JO t 



DYNAMICS. 



of the second (by supposing *= r) will be o x an 



. . 

 arch whose cosine is = = -- . 



( " * \ 1 

 cos.= - J rand this case requires no correction, 



because * must be =0 when /=0. 



a 

 COR. 1. When s=a, then v is infinite, and t^\/ 



X where c is the circumference of a circle whose 



diameter is unity ; hence the whole time of descent to 

 the centre of attraction by a body setting out at differ- 

 ent distances is == a t/ a = % tliat is proportion;d to 

 the square root of the cube of the distance. 



If the force varies as some given power n of the 

 distance from a given point tiirther back than that 

 from which the body sets out, the equation that ex- 

 presses the relation of v and x will be the same as be- 

 fore, only having a-\-s instead of a ,r. 



COR. 2. Hence, if the body is acted on by several 

 pressures in the same or opposite directions, the effect 

 may be found by calculating the separate effects, and 

 then taking the sum or difference. 



PROBLEM II. Suppose that the relation betwixt / and 

 is given ; suppose, for instance, that t = a *| a , and 

 consequently =r my.a jj", m being a known quanti- 

 ty, required the law of the force expressed according 

 to 4? 



Since/ =( 3 .) 



if we regard s as constant, we get/ = 



t* 



Now f = in x (a *) n f = nm X (a s) n -' 

 hence /' = n 3 m 3 x (a 



X s-= 



hence f = 



X ( "~ ^ X ("*)" 



( 



Here/ ==(-*) 



The great object in this case of dynamics is, given 

 the relation of the force to *, v, or /, to find the rela- 

 tion of these to one another, or, given tlie relations of two 

 of these to find the relation of the force to them. The 

 relation most easy to observe, is that of the force to s, 

 or of j to t. 



CASE 6. A body acted on by impulse and pressure at 

 the tame time. It is necessary to bear in mind, that the 

 impulse ceases to act almost immediately, but that the 

 pressure continues to operate. It is evident that the 

 effect will be the same, as if the body had acquired a 

 finite velocity in any other manner, and then begun to 

 be acted on by the pressure. 



Variety 1. When the impulse and prcssHre both act 

 in the same direction. It is clear that the motion will 



lie in thnt direction, and that the effect in any given Application. 

 time will IH; found, by calculating separately the ef- < "~""Y"^ 

 ferts of the impulse anil pressure, and adding them to- 

 gether. 



Variety -. If'/ifn the tiro forces act in opposite direc- 

 It is evident, that the motion will be continual- 

 \\ ivtardcd, and at last de.stroycd. To find the rela- 

 tion-, of s, v, and /, you have only to calculate the ef- 

 fect of the initial velocity, supposing it to remain the 

 .same, and then subtract the effect of the pressure. 'I'd 

 find the time in which the whole initial velocity will be 

 destroyed, or the body brought to rest, we have only 

 to ealeulate in what time the pressure would be able to 

 generate that velocity. It is evident, that if the pres- 

 sure acts always with the same intensity, the motion 

 will be uniformly retarded, or the body will lose equal 

 portions of velocity in equal times; also, the whole 

 time will be proportional to the initial velocity, and tin- 

 whole space proportional to the square of the whole 

 time, or of the initial velocity. Thus, it is evident, 

 this variety of Case 6'tli will come within the range of 

 former problems. 



Variety S. When the impulse and pressure act at an 

 angle. This is by far the most important and most dif- 

 ficult variety, and is usually discussed under the title 

 of Central FORCES. 



If a body, already proceeding with a finite velocity, 

 in consequence of an impulse, or some other cause, is 

 acted on at finite intervals of time by new impulses, 

 each of which differs in its direction from that of the 

 previous motion, its path will evidently be a succession 

 of straight lines making angles with one another. If, 

 instead of being an impulse, the force be pressure, and 

 act incessantly, the straight lines will l>ecome infinitely 

 short, and the path consequently become curvilineal. 

 The curve will lie wholly in one plane, if the direction 

 of the pressure be always in the same plane with the 

 original direction of the motion, and it will be concave 

 on that side toward which the pressure is directed. 

 Again, since a body, if left to itself, moves in a straight 

 line, we may conclude, when it moves in a curve, with- 

 out being compelled to it by a fixed obstacle, that there 

 is a force of pressure constantly deflecting it from the 

 direction of the tangent. 



The arch, through which the body moves in an in- 

 stant, or through which it would move in any unit of 

 time in the direction of the tangent, if the pressure 

 were to cease, is the projectile velocity at that point 

 of the curve. 



In the cases that most commonly occur, the pressure 

 is an attraction ; it is commonly directed to the same 

 point during a considerable part of the motion ; that 

 point is called the centre of attraction, and the force is 

 called a ce ntripetal force. 



The force with which the body, in consequence of 

 its projectile motion, endeavours to recede from the 

 centre of attraction, is called a centrifugal force. These 

 two forces have the general name of central forces. 



A straight line, drawn from any point o'f the curve 

 to the centre of attraction, is called the radius vector. 



The angular velocity at any point of the curve, is the 

 velocity with which the radius vector at that point de- 

 scribee an angle, that is, the angle which it passes over 

 in an instant, or that which it would pass over in any 

 unit of time, were its angular motion to remain uni- 

 form. 



When the line which the Ixxly describes returns into 

 itself, like a circle or an oval, it is called an orbit, and 



