This may be proved in precisely the same manner as the cone, or thus by Simpson's method This method is only approximately true when the taper is very slight. For such a body as a pyramid it would be seriously in error; the volume obtained by this method would be H3 instead of H3. The diameter D is measured from centre to centre of the sections of the ring, i.e. their centres of gravity— Volume = area of surface of revolution × length of path of c. of g. of section CHAPTER III. MOMENTS. THAT branch of applied mechanics which deals with moments is of the utmost importance to the engineer, and yet perhaps it gives the beginner more trouble than any other part of the subject. The following simple illustrations may possibly help to make the matter clear. We have already (see p. 12) explained the meaning of the terms "clockwise" and "contraclockwise" moments. In the figures that follow, the two pulleys of radii R and R, are attached to the same shaft, so that they rotate together. We shall assume that there is no friction on the axle. Let a cord be wound round each pulley in such a manner that when a force P is applied to one cord, the weight W will be lifted by the other. Now let the cord be pulled through a sufficient distance to cause the pulleys to make one complete revolution; we shall then have The work done by pulling the cord = P x 2πR These must be equal, as it is assumed that no work is wasted in friction; hence or PR = WR1 or the contra-clockwise moment = the clockwise moment It is clear that this relation will hold for any portion of a revolution, however small; also for any size of pulleys. The levers shown in the same figures may be regarded as small portions of the pulleys; hence the same relations hold in their case. It may be stated as a general principle that if a rigid body be in equilibrium under any given system of moments, the algebraic sum of all the moments in any given plane must be zero, or the clockwise moments must be equal to the contraclockwise moments. In speaking of moments, we shall always put the units of force, etc., first, and the length units afterwards. For example, we shall speak of a moment as so many pounds-feet or tonsinches, to avoid confusion with work units. Neglecting weight of lever. Second Moments.-The product of a force (f) mass (m) by the square or second power of the length (4) of its arm, viz. (2) al2 v12 is termed the second moment of the force mass area The volume second moment of a volume or an area is sometimes termed the "moment of inertia" (see p. 78) of the volume or area. Strictly, this term should only be used when dealing with questions involving the inertia of bodies; but in other cases, where the second moment has nothing whatever to do with inertia, the term "second moment" is preferable. |