244 PRACTICAL MATHEMATICS 



as the difference between the moments of inertia of the external 

 and internal spheres. 



i = |M R 2 - f M R 2 



OY 522 511 



EXAMPLES XVI 



(1) A rectangular lamina 5 ft. x 7 ft. is immersed in water with 

 its plane vertical and the smaller edge horizontal ; the centroid 

 of the lamina is 12 ft. below the surface of the water. Find the 

 depth of the centre of pressure. 



(2) A circular lamina, 3 ft. radius, is immersed in water with 

 its plane vertical and its centre 7 ft. below the surface of the 

 water. Find the depth of the centre of pressure. 



(3) Find the moment of inertia of a trapezium whose parallel 

 sides are a and b respectively and whose height is h< 



(1) About the side b 



(2) About an axis parallel to the side b and passing 



through the centroid. 



(4) A trapezoidal lamina whose parallel sides are 8 ft. and 5 ft. 

 respectively and whose height is 6 ft. is immersed in water with 

 its plane vertical and its parallel sides horizontal. The larger of 

 the two parallel sides is situated at a depth of 10 ft. below the 

 surface of the water. Find the depth of the centre of pressure 

 when the smaller of the two parallel sides is situated (1) below the 

 larger, and (2) above the larger. 



(5) Find the moments of inertia of the section, Fig. 73, No. 1, 

 about axes parallel to OX and OY respectively and passing 

 through the centroid. 



(6) Find the greatest and least moments of inertia of the 

 section, Fig. 73, No. 2. 



(7) Find the principal moments of inertia of the section, Fig. 73, 

 No. 3, and find the lengths of the major and minor axes of the 

 momenta! ellipse. 



(8) Find the greatest and least moments of inertia of the 

 section, Fig. 73, No. 4 , and then find the moment of inertia about 

 an axis which passes through the centroid and makes an angle 

 of 30 with the side AB. 



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

 Fig. 73, No. 5. 



(10) Find the moments of inertia of the section, Fig. 73, No. 6, 

 about the axes OX and OY, and hence find the moments of 

 inertia about axes parallel to OX and OY respectively and passing 

 through the centroid. (The full depth is 6".) 



(11) Find the moment of inertia of the section, Fig. 73, No. 7, 

 about the side AB, and hence find the moment of inertia about 

 an axis parallel to AB and passing through the centroid. 



