RESISTANCE. 



All which rules may be of ufe in the conftruftion of (hips, 

 and in perfecting the art of navigation univerfally ; as alfo 

 for determining the figures of the balls of pendulums for 

 clocks, &c. 



Resistance of Fluid Mediums to the Motion of Falling 



Bodies. A body freely defcending in a fluid is accelerated 



by the refpeftive gravity of the body, which continually afts 

 upon it, yet not equably, as in a vacuum : the rcfiftance of 

 the fluid occafions a retardation, that is, a diminution of 

 acceleration, which diminution increafes with the velocity of 

 the body. Now there is a certain velocity, which is the 

 greateft a body can acquire by falling ; for if its velocity be 

 fuch, that the refinance anting from it becomes equal to 

 the refpeftive weight of the body, its motion can be no 

 longer accelerated ; for the motion here continually gene- 

 rated by the refpeftive gravity, will be deftroyed by the re- 

 fiftance, and the body forced to go on equably. A body 

 continually comes nearer and nearer to this greateft celerity, 

 but can never attain to it. 



When the denfities of a fluid body are given, the 

 refpeftive weight of the body may be known ; and by know- 

 ing the diameter of the body, it may be found from what 

 height a body falling in vacuo can acquire fuch a velocity as 

 that the refiftance in a fluid (hall be equal to that refpeftive 

 weight, which will be that greateft velocity above-men- 

 tioned. If the body be a fphere, it is known, that a fphere 

 is equal to a cylinder of the fame diameter, whofe height is 

 two-third parts of that diameter ; which height is to be in- 

 creafed in the ratio in which the refpeftive weight of the 

 body exceeds the weight of the fluid, in order to have the 

 height of a cylinder of the fluid, whofe weight is equal to 

 the refpeftive weight of the body ; but if you double this 

 height, you will have a height from which a body falling in 

 vacuo acquires fuch a velocity as generates a refiilance equal 

 to this refpeftive weight, and which therefore is the greateft 

 velocity which a body can acquire, by falling in a fluid, 

 from an infinite height. Lead is eleven times heavier than 

 water ; wherefore its refpeftive weight is to the weight of 

 water, as ten to one ; therefore a leaden ball, as appears 

 from what has been faid, cannot acquire a greater velocity, 

 in falling in water, than it would acquire in falling in vacuo, 

 from a height of 13} of its diameters. 



A body lighter than a fluid, and afcending in it by the 

 aftion of the fluid, is moved exaftly by the fame laws as a 

 heavier body falling in the fluid. Wherever the body is 

 placed, it is fuftaincd by the fluid, and carried up with a 

 force equal to the difference between the weight of a quan- 

 tity of the fluid of the fame bulk as the body, and the body 

 itfelf ; by which not only the aftion of gravity of the body- 

 is deftroyed ; but the body is alfo carried upwards by a mo- 

 tion equably accelerated, in the fame manner as a body heavier 

 than a fluid defcends by its refpeftive gravity : but the equa- 

 bility cf the acceleration is deftroyed in the fame manner by 

 the rcfiftance, in the afcent of a body lighter than the fluid, 

 as it is deftroyed in the defcent of a body heavier. 



When a body fpecifically heavier than a fluid is thrown 

 in it, it i» retarded upon a double account ; on account of 

 the gravity of the body, and 011 account of the refiftance of 

 the fluid ; confequently, a body rifes to a lefs height than 

 it would rife to in vacuo with the fame celerity. But the 

 defefts of the height in a fluid from the height to which a 

 body would rife in vacuo with the fame celerity, have a 

 greater proportion to each other than the heights them- 

 felves ; and in lefs heights the defefts are nearly as the 

 fquares of the heights in vacuo. 



In order to fubmit the above principles to accurate com- 

 putation, we muft refer back to cur preceding determination 



of the retardative force of a fluid to a body moving it, which 

 we found to be 



f 



•x nv~ d' 



1 6 



giu 



6 



_ n7' 



from the two latter terms of which we have s = — ' x 4 d; 



which is the fpace that would be defcribed by the globe, 

 while its whole motion is generated or deftroyed by a con- 

 ftant force, which is equal to the forces of refiftance, if no 

 other force afted on the globe to continue its motion. 

 And if the denfity of the fluid were equal to that of the 

 globe, the refiiting force is fuch as, afting conttantly on 

 the globe without any other force, would generate or deilroy 

 its motion in defcribing the fpace \d, or 4 of its diameter, 

 by that accelerating or retarding force. 



Hence the greateft velocity that a ball will acquire by 

 defcending in a fluid by means of its relative weight in that 

 fluid, will be found by making the refilling force equal to 

 that weight. For, after the velocity has arrived at fuch a 

 degree, that the refilling force is equal to the weight that 

 urges it, it will increafe no longer, and the globe will then 

 continue to defcend with an uniform velocity. 



Now N and n being the feparate fpecific gravities of the 

 globe and fluid, N — n will be the relative gravity of the 

 globe in the fluid ; and, therefore, *u — £ w d (N — n) 



nv'd' . , 



— is the 



16^ 



</• (N - n) 



is the weight by which it is urged, m 



refinance ; confequently ■ — ; ■ = 



when the velocity becomes uniform ; whence we obtain 



® = V ( 2 8 x 3 d x — — ~\ 



for the uniform or greateft velocity of the globe. 



Thus, for example, if a leaden ball one inch in diameter 

 defcend in water, and in air of the fame denfity as at the 

 earth's furface ; the three fpecific gravities being, lead 



ill, water 



and air = — - i 



2JOO 



4 



then 



v = */ (a x 32J- x -— x ic4) = 8.5944 f eet 



per feccmd for the greateft velocity in water ; and 



(193 4 34 2Coo\ „ , 



4 X 17 x 36 x 7 * -J") = 2 & 82 fe « 



per feeond for the greateft velocity in air. 



But as this velocity, all other things being the fame, 

 varies as v / d; it follows that a ball of -nj^th of an inch dia- 

 meter would only acquire velocities -,-^th of thofe given above. 

 Hence it appears, how foon fmall bodies come to their 

 greateft or uniform velocity in defcending in fluids, and 

 how very fmall that velocity is ; which explains the reafon 

 of the flow precipitation of mud and fmall particles in 

 water, as alfo why, in precipitations, the larger and grofs 

 particles defcend fooneft and loweft. 



It appears alfo, from the preceding formula, that where 

 N — h, or the denfity of the body is equal to that of the 

 fluid ; then N — n = o, and confequently the velocity and 

 fpace are in this cafe both equal to zero, as they ought 

 to be. 



Again, when the body is lighter than the fluid, then 



N — k becomes negative, and the motion and force both 



tend the contrary way ; that is, the ball will afcend by the 



6 fam? 



