WHEEL AXD AXLE.] 



MECHANICAL PHILOSOPHY. STATICS. 



727 



In constructing balances on this principle, a and 6 are 

 taken equal to each other, in which case E and F bisect 

 A B and C D, and P and Q are equal. 



WHEEL AND AXLE. The second mechanical power 

 is the wheel and axle ; this machine, in its simplest form, 

 consists of a circular wheel A B (Fig. 114) firmly fixed 

 at right angles to a cylinder C D E F, so that both re- 

 volve together round a common axis G H. 



The forces P and \V F ig. 114. 



are supposed to be ap- 

 plied by weights sus- 

 pended from one ex- 

 tremity of the cord 

 wrapped round the 

 wheel or cylinder, to 

 which the other extre- 

 mity is attached. The 

 forces, by this contriv- 

 ance, always act at 

 right angles to the cir- 

 cumferences of the 

 wheel and cylinder. 



The cords are sup- 

 posed to be perfectly 

 flexible, inextensible, 

 and destitute of weight. 

 Their friction on the 

 surface of the wheel 

 and axle, as well as 

 the magnitude of their diameters, is also neglected. 



Fig. 115 represents a section of the wheel and cylinder 

 perpendicular to their common axis. 



Let A B be the radius rig. 115. 



of the wheel, and A C 

 that of the cylinder or 

 axle. Since the forces 

 P and Q always act at 



right angles to the / / \ * * 



circumference of the 

 wheel and axle, the 

 force P may be repre- 

 sented in magnitude 

 and direction by B P 

 at right angles to A B, 

 the force W by C Wat 

 right angles to A C. 



\Vhen these forces 

 produce equilibrium, 

 taking moments about A, we have 



that is 



The force acting on the surface of the wheel 

 The force acting on the surface of the axle 

 radius of axle 



radius of wheel 



Tliis machine is only a modification of the lever ; for 

 referring to Fig. 115, we may consider it in its position of 

 equilibrium as a bent lever, whose arms are A B and A C, 

 and whose fulcrum is A. 



Fig. 118. 



When it is in motion, we may regard the wheel and 

 xle as made up of a number of spokes ; these spokes 

 some successively into action as levers, and thus the ad- 



vantage of an endless leverage is produced, and the 

 power and weight each act constantly in a straight line, 

 instead of describing circular arcs as in the common 

 lever 



Instead of the wheel, one or more bars are sometimes 

 fixed at right angles to the axle, and these are often so 

 disposed as to allow several men to act at once on the 

 machine. When the axis of the axle is horizontal, a bar 

 fixed at right angles to the extremity of the bar, form- 

 ing what is called a winch, affords a convenient means 

 for applying the force. 



When the axis is horizontal, as in Fig. 116, the 

 machine is called a windlass ; when vertical, as in Fig. 

 117, a capstan. 



Fig. 117. 



TOOTHED WHEELS. The third mechanical power 

 is the toothed wheel ; it is extensively used in the con- 

 struction of cranes, clock and watch-work, and almost 

 every variety of machinery. 



Toothed wheels consist of thin cylinders, having their 

 circumferences indented or covered with projections 

 called teeth or cogs, set at equal distances from each other. 



If two such wheels have their axes placed in such a 

 position that the surfaces of the wheels may be in the 

 same plane, and the edges of their teeth touch each other, 

 as in rig. 118, and one of the wheels be made to revolve 



Fig. IIS. 



round its axis, its teeth will act in succession on the 

 teeth of the other, and cause it to revolve round its axis 

 in an opposite direction. 



To determine the conditions of equilibrium for this 

 machine, we will suppose A and A' (Fig. 118) to be the 

 axes about which the wheels revolve, and the weights P 

 and W, which produce equilibrium, to be attached to the 

 extremities of cords wrapped round, and having their 

 other extremities fastened to, cylinders fixed perpen- 

 dicularly to the wheels, and having a common axis with 

 them. 



Let A B and A' B' be the radii of these cylinders. 



The weight P will communicate a tension to the rope 

 1 P, which will produce a pressure on the cylinder A B ; 

 this pressure will be communicated from the cylinder to 

 the point C, where the tooth of the wheel C E'is in con- 

 tact with the tooth of the wheel CE'. In a similar 

 manner, the pressure produced by W on the cylinder 

 A' B' will be communicated to the same point. 



These two pressures at C will act perpendicul.u-ly to 



