1) How thick is the tape?
1) How thick is the tape?
We first convert all unit lengths to inches and unit times to seconds.
Input := TapeSpeed = 1.3125
Output = 1.3125
Input := PlayTime = 2 3600
Output = 7200
Input := RadSpind = 12/25.4
Output = 0.472441
Input := RadWound = 43/25.4
Output = 1.69291
Input := TapeLength = TapeSpeed PlayTime
Output = 9450.
Approach 1: Model with cylindrical shells.
The number n,
Input := n = (RadWound - RadSpind)/Thick
Output = 1.22047 ------- Thick
is the approximate number of shells required.
We add up the circumference of each shell to solve for the thickness.
Note that the first shell has circumference 2 p (RadSpind + Thick) and each shell around that has circumference given by 2 p (RadSpind + i*Thick) where i goes from 1 to n. Using the fact that Sum[i, {i, 1, n}] = n(n+1)/2, we can simplify the sum of circumferences
Sum[2 p (RadSpind + i*Thick), {i, 1, n}]
as
2 p (RadSpind n + Thick n(n+1)/2) = 2 p n (RadSpind + (n + 1) Thick/2)
Input :=
thickness = Flatten[
Solve[2 Pi n(RadSpind + (n+1) Thick)/2 ==
TapeLength, Thick]];
Input := TapeThickness1 = Thick /. thickness
Output = 0.000687159
Approach 2: Model as a spiral -- use polar coordinates.
As theta: 0 -> 2 p, we want the spiral to increase in radius by Thick. We also require that when theta = 0, that the radius is RadSpind. Finally, we want the length to be TapeLength when the radius is RadWound.
We know the total arc length. So, our goal is to determine the arc length of a spiral as a function of the spacing until it becomes the correct length.
Input := r[theta_] = RadSpind + theta Thick/(2 Pi)
Output =
theta Thick
0.472441 + -----------
2 Pi
Input := rprime = r'[theta]
Output = Thick ----- 2 Pi
Input :=
PolarArcLength = Integrate[Sqrt[r[theta]^2 + rprime^2],
{theta, 0, 2 Pi n}];
Input :=
thickness =
FindRoot[PolarArcLength == TapeLength, {Thick, 0.0006}];
Input := TapeThickness2 = Thick /. thickness
Output = 0.000878567
Input :=
m = n /. Thick -> TapeThickness2;
PolarPlot[RadSpind + theta TapeThickness2/(2 Pi),
{theta, 0, m 2 Pi},
PlotPoints -> 400]
Output = -Graphics-