unequal loading of the pans, the center of gravity is raised, moving from g to g' (Fig. 83). The work done in tipping the beam is equal to the weight of the beam times the height Dg. By the principle of work, this is equal to the excess weight in the pan times the vertical distance the pan moves. If the center of gravity is not near the knife-edge, tipping the beam through a small angle will raise the center of gravity a relatively large distance. In that case the work done is relatively large; hence, to tip the balance, a larger excess weight would be C required in the pan. But if the center of gravity is very close to the knife-edge, it will be lifted a short distance, and the work done will be small. In that case only a small excess of weight will be needed. Therefore, in order to have the balance sensitive (that is, tipped by a small excess of weight), the center of gravity must be close to the knife-edge. Usually there is a small weight on the beam which can be moved D up or down, so that the center of gravity can be raised or gtlowered and thus the sensibility be changed at will. In a FIG. 83 somewhat similar way it can be shown that increasing the weight of the beam lowers the sensibility. It can be proved that the three knife-edges, A, C, and B in Fig. 82, should lie on the same straight line. P' B In a familiar form of scales commonly seen in drug stores the two scalepans are placed above the beam. An interesting thing about this balance is that it does not make any difference where the load is placed on the scalepan. One would naturally infer, on first examination, that if equal weights were placed on the pans, with one weight at the extreme outer edge of the right-hand pan and the other on the inside edge of the left-hand pan, they would not be in equilibrium. The balance has two beams, AB and CD of Fig. 84, connected by knife-edges or pivots at A, B, C, and D to the rods which carry the scalepans. This gives a well-known geometrical device for parallel motion. The beams AB and CD always stay parallel, and the rods P A FIG. 84 AC and BD are always parallel and vertical. Hence the scalepans remain horizontal. When a weight W is placed on the pan P, all points of P move equal distances downward. By the aid of the principle of work it can be seen that since the distance moved downward is always the same, the weight required to balance a given load will be the same for all positions on the pan. Owing to the fact that there is sometimes "lost motion" at the pivots A, B, C, and D, these conditions are not always exactly fulfilled. For this reason it is often possible to detect changes in the equilibrium if the weight is placed first on one side of the pan and then on the other. 102. The hydraulic press. The principle of the hydraulic press has been explained in section 27, but we can look at it from another point of view: Let a be the area of the small piston (Fig. 85) and A that of the large one. If the smaller piston is pushed down a distance s, it will displace a volume of water sa, which will flow over into the other chamber and raise the larger piston a height d. The increase in volume of the water in this chamber will be dA. Since the two volumes are equal, w FIG. 85 which gives the ratio of the distances moved. We may now apply the principle of work. The work done by the force F is Fs, and the work done on the weight w is wd. As these must be equal, The same result can be obtained from the expression developed in section 27, where an entirely different line of reasoning is used. Agreements of this kind are common throughout the entire subject of physics. Such numerous cases of checking make us certain of the truth of the fundamental assumptions of mechanics. 103. The effect of friction. It should be clear to the student that while we have neglected friction in deducing the laws of different types of machines, in many cases friction must be taken into account in practice. In any practical case the following steps can be taken: first, the force is computed, neglecting friction; second, the effect of friction is estimated, and a correction applied. In the next chapter, methods of estimating the magnitude of frictional forces are discussed. In general, the effect of friction is to make the work done by a machine less than the energy or work supplied to it. Friction tends to lower what is called the efficiency of the machine. 