MECHANICS 699 time, generates the unit of velocity. The dimensions of force are therefore [MLT" 2 ]. solute 105. Hence the British absolute unit force is the force
- -t which, acting on one pound of matter for one second,
ce> generates a velocity of one foot per second. itisli [According to the system followed till lately in treatises stem, on dynamics, the unit of mass is g times the mass of the standard weight, g being the numerical value of the acceleration produced (in some particular locality) by the earth s attraction. This definition, giving a varying unit of mass, is exceedingly inconvenient. In reality, standards of weight are masses, not forces. They are employed primarily for the purpose of measuring out a definite quantity of matter, not an amount of matter which shall be attracted by the earth with a given force.] 106. To render our standard intelligible, all that has to be done is to find how many absolute units will produce, in any particular locality, the same effect as does gravity. The way to do this is to 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 acquired by a body falling unresisted for one second is at that place 32 207 feet per second. The variation in gravity for one degree of difference of latitude about the latitude of Leith is only 0000832 its own amount. The average value for the whole of Great Britain differs but little from 32 2 ; that is, the attraction of gravity on a pound of matter in the country is 3 2 2 times the force which, acting on a pound for a second, would generate a velocity of one foot per second. Thus, speaking very roughly, the British absolute unit of force is equal to the weight of about half an ounce. The quantity of 32*2 feet per second per second is usually called g. Its dimensions are obviously [LT" 2 ]. And, if M be the mass . G. S. of a body, its weight is IVfy. In the Centimetre-Gramme- -stem. Second system of units, the absolute unit of force pro duces in one second, in a mass of one gramme, a velocity of one centimetre per second.
- pre- 107. Forces (since they involve only direction and
ntation magnitude) may be represented, as velocities are, by force. vec t orgj that is, by straight lines drawn in their direc tions, 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 presently show ( 117), the same as those which we have already proved to hold for velocities ; so that, with the substitution of force for velocity, 30 is still true. om- 108. The "component" of a force in any direction is ments therefore found by multiplying the magnitude of the force r force, ^y ^g cosine of the angle between the directions of the force and the component. The remaining component in this case is perpendicular fro the other. It is very generally convenient to resolve forces into components parallel to three lines at right angles to each other, each such resolution being effected by multiplying by the cosine of the angle concerned. The magnitude of the resultant of two or of three forces in directions at right angles to each other is the square root of the sum of their squares. entreof 109. The "centre of inertia or mass" of any system iass. of material particles 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. The distance from the plane yz of the centre of inertia of masses m v m. 2 , &c., whose distances from the plane are x v x. 2 , &c., is therefore and similarly for the other coordinates. Hence its distance from the plane {m(x + fj.y + ic
D = x + py + vz - a = , , 2(m) as stated above. And its velocity perpendicular to that plane is
- ("%}
dt J
dD dx dy from which, by multiplying by 2m, and noting that 8 is the dis tance of x, y, z from 8 = 0, we -see that the sum of the momenta of the parts of the system in any direction is equal to the momentum in that direction of the whole mass collected at the centre of mass. The problem of finding the centre of inertia of any given distribution of matter is a question of mere mathematics. We must confine ourselves to a few -examples only. And, first, we may note that when a body is symmetrical about a plane the centre of inertia must obviously lie in that plane. Thus, as an ellipsoid and a rectangular parallele piped have each three planes of symmetry, their centres of inertia lie at their centres of figure, where these planes meet. Again, it is obvious that, if a body can be divided into parts the centres of inertia of which lie on a straight line, the centre of inertia of the whole is in that line. Thus, as a triangular plate may be divided into strips parallel to one side, every one of which has its centre of inertia at its middle point, the centre of inertia of such a plate is the point of intersection of the bisectors of the sides. Its distance from any one side, treated as base, is therefore one-third of the height. Again, if a triangular pyramid (or tetrahedron) be divided into triangular slices by planes parallel to one face treated as base, the centres of inertia of all the slices lie in a straight line. Hence the distance of the centre of inertia from the base is one-fourth of the height. If the base be of any other form, it may be divided into triangles, and thus the whole pyramid (or cone) into tetrahedra, for each of which the same property holds. Hence the centre of inertia of a pyramid divides the line joining the vertex to the centre of inertia of the base in the ratio 3:1. All this is on the supposition that the solids treated of are of uniform density. When we deal either with more complex forms or with heterogeneous bodies, we must in general have recourse to integration. For a continuous body we must take an element of mass, say pSxSytiz, at the point x, y, z instead of the mass m in our original formula. The sums then become integrals, and we have three expressions of the form ffj pxdxdydz f/Ypdxdydz Here p represents the density at x, y t z ; and the integration extends through the whole volume of the body. Thus, for a homogeneous hemisplicre of radius a we have, taking the base as the plane of yz, f a 7r(a" - x-}xdx , The same value would be obtained for any semiellipsoid, whatever be the diametral section, provided a be the height measured per pendicular to the base ; and, in general, from the position of the centre of inertia of any body we may at once find that of the same body homogeneously strained. Recurrin^ to the hemisphere, suppose its density to be at e point proportional to the distance from the centre. Then ^yo have, omitting common constant factors of numerator and denominator, /xdx I Jo r-dr " r a dx r^
jo jo