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See grazing cow.

Now keep the restriction that the length of the rope is $l\leq\pi r$ where $r$ is the radius of the barn, (I like to think of this as a goat tied to a silo) but now suppose the cow can fly and eats bugs. What would be the total volume of its grazing space. I guess also assume the barn is tall enough that it can not fly over the top.

Again the quarter sphere of grazing area out front is not a problem to calculate. My idea was to think of it as a bunch of involutes stacked up. Something like thisenter image description here

Forming the sum of these involute slices as they get thinner and thinner we have

$$ \displaystyle{\lim_{n \rightarrow \infty}\frac{l^3}{6r}\cdot \frac{l}{n}+\frac{\left( l\sqrt{1-\left(\frac{1}{n}\right)^2}\right)^3}{6r}\cdot \frac{l}{n} +\cdots +\frac{\left( l\sqrt{1-\left(\frac{n-1}{n}\right)^2}\right)^3}{6r}\cdot \frac{l}{n} }$$ $$\displaystyle{=\frac{l^4}{6r} \lim_{n \rightarrow \infty}\sum_{k=0}^{n-1}\frac{1}{n}\left(\sqrt{1-\left( \frac{k}{n} \right)^2} \right)^3 }$$ $$\displaystyle{=\frac{l^4}{6r}\int_{0}^{1}\left( \sqrt{1-x^2} \right)^3dx }$$ $$\displaystyle{= \frac{l^4\pi}{32r} }$$ adding the quarter sphere and the two side volumes I get a total grazing volume of $\frac{l^4\pi}{16r}+\frac{1}{3}\pi r^3$.

Wondering if this works or if I made some false assumptions?

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coffeebelly
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  • Your method for the wall-contacting-rope exterior of the barn assumes the rope must first go vertically up the wall of the barn, then be bounded by the involute of the circle with the remaining rope. The slabs in your model should be bounded above and below by cones with apexes at the attachment point of the rope since every path the rope takes should (in the limit) live in one slab. – Eric Towers Jul 12 '14 at 17:47
  • The path of the rope, projected onto the horizontal plane, will follow the base of the silo for some distance and then will follow a tangent to the circle. Moreover the rope will make a constant angle with the horizontal plane, so the length of the projected path is a function of the height at the end of the rope. The boundary points at a given height therefore form an involute. The diagram is essentially correct; the "slabs" give an upper bound on the volume, and the difference between cylindrical slabs and conical slabs is irrelevant since it goes to zero as $n$ goes to infinity. – David K Jul 12 '14 at 17:59
  • Nice. I have been making a mess trying to write an equation for a sector that wraps around following an involute curve and gets pinched at the same time. That may not be clear without a picture. I may keep working on it to insure I get the same answer. – coffeebelly Jul 12 '14 at 18:08

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