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MK--II \MCAL IMIII.nS.ll'IIY. STATICS 



[MOMENT or romom 



A circular hoop u supported in a horizontal position, 



and three weight* of 4, 5 and 6 Iba, respectively are 



suspended over its circumference by three string* 



d together at the centre of the hoop. Neglect- 



ing the friction of the edge of the hoop, ii.,.1 t ho angles 



>een the string! when there is an equilibrium. 

 Three equal forces, each equivalent to 6 Ib>. t act on a 

 : the first two are inclined to each other n' 

 an angle of 7.~>, and tho third is inclined at an angli 

 to the first. Find the magnitude and direction 

 of the result., 



The resultant of two force* is 60 Ihe., and the angle* 

 which it make* with their direction* is 20 and 30' 

 find the component forces. 



A boat, fastened to a fixed point by a rope, i* acted on 

 at the same time by the wind and the current. Sup- 

 pose that the wind was 8.E., the direction of the 

 current S., and the direction of the boat from the 

 fixed point S. 20 W., and also that the pressure on 

 the point was 100 Ibs., it is required to find the forces 

 of the wind and the current. 



In pulling a weight along the ground by a rope, inclined 

 to the horizon, at an angle of 45, a power of 40 Ibs 

 wa* exerted ; required the force with which the body 

 was dragged horizontally. 



Four forces are in the same plane, which are to each 

 other as 6, 8, 10, and 12, act upon a given point, and 

 are inclined to a given line at the angles 20, 40, 80, 

 and 150 respectively i find the magnitude and direction 

 of a fifth force which shall balance the others. 



:i.ll;i:ll M OF A RIGID BODY. Having 

 considered the conditions of equilibrium of a material 

 particle acted on by any number of forces acting in the 

 same plane, we next proceed to consider the conditions 

 of equilibrium of a rigid body under the same circum- 

 stances. Here it may be as well to repeat what has 

 been stated before, that by the term r'xjul body, we un- 

 derstand a body composed of material particles held to- 

 gether by unknown molecular forces of such intensity 

 that the body cannot be altered in shape, or its particles 

 in anv way displaced by any forces which can act upon 

 it. This rijjid body also possesses the property, that if 

 any force be applied to it, its particles will transmit 

 that force, unimpaired in intensity, to any point in the 

 body, which lies in the line of direction in which the 

 force i* acting ; consequently, tho effect of a force on a 

 rigid body will be the same, if we transfer its point of appli- 

 from any one point in the rigid body to any other, 

 provided these two points are in the line in which the 

 force is supposed to act. 



Unless it is otherwise stated, this hypothetieal rigid body 

 is also considered as being destitute of weight. 



If two or more forces acting on a rigid body are ap- 

 lanio point of the body, the conditions of 

 equilibrium will be the same a* those for a material point 

 mi.!, r the same circu instances, for the same force w hi.-li. 

 when applied to the material point, would counteract 

 the effect of these forces, would also keep the body in 



Xliliriiim, when applied to the point in the body upon 

 h the forces act 



PROl'i -.ITION X. 



To And the magnitiulr. ami ilin-ftum of the randtant of two 

 itg on different point* of a ri-ii-l body, the 

 dine- s being in the tame plane, but not 



/.. . ial MM* 



Let a furce P (Fig. 88X represented in magnitude and 



direction by the line A P, act upon the point A of a 



rigid body, and another force Q, represented by H Q, 



ti the point B of the came body. B Q and A P 



(-.til I.. I!,;; !>,. -.HIH- j.Uli.- 



\ r. Produce 1'A and QB to meet in the 

 point C. 



Along C R take C Q'-BQ, and along C A, C P'- AP. 

 Draw Q' II' parallel to C 1", and V 11' parallel to C Q', 



III'. MSJg :; If. 



Join C R', and produce C R' to U, catting AB 

 in D. 



MakeDR-CR'. 



Then 1) R will re- 

 present the resul- 

 tant of the forces 

 P and Q in magni- 

 tude and ili: 



By the prinei|.hi 

 of the traiiM 

 of forces, the points 

 A H and Care wip- 

 posed to be ri-i.liy 

 connected with 

 each other. 



The point of ap- 

 plication of the 

 force P may be 

 transferred from 

 A to C, and the 



force Q from B to J/ 



C. 



Then the forces P and Q acting at C may be replaced 

 by the single force represented in magnitude and direc- 

 tion by C R', Prop. 1 . , and this force may be transferred 

 from C to D, and be represented in magnitude and 

 direction by D R. 



This construction will enable us to represent, graphi- 

 cally, the resultant of any two forces acting on a rigid 

 body in the same plane, but in directions not parallel 

 to one another. 



For the purpose of calculation, it will be convenient, 

 however, to determine tho geometrical relation of the 

 point D to the forces P and Q, and their directions. 



Through D draw DE (Fig. 38), perpendicular to CA, 

 and 1) F perpendicular to C B ; also through P' draw 

 P' G perpendicular to C D. 



Let n represent the angle A C D, and /3 the angle H C D. 



Then since by construction P' R' is parallel to C Q', 

 therefore (Euc., B. I., Prop. XXVII. ), angle CR'P' = 

 angle Q'CR'=/J. 



The angles at E, F, and G, are right angles by con- 

 struction. 



Hence the figure will afford us two pairs of similar 

 triangles. (See Fig. 39). 



Since in the triangles C D E, C G P' the anles D C E 

 and P'CG are both = a, and those at E and <!' aro right 

 angles (as will probably be more readily perceived in 

 ;he above figures than in tho more complicated figure), 

 it follows that tho triangles C D E and C 1" G are 

 equiangular triangles and similar to one another, and 

 therefore (Euc., B. VI., Prop. IV.), 



C D : D F. : : C P' : G P* 



CD_C I 1 

 or 'DE~<; l" 



Similarly, since die angles F C D and G R' P' are both 

 equal to ft, and the angles at F and G right angles, tho 

 .rianglos C F D and 11' G I" are similar triangles, and 



D F : C D : : G P' : P 7 R' 



D F G I" 

 or> C D = P'R' 



Multiplying equals together, we have 

 CD DF C_F <: 1- 



in: CD"" OP*' 



DF OP' 



and therefore 



