RESISTANCE OF MATERIALS 



75 



and the work will be equal to that of the fall of the weight Q 

 from a height h. 



53. (3) Winch. With the aid of handle AB a power Pis exerted 

 perpendicularly to the arm BB', that is to say, tangential to the 

 circumference of the radius BB' = R. The resistance Q is at 

 the extremity of a rope rolled on the cylinder K, whose radius 

 is r (fig. 91): 



For a virtual rotation a of the handle, the work done by the 

 power exerted is PR X a ; that of the resistance is Qr X a 

 (negative work). The condition of equilibrium is : 



Pr = Qr whence P = Q^ . 

 R 



Q 



FIG. 91. 



A winch has only one degree of liberty, it turns round a fixed 

 axis. Instead of a handle other means may be employed for pro- 

 ducing revolutions such as the treadmill and the capstan, both of 

 which are identical in principle with the winch. 



54. (4) The Pulley. Consider first the fixed pulley, a wheel 

 turning round an axis 0, which passes 

 through its centre. The axis is sup- 

 ported by an inverted U-shaped 

 hanger, C. The circumference of the 

 wheel is hollowed out to form a groove 

 in which the rope is passed. 



For a virtual rotation a of the wheel, 

 the work done by the power is Pra ; that 

 resistance is Qra ; the condition of 

 equilibrium being therefore : 



P = Q^(fig. 92). 



The power must, therefore, be equal 

 to the resistance. 



