Equilibrium Under Grav ity.-48. The motion of centre of gravity—as a point, fixed in re lation to any given body, through which its weight may be taken to act—enables us to bring many problems within range of the theorem stated in § 42. For example, suppose that we re quire to know the slope at which a heavy rod AB (fig. i8) can rest in equilibrium with its ends on two smooth plane surfaces CA, CB. Since the effect of gravi tation may be represented by a single force through G, the centre of gravity of the rod, this is in effect a problem of equilibrium under three forces. We know that RA and RB, the forces exerted by the plane surfaces, must intersect in a point vertically above G; moreover, since these surfaces are smooth, RA and RB must be as a differential equation from which the form of the curve may be deduced when w and H are specified. When the curve has been found, the tension T at any point can be found from (59) ; for we have 5o. When the weight per unit horizontal run is uniform—as will be very approximately true of a uniform wire stretched in a flat curve—the centre of gravity of OP must lie above the middle point of OH. Hence we shall have or, by (59) Since W is now proportional to x, we see that y will be propor tional to i.e., the chain will hang in a parabola.
(1821), according to which the force exerted between two sur faces may be inclined to their common normal at any angle which does not exceed some definite value X; this limiting angle X, termed the angle of friction, depends upon the nature of the surfaces in contact, but is independent of the intensity of the reaction between them. On the basis of this law we may say, in the problem con sidered, that the rod can rest in any position (as shown, e.g., in fig. 2o), provided that a point D can be found, vertically above G, such that neither of the angles DBO, DA0 exceeds X, where AO, BO are perpendicular to CA, CB respectively.
52. Coulomb's law of friction may be stated in another way. If R is the normal pressure be tween two surfaces in contact, the resultant action is found by combining R with a tangential component S (due to friction) ; it will thus be inclined at an angle 0 to the normal direction, where Hence, if 0 cannot exceed the angle of friction X, it follows that the ratio S/R cannot exceed a definite limit, tan X, which is called the coefficient of friction, and is commonly denoted by the sym bol ,u . Thus, according to Coulomb's law, a tangential force will be exerted between two rough surfaces in contact, which cannot exceed a definite fractionµ of the normal pressure between them, but will assume, within these limits, any magnitude and direction that may be required to maintain equilibrium. As was shown by the preceding example, the exact nature of the forces which act to maintain equilibrium in a given position will generally be indeterminate.