Strength of Materials. a Text-Book for Manual Training Schools

Cover Strength of Materials. a Text-Book for Manual Training Schools
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Ch. Ill, rectangle, and normal to gh let lines be drawn equal to the squares of z ; thus ee is the square of CE, and ^^ is the square of CG. Now the elementary product a^ is the elementary area EE multiplied by the ordi- nate ee; hence "Eaz^ is represented by a solid standinp on ^(/ whose variable height is shown by the shaded area ghhcg. But the volume of this solid is the product of its length b and this shaded area. The curve ceg\% a parabola because each line ee is the square of the '9 E 'C
... -E-- e-' Fig. 13.
corresponding altitude ce ; accordingly the shaded area is one-third of ghhg. But gli is equal to d, and gg is eq>ual to (^)' ; thus the shaded area is represented by \ . d. ^d\ or r^^d'. Hence is the moment of inertia of a rectangle about an axis through its center of gravity and parallel to its base.
The moment of inertia is a compound quantity result- ing by multiplying an area by the square of a distance ; it thus contains the linear unit four times. If 3 = 3 inches and d = 4.


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