CHAPTER X. FRICTION. 171. In the investigations of the preceding chapter, we have supposed that the surfaces of the bodies in contact are perfectly smooth. By a smooth surface is meant a surface which opposes no resistance whatever to the motion of a body upon it. A surface which does oppose a resistance to the motion of a body upon it is said to be rough. In practice it is found that all bodies are more or less rough. The friction of a body on a surface is measured by the least force which will put the body in motion along the surface. 172. Coulomb made a series of experiments upon the friction of bodies against each other and deduced the following laws. Mémoires des Savans Etrangers, tom. x. (1) The friction varies as the pressure when the materials of the surfaces in contact remain the same. When the pressures are very great indeed, it is found that the friction is somewhat less than this law would give. (2) The friction is independent of the extent of the surfaces in contact so long as the pressure remains the same. When the surfaces in contact are extremely small, as for instance a cylinder resting on a surface, this law gives the friction much too great. These two laws are true when the body is on the point of moving and also when it is actually in motion: but in the case of motion the magnitude of the friction is much less than when the body is in a state bordering on motion. (3) The friction is independent of the velocity when the body is in motion. It follows from these laws that if P be the normal pressure between two surfaces, then the friction is μP, where μ is a constant quantity for the same materials and is called the coefficient of friction. In the state bordering on motion and when the surfaces in contact are of finite extent, we have the following results from experiment: μ , surfaces wood, the grain being in the same direction, = 1 = , one surface wood and the other metal. Oil and grease considerably diminish friction; fresh tallow reduces it to half its value. = for wood. In the state bordering on motion and when the surfaces in contact are single lines, then μ When the surface in contact is a physical point, the statical friction is inconsiderable. For full particulars on this subject, we refer the reader to Coulomb's papers, and to the Memoirs published in the Mémoires de l'Institut. by M. Morin. 173. Angle of Friction. Suppose a body acted on only by its weight to be placed upon a horizontal plane and the plane to be turned round a horizontal line until the body begins to slide. Let W be the weight of the body and a the angle the plane makes with the horizon. The pressure of the body on the plane will be equal to the resolved part of its weight perpendicular to the plane, that is to W cosa. When the body is on the point of sliding the friction is equal to the resolved part of the weight parallel to the plane, that is to W sina. If μ be the coefficient of friction, we have W sina = μ W cosa; This experiment will enable us to determine the value of the coefficient of friction for different substances. 174. In Art. (32) we have found the condition of equilibrium of a particle constrained to rest on a smooth curve; we proceed to the case of a particle on a rough curve. Suppose the curve a plane curve; let X, Y be the forces which act on the particle parallel to the axes of x and y exclusive of the action of the curve. The sum of the resolved parts of X and Y along the tangent to the curve is The sum of the resolved parts along the normal is μ If be the coefficient of friction, the greatest friction capable of being called into action is Hence, the condition of equilibrium will be that the numerical dx value of X + y dy ds must be less than the numerical value of μ (xdy - Y dx ds ds without regard to sign in either case. dx dy ds ds This may be conveniently expressed thus, 2 X + Y must be less than μ2 (X We may exhibit this condition in a different form, as will be seen in the following article. 175. Next, let the curve be of double curvature. Let P denote the resultant force acting on the particle exclusive of the action of the curve; X, Y, Z the components of P parallel to the axes; l, m, n the direction cosines of the tangent to the curve at the point where the particle is placed; the angle between this tangent and the direction of P. The resolved part of P along the tangent is P cose, and that perpendicular to the tangent is P sine. Hence, if μ be the coefficient of friction, we must have for equilibrium P cose μP sine; cose<μ (1-cos" 0); therefore It is easy to shew that this result includes that of the former 176. A particle is constrained to remain on a rough surface; determine the condition of equilibrium. Let P be the resultant force on the particle exclusive of the action of the surface; the angle between the direction of P and the normal to the surface at the point where the particle is placed; u = 0, the equation to the surface; x, y, z the coordinates of the particle. The resolved part of P along the normal is P coso, and that perpendicular to the normal is P sino. Hence, for equilibrium we must have 177. In the following articles of this chapter we shall investigate certain equations which hold when the equilibrium of different machines is on the point of being disturbed. The equations in such cases will involve the forces acting on the machine and μ the coefficient of friction. When we have found one of these limiting equations, we can draw the following inferences: (1) If in order to satisfy the equation for a given set of forces it is necessary to ascribe to a value greater than its extreme value for the substances in question, which is known by experiment, equilibrium is impossible. |