2. In a combination of pulleys in two blocks, when there are six pulleys in each block, what weight can a power of 12 lbs. sustain in equilibrium? Ans. 144 lbs. 3. In a combination of separate movable pulleys, the resistance is 576 lbs., and the power which keeps it in equilibrium is 9 lbs. How many pulleys are there in the combination? Ans. 5. 4. In a combination of pulleys in two blocks, with a single rope, the power is 62 lbs., and the resistance 496 lbs. How many pulleys are there in each block? Ans. 4. 5. In a combination of two movable pulleys, the inclinations of the ropes at each pulley is 60°. What is the power required to support a weight of 27 lbs. ? The Wheel and Axle. Ans. 9 lbs. A B 91. The wheel and axle consists of a wheel A, mounted on an axle or arbor B. The power is applied at one extremity of a rope wrapped around the wheel, and the resistance at one extremity of a second rope, wrapped around the axle in a contrary direction. The whole instrument is supported by pivots projecting from the ends of the axle. In deducing the conditions of equili R P Fig. 80. brium of the power and resistance, we shall suppose them to be situated in planes, at right angles to the axis. Denote the power by P, the re sistance by R, the radius of the wheel by, and the radius of the axle by r'. We shall have, in case of an equilibrium (Art. 49), Pr-Rr', or P: R:: r' : r. (44.) That is, the power is to the resistance as the radius of the axis is to the radius of the wheel, rr P Fig. 81. By suitably varying the dimensions of the wheel and axle, any amount of mechanical advantage may be obtained. If we draw a straight line from the point of contact of the first rope and the wheel, to the point of contact of the second rope and the axle, the power and resistance being parallel, it can readily be shown that it will cut the axis of revolution at a point which divides the line through the points of contact into two parts, which are inversely proportional to the power and resistance. Hence, this is the point of application of the resultant of these two forces. The resultant will be equal to the sum of the forces, and by the aid of the principle of moments, the pressure on each pivot may be computed. When the weight of the machine is to be taken into account, we must regard it as a vertical force applied at the centre of gravity of the wheel and axle. The pressures upon each pivot due to this weight, may be computed separately, and added to those already found. Combinations of Wheels and Axles. 92. If the rope of the first axle be passed around a second wheel, and the rope of the second axle around a third wheel, and so on, a combination will result which is capable of affording great mechanical advantage. The figure represents a combination of two wheels and axles. To deduce the conditions of equilibrium, denote the power by P, the resistance by R, the radius of the first wheel by r, that of the first axle by r', that of the second wheel by r', and that of the second axle by r'". If we denote the tension of the connecting rope by t, this may be regarded as a power applied to the second wheel. From what was demonstrated for the wheel and axle, we shall have, Pr= tr', and tr' = Rr'"'. Fig 82. Multiplying these equations together, member by member, and reducing, we have, Pry" = Rr'r'"'; or, P: R :: r'r''' : In like manner, were there any number of wheels and axles in the combination, we might deduce the relation, That is, the power is to the resistance as the continued product of the radii of the axles is to the continued product of the radii of the wheels. The principle just explained, is applicable to those kinds. of machinery in which motion is transmitted from wheel to wheel by the aid of bands, or belts. An endless band, called the driving belt, passes around one drum mounted upon the axle of the driving wheel, and around another on that of the driven wheel. When the radius of the former is greater than that of the latter, there is a gain of velocity, and a corresponding loss of power; in the contrary case, there is a loss of velocity, and a corresponding gain of power. In the first case, we are said to gear up for velocity; in the second case, we are said to gear down for power. These remarks admit of extension to combinations of any number of pieces, in which motion is transmitted by belts, cords, chains, or, as we shall see hereafter, by trains of toothed wheels. The Crank and Axle, or Windlass. 93. sists of an axle AB, and a crank BCD. The power is applied to the crank-handle DC, and the resistance to a rope wrapped around the axle. The distance from the handle DC to the axis, is called the crank-arm. This machine con DE B Fig. 83. A The relation between the power and resistance, when in equilibrium, is the same as in the wheel and axle, except that we substitute the crank-arm for the radius of the wheel. Hence, the power is to the resistance as the radius of the axle is to the crank-arm. This machine is used in drawing water from wells, raising ore from mines, and the like.. It is also used in combination with other machines. Instead of the crank, as shown in the figure, two holes are sometimes bored at right angles to each other and to the axis, and levers inserted, at the extremities of which the power is applied. The condition of equilibrium remains unchanged, provided we substitute for the crank-arm, the distance from the point of application of the power to the axis. The Capstan. 94. The Capstan differs in no material respect from the windlass, except in having its axis vertical. The capstan consists of a vertical axle passing through strong guides, and having holes at its upper end for the insertion of levers. It is much used on shipboard for raising anchors. The conditions of equilibrium are the same as in the windlass. 95. The Differential Windlass. D B This differs from the common windlass in having an axle formed of two cylinders, A and B, of different diameters, but having a common axis. A rope is attached to the larger cylinder, and wrapped several times around it, after which it passes around the movable pulley C, and, returning, is wrapped in a contrary direction about the smaller cylinder, to which the second end of the rope is made fast. The power is ap Fig. 84. t t-R plied at the crank-handle FE, and the resistance to the hook of the movable pulley. When the crank is turned so as to wind the rope upon the larger cylinder it unwinds from the smaller one, but in a less degree, and the total effect of the power is to raise the resistance R. To deduce the conditions of equilibrium between the power and resistance, denote the power by P, the resistance by R, the crank-arm by c, the radius of the larger cylinder by r, and that of the smaller cylinder by r'. The resistance acts equally upon the two branches of the rope from which it is suspended, hence the tension of each branch may be represented by R. Suppose that the power acts to wind the rope upon the larger cylinder. The moment of the power will be Pc; the moment of the tension of the branch A will be equal tor', this acts to assist the power; the moment of the tension of the branch B will be equal to Rr, this acts to op pose the power. From the principle of moments, we have, That is, the power is to the resistance as the difference of the radii of the two cylinders is to twice the crank-arm. By increasing the crank-arm and diminishing the difference between the radii of the cylinders, any amount of mechanical advantage may be obtained by the use of this machine. Wheel-work. 96. The principle employed in finding the relation between the power and resistance in a train of wheel-work is the same as that used in discussing the wheel and axle and its modifications. To illustrate the method of proceeding, we have taken the case in which the power is applied to a crank-handle which is attached to the axis of a cogged wheel R |