104. The efficiency of machines. The efficiency of a machine is the ratio of the work the machine does to the total energy supplied to it. This term is used not only with the simple machines described here but with all classes of machines-engines, turbines, motors, dynamos, etc. The efficiency of a machine is commercially very important. When an engineer is deciding what type of machine to install, the efficiency, the initial cost, and the cost of maintenance are the most important items. It is often so necessary to know the numerical value of the efficiency that elaborate experiments are carried on to measure the work done and the energy supplied to the machine. The term efficiency is used a great deal in a very loose sense; but there is nothing vague or indefinite about the term as applied to machinery, for it always has the definite meaning given in the definition above. The advantages of the method of precise measurement are obvious. When an attempt is made to improve the efficiency of a machine, there need be no guessing; the efficiency is measured, and the success or failure of the change is known. We hear a great deal about "efficiency methods" and "efficiency engineering." Often this is merely an attempt to apply the precise quantitative methods of mechanics to all sorts of problems. PROBLEMS 1. A weight of 200 lb., 6 in. from the fulcrum of a lever, is balanced by a force 4 ft. from the fulcrum. Find (a) the applied force and (b) the mechanical advantage. 2. A bucket of water weighing 48 lb. is lifted by a wheel and axle. (a) If the axle is 6 in. in diameter and the wheel 3 ft., what force must be supplied to the rim of the wheel? (b) What is the mechanical advantage? 3. Two pulleys 24 in. and 8 in. in diameter are mounted on the same shaft. Power is supplied to the larger one, and a machine is belted to the smaller. What is the mechanical advantage of the shaft and pulleys? 4. A system of pulleys is so arranged that when the load is lifted 1 ft., the applied force moves 6 ft. (a) What is the mechanical advantage ? (b) Draw a diagram of the arrangement of the pulleys. 5. How much force in dynes, parallel to the plane, would be required to support 10 kg. on a smooth plane 6 m. long and 1.5 m. high? 6. A wagon and contents weighing 4000 lb. is pulled up a hill 1000 ft. long and 50 ft. high. (a) How much extra work was done on account of the hill? (b) What was the extra force needed? 7. An inclined plane is 4 ft. high and 10 ft. long. How much force must be exerted parallel to the plane to roll a 300-pound barrel up the plane with a slow, uniform motion? 8. Given a block and tackle, each with two pulleys, what is the largest load that a man can lift (neglecting friction) if he can apply a force of 100 lb.? Make a diagram showing the arrangement. 9. Why is it more difficult to turn a screw with a small-handled screwdriver? Is it length or diameter of the handle that is important? 10. A painter hoists himself to the top of a structure by a block and tackle having two wheels in the upper (fixed) block and one in the lower (movable) block. Man and equipment weigh 200 lb. (a) What is the tension in the rope? (b) in the support of the fixed block? 11. What difference, if any, would it make if the painter of Problem 10 were hauled up by another man, standing on the ground below ? 12. If the forces required to overcome friction in driving a 3000-pound car at 30 mi./hr. amounted to 300 lb. applied parallel to the road, how much force would be required to take the car up a 4 per cent grade at that speed? 13. A screw operated by a lever arm 36 in. long (measured from the center of the screw) has four threads to an inch. Neglecting friction, how large a load can be raised by a force of 10 lb.? 14. Water is supplied to a piston, which has an area of 30 sq. in., at a pressure of 40 lb./sq. in. Draw a diagram of a system of pulleys that will enable a total load of 2 tons to be lifted by the water pressure. CHAPTER IX FRICTION Starting friction, 105. Sliding friction, 106. The angle of repose, 107. Power expended on account of friction, 108. Rolling friction, 109. Fluid friction: viscosity, 110. Viscosity in solids, 111. Viscosity in another sense, 112. 105. Starting friction. It is a matter of common knowledge that it requires a force to drag a body along a surface even when there is no acceleration. If we should measure the required force, we should find that it requires (1) a fairly definite force to start any particular body moving and (2) a smaller force to keep the body moving at a constant speed. The retarding forces which in one case tend to prevent the starting and in the second case tend to stop the motion are called forces of friction, or often merely friction. These retarding forces can be determined by measuring the forces which must be applied to the body to overcome friction; thus the force required to keep a railroad train moving at a uniform speed on a straight level track is equal to the retarding forces of friction. The force which is just sufficient to start a body moving is equal to what is called starting friction. The following facts regarding starting friction have been found by experiment : 1. It is different for different kinds of materials. 2. It is different for different conditions of the rubbing surfaces. 3. For any given body moving on a given surface it is approximately proportional to the force holding the body against the surface. In Fig. 86, F is the force which is just sufficient to start a body moving, and F' is the force pressing the body against the surface. Usually F' will be the weight of the body. The third fact of the last paragraph can be expressed by the relation |