Example 11: The man shown in Figure (a) pulls on the cord with a force of 70

Transcription

1 Chapte Tw ce System 35.4 α α 100 Rx cs R β β 100 Ry cs R 111 Example 11: The man shwn in igue (a) pulls n the cd with a fce f 70 lb. Repesent this fce actin n the suppt A as Catesian cmpnents and detemine its diectin. Slutin: ind the Catesian pjectins f the : x x ft 1 y y ft 1 z 6 30 z 1 4 ft (1) ( 8) ( 4) 8 ft and the Catesian cmpnents equal: x y z 1 * lb 8 *70 0 lb 8 4 *70 60 lb 8 The cdinate diectin angles ae measued between ( ) and the psitive axes f a lcalized cdinate system with igin placed at A, igue (b). 1 cs 1 α cs 1 β cs 1 γ Univesity f Qadisiyah\aculty f Eng.\Civil Dep 7

2 Chapte Tw ce System Example 1: The fce in igue (a) acts n the hk. Expess it as Catesian cmpnents. Slutin: As Shwn in igue (b), the cdinate f pints A and B ae and A( m, 0, m) 4 B 5sin m, 5cs30 5 B[- m, m, 3 m ] 3 m, 5m 5 ind the Catesian pjectins f the : x x 1 y y z 3 z 1 1m 4 m 3.464m ( 4) S, the Catesian cmpnents f : m x 4 * N y * N z * N Univesity f Qadisiyah\aculty f Eng.\Civil Dep 8

3 Chapte Tw ce System Example 13: Tw fces act n the hk shwn in igue (a). Specify the magnitude f and its cdinate diectin angles f that the esultant fce R acts alng the psitive y-axis and has a magnitude f 800 N. Slutin: T slve this pblem, the esultant f fce R and its tw cmpnents 1 and, will each expessed in Catesian cmpnents. Then, as shwn in igue (a), it is necessay that R 1. The Catesian cmpnents f 1, 1x 1 csα 1 300cs45 1.1N 1y 1 csβ 1 300cs60 150Ν 1z 1 csγ 1 300cs10 150Ν Since R has a magnitude f 800 N and acts in psitive y-diectin, the diectins csines α and γ ae equal 90 while β 0, s the cmpnents Σ x Σ z 0 and Σ y 800. S the cmpnents f, x x -1.1 N y y 650 N z z 150 N The magnitude f is thus, ( 1.1) N and its cdinate diectin angles, 1.1 csα α cs β β Univesity f Qadisiyah\aculty f Eng.\Civil Dep 9

4 Chapte Tw ce System 150 csγ γ Example 14: The f is suppted by cables as shwn in the pht. If the cables exet fces 100 N and AC 10 N n the wall hk at A as shwn in igue (a), detemine the esultant fce acting at A. Expess the esult as Catesian cmpnents. Slutin: ind the Catesian pjectins f the : : x x y y m 0 m z z m ( 4) 5.65 m AC: 4 0 x x 1 y y m m z z m 1 AC ( ) ( ) ( ) 4 ( 4) 6.0 m The cmpnents f esultant fce: x y z ix AC 4 4 *100 * AC iy AC 0 *100 * ix AC And the esultant 151N 40 N ( 4) ( 4) *100 * AC 151N x y z ( 151) 16N.Univesity f Qadisiyah\aculty f Eng.\Civil Dep 30

5 Chapte Tw ce System.Univesity f Qadisiyah\aculty f Eng.\Civil Dep 31

6 Chapte Tw ce System Mment f ce When a fce is applied t a bdy it will pduce a tendency f the bdy t tate abut a pint that is nt n the line f actin f the fce. This tendency t tate is smetimes called a tque, but mst ften it is called the mment f a fce simply mment. The magnitude f the mment is: M d (-8) Whee d is the mment am pependicula distance fm the axis at pint O t the line f actin f the fce. Units f mment magnitude cnsist f fce times distance, i.e., N.m lb.ft. Nte 1: If the fce is applied at an angle θ 90, igue b, then it will be difficult t tun the blt since the mment am d' d sinθ will be smalle than d. Nte : If is applied alng the wench, igue c, its mment am will be ze since the line f actin f will intesect pint O (the z-axis). As a esult, the mment f abut O is als ze and n tuning can ccu The mment M abut O, abut an axis passing thugh O and pependicula t the plane, is a vect quantity since it has a specified magnitude and diectin..univesity f Qadisiyah\aculty f Eng.\Civil Dep 3

