36. DEF. By the VELOCITY of a body in motion is meant the degree of quickness with which the body is moving. [This "degree of quickness" is described, or measured, by saying how long the line is that the body moves through in some given portion of time. Thus a clear notion would be conveyed of the Velocity of a coach, if it were said to be nine miles in an hour, the space moved over by the coach being nine miles, and an hour being the portion of time during which the motion takes place. If only the space that is described were mentioned, and nothing were said about the time taken up in describing that space, -or if the time alone were given,-it is evident that no idea at all could be formed of the degree of quickness, that is, of the Velocity,-of the coach's motion. The Velocity of a body is measured, in mathematical investigations, by the number of feet passed over by the body in a second of time. Since the quicker a body moves the more space it will pass over in a given time, it will follow from the observations just made, that The Velocities of two bodies that move during any (THE SAME) time, are as the spaces the bodies respectively describe in that time.] 37. PROP. XV. Assuming that the arcs which subtend equal angles at the centers of two circles are as the radii of the circles, to shew that if P and W balance each other on the Wheel and Axle, and the whole be put in motion, P: WW's velocity: P's velocity. m Suppose P and W to act at the circumferences of the Wheel and the Axle in the same plane perpendicularly to the radii OM and ON (as in Prop. x.); M and let the whole be put into B motion round the axis SO that m On becomes the horizontal diameter. P will now act at m at right n N YW angles to Om, and W will act at n at right angles to On; and the velocities of P and W will be as Mm to Nn, since the former of these arcs is the length through which P will have descended in the same time that W has ascended through Nn. Therefore vel. of P : vel. of W :: Mn : Nn :: OM : ON [by the assumption made in the enunciation], :: W: P, by Prop. x; because P and W originally balanced each other. Wherefore, "Assuming, &c." Q. E. D. 38. PROP. XVI. To shew that if P and W balance each other on the Machines described in Propositions XI, XII, XIII, and XIV, and the whole be put in motion, P: W: W's velocity in the direction of gravity: P's velocity [in the direction in which it acts.] (1) Let C the center of the moveable pulley (see Fig. Prop. x1), be raised through any height, as an inch, W will be raised through an inch, and each of the strings RB and AP will have been shortened an inch, so that if P continue to keep the string tight, it will have moved through two inches in the time that W has been raised one inch. Now P: W:: 1 2 by Prop. XI, : : 1 inch 2 inches :: W's velocity: P's velocity; by Art. 36. Q.E.D. (2) In the system when the same string passes round all the pulleys, and the parts of it between the pulleys are parallel, as in Prop. x11, (see Fig.), if the lower block be raised an inch, each of the strings between the pulleys will be shortened an inch, and (there being 2n of such strings) P must have moved through 2n inches in the time that W moved through one inch, in order to have kept the string tight. :: W's velocity: P's velocity; by Art. 36. Q. E. d. (3) In the system where each pulley hangs by a separate string and the strings are parallel (Prop. XIII), if W be raised through an inch, then if P have moved through such a space so as to keep the strings tight, A will have been raised through one inch; B through two inches (by the first case proved in this Proposition); and B having been raised through two inches, C (by the first case) will have moved through 2 × 2, or 22, inches; the next moveable pulley (the fourth) would have been raised through 2 × 2o (or 23) inches; and by the same kind of reasoning, if n were the number of moveable pulleys, the highest of them would have moved through 2-1 inches. The end therefore of the string by which P acts would have moved through 2′′ inches; and P : W :: 1 : 2", by Prop. XIII.; ::1 inch : 2" inches :: W's velocity: P's velocity. Q. E. D. (4) Let the weight (W) be kept at rest at D on the inclined plane AB by the power (P) which acts parallel to the plane by means Р_Б P H G draw GH horizontal, and DH through D vertical. Then by being moved through DG, W has been raised through a vertical height DH,-that being the vertical line cutting the horizontal lines passing through the two positions of W. In the time therefore that P moves through a space equal to DG, the vertical height that W moves through is DH, and therefore these two lines are to each other as the velocity of P in the direction of its action is to the velocity of W in the direction of gravity. Now, in the triangles GDH and ABC, since GH is parallel to AC, and AG meets them, .. angle DGH = alt. angle_BAC; and since DH, being vertical, is parallel to BC, and DB meets them, ... angle GDH = alt. angle ABC ; .. also angle DHG = angle BCA, and the triangles are equiangular. And P: W:: BC: AB; by Prop. XIV.; :: DH: DG, by equiangular triangles. :: DH : Pp, :: W's vel. in direction of gravity : P's vel. in the direction of its action. Wherefore, “ If Pand W balance, &c.” Q. E. D. |