When determining the moment of inertia of an area by integration it will usually be found that while it may be relatively easy to perform the integrations for certain positions of the axis, other positions may be difficult because of the shape of the boundary of the area. For example, the moment of inertia of a rectangular area may easily be found by integration for any axis which is parallel to one of the sides. For an axis which is inclined at an angle to the sides, however, the integration and the substitution of the boundary conditions become more difficult. We should like, therefore, a method of determining the moment of inertia of an area about any inclined axis in an area, given the moment of inertia about some other axis.
In Fig. 1 suppose that the moments of inertia
The coordinates of an element of area
Example. Find the moment of inertia of the area of a rectangle about one of the long diagonals (Fig. 2).
Solution. We shall take the center of the rectangle as the origin of an
4.8.1 PROBLEMS
1. Find by integration the product of inertia
2. Find the product of inertia
3. Find the product of inertia
4. Show that the sum of the moments of inertia about two perpendicular axes in the plane of an area is a constant independent of the angle of rotation of the axes in the area, e.g., in Fig. 1,
5. Show that the product of inertia of an area with respect to two inclined axes (as in the following figure) is given by:
6. Show that the angle defining the axis about which the moment of inertia is a maximum or a minimum is given by (see the figure of the previous problem):
7. Show that the product of inertia of an area with respect to the principal axes is equal to zero (refer to Problem 108 above).
8. The Structural Steel Handbook states that the least radius of gyration of the cross-sectional area of a