7 Chapte Tw ce System Diectin: The diectin f M is defined by its mment axis, which is pependicula t the plane that cntain the fce and its mment am d. The ight-hand ule is used t establish the sense f diectin f M. Accding t this ule, the natue cul f the finges f the ight-hand, as they ae dawn twads the palm, epesent the tendency f tatin caused by the mment. As the actin is pefmed, the thumb f the ight-hand will give the diectin sense f M. Ntice that the mment vect is epesented in thee-diectinally by a cul aund an aw as in igue b. Since in this case the mment will tend t cause a cunteclckwise tatin, the mment vect is actually diected ut f page. If the fce des nt lie in a plane pependicula t the mment axis, it may be eslved int tw cmpnents, ne being paallel t the mment axis and the the lying in a plane pependicula t the axis. The cmpnent f paallel t the efeence axis has n tendency t tate the bdy abut the axis and has n mment with espect t this axis. The mment f the the cmpnent is thus the mment f the fce with espect t the line axis. M d (-9) Resultant mment: tw-dimensinal pblems, whee all the fces lie in the x-y plane the esultant (M R ) abut pint O (the z-axis) can be detemined by finding the algebaic sum f the mments caused by all fces in the system. As a cnventin, we will geneally cnside psitive mment as cunteclckwise since they ae diected alng the psitive z-axis (ut f page). Clckwise mment will be negative. Ding this, the diectinal sense f each mment can be epesented by a plus minus sign..univesity f Qadisiyah\aculty f Eng.\Civil Dep 33

8 Chapte Tw ce System theefe: Using this sign cnventin, the esultant mment in figue belw is (M R ) O d; (M R ) O 1 d 1 - d 3 d 3 If the numeical esult f this sum is psitive scala, (M R ) O will be cunteclckwise mment (ut f page); if the esult is negative, (M R ) O will be clckwise mment (int the page). Pinciple Mments f ces: When detemining the mment f a fce abut a pint, it is ften cnvenient t use the pinciple f mments, als knwn as Vaignn s theem which indicates that: The mment f a fce abut a pint is equal t the sum f the mments f its cmpnents abut that pint. Example 15: each case illustated in igues belw, detemine the mment f the fce abut pint O. Slutin: igue (a) igue (b) igue (c) igue (d) igue (e) M O -(100)() N.m M O -(50)(0.75) N.m M O - (40)(4 cs30 ) -9 9 lb.ft M O (60)(1 sin 45 ) 4.4 lb.ft M O (7)(4-1) 1.0 kn.m.univesity f Qadisiyah\aculty f Eng.\Civil Dep 34

9 Chapte Tw ce System Example 16: Detemine the esultant mment f the fu fces acting n the d shwn in igue abut O. Slutin: M R Σd; M R -50() 60(0) 0(3 sin30 ) -40(4 3 cs30 ) -334 N.m M R 334 N.m Example 17: Detemine the mment f the fce in figue abut O. Slutin I: The mment am d in igue (a) can be fund fm tignmety. Thus, d (3) sin m M O d -(5)(.898) kn.m 14.5 kn.m Since the fce tends t tate bit clckwise abut pint O, the mment is diected int the page. Slutin II: The x and y cmpnents f the fce ae indicated in igue (b). M O x d y y d x -(5 cs45 )(3 sin30 )-(5 sin45 )(3 cs30 ) kn.m 14.5 kn.m Slutin III: The x and y axes can be set paallel and pependicula t the d's axis as shwn in igue (c). Hee x pduces n mment abut pint O since its line f actin passes thugh this pint. Theefe,.Univesity f Qadisiyah\aculty f Eng.\Civil Dep 35