EQUATION OF SPIRAL

Replies

• 

  Ramani Aswath

  A rigorous analytical solution can be got using integration and the equation for a log spiral. Do not have the time or the inclination to attempt.
  A much simpler solution good enough for all practical purposes can be worked out quickly.
  
  Based on data given, the first round will have a length of Pi*(X+D) taking the axis of the wire as reference.
  Each successive round will have a length Pi*2*D more than the previous round.
  
  For the given total length of L, let us say n rounds can be made.
  
  Then, the total length can be given as:
  L = Pi*[(X+D) + (X+3D) + (X+5D) + …+ (X+(2*n - 1)*D)
  = Pi*[n*X + D*(1+3+5….+(2*n - 1))]
  = Pi*(n*X + n^2*D) {sum of first n odd numbers = n^2}
  
  Rearranging: D*n^2 + X*n - (L/Pi) = 0
  
  Solve the quadratic for n. Of the two roots only one will be physically feasible.
  Since Pi and a variable quantity L are involved, n may not be an integer. The Integral part of n = [n], which means that there would be [n] full turns and a little left over.
  The largest diameter measured over the last partial coil portion of the wire will be:
  Dia(max.) = X + [n]*D + D
• 

  Ramani Aswath

  If a rubber yoga or exercise mat is available, this can be verified by winding it tightly over different diameter rods and measuring the diameters. Trial even on one rod will validate the solution.
• 

  MURTUZA ZAVERI

  THANX A LOT, RAMANI SIR
• 

  Ramani Aswath

  A.V.Ramani

  Dia(max.) = X + [n]*D + D

  I apologise. There is an error.
  The above should be:
  Dia(max.) = X + 2*[n]*D + D
  Each round adds 2*D to the total diameter.