2. General Physics

Legal Notice, Disclaimer, and Book Ordering Information:

Null

Setup

\(Clear["Global`*`"]; 
Off[General::spell]; 
Off[General::spell1]; 
    \)

Get["D:\Mma\Inits\ch2init.m"]

2.4 Problems

2.4.2  Projectile Motion in a Rotating Reference Frame

Problem 2:  The Foucalt Pendulum

Needs["Calculus`VectorAnalysis`"]

Part (a)  Find equations of motion for a pendulum in a rotating reference frame on the Earth's surface.

ω[t]  = {-(Subscript[ω, 0]) Sin[θ], 0, 
        (Subscript[ω, 0]) Cos[θ]}

force1 = {0, 0, -m g}

\(force2 = {tension \((\(-x[t]\)/L)\)
		\ \ \ \ \ \ \ \ \ \ \ \ \ , 
        tension \((\(-y[t]\)/L)\)
	\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ , 
        tension \((1\  - \ z[t]/L)\)}\)

force = force1 + force2;

r[t_] := {x[t], y[t], z[t]}

leftHandSide =  m D[r[t], {t, 2}]

\(\(rightHandSide = 
	
      \((\((force
			 - \ \ \ m\ CrossProduct[D[ω[t], t], r[t]]

              			 - 2  m\ \ CrossProduct[ω[t], D[r[t], t]]

              			 - \ \ \ 
              m\ \ CrossProduct[ω[t], 
                  \((CrossProduct[ω[t], r[t]])\)])\) // ExpandAll)\)

    \)\)

eq1 = Thread[rightHandSide == leftHandSide]

constraint1 = Thread[Table[D[z[t], {t, n}], {n, 0, 2}] -> 0]

constraint2 = {((Subscript[ω, 0])^2) -> 0,  Sin[θ] -> 0}

constraint = Join[constraint1, constraint2]

eq2 = eq1 /. constraint

{eq3x, eq3y} = 
      Solve[eq2, {x''[t], y''[t]}, tension] /. 
          Rule -> Equal // Flatten

Part (b)  Solve equations of motion for x[t] and y[t].

eq4 = I eq3y // Thread[#, Equal]&

eq5 = eq3x + eq4 // Thread[#, Equal]&

eq6 = eq5 /. {x -> ((u[#] - I y[#])&)} // ExpandAll

initial = {u[0] == I amp,  u'[0] == 0}

\(dsol = 
      \(\(DSolve[Join[{eq6}, initial], u[t], t] //. constraint
 // ExpandAll
            \) // ExpToTrig\) // Simplify\)

Part (c)  Plot solution over one-fourth rotation period (τ=2π/(Subscript[ω, 0]))
for θ=0 (North Pole), π/2 (Equator), and π (South Pole).

values = {g -> 9.81,  (Subscript[ω, 0]) -> 1/10, L -> 1,  amp -> 1}

τ = 2  π/(Subscript[ω, 0]) /. values

\(\(ParametricPlot[
      Evaluate[\(\({Re[u[t]], Im[u[t]]} /. dsol\) //. values\)

          	 /. {θ -> 0}], {t, 0, τ/4}
, Frame -> True
, 
      FrameLabel -> "North Pole"
, \ 
      AxesLabel -> {"x-axis, south", "y-axis, east"}]; \)\)

\(\(ParametricPlot[
      Evaluate[\(\({Re[u[t]], Im[u[t]]} /. dsol\) //. values\)

          	 /. {θ -> π/2}], {t, 0, τ/4}
, Frame -> True
, 
      FrameLabel -> "Equator"
, \ 
      AxesLabel -> {"x-axis, south", "y-axis, east"}]; \)\)

\(\(ParametricPlot[
      Evaluate[\(\({Re[u[t]], Im[u[t]]} /. dsol\) //. values\)

          	 /. {θ -> π}], {t, 0, τ/4}
, Frame -> True
, 
      FrameLabel -> "South Pole"
, \ 
      AxesLabel -> {"x-axis, south", "y-axis, east"}]; \)\)