Measuring Spacing with Shortcut Technique: Is Uniformity Achieved?

[erobz]

I saw on a home improvement show some "shortcut" for measuring out a series of fixed measurements over a distance (it pertained to spacing baluster for porch railing). The contractor places a series of lines of some small incremental distance (1 or 2 inches say) on an elastic fabric band, and the stretches the band until the desired spacing is achieved on the last post. I thought, "is that going to yield increments of uniform spacing over the entire length or is this just a close enough technique?"

So, I start with:

$$ \frac{x}{l} = \frac{x + dx}{ l + dl} $$

Where $x$ is a marked position on the band, and $l$ is the length of the band.

This yields

$$ dx = \frac{x}{l}dl \tag{1} $$

I am going to say that $x = fl$, where $0 \leq f \leq 1$

From (1)

$$ \begin{align} \int_{{{x_a}}}^{ {x_a}'} dx &= f_a \int_{L_o}^{L} dl \tag*{} \\ \quad \tag*{} \\ {x_a'} &= f_a \Delta l + x_a \tag*{} \end{align}$$

${x_a}'$ is the new coordinate of $x_a$ after the band undergoes a change of length $\Delta l = L - L_o$

So next imagine $x_2 - x_1$ defines some increment. After a stretch of $\Delta l$ the new coordinates of the markings are given by:

$$ {x_2}' - {x_1}' = \left( f_2 - f_1 \right) \Delta l + x_2 - x_1 = \left( f_2 - f_1 \right) \left( \Delta l + L_o \right) $$

Similarly let $x_4 - x_3$ define an increment at some other location on the band.

$$ {x_4}' - {x_3}' = \left( f_4 - f_3 \right) \Delta l + x_4 - x_3 = \left( f_4 - f_3 \right) \left( \Delta l + L_o \right) $$

We can now see given some arbitrary $\Delta l$ as long as $f_4 - f_3 = f_2 - f_1$ (that is to say the increments initially made are of uniform length), we have that:

$$ {x_4}' - {x_3}' = {x_2}' - {x_1}' $$

and the "shortcut" the contractor used should, indeed create increments of uniform length on the band after stretched to its new length.

I think I have it right? This ignores the width of the marks (assumed very thin), and length contraction in the band in orthogonal directions.

 

[kuruman]

I think you are overcomplicating things. The assumption at work here is that if you stretch the band by pulling its ends farther apart, the band will stretch uniformly. It is expressed mathematically with your equation (1) but wether the band actually does that is a property of the band material.

Say you subdivide the length $l$ of the unstrectched band into $N$ equal increments separated by $\delta$ so that $l=N\delta$. Under the assumption, if you stretch the band to a new length, you will still have $N$ subdivisions equally separated. It's like taking a photograph of a 12" ruler. If you print it enlarged by a factor of 1.5, the one-inch markings will be equally spaced 1.5'' apart.

However, If the band stretches non-uniformly, say slightly more near the ends than in the middle, the marks will not be equally separated. My sense is that a real band (I have never used one) will not stretch uniformly and that this method is close enough. How close is close enough for a real band cannot be derived mathematically but can be investigated experimentally.

 

[erobz]

I think you are overcomplicating things. The assumption at work here is that if you stretch the band by pulling its ends farther apart, the band will stretch uniformly. It is expressed mathematically with your equation (1) but wether the band actually does that is a property of the band material.

Say you subdivide the length $l$ of the unstrectched band into $N$ equal increments separated by $\delta$ so that $l=N\delta$. Under the assumption, if you stretch the band to a new length, you will still have $N$ subdivisions equally separated. It's like taking a photograph of a 12" ruler. If you print it enlarged by a factor of 1.5, the one-inch markings will be equally spaced 1.5'' apart.

However, If the band stretches non-uniformly, say slightly more near the ends than in the middle, the marks will not be equally separated. My sense is that a real band (I have never used one) will not stretch uniformly and that this method is close enough. How close is close enough for a real band cannot be derived mathematically but can be investigated experimentally.

Well, I overcomplicated it because I under thought it. What you say makes perfect sense (and seems obvious) now that you've pointed it out!

I was driven that way a because I remembered from the classic "ant on a stretching rubber band" that the absolute velocity of the ant depends on where the ant is on the rubber band. So, I was initially thinking that if each point on the stretching band had a different velocity, then the distances between two points, at different points would be different (I obviously proved that intuition false). I'm guilty of being lazy. I knew how to get to that point, so I just went with what I know, instead of thinking originally about this particular problem.

 

[Baluncore]

My sense is that a real band (I have never used one) will not stretch uniformly and that this method is close enough.

Most of the variation is eliminated when you chain many rubber bands together from the same pack. That also eliminates the need to mark the band as the knots are equally spaced.
How good is good enough in home improvement? If it looks right, it is correct.