for Absolute the length of the seconds pendulum, whose period is two Suggestions seconds. Hence taking the earth's radius as 6,370,000 metres, Unit of and its density as 51⁄2 times that of our standard globe, Time. 224. The absolute unit depends on the unit of matter, the unit of time, and the unit of velocity; and as the unit of velocity depends on the unit of space and the unit of time, there is, in the definition, a single reference to mass and space, but a double reference to time; and this is a point that must be particularly attended to. 225. The unit of mass may be the British imperial pound; the unit of space the British standard foot; and, accurately enough for practical purposes for a few thousand years, the unit of time may be the mean solar second. solute unit. We accordingly define the British absolute unit force as "the British abforce which, acting on one pound of matter for one second, generates a velocity of one foot per second." Prof. James Thomson has suggested the name "Poundal" for this unit of force. with 226. To illustrate the reckoning of force in "absolute measure," Comparison find how many absolute units will produce, in any particular gravity. locality, the same effect as the force of gravity on a given mass. To do this, measure the effect of gravity in producing acceleration on a body unresisted in any way. The most accurate method is indirect, by means of the pendulum. The result of pendulum experiments made at Leith Fort, by Captain Kater, is, that the velocity which would be acquired by a body falling unresisted for one second is at that place 32 207 feet per second. The preceding formula gives exactly 32.2, for the latitude 55° 33', which is approximately that of Edinburgh. The variation in the force of gravity for one degree of difference of latitude about the latitude of Edinburgh is only 0000832 of its own amount. It is nearly the same, though somewhat more, for every degree of latitude southwards, as far as the southern limits of the British Isles. On the other hand, the variation per degree is sensibly less, as far north as the Orkney and Shetland Isles. Hence Gravity of the augmentation of gravity per degree from south to north throughout the British Isles is at most about Unit weight or mass in terms of Kinetic Unit. of its whole amount in any locality. The average for the whole of Great Britain and Ireland differs certainly but little from 32.2. Our present application is, that the force of gravity at Edinburgh is 322 times the force which, acting on a pound for a second, would generate a velocity of one foot per second; in other words, 32.2 is the number of absolute units which measures the weight of a pound in this latitude. Thus, approximately, the poundal is equal to the gravity of about half an ounce. Effective component of a force 227. Forces (since they involve only direction and magnitude) may be represented, as velocities are, by straight lines in their directions, and of lengths proportional to their magnitudes, respectively. Also the laws of composition and resolution of any number of forces acting at the same point, are, as we shall show later (§ 255), the same as those which we have already proved to hold for velocities; so that with the substitution of force for velocity, SS 26, 27, are still true. 228. In rectangular resolution the Component of a force in any direction, (sometimes called the Effective Component in that direction,) is therefore found by multiplying the magnitude of the force by the cosine of the angle between the directions of the force and the component. The remaining component in this case is perpendicular to the other. It is very generally convenient to resolve forces into components parallel to three lines at right angles to each other; cach such resolution being effected by multiplying by the cosine of the angle concerned. Geometrical 229. The point whose distances from three planes at right preliminary angles to one another are respectively equal to the mean dis Theorem to definition of centre of tances of any group of points from these planes, is at a distance inertia. from any plane whatever, equal to the mean distance of the group from the same plane. Hence of course, if it is in motion, its velocity perpendicular to that plane is the mean of the velocities of the several points, in the same direction. Theorem Let (x, y, z), etc., be the points of the group in number i; Geometrical and x, y, be the co-ordinates of a point at distances respectively preliminary to definition equal to their mean distances from the planes of reference; that of centre of is to say, let inertia. Thus, if p1, P, etc., and p, denote the distances of the points in Y1+y,+ etc. 2 P1 = lx, + my1 + nz, − a, etc.; and similarly, Substituting in this last the expressions for x, y, z, we find P1+ P2+ etc. 1 2 = == 2 + which is the theorem to be proved. Hence, of course, dp_1/dp, dp: 1 + dt 2 dt dt 21+ 2+ etc. etc.). inertia. 