Winding a wire of definite width around a right cylinder

Question

The original question is presented like this:

A copper wire, $3~mm$ in diameter is wound about a cylinder whose length is $12~cm$ and diameter $10~cm$, so as to cover the curved surface area of the cylinder. Find the length of the wire.

Answer: $400~\pi$

---

I am a highschool student and with my teacher's help and my own start I came to the bookish answer. The process was like you count the no. of loops to be placed around the cylinder and then multipy the length of wire in 1 loop and no. of loops to get the final length of wire. Everthing was good, it came out to be 40 loops and now the problem starts.

According to my teacher and my own second start (At first I thought the same that I am thinking right now but we need to follow our teacher so that's why I stuck with his reasoning.), the length of wire in 1 loop is 10 $\pi$ cm , which is the circumference of the cylinder.

My reasoning: I feel that the length of the circular loop of wire will be $2\pi (R+r)$ where R is the radius of cylinder and r is the radius of the wire but according to my teacher it is just the circumference of the circle.

To try and to justify this I tried using some items I found at my home and measuring the length of 2 different wires around a cylindrical pencil holder. Although I am getting a different length each time when I change the thickness but I cannot justify this to my teacher. I tried explaining him but he says that it will be same no matter what is the radius of the wire and I cannot explain him how.

He is quite understanding btw, just the thing is I don't have any formal explanation to give him and whenever I try I come back with one explanation from him in hand. He says that the wire is touching the circumference so the length will be same as that. I saw that Veritasium's recent video on youtube about the SAT circle problem and can relate this with it which is how I came us to my reasoning part, but I still don't know if I am correct or not or the length difference is just due to rough measuring?

Any kind of 3D graphic visualizations would be appreciated very much and feel free to explain a little deeply.

Who is correct? Me or my teacher? How?

Sorry for bad English and the way I have put up the matter.

Related Video: https://youtu.be/FUHkTs-Ipfg?si=3rTKZjwRwtgfRehg

Answer 1

Suppose that instead of covering the outside of a cylinder, we cover the inside of a cylinder. If we measure the length of wire along the surface of the cylinder as your teacher did, and it takes $40$ turns of the wire to cover it, the length of wire would be $40d\pi$, where $d$ is the diameter of the cylinder.

Now suppose $d= 10.6\ \text{cm}$. Then the length of the wire is $424\pi \ \text{cm}$.

Now that we have used this length of wire to cover the inside of this cylinder, notice that the inner diameter of the coil of wire is just exactly $10\ \text{cm}$. In other words, this exact same coil of wire would tightly wrap a cylinder of that diameter and just barely cover it.

So the length of the coil is $400\pi\ \text{cm}$.

How can one wire have two different lengths? This is a consequence of a poorly written problem. There is no fundamental reason to measure the length along the shorter side of the wire; it is much more typical to measure the longer side. (I would measure along the midline of the wire, which is more likely to represent the unwound length.)

This discrepancy could have been avoided by wrapping the cylinder in something like tape, which we can assume has width but no thickness.

There are other flaws in the problem, such as the fact that in order to cover all of a cylinder with a helical coil, some of the wire will only partially overlap the cylinder and you will end up needing $41$ turns rather than $40$.

You gave some very good thought to this problem before you talked with your teacher. I agree with your formula $2\pi(R+r)$. You are in a bad position to argue with your teacher, so you may have to give this one up, but after you finish this class please go back to using your original, better formula.

If you continue studying math in college, you will probably find that the textbooks are much better written. Keep thinking for yourself and don’t let anyone destroy the insight and intelligence that you clearly have.

Answer 2

Your teacher’s answer is a classic case of neglecting those values which are too small to change the result. Imagine you are wearing a ring on your finger. What would you say was the length of the ring was? It’s outer perimeter (a variant of your type of answer) or its inner perimeter (your teacher’s type of answer)?

In the end it all comes down to how you define length for a curved pipe like structure. Our inherent definition of length depends on 1-D objects and to extend it to larger scales, you will have to explicitly explain the definition, or stick to a sub-optimal compromise.

Food-for-thought: If you are winding the rope on the cylinder, you are actually tracing out a helix. Considering the helix as a set of circles is also another approximation you are using.

Answer 3

I am also a high school student, and although I haven't taken Precalculus since 6th grade, but I will attempt to answer this question. Your question is on a basis of whether the magnitude of the wire is even accounted for in the circumference. There are many ways this can change due to different situations. For example, a small change of magnitude to an exponential function will drastically change the output of the function, but a slight change in magnitude to a function will not change the first derivative as the (x^' ) as adding a constant c to the function will not affect the first derivative. This may be a little advanced for precalculus, but I would say if the radius of the wire is more than 1% of the radius of the cylinder, which is not the case here, then you should include it. Hope this helps!