Pulley. 105 



IA : ID : : sin IDA : sin IAD, 

 sin Flq : sin OAD, 

 sin FI q : sin G7 q. 



We may say, therefore, universally, that when there is an equili- 

 brium by means of the simple pulley, freed or moveable ; (1.) The 

 tensions of the two parts of the rope which passes round the polity, 

 or the powers applied to them, are equal', (2.) That each of these 

 powers is to the force exerted at the centre of the pulley, as the sine of 

 half the angle formed by the two parts of the ropes in question, is to 

 to the sine of the whole of this angle. 



Thus in the fixed pulley, there is no other advantage gained Fi g- 82, 

 by the agent q, except that of being able to change at pleasure 

 the direction in which the action shall be employed. But in the 

 moveable pulley, there is possessed by the agent 9, the double pig. 84. 

 advantage of a change of direction and an augmention of the 85j 86- 

 effect of the action. But it is to be remarked, that according as 

 the direction is changed, the force exerted upon the centre va- 

 ries, so that there is a direction in which the effect produced by 

 a given power is the greatest possible ; and this is when the two 

 parts of the rope GO, Tq, are parallel, as will be readily per- 

 ceived. 



178. If we draw the radii OF, FT, and the chord OT, the Fig. 84. 

 the triangle OFT, having its sides perpendicular respectively to 

 those of the triangle BID, will be similar to BID ; whence Geom. 



IB : ID : : FT : OT, 

 or 



q : g :: FT : OT '; 



that is, the tension of either part of the rope is to the force exerted 

 against the centre F, as the radius of the pulley is to the chord of 

 the arc embraced by the rope. 



Now it is evident that this last ratio is the greatest possible 

 when the two parts of the rope are parallel ; hence in the move- 

 able pulley the power is the least possible, or is exerted to the 

 greatest advantage, when the two parts of the rope are parallel ; 

 and it is then half of the force exerted against the centre of the 

 pulley. This second kind of pulley is made use of in tightening 

 Mech. 14 



