Ex. 8. The numerical value of a certain force is 56 when the pound is the unit of mass, the foot the unit of length, and the second the unit of time; what will be the numerical value of the same force when the hundredweight is the unit of mass, the yard the unit of length, and the minute the unit of time? Denoting the required value by a we have Dimensions of Mechanical and Geometrical Quantities. 14. In the following list of dimensions, we employ the letters L, M, T as abbreviations for the words Length, Mass, Time. The symbol of equality is used to denote sameness of dimensions, Area L2, Volume L3, Velocity = = L T' tum which it generates per unit of time, and is therefore the quotient of momentum by time-or since a force is measured by the product of a mass by the acceleration generated in this mass. arc The dimensions of angle,* when measured by radius' are zero. The same angle will be denoted by the same number, whatever be the unit of length employed. In fact we have arc = L radius L = Lo. The work done by a couple in turning a body through any angle, is the product of the couple by the angle. The identity of dimensions between work and couple is thus verified. * The name radian has been given to the angle whose arc is equal to radius. "An angle whose value in circular measure is being the product of moment of inertia by angular velocity, or the product of momentum by length. Intensity of pressure, or intensity of stress generally, being force per unit of area, is of dimensions force area ; that Intensity of force of attraction at a point, often called simply force at a point, being force per unit of attracted mass, is of dimensions force L or mass T2 It is numerically equal to the acceleration which it generates, and has accordingly the dimensions of acceleration. The absolute force of a centre of attraction, better called the strength of a centre, may be defined as the intensity of force at unit distance. If the law of attraction be that of inverse squares, the strength will be the product of the intensity of force at any distance by the square of this distance, and its dimensions will be 1 L3 T2 Curvature (of a curve) =, being the angle turned by the tangent per unit distance travelled along the curve. 1 Tortuosity = being the angle turned by the osculatI' ing plane per unit distance travelled along the curve. The solid angle or aperture of a conical surface of any form is measured by the area cut off by the cone from a sphere whose centre is at the vertex of the cone, divided by the square of the radius of the sphere. Its dimensions are therefore zero; or a solid angle is a numerical quantity independent of the fundamental units. B The specific curvature of a surface at a given point (Gauss's measure of curvature) is the solid angle described by a line drawn from a fixed point parallel to the normal at a point which travels on the surface round the given point, and close to it, divided by the very small 1 area thus enclosed. Its dimensions are therefore The mean curvature of a surface at a given point, in the theory of Capillarity, is the arithmetical mean of the curvatures of any two normal sections normal to each other. Its dimensions are therefore 1 L 19 CHAPTER II. CHOICE OF THREE FUNDAMENTAL UNITS. 15. NEARLY all the quantities with which physical science deals can be expressed in terms of three fundamental units; and the quantities commonly selected to serve as the fundamental units are a definite length, a definite mass, a definite interval of time. This particular selection is a matter of convenience rather than of necessity; for any three independent units are theoretically sufficient. For example, we might employ as the fundamental units a definite mass, a definite amount of energy, a definite density. 16. The following are the most important considerations which ought to guide the selection of fundamental units: (1) They should be quantities admitting of very accurate comparison with other quantities of the same kind. |