APPLICATION OF GRAPHIC METHODS 291 



a rope, it cannot be safely neglected, and leads to sensible dimi- 

 nution of efficiency, as where power is transmitted by belts and 

 pulleys. Coming to the second class of cases, it is clear that when 

 a heavy element is being lifted, lowered, accelerated, or retarded 

 it is not in equilibrium under the action of the external forces 

 at the joints calculated in the manner hitherto described ; we 

 shall, however, hereafter include the forces due to these causes 

 in calculating the forces at the joints, and there remains only 

 one mode in which a loss of energy occurs in the course of its 

 transmission by an element, namely, its dissipation in over- 

 coming an internal couple. This case finds an illustration in the 

 case of a rope wound on to a pulley, or unwound from one ; the 

 pull on the rope is not transmitted in a direct line, as we have 

 hitherto supposed, but in consequence of the couple required to 

 bend or unbend the rope, the line of action is shifted sideways 



through a length ^, where m is the moment of the couple, and 



F the force transmitted. This translation of the force affects the 

 values of / for all the subsequent joints. It can be represented 

 in the dynamic frame by showing the line of action of the 

 force shifted parallel to itself in a disadvantageous direction. 

 If in the given problem we know the useful resistance, we must, 

 in constructing the diagram, shift the force at the driving end ; 

 vice versa, if the driving effort is known, we must shift the force 

 at the resisting end. The fraction expressing the loss of efficiency 

 due to this cause is not, like that due to friction or stretching, 

 independent of the magnitude of the forces involved, but will, on 

 the contrary, always involve complete inefficiency when the 

 force is very small, and implies a gradually increasing efficiency 

 as the force transmitted increases. Thus, a small force exerted 

 on a stiff rope passing over a pulley produces no effect on the 

 further side, because it is insufficient to bend the rope. The 

 loss by an internal couple always diminishes the resistance which 

 a given driving effort can overcome, whereas the loss of internal 

 work done in overcoming a single force has not this effect. The 

 case of the transmission of power by fluids in pipes will be ex- 

 amined in a subsequent paper, after machines composed of solid 

 parts have been analysed. 



13. Simple Machines. Lever. Let a. Fig. 11, be a lever to 



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