32 INTRODUCTION TO DYNAMICS. [53. 



(39) Find the centroid of a solid segment of a sphere of radius a, 

 the height of the segment being h. 



(40) Show that, both for a triangular area and for a tetrahedra' 

 volume, the distance of the centroid from any plane is the arithmetic 

 mean of the distances of the vertices from the same plane. 



(41) Find the centroid of the paraboloid of revolution of height 

 generated by the complete revolution of the parabola y 2 = ^ax about 



;its axis. 



(42) The area bounded by the parabola y 2 =^ax, the axis of x, 

 and the ordinate y=y if revolves about the tangent at the vertex. Find 



-the centroid of the solid of revolution so generated. 



(43) The same area as in problem (42) revolves about the ordinate y lf 

 Find the centroid. 



(44) Find the centroid of an octant of an ellipsoid xP/ 



(45) The equations of the common cycloid referred to a cusp as 

 origin and the base as axis of x are x = a (6 sin#), y = a(i cos#) 



Find the centroid : (a) of the arc of the semi-cycloid (i.e. from cusp 

 to vertex) ; (b) of the plane area included between the semi-cycloid and 

 the base ; (c) of the surface generated by the revolution of the semi- 

 cycloid about the base ; (d ) of the volume generated in the same case ; 

 (e) of the surface generated by the revolution of the whole cycloid 

 (from cusp to cusp) about its axis, i.e. the line through the vertex at 

 right angles to the base ; (/) of the volume so generated. 



(46) Find the centroid of a solid hemisphere whose density varies 

 . as the nth power of the distance from the centre. 



(47) From out of the right cone ABC a cone ABD is cut of the 

 same base and axis, but of smaller height. Find the centroid of the 



remaining solid. 



(48) A triangle ABC, whose sides are a, b, c, revolves about an axis 

 situated in its plane. Find the surface area and volume of the solid so 

 generated, if/, ^, r are the distances of A, B, C from the axis. 



(49) " Water is poured gently into a cylindrical cup of uniform thick- 

 ness and density. Prove that the locus of the centre of gravity of the 

 water, the cup, and its handle is a hyperbola." (Routh.) 



