial angular velocities. The right graphic shows the positions and the velocity and the angular velocity (as dashed lines) for the same four initial angular velocities. The graphics show clearly that a spinning disk moves further. They also show that the translational motion and the rotational motion stop at the same moment, meaning the disk cannot end its translational motion first and just spin or end spinning before ending its translational motion [✶].

Module[{𝓋0 = 2, ndsol, Τ = 4}, 
Show[GraphicsArray[Show /@ Transpose[ Function[{ω0, color},
(* solve differential equations *)
ndsol = NDSolve[{𝓍'[τ] == 𝓋[τ], φ'[τ] == ω[τ], 
                 𝓋'[τ] == -ℱ[𝓋[τ]/ω[τ]], ω'[τ] == -2 𝒯[𝓋[τ]/ω[τ]],
                 𝓍[0] == 0, 𝓋[0] == 𝓋0, φ[0] == 0, ω[0] == ω0},
                {𝓍, 𝓋, φ, ω}, {τ, 0, Τ}, StoppingTest -> 𝓋[τ] < 0];
Τ1 = ndsol[[1, 1, 2, 1, 1, 2]];
(Plot[Evaluate[{#1[τ], #2[τ]} /. ndsol[[1]]], {τ, 0, Τ1},
     PlotStyle -> {{color, Thickness[0.002]},
                   {color, Thickness[0.002], Dashing[{0.02, 0.02}]}},
     DisplayFunction -> Identity, AxesLabel -> {τ, {#1, #2}}]& @@@ 
     (* show 𝓍, 𝓋, and φ, ω *) {{𝓍, φ}, {𝓋, ω}})] @@@
  (* various initial velocities *)
  {{0.010, Hue[0.00]}, {0.500, Hue[0.12]}, 
   {1.000, Hue[0.22]}, {3.000, Hue[0.78]}}]]]]