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Barrel


Barrel

A barrel solid of revolution composed of parallel circular top and bottom with a common axis and a side formed by a smooth curve symmetrical about the midplane.

The term also has a technical meaning in functional analysis. In particular, a subset of a topological linear space is a barrel if it is absorbing, closed, and absolutely convex (Taylor and Lay 1980, p. 111). (A subset S of a topological linear space X is absorbing if for each x in X there is an r>0 such that ax is in X if for each a such that |a|>=r. A subset S of a topological linear space is absolutely convex if for each x and y in S, ax+by is in S if |a|+|b|<=1.)

When buying supplies for his second wedding, the great astronomer Johannes Kepler became unhappy about the inexact methods used by the merchants to estimate the liquid contents of a wine barrel. Kepler therefore investigated the properties of nearly 100 solids of revolution generated by rotation of conic sections about non-principal axes (Kepler, MacDonnell, Shechter, Tikhomirov 1991).

BarrelElliptic

For sides consisting of an arc of an ellipse, the equation of the side is given by

 x(z)=r_2sqrt(1-((z-1/2h)^2)/(a^2)),
(1)

with x(0)=r_1. Solving for a gives

 a=(hr_2)/(2sqrt(r_2^2-r_1^2)),
(2)

so the sides have equation

 x(z)=sqrt(r_2^2+((r_1-r_2)(r_1+r_2)(h-2z)^2)/(h^2)).
(3)

Using the equation for a solid of revolution then gives

V=piint_0^h[x(z)]^2dz
(4)
=1/3pih(2r_2^2+r_1^2),
(5)
BarrelParabolic

For sides consisting of a parabolic segment, the equation of the side is given by

 x(z)=r_2+a(z-1/2h)^2
(6)

with x(0)=r_1. Solving for a gives

 a=(4(r_1-r_2))/(h^2),
(7)

so the sides have equation

 x(z)=r_2+((r_1-r_2)(h-2z)^2)/(h^2).
(8)

Using the equation for a solid of revolution then gives

V=piint_0^h[x(z)]^2dz
(9)
=1/(15)pih(3r_1^2+4r_1r_2+8r_2^2).
(10)

See also

Cylinder, Hyperboloid, Solid of Revolution

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References

Harris, J. W. and Stocker, H. "Barrel." §4.10.4 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 112, 1998.Kepler, J. New Solid Geometry of Win Barrels.MacDonnell, J. "The Mathematician's Quest for Superlatives from Geometrical and Calculus Considerations." http://www.faculty.fairfield.edu/jmac/ther/superlatives.htm.Shechter, B.-S. "Kepler's Wine Barrel Problem in a Dynamic Geometry Environment." http://www.math.uoc.gr/~ictm2/Proceedings/pap420.pdf.Taylor, A. E. and Lay, D. C. Introduction to Functional Analysis, 2nd ed. New York: Wiley, 1980.Tikhomirov, V. M. "New Solid Geometry of Wine Barrels." Stories About Maxima and Minima. Providence, RI: Amer. Math. Soc., 1991.

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Barrel

Cite this as:

Weisstein, Eric W. "Barrel." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Barrel.html

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