ROTATION. 



pended ; then, having given the radius S D^ the power P, formula , for acce lerated motion, by making /= tL l L 



and the weight of the wheel, or cvhnder, u £. t , it is re- ' w r >- , . 



and ttie weight 01 me wneei, or cyimuer, xj i- x , >«■_« • »■- to r~ 4- p d'~ 



quired to determine the angular velocity generated in the the accelerating force> as determined in the precedln „ part 



fyftem m a given time. of this propofition. 



Let Rreprefent the centre of gyrat.on, and make bK-^, In ^ aW ; nvefti tion we hav£ confldL . rcd the „. 



SD = </, the weight of the cylinder E Dl = w, and the ^ m t(> bc g ^.^ % ^ ^ fe 



given power = / ; then the force which accelerates the is dl)v f 0US) the f amewi ll have place for any /yftem of bodies, 



r • , c • u„f ;f provided only that the axis of motion paftes through its 



uDDofe 6 void or inertia ; but it r f , , „ , , f <? . 



"rr' 1 r u '" centre of gravity, and that R be the centre of gyration of 



the inertia of p be confidered, the force of acceleration is the fyftem 



C , the force of gravity being affumed unity ; but 



iur 1 + p li- 

 lt the force of gravity be taken i6 T V = 193 inches = / ; 

 and 3. 1 41 59 

 in the point 



„ . pd- . 

 point D is - — , it we ii 



Prop. II. 



: ot gravity be taKen 10^= 193. lnc[l1 - 5 - ' ' t ] ie weight a applied to the cir 

 q = c, then, m the time / the velocity generated & J^ circumfcrence Q 



D,or defending weight P, will be reprefented by r ig ^ ^ tQ affi ^ f 



2/tpd' 



inches in a fecond. 



p d ' + zu r- 



And fince the circumference of the circle EFD= led, 

 the angular velocity generated in the time /, will be 



360° x 2/tpd'- 360 1 tpd 



zed (pd z + iur') c(pd- + nvr) 



degrees, 



Let ABC {jig. 5.) reprefent a wheel and axle, and 

 let the axis be horizontal, having given its weight iv, and 



circumference of the axle, and 

 f the wheel, in order to raiie 

 equired to affign the fpace defcribed by the elevated 

 weight q from reft in any given time ; the proportion of 

 the radii of the wheel and axle being alfo known. 



Here the abfolute force which impels D is p, and fince 



q aft s in a direftion contrary to p, with a force = ~Y\ ' 



S A 



Itpd 



this muft be fubufted from p, which gives p — 



revolutions in a fecond. 



SD 



cpd z + civr' 

 1. On the fame principles it will be found, that the angular 



a.SD - ? SA , , .'- , ,. , . , 



- „ ... tor the motive force which impels 



velocity generated" in the fyftem during the defcent of the D Let the centre of gyration of the whed and axle be 

 weight p, through any fpace S, is R . then r uppo f e the mafs of matter in the whole fyf- 



. vel. = >/(— -, '•?-' i J x 3 6o° degrees, or «. . S R 1 4- q . S A"- + * . S D* 



\' \Sd'p +- r r x iv/ ° tern removed, if the mafs 



ang 

 ang 



SD' 



vel. = 4 / ( — ) revol. in a fecond. be concentrated in D, the point D will be accelerated in 



V \c z d*p + c* r iv / the fame manner, as when the parts of the fyftem are dif- 



The fpace defcribed by the weight p in its defcent pofed as defcribed in the problem. Since, then, the force 



from reft during- / feconds, is ; = -r - 



° d- p -+- r iv 



confequently, the time of defcribing s is, 



— ; — ' and which impels D 



SD 



SA 



SD 



it follows that 



= v/ 



<d*p + 



Ipd' 



feconds. 



3. The fpace defcribed by/ from reft, while an angular 

 velocity of n revolutions in a fecond is generated, is 



n* c 1 d' p -f- n c r' 10 

 < = JJ- * 



4. The force which accelerates the centre of gyration R, 



pdr 

 J d'p + r- tv 



5. The abfolute velocity generated in the weight p, 

 while it defcends from reft through the fpace S, is 



/ *i*p<{' 



V= V d : p--'r r av 



v = * / — p— , and the velocity generated in the 



\/ d~ p 4- r 7 iv 



point R, in the time /, is 



2 ltd pr 



v = — j-t- *— — . 



p + f iv 



All thefe refults are drawn immediately from the known 



this force, dirid<;d by the whole inertia, or the mafs fup- 

 pofed to be concentrated in D, will give for the accele- 



. , ^ p.SD' -q.SA. ST> 



rative force on D = — c - D - T ^^ -— -r-, and 



iv . S R- -r p . S D- -t- q . S A' 



, . , /.SD.SA-9SA 2 



the accelerative force on q — — ^ — ■ ptnJ— - o « . 



2 iv. S R -+/>. S D' 4- q.S A'- 



Or, if we make S D = », S A = m, S R = r, then the 



. p n — a m n 



accelerating force on D is — = — ■ r. 



■iv r "4- p n + q nr 



If the inertia of the wheel and axle is not confidered, 



p n~ — q v.i n 



the above becomes 



, and the velocity of the point R, is 



p «" 



And if the inertia 



of p alfo = o, then we have 



p n" — qmn 

 q m' 



; or, if the mats 



moved have no weight, but poffefles inertia only, as when it 

 is drawn along a perfeftly pohfhed plane, as reprefented alfo 



\nfg. 5, then the accelerative force is fimply 



pn- 



p n ' -{■ q nr 



Let this accelerative force in any of the cafes we have 



fuppofed 



