BOOK OF MATHEMATICAL PROBLEMS. 



1"71. A hollow cone very nearly filled with fluid rotates 

 uniformly about a generating line which is vertical: prove that 

 the pressure on the base is 



3TFtana fer-co 9 



tan a Caw ,- _ - x _ . ") 

 < (1+5 cos 9 a) + 8 sin a V ; 



W being the weight of the fluid, ^^ tte vertical angle, a 

 the radius of the base, and <o the angular velocity. 



1572. A right circular cone, the length of whose axis is h 

 and the radius of its base a, floats in a heavy uniform fluid with 



27 

 =^ of its volume below the surface : prove that, when the , fluid 



revolves about the axis of the cone with angular velocity 



'525 gh 

 592 a*' 



O7 



the cone will float with a length h or -j of its axis immersed. 

 Investigate which of the two positions is stable. 



1573. A sphere, radius a, floats in a mass of fluid, which is 

 revolving uniformly about a vertical axis, with its centre at the 

 vertex of the free surface of the fluid : prove that 



4 (p* + 4a 2 ) (a -pq) = a(p + aq)*', 



where p = 8 , and 1 + q : 2 is the ratio of the densities of the 

 sphere and fluid. 



1574. A hollow cone, very nearly filled with uniform heavy 

 fluid, rotates uniformly about a horizontal generating line : prove 

 that the whole pressure on the base in its lowest position is 



3 + -cos a + 5 cos 2 a 



) 



