296 



DYNAMICS. 



PROP. IX. 



If tho body move uniformly in a circle, the centri- 

 petal force will l>e directly as the radius, and inversely 

 as the square of the periodic time. 



For in uniform motion, the velocity is proportional 

 to the space divided by the time. Hence, let c denote 







the circumference, &c. then v == , but c == r, there- 

 fore v = , and w 1 = . Substituting this value of 

 v* in last Prop, we get/ = . 



PROP. X. 



If the body move uniformly in a circle, the centri- 

 petal force will be directly as the product of the dis- 

 tance by the square of the angular velocity. 



For let a denote the angular velocity. It is evident 



= . Substituting this value of /' 

 ' o 8 



in last Prop, we gety == r a*. 



The four last propositions give us proportional equa- 

 tions, with regard to the revolution in a circle. 



The two next afford us absolute equations, which are 

 also of frequent use. 



that t = and 

 a 



PROP. XI. 



If the body revolve uniformly in a circle, the space 

 through which it would move by the action of the cen- 

 tripetal force alone in any unit of time, such as a se- 

 cond, will be equal to the square of the arch described 

 in the same unit, divided by the diameter, or twice the 

 radius. 



It appeared in Prop. 7- that the space through which 

 the body would move in an instant by the action of the 

 centripetal force, is equal to the square of the arch de- 

 scribed in an instant, divided by the deflective chord ; 

 which, in this case, is the diameter. But the space de- 

 scribed in consequence of an attraction acting always 

 with the same intensity, is proportional to the square 

 of the time; and the motion in the circle being uni- 

 form, the arch described is proportional to the time ; 

 and, therefore, the square of the arch proportional to 

 the square ef time. Hence, the effect of the attractive 

 force in any time is proportional to the square of the 

 arch described in the same time ; and, therefore, since 

 the proposition holds in the case of an instant, it must 



hold in the case of any other unit ; or f = , the 



unit of time being the same for f and v, but of any 

 magnitude. 



COR. 1 . Of these three, viz. f, v, r, any two being 

 given, the third may be found. 



COR. 2. The projectile velocity is equal to that which 

 a body would acquire in falling through half the ra- 

 dius by the uniform action of the centripetal force-. 



For by the doctrine of accelerated motion, the ve- 



locity so acquired would be = 2 J-' = ,J ^ r , 

 = */'2 rf, and by the formula in this proposition, v = 



prri-cdiiig corollaries, will apply to a circle described 

 with a variable motion, if you substitute half the de- > 

 fleet ive chord instead of radius, and the space that would 

 be descril>ed in the same unit with an unaltered velo- 

 city, instead of the arch actually described. 



PROP. XII. 



If the body revolve uniformly in a circle, the space 

 through which the body would move by the action of 

 the centripetal force alone, in any unit of time, will be 

 equal to twice the radius, multiplied by the square of 

 the circumference of the circle whose diameter is unity, 

 and divided by the square of the periodic time, orf= 



j , where m denotes the circumference of a circle- 

 whose diameter is unity, and therefore = 3.1416 near- 



J y- 



For the circumference of the given circle will be 

 2 r m ; and the motion being uniform, v = , and 



v = t^l ; but by last Prop. /= |A therefore /= 



1^1 = 19-739 X -^ nearly. 



COR. 1. Of these three, \\7..f, r, t, any two being 

 given, the third may be found. 



COR. 2. The periodic time is to the time of falling 

 along half the radius by the uniform action of the cen- 

 tripetal force, as the circumference of a circle to the 

 radius. 



For, by the doctrine of accelerated motion, the time 



of falling along half the radius will be = ,J > and 

 by the formula of this Prop. / = m X J'~ m X 



COR, 3. It is evident tliat this proposition, and die 



_ = 2 m x / Hence the latter to the former 



2/ V 2/ 



as 2 m : I : : m : -J, that is, as the circumference of a 

 circle to radius. 



PROP. XI 1 1. 



The centripetal force necessary to make a body de- 

 scribe a circle with the same angular velocity as it has 

 in the different points of its orbit, of whatever form, 

 is inversely as the cube of the distance from the centre 

 of attraction. 



For in circular motion f = r a 1 (Prop. 10.) and in 



any curve a == (Prop. 4. ) and cfl == , where r de- 

 notes the radius vector. Hence, /"== =?=^y 



PROP. XIV. 



If one body describe a curve (BE) around a centre p,. AT , : 

 of attraction (c), and another descend toward that cen- ccjil.r. 

 tre in a str-ii^ht line (AC) by the mere action of the Fig- 1?. 

 centripetal force, which is supposed to be the same at 

 equal distances ; and if in any two points (P., A) equi- 

 distant from the centre, the bodies have equal veloci- 

 ties, they will also have equal velocities in any other 

 two points equidistant from it. 



Take the indefinitely small arch BE. From C, as a 

 centre, with the radii CB, CE, describe the nrclu-s BA, 

 El), the velocities at E and D shall be equal, for draw 

 FG4-BE. 



