CHAPTER XVI 291. Mechanical Advantage. In Art. 18, it was found that when a heavy body is raised through a vertical height by pushing it upward along an inclined plane of a certain length, Resistance X Height = Effort X Length. If we divide both members of this equation by the height of the plane, the result is Length Supposing the length of the plane to be 12 feet and the height Length 3 feet, the quantity, „ . , = V = 4. For this particular plane, then, Resistance = Effort X 4; or the greatest resistance that can be overcome with this plane is equal to 4 times the effort applied. If then we wish to know the resistance that may be overcome with this plane when an effort of 100 pounds is applied, we have only to make the mental calculation, Resistance = 100 X 4 = 400 pounds. Conversely if we are to raise a weight of 600 pounds with this plane, the effort = fifa = 150 pounds. This number, which tells relatively how great a resistance can be overcome with a machine when a given effort is applied to it, is called the mechanical advantage of the machine. The mechanical advantage of a machine is the number by which the effort must be multiplied to get the resistance. From this definition it follows that Effort X Mechanical Advantage = Resistance. 292. Mechanical Advantage of the Simple Machines. By carefully considering the discussions of the simple raa- 303 chines in Arts. 18-28 and the work equation (Art. 261 in connection with the discussion of Art. 291, the following statements may be clearly understood. (a) The mechanical advantage of the inclined plane when the effort acts along the plane is equal to —. Height (b) The mechanical advantage of the wheel and axle is , circumference of wheel diameter of wheel equal to . , or to or to circumference of axle diameter of axle radius of wheel radius of axle (c) For any machine whatever Resistance Displacement of Effort -. . . , = —^ := Mechanical Effort Displacement of Resistance Advantage. 293. Compound Lever. Nearly all composite machines can readily be seen to be made up of parts that act in accordance with the principles of the simple machines. Thus, the compound lever (Fig. 165) illustrates a principle which is applied in the hardware merchant's platform scales. It , consists of a lever def of the first kind, which turns about a fulcrum at e, and acts at a on another lever abc of the second kind so Wp c as to turn '"A about the fulcrum c i and make it balance the weight of R Fig. 165 Compound Lever . _ , -, , at b. ror the first lever the effort fyr arm ef = 12 inches and the resistance arm de = 2 inches, so its mechanical advantage is V" = 6, i. e., the force exerted at a by d is 6 times the force F. But for the second lever the effort arm ac = 48 inches and the resistance arm bc = 4 inches, therefore its mechanical advantage is -V = 11 Hence, since the pull by F at a = F X 6, and since the resistance that can be lifted at R by any force at a is 12 times that force, the resistance R = F X 6 X 12, or R = F X is thus found to be the product of the mechanical advantages cf the two simple machines that compose it. Similar reasoning with any composite machine will be found to give a similar result; so we may say that the mechanical advantage of a composite machine is the product of the mechanical advantages of its component parts. 294. Geared Windlass. This combination (Fig. 166) consists of a windlass AB and a wheel and an axle CD. AB drives CD by means of teeth or cogs which fit smoothly into the teeth of C. The resistance Wis thus overcome by winding the rope on the, axle D. Suppose the circumference of the circlein which the crank handle A moves is 36 inches, and that the circumferences of the wheels B and C and of the axle D are 9, 48, and 12 inches respectively. Then for one revolution of the crank, the displacements of A and B are 36 inches and 9 inches respectively; so the mechanical advantage of AB is V = 4. In like manner the mechanical advantage of CD = f-f "■ 4. Hence the mechanical advantage of the combination = 4 X 4 = 16, i. e. a force of 100 pounds at A would balance a pull of 1600 pounds at D. Fig. 166 Geared Windlass Such a machine is often combined with a pulley in the derrick and in the crane, which are used for hoisting heavy stones and steel girders in the construction of buildings and bridges. 295. Train of Cog Wheels. Instead of only two cog wheels we may have several, geared one into another as in Fig. 167. The mechanical advantage for a combination of any number of wheels is found by the same principle as for two (Arts. 293 and 294). If a, b, c represent the circumferences of the larger wheels, and a', b', c' those of the corresponding smaller wheels as in the diagram, the mechanical advantage = . Since the numbers of teeth in the a'Xb'Xc' wheels are proportional to their circumferences, we may let the letters a', b, b', c represent the numbers of teeth in the wheels instead of the numbers of inches in their circumferences; and the calculation for the mechanical advantage will give the same result. It is sometimes more convenient to count the teeth than to measure _ _ " „, the circumferences. Such trains Train Of Cog Wheels of cog wheels are used in the running gears of trolley cars, elevators, and turning-lathes to increase the speed and reduce the necessary force, or vice versa. Gear wheels are also used in clocks. The minute hand and the hour hand are attached to cog wheels that are so geared to the driving wheel that the former goes around once an hour while the latter goes around once in 12 hours, or iV as fast. The speed of the driving wheel is controlled by the regular swings of the pendulum, which in its turn is kept going by little pushes communicated to it at each swing by a toothed wheel that is properly geared to the driving wheel. The driving wheel is kept going either by a weight (as at |