DYNAMICS. 



291 



Resolution 

 of Forces. 



SECTION IV. 

 Resolution of Forces. 



THIS consists in resolving a single force into others 

 to which it is equivalent, and which are to have given 

 directions. 



1 . If the given directions are all in the same plane 

 with that of the given force, we can always perform the 

 resolution into any number of forces, by inverting the 

 process of Case 1 . last Section. 



Thus, if two, we will make a parallelogram, of which 

 the diagonal represents the given force, and the sides 

 are parallel to the given directions. If three, we may 

 first resolve a part of the given force into two of the 

 given directions, and then the remainder into one of 

 these directions, and the third direction. 



2. When the given directions are not in the same 

 plane with that of the given force, we cannot resolve it 

 into two, but we can always resolve it into three given 

 directions, provided two of them are in one plane. We 

 can first resolve it into two forces, one of which is pa- 

 rallel to one of the given directions, and the other pa- 

 rallel to the plane of the other two ; and then resolve 

 this other into the other two given directions. 



3. One important application of the resolution offer- 

 ees, is to find the effect of a force in a given direction, 

 or in other words, to reduce it to a given direction. 



It is evident that, in order to do this, we have only 

 to resolve the force into two, one of which is parallel to 

 the given direction, and the other at right angles to it 

 The latter can have no effect in the given direction, and 

 therefore the other will express the whole effect. 



4>. Another important application of this subject is to 

 the composition of forces. 



The last cose of composition cannot be managed un- 

 '-> I >y resolution. Y r ou may fix on three directions, 

 i\v<> of them in one plane; resolve each force into three, 

 parallel to these directions, find the equivalent of all 

 those parallel to one line, then of those parallel to ano- 

 ther, &c. then compound the three equivalents, if they 

 pass through the same point ; if not, the problem is im- 

 possible, that is, there is no single force equivalent to 

 them all. 



The most convenient directions to fix on will be three 

 straight lines passing through the same point at right 

 angles to one another, and the most convenient way of 

 ri-i living, will be to calculate on the principle mention- 

 ed in Case 2d of last Section, that the three adjacent 

 edges of a parallelepiped compose a force equal to the 

 diagonal. Hence, since in this case the parallelepiped 

 is rectangular, let J denote any force. 



a the angle it makes with a parallel 

 to one of the assumed lines. 



Then R : cos. a : :/ :l2!L? = the force estimated 



in that direction. 



Let It, c, (I, denote the three equivalents found; then, 

 if they pass through the same point, they will be three 

 edges of a rectangular parallelepiped, of which the dia- 

 gonal is the equivalent sought, which call m; then 

 m=\^a' + 6*4-c s . The same method may be applied 

 with advantage to Case 2d of composition, when the 

 forces are not all in one plane ; ami even when the for- 

 ces are all in one plane, a similar method will some- 

 times be convenient, only in that case it will be- suffi- 

 cient to resolve each force in the direction of two per- 



pendicular lines. It is evident that in Case 2d the Application, 

 composition can always be effected, because the three ^ "Y"^^ 

 equivalents will always pass through the point to which 

 all the given forces are supposed to be applied. 



SECTION V. 



Application* of the General Principles of Dynamics to 

 some Cases in which the bulk of the Body is over- 

 looked, or rather in which the Body is regarded as 

 having its whole Matter collected in a Point. 



CASE 1. A body acted on by an impulsive force. By 

 the first law, the motion will be uniform and rectili- 

 neal. 



Hence, if v = the velocity, or space through which 

 the force will make the body go in a certain unit of 

 time, / = the whole time or number of units during 

 which the body moves, s = the whole space or line pas- 

 sed over, it is evident that 



*=/. 



This is the fundamental equation, which involves all 

 the rest. 



Thus -=r. 

 v 



Hence also s=t v, ==t when v the same, and =v when t 

 the same. 



t= , ==s when v the same, and == when 



v ' v 



* the same. 

 r== , =* when t the same, and = when 



the same. 



CASE 2. A body acted on by Into impulsive forces. 

 If they act in the same direction, the velocity will be 

 equal to the sum of the separate velocities. 



If in opposite directions, it will be equal to their dif- 

 ference. 



If at an angle, the velocity and direction will be 

 found by the rule given for finding the equivalent of 

 two forces, in last Section ; that is, by finding the dia- 

 gonal of a parallelogram, of which the sides represent 

 the separate velocities. 



CASE 3. A body acted on by an equal impulsive force 

 at equal intervals of time in the same direction. Sup- 

 pose the force is such, that during the first interval of 

 time the body will pass over the space a. 



With this velocity it would go on uniformly, but at 

 the end of the interval it receives an equal impulse, 

 which, according to the 2d law, will just add as much 

 to its velocity, and so on : hence during the second in- 

 terval, it will pass over the space 2 a ; during the third, 

 3 a, &c. ; so that if t denote the whole time, or whole 

 number of intervals, it will, during the /"' interval, pass 

 over the space t a. 



Hence it is evident, that the velocity at any period 

 of the motion will be in proportion to the number of 

 intervals, or to the time that has elapsed. 



The whole space will be := the sum of the arithmeti- 



ft-l-ta ta + fa 

 cal progression a, 2o, 3fl . . . . /a= - X t -^ ^ 



t+f 

 X 2 



