A little confused on this question on how to set the equation up. Any help is ap

Question

A little confused on this question on how to set the equation up. Any help is appreciated

Ten turns of copper wire are to be wrapped helically around a cylinder of length 20mm and radius 2.5mm. The turns are to be wrapped evenly from one end of the cylinder to the other. Assume the cylinder axis is on the +z-axis with the bottom at z = 0 and the top at z = 20. Find r ( t) = [x(t) y(t) z(t)]. Hint # 1: Use the parameterization formula for a helix discussed in class.) (Hint #2: For 10 turns, let 0 t 20pi) Use the arc length formula from section 9.5 along with your answer from (a) to calculate the length of wire required.

Explanation / Answer

we know for a helix

r(t) = [x(t), y(t) ,z(t)]

where x(t) = a cost

y(t) = asint

and z(t) = bt

here a is the radius of the helix and for the given curve a= 2.5mm

and b is given as the distance helix rises per turn/2pi

so in this given case we can see that the helix rises 20/10 = 2mm/turn

as 1 turn = 2pi angles

so a= 2.5

b = 2mm/turn

so x(t) = 2.5cost

y(t) = 2.5sint

z(t) = 2t

r(t) = [2.5cost,2.5sint, 2t]

and the lenght of arc is given by

T.sqrt(a2+b2)

where T is the maximum value of t, here it is 20pi as there are ten turns

so length of arc = 20*3.14*sqrt(2.5^2 + 2^2) = 201.05 mm

the length of wire required = 201.05mm

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