ILLUSTRATIVE EXAMPLE 



65 



ILLUSTRATIVE EXAMPLE 



Wheel and axle. The apparatus known as the "wheel and axle" consists of 

 a circular axle free to turn about its central axis, to which a circular wheel 

 is rigidly attached, so that its center is on the center of the axle. A rope or 

 string is wound round the axle, and has a weight attached to its end. A second 

 rope or string is wound round the circumference of the circle in the opposite 

 direction, and this again has a weight attached to its end. By a suitable choice 

 of the ratio of these two weights, the apparatus may be balanced so that there 

 is no tendency for it to turn about its axis. 



Let us consider the equilibrium of the system consisting of the wheel and 

 axle and of those parts of the strings or ropes which are wound round them. 

 To simplify the problem, let us disregard altogether the weight of the system. 

 Then the externally applied forces are 



(a) the tension of the rope wound round 

 the wheel ; 



(6) the tension of the rope wound round 

 the axle ; 



(c) the action of the supports which keep 

 the wheel and axle from falling. 



Let the weights be denoted by P and Q, so 

 that these are also the tensions of the strings, 

 and let the rad'ii of the wheel and axle be a, 

 b respectively. Let us express mathematically 

 that the sum of the moments of the externally 

 applied forces about the axis is nil. 



The moment of the tension of the string 

 on the wheel is Pa, for P is the amount of 

 the tension, which acts at right angles to the 

 axis, and a is the shortest distance from the 

 axis to the line of action of this tension. 



Similarly the moment of force (b) is Q6, 

 the negative sign being taken because this tends to turn the system in the 

 direction opposite to that in which the first tension tends to turn it. 



If we imagine the system to be supported by forces acting on the axis itself, 

 the moment of forces (c) vanishes, for the lines of action of these forces intersect 

 the line about which we are taking moments. Thus the required equation is 



Pa - Qb = 0. 



This equation simply expresses that 



[tendency of P to turn system] [tendency of Q to turn system] =0. 



Thus when the system is balanced so as to remain at rest we must have 



so that the weights must be inversely as the radii. Practical examples of the 

 principle of the wheel and axle are supplied by the windlass and capstan. 



FIG. 33 



