in a circular path, the string will be put into tension,the amount of which will depend upon (1) the mass of the body, (2) the length of the string, and (3) the velocity with which the body moves. The tension in the string is equal to the centrifugal force. We will now show how the exact value of this force may be calculated in any given instance.1 Path of rotating Hodograph body Let the speed with which the body describes the circle be constant; then the radius vector of the hodograph will be of constant length, and the hodograph itself will be a circle. Let the body describe the outer of the two circles shown in the figure, with a velocity v, and let its velocity at A be represented by the radius OP, the inner circle being the hodograph of A. Now let A move through an extremely small space to A,, and the corresponding radius vector to OP,; then the line PP, represents the change in velocity of A while it was moving to A. (The reader should never lose sight of the fact that change of velocity involves change of direction as well as change of speed, and as the speed is constant in this case, the change of velocity is wholly a change of direction.) FIG. 8. As the distance AA, becomes smaller, PP, becomes more nearly perpendicular to OP, and in the limit it does become perpendicular, and parallel to OA; thus the change of velocity is radial and towards the centre. We have shown on p. 17 that the velocity of P represents the acceleration of the point A; then, as both circles are described in the same time— But OP was made equal to the velocity of A, viz. v, and OA is the radius of the circle described by the body. Let OA R; then 1 For another method of treatment, see Barker's "Graphic Methods of Engine Design." This force acts radially outwards from the centre. Sometimes it is convenient to have the centrifugal force expressed in terms of the angular velocity of the body. We have Change of Units. It frequently happens that we wish to change the units in a given expression to some other units more convenient for our immediate purpose; such an alteration in units is very simple, provided we set about it in systematic fashion. The expression must first be reduced to its fundamental units; then each unit must be multiplied by the required constant to convert it into the new unit. For example, suppose we wish to convert foot-pounds of work to ergs, then I foot-poundal = 453'6 × 30°482 = 421,390 ergs and I foot-pound = 32'2 foot-poundals = = 32 2 X 421,390 13,560,000 ergs CHAPTER II. MENSURATION. MENSURATION consists of the measurement of lengths, areas, and volumes, and the expression of such measurements in terms of a simple unit of length. Length. If a point be shifted through any given distance, it traces out a line in space, and the length of the line is the distance the point has been shifted. A simple statement in units of length of this one shift completely expresses its only dimension, length; hence a line is said to have but one dimension, and when we speak of a line of length 7, we mean a line containing / length units. Area. If a straight line be given a side shift in any given plane, the line sweeps out a surface in space. The area of the surface swept out is dependent upon two distinct shifts of the generating point: (1) on the length of the original shift of FIG. 9. the point, i.e. on the length of the generating line (); (2) on the length of the side shift of the generating line (d). Thus a statement of the area of a given surface must involve two length quantities, / and d, both expressed in the same units of length. Hence a surface is said to have two dimensions, and the area of a surface ld must always be expressed as the product of two lengths, each containing so many length units, viz. Area length units x length units = =(length units)" Volume. If a plane surface be given a side shift to bring it into another plane, the surface sweeps out a volume in space. The volume of the space swept out is dependent upon three distinct shifts of the generating point: (1) on the length of the original shift of the generating point, i.e. on the length of the generating line 7; (2) on the length of the side shift of the generating line d; (3) on the side shift of the generating surface t. Thus the statement of the volume of a given body or space must involve three length quantities, 4, d, t, all expressed in the same units of length. FIG. 10. Hence a volume is said to have three dimensions, and the volume of a body must always be expressed as the product of three lengths, each containing so many length units, viz. Volume length units length units length units = = (length units)3 |