SQUARES, CUBES, SQUARE ROOTS, AND CUBE ROOTS OF NUMBERS.-Continued No. Square Cube Square Cube No. Square Cube Square Cube root root 9.9900 991 982081 973242271 31.4802 9.9699 996 992016 988047936 31.5595 9.9866 992 984064 976191488 31.4960 9.9733 997 994009 991026973 31.5753 993 986049 979146657 31.5119 9.9766 998 996004 994011992 31.5911 994 988036 982107784 31.5278 9.9800 999 998001 997002999 31.6070 9.9967 995 990025 985074875 31.5436 9.9833 1000 1000000 1000000000 31.6228 10. 9.9933 CHAPTER XIII LEVERS 90. Types of Machines.-All machines consist of one or more of the three fundamental types of machines-the Lever, the Cord, and the Inclined Plane or Wedge. Any piece of mechanism can be proved to be of one or more of these types. Pulleys, gears, and cranks will be shown to be forms of Levers; belts and chains come under the type called the Cord; while screws, worms, and cams are forms of Inclined Planes. They are all used to transmit power from one place to another and to modify it, as desired. 91. The Lever. The lever is probably the most used and the simplest type of machine. We are all familiar with it in its simplest forms, such as crow bars, shears, pliers, tongs, and the numerous simple levers found on machine tools. A lever is a rigid rod or bar so arranged as to be capable of turning about a fixed point. This fixed point about which the lever turns is called the Fulcrum. In Fig. 37 the fulcrum is W FIG. 37. represented by the small triangular block F. The position of this fulcrum determines the effect which the force P applied at one end has toward lifting the weight W at the other end. If F is close to W, a comparatively small force P may be able to raise the weight W, but if F is moved away from W and placed close to P, then a greater force will be required at P. If F is in the middle, P and W will be just equal. In every lever there are two opposing tendencies: first, that of the load or weight W tending to descend; and second, that of the force P tending to raise W. The ability of W to descend or to resist being lifted depends on two things-its weight and its distance from the fulcrum F. The product of these two is the 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 called the force arm and the length from W to F, the weight arm. Then, for balance, we have the equation: Force X force arm = Weight weight arm as shown in Figs. 39, 40, and 41, we will have the formula PXa=Wxb 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 If it is the length a that is wanted, we can see that P would lift W when We have a lever 14 ft. long, with the fulcrum placed 2 ft. from the end, as shown in Fig. 38; how much force must we exert to lift 1800 lb.? In this problem a is 12 ft., b is 2 ft., and W is 1800 lb. PXa=Wxb PX12 1800 X 2 = 3600 Then, if PX 12(or 12×P) is 3600 P will be 3600÷12 P 3600÷12-300 lb., Answer. It will be seen that the relation between force, weight, force arm, and weight arm, can be written as an inverse proportion. This form of expressing the relation is not generally as useful as the other form, PXa=Wxb. It is very useful, however, in cases where neither the force arm nor weight arm are 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? P: W ba 150 750 ba 150 750 1:5 b: a 5 6 a= b = = =1:5 of 12 10 ft. = 1 6 of 12-2 ft. Explanation: The total length of the timber (12 ft.) is the sum of a and b (see Fig. 39). We can find the ratio of b to a which is the same as P:W and reduces to 1:5. If the ratio is 1:5, then the whole length is 6 parts of which a is 5 parts and b, 1 part. Hence a=10 ft. and b=2 ft., and the fulcrum must be placed 2 ft. from the weight. 92. Three Classes of Levers.-Levers are divided into there kinds or classes according to the relative positions of the force, fulcrum, and weight. Those shown so far are of the first class, Fig. 39, 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. 40, the weight is between the fulcrum and the force, and the weight is lifted by pulling up at P. In the third class, Fig. 41, 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. In class 3, the force arm is shorter than the weight arm. This type may be seen on the safety valves of many 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 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. 93. Compound Levers.-We frequently meet with compound levers; 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: |