230. The Centre of Inertia of a system of equal material Centre of points (whether connected with one another or not) is the point whose distance is equal to their average distance from any plane whatever (§ 229). A group of material points of unequal masses may always be imagined as composed of a greater number of equal material points, because we may imagine the given material points divided into different numbers of very small parts. In any case in which the magnitudes of the given masses are incommensurable, we may approach as near as we please to a rigorous fulfilment of the preceding statement, by making the parts into which we divide them sufficiently small. On this understanding the preceding definition may be applied to define the centre of inertia of a system of material points, whether given equal or not. The result is equivalent to this: Centre of The centre of inertia of any system of material points whatever (whether rigidly connected with one another, or connected in any way, or quite detached), is a point whose distance from any plane is equal to the sum of the products of each mass into its distance from the same plane divided by the sum of the masses. We also see, from the proposition stated above, that a point whose distance from three rectangular planes fulfils this condition, must fulfil this condition also for every other plane. The co-ordinates of the centre of inertia, of masses w,, w2, etc., at points (x ̧‚ Y ̧, ≈ ̧), (X2, Y21 ≈), etc., are given by the following formulæ : : Σω These formulæ are perfectly general, and can easily be put into the particular shape required for any given case. Thus, suppose that, instead of a set of detached material points, we have a continuous distribution of matter through certain definite portions of space; the density at x, y, z being p, the elementary principles of the integral calculus give us at once Σwz Στο etc., where the integrals extend through all the space occupied by the mass in question, in which has a value different from zero. P The Centre of Inertia or Mass is thus a perfectly definite point in every body, or group of bodies. The term Centre of Gravity is often very inconveniently used for it. The theory of the resultant action of gravity which will be given under Abstract Dynamics shows that, except in a definite class of distributions of matter, there is no one fixed point which can properly be called the Centre of Gravity of a rigid body. In ordinary cases of terrestrial gravitation, however, an approximate solution is available, according to which, in common parlance, the term "Centre of Gravity" may be used as equivalent to Centre of Inertia; but it must be carefully remembered that the fundamental ideas involved in the two definitions are essentially different. Inertia. The second proposition in § 229 may now evidently be centre of stated thus: The sum of the momenta of the parts of the system in any direction is equal to the momentum in the same direction of a mass equal to the sum of the masses moving with a velocity equal to the velocity of the centre of inertia. 231. The Moment of any physical agency is the numerical Moment. measure of its importance. Thus, the moment of a force round a point or round a line, signifies the measure of its importance as regards producing or balancing rotation round that point or round that line. a force point. a force 232. The Moment of a force about a point is defined as the Moment of product of the force into its perpendicular distance from the about a point. It is numerically double the area of the triangle whose vertex is the point, and whose base is a line representing the force in magnitude and direction. It is often convenient to represent it by a line numerically equal to it, drawn through the vertex of the triangle perpendicular to its plane, through the front of a watch held in the plane with its centre at the point, and facing so that the force tends to turn round this Moment of point in a direction opposite to the hands. The moment of a force round any axis is the moment of its component in any plane perpendicular to the axis, round the point in which the plane is cut by the axis. Here we imagine the force resolved into two components, one parallel to the axis, which is ineffective so far as rotation round the axis is concerned; the other perpendicular to the axis (that is to say, having its line in any plane perpendicular to the axis). This latter component may be called the effective component of the force, with reference to rotation round the axis. And its moment round the axis may be defined as its moment round the nearest point of the axis, which is equivalent to the preceding definition. It is clear that the moment of a force round any axis, is equal to the area of the projection on any plane perpendicular to the axis, of the figure representing its moment round any point of the axis. about an axis. on projec 233. The projection of an area, plane or curved, on any Digression plane, is the area included in the projection of its bounding tion of line. areas. |