ROTATION. 



= i'y=gfr = 



the accelerative 



fuppofed be put = f; then the fpace, velocity, time, &c. 

 will be found immediately from the general formula 



v' 



Suppofe, for example, S D = n = 6, S A = m = 2, 

 then SR = r = 6 <J \. (See Center of Gyration.) Let 

 alfo 10 = 10, ^ = 100, q = 40. Then, by our lirlt formula, 



n'p — nmq 36 . 100 — 6 . 2 . 40 



Bf' + pn' + q m' 10 . 18 4- IOO . 36 4- 40 . 4 



3600 — 480 3120 156 



180 4- 3600 4- 160 " 3940 197' 



force on D, that of gravity being 1. Whence the fpace 

 paflcd through by the defcending weight p in i", is s = 



156. i6 T ' r . 103. ic6 . . 

 WV = -^ — feet, or -^ i- inches. 



SJ 197 '97 



Whenever motion is communicated to any body, a certain 

 refiltance mult have been overcome by the moving force : 

 this refiftance is of various kinds. 1. The inertia of the 

 mafs moved, whereby it endeavours to perfevere in its ltate 

 of quiefcence, or of uniform motion in a right line. 2. That 

 of a weight, or other abfolute force, oppofed to the action 

 of the moving power. 3. Obftacles upon which the moving 

 body impinging is retarded in its progrefs ; fuch, for ex- 

 ample, is the refiltance which arifes from the particles of a 

 fluid through which a body moves. The ellimation of thefe 

 refiftanccs, and their effects in retarding the motion of bodies 

 acted on by a given force, are deducible from the laws of 

 motion, and conititute a part of the folution of almolt every 

 problem relating to the motion of bodies. 



The moving Forces alfo are of various kinds ; vix. gravity, 

 mufcular power, the impaft of bodies folid or fluid, &c. ; 

 and the effect of thefe moving forces, which are exerted on 

 bodies in order to create motion, exclufivc of the refinance 

 oppofed to them, depend on the various circumnances of the 

 time in which they aft, and on the fpaces through which the 

 bodies moved are impelled, &c. Whence it follows, that, 

 from the great variety of undetermined conditions which may 

 enter into mechanical problems, there mult be of courfe va- 

 rious ways of producing the fame mechanical effect ; and it is a 

 very material part of the art, confidered either in a theoretical 

 or practical point of view, to proportion the means to the 

 end, and to eflcft this with all the advantages that the nature 

 of the cafe is capable of. It is the due oblervation of thefe 

 particulars, which contribute to render the mechanic inftru- 

 ments complete, and the neglect of them defective, in their 

 confiruftion. This proper choice of means to produce me- 

 chanical effect, is frequently the refult of long and con- 

 tinued experience, independent of theory ; the know- 

 ledge of which, however, when immediately applied to 

 practice, would fave the artift much time and trouble ; and, 

 at the lame time, be productive of other advantages which 

 experience alone mult be deltitute of. 



Phot. III. 



In order to illufirate this application of theory to prac- 

 tice, let A B C ( fig. 5.) be a wheel and axle moveable 

 round an horizonal axis, pafliug through S, and a given 

 weight q, which applied to the dreamier "nee of the axle, to 

 be raifed by the application of a given moving force, / . 

 which is applied to the circumference of the wheel : let it 

 be required to afiign the proportion of the radii of the 

 wheel and axle, fo that the time in which the weight iv 

 afcends through any given fpace (hall be the leall pufiible. 



Let the given radius of the axle S A = m, S R = r, 

 S D = * j then, by the lalt propofition, the accelerative 



force on p is 

 rates q is 



q m x 



iv r + p x -\- q m 

 m x p — m q 



and that which accele- 



, the fquare of the time, 

 iv r • — p x -r q m : 



therefore, in which any fpace S is defcribed by the amend- 

 ing weight q, is 



+ p -V* + qi 



f = 



I 



mp x 



(/ being =193 in. J 



and which, by the quenion, is to be a minimum. And 

 which, therefore, put into fluxions, and reduced, gives 



m q -t- ./ (hi 1 q' 4- p iv r' 4- q p m') 



the radius required. 



Or, if we do not confider the inertia of the wheel, then 



m q 4- v / (m~ q* + qpnf) 



<h 



then m = 



m + m y/ 2 = m (1 4- */ 2), the radius of the wheel 

 in this particular cafe. 



To give an example in numbers, let q = 1 00, ^ = 33, 

 iv = 20, r = A / 50, or r- = 50, m = 1, then the diftance 



fought is - 



100 4- A / (10000 4- 33000 4- 3300) 



33 



33 



= 9.55 inches. 



This, therefore, will be the molt convenient dirtance to 

 apply the given moving force, when the chief object is to 

 leffen the time of afcent. If it be required to alfign the 

 diftance S D, when the momentum communicated to iv, 

 while it afcends through a given fpace, is the greateft pof- 

 fible, the folution will be the fame as before, which, therefore, 

 anfwers to two conditions ; that is, it will render the time 

 in which q afcends through a given fpace the lealt, and the 

 momentum generated during the fame afcent the greateft 

 pollible. 



If the weight q, inftcad of afcending in a vertical direction, 

 is drawn along an horizontal plane, as in Jig. 5, the 

 furface of which is fuppofed perfectly polifhed ; then the 

 weight q, as oppofed to the moving force = o, poflefling 

 in this cafe only inertia, we lhall have for the radius of the 

 wheel 



'«"'' 4- p qm') 



• (A< 



= J 



iv r + q /»* 



which, if the inertia of iu be rejected, becomes firnply 



x = / - . 

 J P 



Hence, if the quantity of matter to be drawn along the 

 plane is four times greater. than that which 1 bed in 



the moving foi -..dins of the axle S A being given, 



in order that it nu> I lied with the greatell velocity 



poflible, and with th)B greatell momentum, the radius of the 

 wheel mould be double that oi the axl< , when tin inertia of 

 the wheel is not confidered. 



Pbop. IV. 



further applii OUT theory, let ABC, 



(Jit. c.) be a wheel and axle; when, having given a 

 moving force or weight />, acting on the circumference ol 



the 



