700 A uniform hemispherical shell gives x = a by the well-known result due to Archimedes. From this, by taking concentric hemispherical elements, we may reproduce the preceding result for a solid hemisphere in the form 2<e 1 5 1-irx-dx . x Here the first factor under each integral sign is the volume of the hemispherical element of radius x, and the second is proportional to its density. If the density of a thin uniform spherical shell be everywhere proportional to the inverse cube of the distance from an internal point, that point is the centre of inertia. For, if a double cone of small angle be drawn, having that point as vertex, the volumes of the portions of the shell which it cuts out are as the squares of their distances from the vertex. Hence their masses are inversely as their distances from the vertex, which is thus their centre of inertia. The whole shell may be divided into pairs of elements for each of which this is true. The reader may easily prove that, if the density of a solid sphere be inversely as the fifth power of the distance from an external point, the "electric image" of that point is the centre of inertia. It may be proved in the last two examples that this point is not merely the centre of inertia of such distributions of matter, but that it is also a true " centre of gravity " in the sense that the whole attracts, and is attracted by, any other body whatever, as if its whole mass were concentrated in this point. Moment HO. By introducing in the definition of moment of m - velocity ( 46) the mass of the moving particle as a n factor, we have an important element of dynamical science, the " moment of momentum." The laws of composition and resolution are the same as those already explained. Its dimensions are [ML 2 ! 1 1 ]. Work. 111. A force is said to " do work " if it moves the body to which it is applied ; and the work done is measured by the resistance overcome, and the space through which it is overcome, conjointly. The dimensions of work are therefore [MLT 2 . L] or [ML 2 T 2 ], the same as those of kinetic energy. Thus, in lifting coals from a pit, the amount of work dons is proportional to the weight of the coals lifted ; that is, to the force overcome in raising them ; and also to the height through which they are raised. The unit for the measurement of work, adopted in practice by British engineers, is that required to overcome the weight of a pound through the height of a foot, and is called a " foot-pound." In purely scientific measurements, the unit of work is not the foot-pound, but the absolute unit force ( 105) acting through unit of length. If the weight be raised obliquely, as, for instance, along a smooth inclined plane, the distance through which the force has to be overcome is increased in the ratio of the length to the height of the plane; but the force to be overcome is not the whole weight, but only the resolved part of the weight parallel to the plane ; and this is less than the weight in the ratio of the height of the plane to its length. By multiplying these two expressions together, we find, as we might expect, that the amount of work required is unchanged by the substitution of the oblique for the vertical path. Generally, if s be an arc of the path of a particle, S the tangential component of the applied forces, the work done on the particle between any two points of its path is /Sds, taken between limits corresponding to the initial and final positions. Referred to rectangular coordinates, it is easy to see, by the law of resolution of forces, 117, that this becomes ds ds ds where X is the component force parallel to the axis of x. 112. Thus it appears tha t, for any force, the work done during an indefinitely small displacement of the point of application is the product of the resolved part of the force in the direction of the displacement into the displacement. From this it follows that, if the motion of a body be always perpendicular to the direction in which a force acts on it, the force does no work. Thus the mutual normal pressure between a fixed and a moving body, the tension of the cord to which a pendulum bob is attached, the attraction of the sun on a planet if the planet describe a circle with the sun in the centre, are all cases in which no work is done by the force, In fact the geometrical condition that the resultant of X, Y, Z shall be perpendicular to ds is dx dy dz -A- . |- 1 - , ~T Lt U , as as ds and this makes the above expression for the work vanish. 113. Work done on a body by a force is always shown Iran by a corresponding increase of kinetic energy, if no other llia ti forces act on the body which can do work or have work v done against them. If work be done against any forces, the increase of kinetic energy is less than in the former case by the amount of work so done. In virtue of this, however, the body possesses an equivalent in the form of " potential energy," if its physical conditions are such Pote: that these forces will act equally, and in the same direc- cner tions, when the motion of the system is reversed. Thus there may be no change of kinetic energy produced, and the work done may be wholly stored up as potential energy. Thus a weight requires work to raise it to a height, a spring requires work to bend it, air requires work to com press it, &c.; but a raised weight, a bent spring, compressed air, &c., are stores of energy which can be made use of at pleasure. As an illustration of the calculation of work, take the following question. Suppose one end of an elastic string to be attached to a mass resting on the ground, what amount of work must be done, in raising the other end vertically, before the mass is lifted ? If x be at any instant the length of the string, I its original length, its tension is x-l E - r . Hence the value of x, when the mass is just lifted, is of we where W is the weight of the mass. The whole work done is the sum of all the elementary instalments of the form X-l These must be summed up from x = l to x = x v so that the result required is U K It is to be observed that this quantity becomes less in proportion as E is greater, i.e., the less extensible is the string. An interesting variation of the question consists in supposing the upper end of the string to be attached to the rim of a wheel, rough enough to prevent slipping. Here the various portions of the string are wound on in a more and more stretched state as the operation proceeds. At any stage of the operation let x be the unstretched length of the part already wound on the wheel. The tension of the free part is then f I X During the next elementary step of the process a portion dx is wound on. But its stretched length is Idx
l-x