1 68 



STATICS. 



[275- 



275. Homogeneous Spherical Shell : Analytical method. Let Q 

 (Fig. 84) be any point on the sphere; PQ = r, CQa, CP=p, 



^-PCQ = 0. Through Q lay 

 a plane at right angles to CP, 

 and take as element the mass 

 contained between this and an 

 infinitely near parallel plane. 

 P This mass element is 



= p 2 ira sin * adO, 



and its attraction at P along 

 CPis 



Fig. 84. 



a 2 sin OdO-> cos CPQ = 



sin OdO . 



P ~ a 





The relation between r and follows from the triangle CPQ, which 

 gives 



zap cos 0, 



hence 



rdr = ap sin 



Substituting for a sin OdO and for a cos their values from the last 

 two relations, the expression for the attraction of the elementary ring. 

 becomes 



rdr 

 2VKpa .-- 





a fi-cf+r> , 

 = ,_.< ^ -.dr. 



(a) For an internal point P, we have p<a, and the limits for r are 

 from a p to a +p. Hence the resultant attraction is 



= o. 



(b) For an external point P, we have p>a, and the limits are from 

 p a to p + a. Hence the attraction becomes 



* 



R = 



]*>+<* 

 ,-.= 



M 



1 . 



'A. 



276. Exercises. 



(i) Show that the attraction exerted by a right circular cone of ver- 

 tical angle 2 a and height h, at the vertex, is = 2 TTK/O ( i 



