27 



TABLE 13— FORMULAS FOR MOMENTS OF INERTIA, RADII OF GYRATION, AND 

 WEIGHTS OF VARIOUS SHAPED SOLIDS 



In each case the axis is supposed to traverse the center of gravity of the body. The axis is one of 

 symmetry. The mass of a unit of volume is zc. 



Square of 



Moment of radius of 



Hody Axis Weight inertia I„ gyration p 2 



4mir 3 %mvr' 2r^_ 



Sphere of radius r Diameter — ~ — jy- 5 



Spheroid of revolution, po- 

 lar axis 2a. equatorial di- 4 2 8nU , ar < 2 r 2 

 ameter 2r Polar axis — — — — -y 



„, „ » ■ o 4mvabc 4mvabc(b 2 + c 2 ) b 2 + r 



Ellipsoid, axis 2a, 2b. 2c. Axis la ~ — 15 5 



Spherical shell, external ra- 4iwt>(r' — r") &mv(r* — r' n ) 2(r 5 — r") 



dins r. internal r' Diameter ^ ^ 5( r 3__ r ' 3 ) 



Ditto, insensibly thin, ra- 8irzur*dr 2r 2 



dius r, thickness dr Diameter 4mvr dr ^ -y 



Circular cylinder, length 2a, Longitudinal r 2 



radius r axis 2a 2mi.>ar 2 invar* y 



Elliptic cylinder, length 2a, Longitudinal mmbc(b" + c 2 ) If + c 2 



transverse axes 2b, 2c. . . axis 2a 2muabc 2 4 



Hollow circular cylinder, 



length 2a. external ra- Longitudinal r 2 + r' 2 



dius r. internal r' axis 2a 2mva(r* — r) mva(r — r) — 2~ 



Ditto, insensibly thin, thick- Longitudinal 

 ness rfr axis 2a 4-irzvardr 4irzvar 3 dr r 



Circular cylinder, length 2a, Transverse o mva^iSr* + 4a 1 ) r a 2 



radius r diameter ^mvar .i_ 



Elliptic cylinder, length 2a, Transverse rwabc(2>c 2 + 4a 2 ) r , a 2 



transverse axes 2a, 2b. . . axis 2b 2mvaoc ^ 4 ^ 3 



Hollow circular cylinder, 



length la, external ra- Transverse „. K , a r 3 ( r > _ r '«) -> ^ + r * 2 



dius r, internal r' diameter 2irwa(r — r ") — — -^ 4. 4 fl * ( r = _ r " ) j ^ r ^ 



Ditto, insensibly thin, thick- Transverse 4 r = a 2 



ness rfr diameter Amvardr mva(2r* + — a 2 r)dr -= + ~r 



Rectangular prism, dimen- 8u>abc(b 2 + c 2 ) If + c 2 

 sions 2d. 2&, 2c Axis 2a 8wabc z ^ — 



Rhombic prism, length 2a 2zvabc(b 2 + r) b 2 + r 



diagonals 2/>, 2c Axis 2a 4zvabc ; — 7 — 



_.. . ... . , 2zvabc(c 2 + 2a 2 ) c a~ 



Ditto Diagonal 2b 4ivabc r ~fi ~T 



For further mathematical data see Smithsonian Mathematical Tables, Becker and Van Orstrand 

 (Hyperbolic, Circular and Exponential Functions); Smithsonian Mathematical Formulae and Tables 

 of Elliptic Functions, Adams and Hippisley ; Smithsonian Elliptic Functions Tables, Spenceley ; 

 Smithsonian Logarithmic Tables, Spenceley and Epperson; Functionentafeln. Jahnke und Emde (xtgx, 

 gf'tgx, Roots of Transcendental Equations, a + bi and re"', Exponentials, Hyperbolic Functions, 



da, j du, I - — du, Fresnel Integral, Gamma Function, Gauss Integral 



\7T 



c **dx, Pearson Function c *'" 



sin r c"dx, Elliptic Integrals and Functions, Spherical and 



Cylindrical Functions, etc.). For further references see under Tables, Mathematical, in the 16th ed. 

 Encyclopaedia Britannica. See also Carr's Synopsis of Pure Mathematics and Mellor's Higher Mathe- 

 matics for Students of Chemistry and Physics. 



SMITHSONIAN PHYSICAL TABLES 



