measure of the tendency of W to descend. This product is called, in books on mechanics, the Moment. Likewise, the force P has a moment, which is the product of the force P and the distance from P to the fulcrum F. If the force and the weight just balance each other, their moments are equal. The length from P to F is from W to F, the weight arm. Force Force arm called the force arm and the length Then, for balance, we can say that, = Weight X Weight arm If we let P stand for the force a stand for the force arm W stand for the weight and b stand for the weight arm as shown in Figs. 45, 46, and 47, we will have the formula Although the force and weight are really balanced when this formula is fulfilled, still we use the formula for calculating the forces necessary to lift weights. The very slightest increase in the force above that necessary for balance will cause W to rise and, therefore, we can say practically that P will lift W if PXa = W × b In Fig. 43 is shown a lever with the force P at one end, the weight W at the other, and the fulcrum F between them. From the figure, the force arm a is 12 ft. and the weight arm b is 2 ft. What force at P will lift a load W of 1800 lb.? Force Force arm = Weight Weight arm Then if P times 12 is 3600, P will be 3600 divided by 12 A simple type of lever known as a "pinch bar" is illustrated in Fig. 44. As shown here it is used chiefly for moving railway cars. A force of 2400 lb. is required at the point of the bar in order to move a certain car, and a man applies a force of 100 lb. P 48 FIG. 44.-Application of lever as used for pinch bar. at P. If the applied force is just sufficient to move the car, how long is the weight arm b? In this case the force is 100 lb., the force arm 48 in., the weight 2400 lb., and the weight arm, b. If 2400 times b equals 4800, then b equals 4800 divided by 2400, It will be seen that the relation between the force, weight, force arm, and weight arm can be written as an inverse propor This form of expressing the relation is not generally so useful as the other form PX a = W X b. It is very useful, however, in cases where neither the force arm nor the weight arm is known. Example: If a man wanted to lift a 750-lb. weight by means of a 12-ft. timber used as a lever, where would he place the fulcrum so that his whole weight of 150 lb. would just raise it? The total length of the timber (12 ft.) equals the sum of b and a (Fig. 45). From the ratio = b 1 5? which reads "b is to a as 1 is to 5,” it is clear that b is one part of the length, a is 5 parts, and the whole length is 6 parts. One part, or a, will, therefore, be of the length, and 5 parts, or b, will be § of the length. 86. Three Classes of Levers.-Levers are divided into three kinds or classes according to the relative positions of the force, fulcrum, and weight. 1. FIG. 45.-Lever of the first class. Those shown so far are of the first class, Fig. 45, in which the fulcrum is between the force and the weight. The weight is lifted by pushing down at P. In the second class, Fig. 46, the weight is between the fulcrum and the force, and the weight is lifted by pulling up at P. In the third class, Fig. 47, the force P is between the weight and the fulcrum and, therefore, P must be greater than the load that it lifts. The weight is lifted by an upward force at P. In all these types the same rule holds that: Particular attention should be given to the fact that the force arm and weight arm are always measured from the fulcrum. In levers of class 2, the force arm is the entire length of the lever. 2. W a FIG. 46.-Lever of the second class. In class 3, the force arm is shorter than the weight arm. This type may be seen on the safety valves of some boilers and is used so that a small weight can balance a considerable pressure at P. Quite often there appear to be two weights, or two forces, on a lever, and it is difficult to decide which to designate as the force W P FIG. 47.-Lever of the third class. and which as the weight. It really makes no difference which we call the force and which the weight; the relations between them would be the same in any case. 87. Compound Levers.-Compound levers are frequently used; but problems concerning them are easily reduced to repeated cases of single levers, the force of one lever corresponding to the weight of the next, etc. To illustrate this we will solve the following example: Example: We wish to lift 8000 lb. with a compound lever as shown in Fig. 48, the first one being 10 ft. long with the weight 2 ft. from the end; the second 16 ft. long with the fulcrum 4 ft. from the end; what will be the neces sary force, P2? PX a = W × b P1 X 10 = 8000 X 2 P1 W 2 2 = = 16,000 Pi P2 X a = P2 = = = 6400 = 1600 lb. 1600 lb. W2 x b 1600 X 4 = 1 533 lb., Answer. Explanation: Taking the first, or lower lever, we find it to be an example of the second class. W has a weight arm b of 2 ft. The force has an arm equal to the whole length of the lever, or 10 ft. The necessary force on the end of this lever we find to be 1600 lb. The second lever must pull upward through the connection with a force of 1600 lb. In other words, the 8000-lb. weight on the first lever is equivalent to a 1600-lb. weight on the short end of the second lever. The first lever pulls downward the same amount that the 24 10' P = W2 12 FIG. 48.-System of compound levers. = second pulls up, or Pı W2. Having this 1600 as the weight, we find that a force of 533 lb. is needed on the end of the second lever. 88. Mechanical Advantage.-The ratio of the weight to the force is often called the Mechanical Advantage of the lever. Thus, if a man exerting a force of 50 lb. on a lever can raise a weight of 500 lb., the lever has a mechanical advantage of 500 = 10. In simple words, by such a lever every pound of 50 force the man exerts can raise a weight of 10 lb. This means that a 1-lb. force will lift 10 lb., a 2-lb. force will lift 20 lb., a 10lb. force will lift 100 lb., etc. The force multiplied by the M.A. (mechanical advantage) gives the weight that can be lifted, or the weight divided by the M.A. gives the force required. From a previous paragraph, we found that the ratio of the weight to the force equals the ratio of the force arm to the weight W a W Since the M.A. equals it must also equal' P a rm, or = P b a P |