Stiffness of Cords. 157 



ble ; so that a certain force is required in order to bring the 

 parts A'O, CC', into the direction of tangents to the points A 

 and C; in other words, a force must be employed which would 

 be unnecessary but for this want of flexibility. 



The pulley being always supposed to move with perfect ease 

 upon its axle /, if instead of a cord a ribbon be employed, a very 

 small increase in the weight q will cause the pulley to turn. But 

 if the cord be replaced, it will evidently be necessary to augment 

 the weight q ; (l.) According as the sum of the weights/) and q, 

 or, in general, the whole force by which the cord is stretched, is 

 more considerable ; because, other things being the same, the 

 resistance occasioned by the weights p and 9, when by the stiff- 

 ness of the cord they take the positions A'Op', CC' <?', will in- 

 crease as the weights themselves increase. 



(2.) The addition to be made to q, must be greater according 

 as the radius of the pulley, (or of the surface over which the 

 cord passes) is less. For the resistance which the power meets 

 with arises from this, that the cord, instead of adapting itself to 

 the revolving surface, remains at a certain distance, forming a 

 curve jo 7 OA' and making with the surface an angle OA'A ; and 

 this resistance will evidently be the greater, according as the 

 curvature A'O of the cord departs more from the curvature of the 

 surface ; that is, according to the smallness of the radius of this 

 surface. 



(3.) The power applied must also be increased in proportion 

 to the diameter of the cord. Indeed it is manifest, that, other 

 things being the same, the cord will bend the less according as the 

 thickness is greater ; but we have just seen that the resistance to 

 the power is greater according as the curve A'O differs more 

 from the curve A' A ; it is therefore the greater according as A'O 

 differs less from a straight line, or the position of an inflexible 

 rod, that is, according as the cord has a greater diameter or 

 radius. 



254. Let us suppose that k is the addition to be made to a 

 power to render it sufficient to overcome the resistance arising- 

 from the stiffness of the cords, when the entire force by which 

 the cord is stretched is />, the diameter of the cord which bears 



