(3) A weight W (fig. 62) is suspended from the middle point of a rigid rod without weight, connecting the centres O, O', of two equal heavy wheels, which rest on a rough inclined plane: the wheel O is locked: to find the greatest inclination of the plane which is consistent with the equilibrium of the carriage. Let P be the weight, and r the radius of each of the wheels; let 00' = 2a, = the inclination of the plane to the horizon; let R, R', be the reactions of the plane on the wheels at right angles to itself; μR the friction on the wheel O, μ being the coefficient of friction; F the action of the plane on the wheel O at right angles to R'; X, Y, the resolved parts, parallel and perpendicular to the plane, of the action of the wheel O' on the rod 00′; and X', Y', the similarly resolved parts of the reaction. For the equilibrium of the wheel O and the rod 00', regarded as one system, we have, resolving forces parallel to the inclined plane, μR=X+(P+ W) sin ........... resolving forces at right angles to the plane, (1); Again, for the equilibrium of the wheel O', we have, taking moments about the point of contact of this wheel with the plane, X'r= Pr sin o, or X' = P sin &........ (4). From the equations (1) and (4), observing that X' is by the nature of action and reaction equal to X, we get μR= (2P+ W) sin þ.. (5). Again, from (2) and (3), μrR+2a (P+W) cos - 2aR = Wa cos &, (2a-pr) Ra (2P+ W) cos p..... From (5) and (6) we obtain for the required inclination of COR. Having ascertained p, we know R from (5) and X' or X from (4), and therefore Y from (2); also, F being the only force acting on the wheel O' which does not pass through its centre, it is evident that Fmust be equal to zero. (4) Two equal beams AC, BC, connected by a smooth hinge at C, are placed in a vertical plane, their lower extremities A and B resting on a rough horizontal plane; from observing the greatest value of the angle ACB for which equilibrium is possible, to determine the coefficient of friction at the ends A and B. 2 If ẞ be the greatest value of ACB, and μ be the coefficient of friction at each of the ends; then μ=tan B. (5) Two equal semicylinders are placed horizontally at the same vertical altitude, their flat faces, which are rough, resting against two vertical and parallel plane surfaces the distance between which is infinitesimally greater than the diameter of either cylinder: a smooth wedge, the vertex of which is downwards, rests between the two semi-cylinders on their curved surfaces: to find the vertical angle of the wedge, supposing the cylinders to be on the point of slipping downwards. Let W be the weight of either cylinder, W' that of the wedge, the coefficient of friction, and 20 the vertical angle of the wedge: then SECT. 3. Systems of Beams. (1) Two uniform rods AC, BC, (fig. 63), are connected together by a smooth hinge-joint at C, their other ends being fastened to two smooth fixed hinges A, B, in a vertical line: to find the magnitudes and directions of the pressures on the hinges and of the mutual action of the rods at the joint. It is frequently convenient in problems of this class, to make use of diagrams in which the several members of the system are represented to the eye in a state of slight detachment; the actions and reactions being indicated by arrowed lines not running into each other. The student will thereby escape falling into errors of sign in writing down the equations of equilibrium, to which he is liable from confounding together actions and reactions. In fact, the problem thereby resolves itself into the consideration of the equilibrium of several distinct bodies. Let AC=2a, BC = 26, and let tan a, tan ß, be represented by m, n, respectively. The horizontal and vertical components of the actions and reactions on the rods are indicated in the diagram, as well as the weights of the rods. For the equilibrium of AC there is, resolving horizontally, In like manner, for the equilibrium of BC, X'= X".... Y' = Q + Y"....... (3). (4), (5), The two components of the pressures exerted at A, C, B, upon each rod having been ascertained, the required directions and magnitudes of these pressures are therefore known. (2) At the middle points of the sides of any polygon ABCDE...... (fig. 64), and at right angles to them, are applied a series of forces P, Q, R, ......, respectively proportional to the sides; the sides of the polygon are perfectly rigid, and capable of moving freely about the angular points A, B, C, D, ...; to determine the form of the polygon that it may be in equilibrium, the lengths of the sides being given. p, q, r, s, Let denote the mutual actions of the sides of the polygon at the angles A, B, C, D, ......, of which the directions will lie in certain straight lines bBß, cCỵ, dDƐ, .............. |