14 



MECHANICS. 



ferred, that the friction is an uniformly 

 retarding force, on exactly the same 

 principles as have already been fully 

 developed in (9) and (10). 



The string used in these experiments, 

 like those described in the last Chapter, 

 should be so flexible as that its stiffness 

 shall produce no sensible effect on the 

 results. 



(20.) By a series of experiments con- 

 ducted as we have described, Coulomb 

 found that, like the other modifications 

 of friction, the law of the proportionality 

 of the friction to the pressure obtained, 

 also in this case, subject however to the 

 exception before mentioned, that in 

 very great pressures the friction is 

 somewhat less in proportion. 



He also found, that, as in the friction 

 of sliding, great advantage was gained 

 by greasing the surfaces. In general, 

 fresh tallow diminishes the friction by 

 one-half. It increases as the grease is 

 wasted away. This effect is, however, 

 more slow than in the friction of sliding. 



This species of friction is also an 

 uniformly retarding force, and is there- 

 fore independent of the velocity. 



Like the other species of friction, the 

 quantity of this depends on the quality 

 of the surfaces. If iron revolve in con- 

 tact with brass, the friction is one- 

 seventh of the pressure. When both 

 surfaces are wood, the friction is one- 

 twelfth of the pressure. 



In general, the same observations 

 which were made respecting the friction 

 of sliding, will also apply to the species 

 of friction which we have considered in 

 this Chapter. 



(21.) The friction of bodies turning on 

 pivots seems to come within the species 

 we are now considering. . This was also 

 examined by Coulomb, and a memoir 

 on the subject was published by him in 

 the Memoirs of the French Academy in 

 1790. A very succinct and clear ac- 

 count of this is given by Dr. Gregory in 

 the second volume of his Mechanics, 

 from which we extract the following 

 particulars : 



" Bodies which are made to turn upon 

 pivots are usually suspended by means 

 of a cheek, socket, or collar, of very 

 hard matter. The collar has its cavity 

 of a conic form, and terminated at its 

 summit by a little concave segment, 

 whose radius of curvature is very small. 

 The point of the pivot which is sustain- 

 ed by this collar forms at its summit a 

 little convex surface, whose radius of 

 curvature should be still smaller than 



that of the extremity of the cheek. Ex- 

 perience evinces that the curvature of 

 the bottom of the socket is irregular, 

 and that the friction of a collar of agate 

 on which a pivot turns, is frequently 

 five or six times more considerable than 

 the momentum of friction of a well- 

 polished plane of agate on which the 

 same pivot turns. 



" These considerations induced M. 

 Coulomb to employ in the course of his 

 experiments, not a cheek or a socket, 

 but a well-polished plane, to support 

 the body on the point of a pivot. To 

 prevent the body from sliding, he took 

 care that its centre of gravity should be 

 very low, with respect to the point of 

 suspension : he then made the body to 

 whirl or spin about its pivot, by im- 

 pressing upon it a rotatory motion. By 

 means of a seconds watch, he observed 

 exactly the time employed by the body 

 in making the first four or five turns, 

 and he thence deduced easily a mean 

 turn to determine the primitive velocity : 

 after this he counted the number of 

 turns which the body made before it 

 stopped. 



" Coulomb took a glass bell of 48 lines 

 in diameter and 60 lines in height, which 

 weighed 5 ounces. He placed it on the 

 point of a pivot ; and after giving it 

 successive degrees of velocity about that 

 pivot, he observed very exactly the time 

 that it employed to make the first turn, 

 which gave him for the mean velocity 

 that which answered to the half of such 

 first turn. He then estimated the num- 

 ber of turns made by the bell before it 

 stopped : the results were as below 



" 1st Trial. The bell made one turn in 

 4", and came to rest after 34 T \j turns. 



" Id Trial. The bell made one turn in 

 6y, and stopped after 14^ turns. 



" 3d Trial. The bell made one turn in 

 11", and stopped after 4^ turns. 



[" Now if b denote the primitive 

 velocity, X the space described between 

 the commencement and the end of the 

 motion, A the constant momentum of the 



retarding force ;/ the sum of the pro- 

 ducts of every particle ^, by the square 

 of its distance r from the axis of rota- 

 tion, divided by the quantity a, measuring 

 the distance from the axis of rotation 

 to the point whose primitive velocity is 

 b, it is easy to find the following analy- 

 tical expression for the constant mo- 

 mentum of the vis retardatrix, viz. 



