Ball ring

This article deals with a geometric figure; the word "ball ring" is also used for the balls of a ball bearing together with the "cage" that holds them.

Ball ring: ball with cylindrical bore (right: longitudinal section)

A ball ring is a part of a solid ball that consists of a ball with a cylindrical bore. It is bounded outside by a symmetrical spherical zone and inside by the lateral surface of a straight circular cylinder .

The volume of a ball ring is

${\ displaystyle V = {\ frac {\ pi h ^ {3}} {6}}}$ ,

where is the radius of the sphere, the height and the radius of the bore (cylinder). ${\ displaystyle r}$ ${\ displaystyle h}$ ${\ displaystyle a}$

Its surface (spherical zone and cylinder jacket) is

${\ displaystyle O = 2 \ pi h (r + a)}$

The relationship between the sizes is: ${\ displaystyle r, a, h}$

${\ displaystyle r ^ {2} = a ^ {2} + {\ frac {h ^ {2}} {4}}}$ .

The volume only depends on the height of the ball ring and not on the ball radius . This becomes plausible when you consider that the spherical ring becomes thinner and thinner as the spherical radius increases. ${\ displaystyle h}$ ${\ displaystyle r}$

Derivation of the formulas

The spherical ring can be imagined to have arisen from a symmetrical spherical layer (i.e. ) of height , from which a straight circular cylinder (height , radius ) is removed from the inside . For the volume this means: ${\ displaystyle a_ {1} = a_ {2} = a}$ ${\ displaystyle h}$ ${\ displaystyle h}$ ${\ displaystyle a}$

{\ displaystyle V = V_ {KS} -V_ {Z} = \ pi a ^ {2} h + {\ frac {\ pi h ^ {3}} {6}} - \ pi a ^ {2} h = { \ frac {\ pi h ^ {3}} {6}}}

.

The surface of the spherical ring is composed of the symmetrical spherical zone and the jacket of the cylinder:

{\ displaystyle O = M_ {KZ} + M_ {Z} = 2 \ pi rh + 2 \ pi ah = 2 \ pi h (r + a)}

.

More ball parts

literature

Gardner , M .: Hexaflexagons and Other Mathematical Diversions: The First Scientific American Book of Puzzles and Games (1959, 1988; University of Chicago Press, ISBN 0226282546 , pages 113-121).
Weisstein, Eric W .: Spherical Ring. From MathWorld - A Wolfram Web Resource; see Spherical Ring .
Bartsch, Hans-Jochen: Mathematical formulas, 10th edition, 1971, Buch- und Zeitverlagsgesellschaft mbH, Cologne, without ISBN.

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