Sfernost

Sfêrnost (tudi sfêričnost) je merilo, kako zelo je oblika objekta podobna obliki popolne sfere. Sfernost kroglic v krogličnem ležaju na primer določa kakovost ležaja, kot je obremenitev, ki jo lahko prenese, ali hitrost, s katero se lahko vrti, ne da bi odpovedal. Sfernost je poseben primer mere kompaktnosti oblike. Sfernost ${\displaystyle \Psi \!\,}$ delca, kot jo je definiral Hakon Wadell leta 1935, je razmerje med površino sfere z enako prostornino kot dano telo in površino telesa:^[1]

${\displaystyle \Psi ={\frac {\pi ^{\frac {1}{3}}(6V_{p})^{\frac {2}{3}}}{A_{p}}}\!\,,}$

kjer je:

• ${\displaystyle V_{p}\!\,}$ – prostornina telesa,
• ${\displaystyle A_{p}\!\,}$ – površina telesa.

Sfernost sfere je enaka 1 po definiciji, in po izoperimetrični neenakosti bo imelo vsako telo, ki ni sfera, sfernost manjšo od 1.

Sfernost velja v treh razsežnostih. Njen analog v dveh razsežnostih, kot so krožnice prečnega prereza vzdolž valjnega objekta, kot je gred, se imenuje okroglost.

Sfernost ${\displaystyle \Psi \!\,}$ sploščenega sferoida (podobnega obliki planeta Zemlja) je:

${\displaystyle \Psi ={\frac {\pi ^{\frac {1}{3}}(6V_{\rm {p}})^{\frac {2}{3}}}{A_{\rm {p}}}}={\frac {2{\sqrt[{3}]{ab^{2}}}}{a+{\frac {b^{2}}{\sqrt {a^{2}-b^{2}}}}\ln {\left({\frac {a+{\sqrt {a^{2}-b^{2}}}}{b}}\right)}}}\!\,,}$

kjer sta ${\displaystyle a\!\,}$ in ${\displaystyle b\!\,}$ velika in mala polos nosilne elipse. Za Zemljo veljata polmera ${\displaystyle a=6378,1366{\text{ km}}\!\,}$ in ${\displaystyle b=6356,7523{\text{ km}}\!\,,}$

in njena sfernost:

${\displaystyle \Psi _{\rm {Z}}={\frac {\pi ^{\frac {1}{3}}(6V_{\rm {Z}})^{\frac {2}{3}}}{A_{\rm {Z}}}}={\frac {2{\sqrt[{3}]{6378,1366\cdot 6356,7523^{2}}}}{6378,1366+{\frac {6356,7523^{2}}{\sqrt {6378,1366^{2}-6356,7523^{2}}}}\ln {\left({\frac {6378,1366+{\sqrt {6378,1366^{2}-6356,7523^{2}}}}{6356,7523}}\right)}}}=0,999997993825555\ldots \!\,.}$

Wadell je definiral sfernost kot površino sfere z enako prostornino delca, deljeno z dejansko površino delca.

Najprej je treba zapisati površino sfere ${\displaystyle A_{\rm {s}}\!\,}$ z izrazi prostornine delca ${\displaystyle V_{\rm {d}}\!\,}$:

${\displaystyle A_{\rm {s}}^{3}=\left(4\pi r^{2}\right)^{3}=4^{3}\pi ^{3}r^{6}=4\pi \left(4^{2}\pi ^{2}r^{6}\right)=4\pi \cdot 3^{2}\left({\frac {4^{2}\pi ^{2}}{3^{2}}}r^{6}\right)=36\pi \left({\frac {4\pi }{3}}r^{3}\right)^{2}=36\,\pi V_{\rm {d}}^{2}\!\,}$

in tako:

${\displaystyle A_{\rm {d}}=\left(36\,\pi V_{\rm {d}}^{2}\right)^{\frac {1}{3}}=36^{\frac {1}{3}}\pi ^{\frac {1}{3}}V_{\rm {d}}^{\frac {2}{3}}=6^{\frac {2}{3}}\pi ^{\frac {1}{3}}V_{\rm {d}}^{\frac {2}{3}}=\pi ^{\frac {1}{3}}\left(6V_{\rm {d}}\right)^{\frac {2}{3}}\!\,,}$

od koder se definira ${\displaystyle \Psi \!\,}$ kot:

