94

GRAPHIC STATICS.

where g t is the centre of gravity of the cone 0 A A', g 2 the centre of gravity of theconical sector 0 A C A', r the radius 0 A, and

_ (2 r-h)(r-h)

°~ 27 -

where h is the height B C of the segment: a is readily constructed as a fourth propor-tional to r — h ; 2 r — h ; and 2 r.

Solids bounded by Irregular Surfaces .—The centres of gravity of such solids maybe found by supposing them cut up by parallel planes into a large number of slices.If the thickness of these slices is small, the centre of gravity of each will be verynearly the same as that of its mean section, i. e. its section by a plane parallel to andbisecting the distance between the two planes by which it is bounded. If the thicknessof the slices is the same, their volumes will be approximately as the areas of theirmean sections. Supposing parallel forces proportional to the volumes of the slices toact at their respective centres of gravity, the centre of these parallel forces can bedetermined as in § 62, or § 63.

CHAPTER IX.

MOMENT OF INERTIA.—CENTRAL ELLIPSE, &o.

67. Moments of Parallel Forces. —The moment of a force with respect to aplane is equal to the product of the force into the distance of its point of applicationfrom the plane, measured in any direction. The total moment of a system of parallelforces is equal to the sum of the moments of the individual forces, provided that thedistances which form one factor of these products are all taken parallel. The sum ofthe moments of the forces is equal to the moment of their resultant.

If the points of application of the forces all lie in one plane, then the moments ofthe forces about any axis lying in that plane are severally equal to the product ofthe individual forces into the distances of their points of application from the axismeasured in any direction, hut all parallel.

Thus if the forces p 1 .. .y> 5 act at a 1 . .. a B (Fig. 90) respectively, their momentsabout an axis X X lying in the plane containing a 1 ...a s are equal to p x . y x , p 2 . y 2 ,....,where yi, y 2 ... are the distances of a 1 , a 2 ... from X X measured in directions makingany angle 6 with X X.

The total moment of the systems of forces is 2 (/? . y), and

S(p.y) = Sjp.T,

when Y is the distance of the centre of the system from X X measured in a direction