friction, and taking moments about the center of the wheel (see Fig. 167), we have, for uniform motions, Taking moments about the point of contact of the wheel and rail, we have, for the position shown, and since we have P(a+r) = R'a, It follows that T-R; that is, the train resistance cannot be greater than the adhesion of the drivers to the rails. This adhesion in American practice is usually taken as to the load on the drivers. Problem 230. What resistance R may be overcome by a locomotive moving at uniform speed, diameter of drivers 62 in., cylinders 16 × 24 in., and a steam pressure on the piston of 160 lb. per square inch? What should be the weight of the locomotive on the drivers? Problem 231. If the diameter of the drivers of a locomotive is 68 in., and the size of the cylinder is 20 x 24 in., what train resistance may be overcome by a steam pressure of 160 lb. per square inch? Problem 232. A locomotive has a weight of 155 tons on the drivers, if the adhesion is taken as , this allows 31 tons for the drawbar pull. The train resistance per ton of 2000 lb., for a speed of 60 mi. per hour, is 20 lb. Find the weight of the train that can be pulled by the locomotive at the speed of 60 mi. per Problem 233. hour. An 80-car freight train is to be pulled by a single expansion locomotive at the rate of 30 mi. per hour. The weight of each car is 60,000 lb., and the resistance for this speed is 10 lb. per ton. What must be the weight on the drivers, if the adhesion is ? 145. Friction. CHAPTER XIV FRICTION When one body is made to slide over another, there is considerable resistance offered because of the roughness of the two bodies. A book drawn across the top of a table is resisted by the roughness of the two bodies. The rough parts of the book sink into the rough parts of the table so that when one of the bodies tends to move over the other, the projections interfere and tend to stop the motion. The bearings of machines are made very smooth, and usually we do not think of such surfaces as having projections. Nevertheless they are not perfectly smooth, and when one surface is rubbed over the other, resistance must be overcome. This resisting force to the motion of one body over another is known as friction. When the bodies are at rest relative to each other, the friction is known as the friction of rest, or static friction. When they are in motion with respect to each other, the friction is known as the friction of motion, or kinetic friction. be acting on it; the downward force G (not shown) and the reaction R inclined back of the vertical through the angle. The reaction R of the plane on the body may be resolved into two components, one horizontal and one vertical. The horizontal force is known as the force of friction, and the normal force, the normal pressure. The tangent of the F angle 0, or, is called the coefficient of friction. This N coefficient of friction, which we shall represent by f, may be defined as the ratio of the force of friction to the normal pressure; it is an absolute number. The coefficient of friction is usually determined by allowing a body to slide down an inclined plane as shown in G SIN N=G cos FIG. 169 Fig. 169. The angle is increased until the force of friction F will just keep the body from sliding down the plane. The angle is then called the angle of repose, and the tangent of is the coeffi cient of friction. It is possible with such an apparatus to determine the coefficient of friction for various materials. It has been found that after motion begins the friction is less, that is, the friction of motion is less than the friction of rest. This is an important law for engineers. 147. Laws of Friction for Dry Surfaces. -Very little was known of the laws of friction until within the last seventyfive years. About 1820 experiments were made that seemed to show that, for such materials as wood, metals, etc., friction varies with the pressure, and is independent of the extent of the rubbing surfaces, the time of contact, and the velocity. A little later (1831) Morin published the following three laws as a result of his experiments on friction: (1) The friction between two bodies is directly proportional to the pressure; that is, the coefficient of friction is constant for all pressures. (2) The coefficient and amount of friction for any given pressure is independent of the area of contact. (3) The coefficient of friction is independent of the velocity, although static friction is greater than kinetic friction. These laws of Morin hold approximately for dry unlubricated surfaces, although it has been found that an increase in speed lowers the coefficient of friction. The coefficient of friction is a little greater for light pressures upon large areas than for great pressures on small areas. The following is a table of some of the coefficients of friction as determined by Morin : COEFFICIENTS OF FRICTION, DUE TO MORIN |