EQUILIBRIUM OF A BODY RESTING ON AN AXLE. 155

CHAPTER VII.

OF THE EQUILIBRIUM OF BODIES RESTING ON AN AXLE, AND OF THE RIGIDITY OF ROPES; WHEEL AND AXLE, PULLEY.

SECTION I.

85. Fundamental Condition of Equilibrium in the state bordering on motion, of a body capable of revolving round an axle. - All the forces acting on the body can

FIG. 105.

0

R

P

be reduced to a single resultant, to which, when the body is at rest, the reaction of the bearing must be equal and opposite; let the annexed figure represent the axle resting on its bearing; let R be the resultant of the forces acting on the body, and let its direction cut the circumference of the bearing at the point P; take o the centre of the bearing and join op; this line is the normal to the point of contact; the body will therefore be in the state bordering on motion when the angle OPR equals the limiting angle of resistance, the motion being about to ensue in the direction indicated by the arrow head. This consideration enables us to give a very simple construction, which will apply to all cases in which the forces act on the body along parallel lines. Take o the centre of the bearing fig. 106, draw a line a o parallel to the directions of the forces; if the body is about to move in the direction indicated by the arrow head, make the angle AOP equal to the limiting

FIG. 106.

R

angle of resistance; then the resultant force must act along the line RP parallel to OA, since this is the only line drawn parallel to O A which will cut the circumference in a point P such that the angle OPR equals the limiting angle of resistance; hence if we measure moments round p, we shall obtain the required relation between the forces, the sum of those moments being equal to zero by Art. 58. Of course if the motion is about to ensue in a contrary direction, the angle AOP must fall on the other side of O Α. It will be remarked that the radii of the axle and its bearing are sensibly equal, so that though in the diagram they are represented as different that difference never enters the question.

A

P

86. Friction of Axles. - When the body is in the state bordering on motion, the values of the coefficient of friction are the same as those given in the last chapter; the same is also true in cases of motion where no unguent is interposed; in nearly all cases of motion, however, an axle is kept well greased, both to prevent wear and to diminish the resistance; the unguent may be supplied at intervals, as in the case of a common cart wheel, or continuously as in the case of the wheel of a railway carriage; as might be expected, a continuous supply of unguent is found to be the most effective means of diminishing the resistance; the following table gives the values of the coefficients of friction, and the limiting angle of resistance for the axles and bearings most commonly used; the coefficients of friction are taken from the experimental determinations of General Morin,* from which the limiting angle of resistance has

* Notions Fondamentales, p. 309. To avoid ambiguity, the means of some of Gen. Morin's results have been taken; thus, instead of 0.07 to 0.08, the above table gives 0.075.

been calculated those cases have been selected in which the unguent is most effective in diminishing friction.

Ex. 405. Let AB (fig. g) be a beam movable about a wrought-iron axle which rests on a cast-iron bearing, and whose axis passes at right angles through the axis of the beam ; * the centre c of the axle is 12 in. from A, and 30 in. from the centre of gravity of the beam and axle, the radius of the axle being 3 in.; the weight of the whole (i.e. of the beam and axle) is 400 lbs.: find the weight which, when hung at a, will just cause the end a

to descend.

Draw the figure to scale; draw through c the vertical line CD, and make the angle DCQ equal to the limiting angle of resistance (10° 45'); draw the

* Of course there are in reality two bearings situated symmetrically with reference to the length of the beam, each of which supports half the united

pressures and w; the plan of the machine being shown in the accompanying figure.