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50 APPLIED MECHANICS 
10. UAB is a quadrant of a circle, the radii OA and OB being 23 inches long. 
CD is a straight line cutting OB at C and OA at D. OC=2 inches, OD=1} 
inches. Determine the centroid of the figure ABCD. . z 
11. The figure shown at Ex. 11, Fig. 47, is subjected to fluid pressure, which 
varies uniformly from } lb. per square inch at the level AB to 1} lbs. per square 
inch at the level CD. Determine the position of the centre of pressure of the 
figure. 
: 12. The figure shown at Ex. 12, Fig. 47, represents the section of a bar 
which is subjected to tensile stress. The stress varies uniformly from nothing 
at AB to 3 tons per square inch at CD. . Determine the position of the centre of 
stress of the section. 
13. A vertical wall is 80 yards long and 42 feet high. The adjoining table 
gives the pressures of the en 
wind on it, p pounds per 
square foot,atvarious heights h 4 10 18 25 33 42 
h feet above the ground. 
Draw a diagram showing 
the relation between p and p 9 12 | 16°7 | 20°3 | 23°5 | 26 
hk. Find the mean pressure 
on the wall in lbs. per square 
foot, and the total wind force on the wall in lbs. Find the line of action of 
this force. Employ scales of 1 inch to 10 feet, and 1 inch to 10 lbs. per square 
foot. [B.E.] 
67. Moment of Inertia.—The sum of the products of the mass of 
each elementary part of a body and the square of its distance from a 
given axis is called the moment of inertia of the body about that axis. 
Thus, if m,, m,, mg, etc., be the masses of the parts of the body, and 
1) %g Tg, etc., be the distances of these parts respectively from the axis, 
then the moment of inertia =I =m,rj + Mors + Mel 3 +ete.... =2Zmr’, 
The moment of inertia of an area and the moment of inertia of a line 
are defined in a similar manner by substituting area or length for mass. 
But since areas and lines have no inertia, they have, strictly speaking, no 
moment of inertia. 
The moment of inertia of a force about an axis perpendicular to the 
line of action of the force is the product of its magnitude and the square 
of the distance of its line of action from the axis. 
The graphic method of determining the moment of inertia of a plane 
area, or of a system of parallel forces, will be understood from the two 
examples worked out in Figs. 48 and 49. 
Fig. 48 shows the application of the method to finding the moment 
of inertia of a force AB about a point 
M or about an axis through M and 
perpendicular to the plane of the paper. 
Through M draw MY parallel to AB. 
Draw MN perpendicular to AB. Ap- 
plying the construction explained in 
Art. 59, a/b’ x oh = ABx MN. Choose 
a pole o’ at a distance o’h’ from a/b’, 
which is a simple multiple or sub- 
multiple of the linear unit. From a 
point »”’ in AB draw n’’a’’ parallel 
to o’a’ and n”b” parallel to 0’b’. Since ) 
the triangle abn’ is similar to the Fic. 48 
triangle a’b’o’, it followsthat ab” x oh’ : erp 
=a'b’ x MN,and thereforea”b” x o’h' x oh =a'b! x oh x MN. Buta’d’ x oh 
