There are short methods of computing the exact values of the moments of inertia of simple fig ures (rectangles, circles, etc.,), but they cannot be given here since they involve the use of difficult mathematics. The foregoing method to obtain approximate val ues of moments of inertia is used especially when the area is quite irregular in shape, but it is given here to explain to the student the meaning of the moment of inertia of an area. He should understand now that the moment of inertia of an area is sim ply a name for such sums as we have just computed. The name is not a fitting one, since the sum has nothing whatever to do with inertia. It was first used in this connection because the sum is very similar to certain other sums which had previously been called moments of inertia.
Unit of Moment of Inertia. The product (area X dis tance') is really the product of four lengths, -two in each factor ; and since a moment of inertia is the sum of such products, a moment of inertia is also the product of four lengths. Now the product of two lengths is an area, the product of three is a vol ume, and the product of four is moment of inertia—unthinkable in the way in which we can think of an area or volume, and there fore the source of much difficulty to the student. The units of these quantities (area, volume, and moment of inertia) are respec. tively: the square inch, square foot, etc., " cubic " , cubic " " , • " biquadratic inch, biquadratic foot, etc.; but the biquadiatic inch is almost exchisively used in this connec tion; that is, the inch is used to compute values of moments of inertia, as in the pre ceding illustration. It is often written thus: Inches'.
52. Moment of Inertia of a Rectangle.
Let b denote the base of a rectangle, and a its altitude; then by higher mathematics it can be shown that the moment of inertia of the rectangle with respect to a line through its center of gravity and parallel to its base, is Example. Compute the value of the moment of inertia of a rectangle 4 X 12 inches with respect to a line through its center of gravity and parallel to the long side.
Here b=12, and a = 4 inches ; hence the moment of inertia desired equals •„1,(12 x4)=64 inches'.
1. Compute the moment of inertia of a rectangle 4X12 inches with respect to a line through its center of gravity and parallel to the short side. Ans. 576 inches'.
53. Reduction Formula. In the previously mentioned "handbooks" there can be found tables of moments of inertia of all the cross-sections of the kinds and sizes of rolled shapes made.
The inertia-axes in those tables are always taken through the cen ter of gravity of the section, and usually parallel to some edge of the section. Sometimes it is necessary to compute the moment of inertia of a "rolled section" with respect to some other axis, and if the two axes (that is, the one given in the tables, and the other) are parallel, then the desired momentof inertia can be easily com puted from the one given in the tables by the following rule: The moment qf inertia of an area with respect to any axis equals the moment of inertia with respect to a parallel axis through the center of gravity, plus the product of the area and the square of the distance between the axes.
0i, if I denotes the moment of inertia with respect to any axis; the moment of inertia with respect to a parallel axis through the center of gravity; A the area; and d the distance between the axes, then Ad' . (5) Example. It is required to compute the moment of inertia of a rectangle 2 X8 inches with respect to a line parallel to the long side and 4 inches from the center of gravity.
Let I denote. the moment of inertia sought, and the moment of inertia of the rectangle with respect to a line parallel to the long side and through the center of gravity (see Fig.
28). Then (see Art. 52); and, since b=8 inches and a=2 inches, (8 X 2')=53 biquadratic inches.
The distance between the two inertia axes is 4 inches, and the area of the rectangle is 16 square inches, hence equation 5 becomes 1=53+16X 4'=-261-i biquadratic inches.
1. The moment of inertia of an "angle" 23X2 X 3 inches (lengths of sides and width respectively) with respect to a line through the center of gravity and parallel to the long side, is 0.64 inches'. The area of the section is 2 square inches, and the dis tance from the center of gravity to the long side is 0.63 inches. (These values are taken from a "handbook".) It is required to compute the moment of inertia of the section with respect to a line parallel to the long side and 4 inches from the center of gravity. Ans. 32.64 inches'.
54. Moment of Inertia of Sections. As before stated, beams are sometimes "built np" of rolled shapes (angles, channels, etc.). The moment of inertia of such a section with respect to a definite axis is computed by adding the moments of inertia of the parts, all with respect to that same axis. This is the method for computing the moment of any area which can be divided into simple parts.