1116

THEORY AND PRACTICE OF ENGINEERING.

Book 11.

semicircle VRQ common to these two solids, and the centre of gravity of the demi-cylinder,that of the frustum and that of its complement being in theradius Y R, perpendicular to the diameter V Q. We shall sup-pose that the first is at the point B, the second at the point C,and tHe third at the point D ; on which we must remark that theposition of the two first is known, for the semicircle V RQ is toits diameter, V Q, as ^ of the radius Y R is to the interval YB :the demi-cylinder being composed of aa infinity of semi-circles,all of whose centres of gravity pass through the same line ofdirection, we may regard all these circles, or the demi-cylinderunited in the weight P : on the other hand, as we have the posi-tion of the centre of gravity of the frustum, by making Y r C equalto | of the radius Y r R, we may also suppose its solidity united inthe weight T, and that of its complement in the weight S. Con-sidering the point B as the fulcrum of a lever D C , round whichthe weights S and T are in equilibrio, the first of which is tothe second as 19 to 14, we shall have 19 : 14 :: BC; BD =

|| x B C. If we make an angle Y C K at pleasure, taking theline C E of any length, it must be divided into 19 equal parts ;make E F equal to 14 of these parts, draw the line E B, and throughthe point F draw a parallel to it, E D, which will give the pointD, the centre of gravity of the complement of the frustum, since C E, E F : C B, B D.

Having a solid, as in fig. 1698., composed of a half cylinder, () E A L M H, and a frustum,O L B M, we shall find in the radius A K the point through which the line of direction ofthe centre of gravity of this solid should pass : for taking the weight P, fig. 1707., for thatof the half cylinder, and the weight, T, for that of the frustum, we say, as the sum of thetwo weights, which is the same as the solid, is to the interval B C, so is the weight T, orthe frustum, to the interval B G of the centre of gravity of the half cylinder to that whichwe seek.

To find it in the same manner, in the radius K D of the base of the solid of fig. 1699.,the point through which the direction line of its centre of gravity should pass, we remarkthat this solid being composed of the demi-cylinder EQ.VRDH and the complementRQOMVof a frustum, we may, by taking in fig. 1707. the weight P for that of thehalf cylinder, and the weight S for that of the complement, say, as the sum of these twoweights is to the interval D B, so the weight S, or the solidity of the complement of thefrustum, is to the interval B H, from the centre of gravity of the frustum, to that of theentire solid.

If in fig. 1696. we take from the cylinder PLN It the equal frustums MX, It O, therewill remain the regular solid POM VRQ, O, whose axis, DC , passing through its centreof gravity, we may suppose this solid united in the weight Y : taking the line D A equalto ^ the radius D L, the centre of gravity of the equal frustums MBLO or MPL O,being at the point A, we may also suppose it united in the weight X : we say, then, as thesum of the weights X and Y, or the solidity of the cylinder PLN R, is to the intervalD A, so the weight X, or the sum of the two frustums, is to the interval D S from thecentre of gravity L N to the centre of gravity of the solid P B R.

Lastly, we may in the same manner find the centre of gravity of the solid represented infig. 1697., by subtracting the double frustum OGB M, and making X \ equal to ^ theradius, in order to say, as the cylinder A L N D is to the interval X Y, so is the sum of thetwo frustums to the interval X S from the centre of the semicircle to the required centreof gravity S.

Specific Gravities of Bodies are determined on the principles of hydrostratics. Thediminution of the apparent weight of a solid, upon immersion into fluid, affords us an easyand ready method of comparing its density with that of the fluid: for the weight of thesolid being known previous to immersion, it is then ascertained what quantity it haslost when plunged into a body of pure water, by measuring the bulk of water displaced :thus we obtain the proportion of the specific gravity of the body to that of water, which isthe usual standard of comparison: for when a solid is weighed in water it will lose asmuch of its weight as is equal to the quantity of fluid displaced ; for bodies so plungedare pressed upwards with a force equal to the weight of the fluid displaced; and asthis force acts in opposition to the natural gravity or absolute weight of the body, itsabsolute weight must be diminished by a quantity equal to the weight of the fluid dis-placed. The weight which the body loses is not destroyed, but is sustained by an equaland opposite force.

When the specific gravity of a body is equal to that of the water, the part immersed isequal to the whole body ; or, it may be said, the solid will be completely immersed, andremain wherever it is placed : when the specific gravity of the body is greater than that ofthe fluid, it will sink to the bottom, and when the specific gravity of the fluid is greater

Fig.1707.