70 Table 38. 



MOMENTS OF INERTIA, RADII OF GYRATION, AND WEIGHTS. 



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

 one of symmetry. The mass of a unit of volume is w. 



Body. 



Sphere of radius r 



Spheroid of revolution, po- 

 lar axis 2a, equatorial di- 

 ameter 2r 



Ellipsoid, axes 2a, 2b, 2c 



Spherical shell, external ra- 

 dius r, internal r' 



Ditto, insensibly thin, ra- 

 dius r, thickness dr 



Circular cylinder, length za, 

 radius r 



Elliptic cylinder, length 2a, 

 transverse axes 2b, zc 



Hollow circular cylinder, 

 length 2a, external ra- 

 dius r, internal r' 



Ditto, insensibly thin, thick- 

 ness dr 



Circular cylinder, length za, 

 radius r 



Elliptic cylinder, length 2(7, 

 transverse axes 2a, zb 



Hollow circular cylinder, 

 length 2.7, external ra- 

 dius r, internal r' 



Ditto, insensibly thin, thick- 

 ness dr 



Rectangular prism, dimen- 

 sions 2a, zb, zc 



Rhombic prism, length 2a, 

 diagonals zb, zc 



Ditto 



Axis. 



Diameter 



Polar axis 



Axis za 

 Diameter 



Diameter 



Longitudinal 

 axis za 



Longitudinal 

 axis za 



Longitudinal 

 axis za 



Longitudinal 

 axis 2a 



Transverse 

 diameter 



Transverse 

 axis 2b 



Transverse 

 diameter 



Transverse 

 diameter 



Axis 2a 

 Axis za 

 Diagonal zb 



Weight. 



i,irwr* 



4n'wa* 



3 



i\iTwr'^dr 



Zitwabc 

 2irie/a{r^ — r'') 



4irwardr 



2irwar^ 



Zirwabc 



27rwa(;-2 — r'^) 



^TTTvardr 

 2>wabc 

 4zvabc 

 ifWabc 



Moment of Inertia lo. 



15 



Zvwar^ 

 IS 



4ir'wabc{b--\-fi^ 



Ts 



Z-Kwir^ — r'6) 



15 



Zmur^dr 



3 



■trwar^ 

 vwabc{b'^-\-c'^) 



irwa(/^ — r"^) 



4irwat*dr 



■ir~var'(y--\- 4a') 

 ^ 6 

 'irLt)abc(y'^-\-4a-) 

 6 



tnoa S 3(r*— r"») ) 



6 \ 4-4^2(r2-r'2) j 



Ttwaizr^A — ah-)dr 

 3 

 Swabc(b^-\-c^) 



2wabc{b"-\-c'^) 



Zwabc((-'^-\-2a") 



Square of Ra- 

 dius of G)'ra- 

 tion p-. 



5 



5 



2(r>—r'^) 



2r2 



3 



^2 



2 

 4 



y2J.^'2 



r^ a^ 



4"^ "3 



c^ a"^ 



7 + 7 



^2-1-^2 

 3 



<J2+r2 



- 4- — 



(Taken from Rankine.) 



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

 (Hyperbolic, Circular and Exponential Functions); Functionentafeln, Jahnke und Emde (xigx, 

 x-^tgx. Roots of Transcendental Equations, a -|- bi and re**'. Exponentials, Hyperbolic Functions, 



/^^^ du, I '^^^ du, I — du, Fresnel Integral, Gamma Function, Gauss Integral 



^~^J o 



e-x^dx, Pearson Function d-if" 



/>' 



n'' t>'xdx. Elliptic Integrals and Functions, Spherical 



and Cylindrical Functions, etc.). For further references see under Tables, Mathematical, in the 

 nth e'd. Encyclopaedia Britannica. See aLso Carr's Synopsis of Pure Mathematics and Mellor's 

 Higher Mathematics for Students of Chemistry and Physics. 

 Smithsonian Tables. 



