246 PRACTICAL MATHEMATICS 



(12) Find the greatest and least radii of gyration of the section, 

 Fig. 73, No. 8. 



(13) Find the principal moments of inertia of the section, 

 Fig. 73, No. 9, and find the lengths of the major and minor axes 

 of the momental ellipse. 



(14) Find the moment of inertia of the section, Fig. 73, No. 1, 

 about an axis passing through the centroid and making an angle 

 of 45 with OX. 



(15) A circular lamina of mass 5 Ibs. and radius 5 ft. rotates 

 uniformly about an axis perpendicular to its plane and just 

 touching its circumference. Find the kinetic energy of rotation 

 if the lamina makes 50 revolutions per minute. 



(16) If the lamina in Question 15 rotates uniformly about a 

 tangent and makes 50 revolutions per minute, what will be the 

 kinetic energy of rotation ? 



(17) Find the radius of gyration, about an axis passing through 

 the centre and perpendicular to its plane, of a circular lamina 

 of radius a, when the density d is such that d = kx where k is 

 a constant and x is the distance from the centre. 



(18) In Question 17, if d = k(a x) where x is the distance from 

 the centre, what will be the radius of gyration about the same 

 axis ? 



(19) Find the moment of inertia of the :'rustum of a cone about 

 its axis, the height being 8 inches, radius of the top 3 inches, 

 and the radius of the bottom 5 inches. A cubic inch of the material 

 weighs 0-26 Ib. 



(20) ABCDE is a figure made up of a square ABCD, 5 inches 

 side, and an equilateral triangle ADE, 5 inches side, the vertex, 

 E, of the triangle lying outside the square. Find the greatest 

 and least moments of inertia of the figure, and hence find the 

 moment of inertia about BD, the diagonal of the square. 



(21) If I is the moment of inertia of an area about a straight 

 line in the same plane passing through its centre, and I is its 

 moment of inertia about a parallel line in the plane, there is a 

 rule which enables us to calculate I if we know I . Prove the 



rule : If for a circle I is r 4 , what is I about a tangent to the 



4 



circle ? (B. of E., 1913.) 