${\displaystyle \Psi ={\frac {A_{\rm {s}}}{A_{\rm {d}}}}={\frac {\pi ^{\frac {1}{3}}\left(6V_{\rm {d}}\right)^{\frac {2}{3}}}{A_{\rm {d}}}}\!\,.}$

ime	slika	prostornina	površina	sfernost
platonska telesa
tetraeder		${\displaystyle {\frac {\sqrt {2}}{12}}\,s^{3}\!\,}$	${\displaystyle {\sqrt {3}}\,s^{2}\!\,}$	${\displaystyle \left({\frac {\pi }{6{\sqrt {3}}}}\right)^{\frac {1}{3}}\approx 0,671\!\,}$
kocka		${\displaystyle s^{3}\!\,}$	${\displaystyle 6\,s^{2}\!\,}$	${\displaystyle \left({\frac {\pi }{6}}\right)^{\frac {1}{3}}\approx 0,806\!\,}$
oktaeder		${\displaystyle {\frac {1}{3}}{\sqrt {2}}\,s^{3}\!\,}$	${\displaystyle 2{\sqrt {3}}\,s^{2}}$	${\displaystyle \left({\frac {\pi }{3{\sqrt {3}}}}\right)^{\frac {1}{3}}\approx 0,846\!\,}$
dodekaeder		${\displaystyle {\frac {1}{4}}\left(15+7{\sqrt {5}}\right)\,s^{3}\!\,}$	${\displaystyle 3{\sqrt {25+10{\sqrt {5}}}}\,s^{2}\!\,}$	${\displaystyle \left({\frac {\left(15+7{\sqrt {5}}\right)^{2}\pi }{12\left(25+10{\sqrt {5}}\right)^{\frac {3}{2}}}}\right)^{\frac {1}{3}}\approx 0,910\!\,}$
ikozaeder		${\displaystyle {\frac {5}{12}}\left(3+{\sqrt {5}}\right)\,s^{3}\!\,}$	${\displaystyle 5{\sqrt {3}}\,s^{2}\!\,}$	${\displaystyle \left({\frac {\left(3+{\sqrt {5}}\right)^{2}\pi }{60{\sqrt {3}}}}\right)^{\frac {1}{3}}\approx 0,939\!\,}$
okrogle oblike
stožec ${\displaystyle (h=2{\sqrt {2}}r)\!\,}$		${\displaystyle {\frac {1}{3}}\pi \,r^{2}h\!\,}$ ${\displaystyle ={\frac {2{\sqrt {2}}}{3}}\pi \,r^{3}\!\,}$	${\displaystyle \pi \,r(r+{\sqrt {r^{2}+h^{2}}})\!\,}$ ${\displaystyle =4\pi \,r^{2}\!\,}$	${\displaystyle \left({\frac {1}{2}}\right)^{\frac {1}{3}}\approx 0,794\!\,}$
polobla		${\displaystyle {\frac {2}{3}}\pi \,r^{3}\!\,}$	${\displaystyle 3\pi \,r^{2}\!\,}$	${\displaystyle \left({\frac {16}{27}}\right)^{\frac {1}{3}}\approx 0,840\!\,}$
valj ${\displaystyle (h=2\,r)\!\,}$		${\displaystyle \pi r^{2}h=2\pi \,r^{3}\!\,}$	${\displaystyle 2\pi r(r+h)=6\pi \,r^{2}\!\,}$	${\displaystyle \left({\frac {2}{3}}\right)^{\frac {1}{3}}\approx 0,874\!\,}$
svitek ${\displaystyle (R=r)\!\,}$		${\displaystyle 2\pi ^{2}Rr^{2}=2\pi ^{2}\,r^{3}\!\,}$	${\displaystyle 4\pi ^{2}Rr=4\pi ^{2}\,r^{2}\!\,}$	${\displaystyle \left({\frac {9}{4\pi }}\right)^{\frac {1}{3}}\approx 0,894\!\,}$
sfera		${\displaystyle {\frac {4}{3}}\pi r^{3}\!\,}$	${\displaystyle 4\pi \,r^{2}\!\,}$	${\displaystyle 1\!\,}$
druge oblike
rombski triakontaeder		${\displaystyle 4{\sqrt {5+2{\sqrt {5}}}}\,s^{3}\!\,}$	${\displaystyle 12{\sqrt {5}}\,s^{2}\!\,}$	${\displaystyle {\frac {\pi ^{\frac {1}{3}}\left(24{\sqrt {5+2{\sqrt {5}}}}\right)^{\frac {2}{3}}}{12{\sqrt {5}}}}\approx 0,9609\!\,}$
disdiakisni triakontaeder		${\displaystyle {\frac {180}{11}}\left(5+4{\sqrt {5}}\right)\,s^{3}\!\,}$	${\displaystyle {\frac {180}{11}}{\sqrt {179-24{\sqrt {5}}}}\,s^{2}\!\,}$	${\displaystyle {\frac {\left(\left(5+4{\sqrt {5}}\right)^{2}{\frac {11\pi }{5}}\right)^{\frac {1}{3}}}{\sqrt {179-24{\sqrt {5}}}}}\approx 0,9857\!\,}$

• ekvivalentni sferni premer
• sploščenost
• indeks sfernosti
• izoperimetrično razmerje
• okroglost
• okroglost (geologija)
• Willmoreova energija

Poglejte si besedo sphericity v Wikislovarju, prostem slovarju.

• Grain Morphology: Roundness, Surface Features, and Sphericity of Grains