5.1 Moment of a Force Scalar Formation

Transcription

1 Outline ment f a Cuple Equivalent System Resultants f a Fce and Cuple System ment f a fce abut a pint axis a measue f the tendency f the fce t cause a bdy t tate abut the pint axis Case 1 Cnside hizntal fce F x, which acts pependicula t the handle f the wench and is lcated d y fm the pint O F x tends t tun the pipe abut the z axis The lage the fce the distance d y, the geate the tuning effect Tque tendency f tatin caused by F x simple mment ( ) z ment axis (z) is pependicula t shaded plane (x-y) F x and d y lies n the shaded plane (x-y) ment axis (z) intesects the plane at pint O 1

2 Case pply fce F z t the wench Pipe des nt tate abut z axis Tendency t tate abut x axis The pipe may nt actually tate F z ceates tendency f tatin s mment ( ) x is pduced Case ment axis (x) is pependicula t shaded plane (y-z) F z and d y lies n the shaded plane (y-z) Case 3 pply fce F y t the wench N mment is pduced abut pint O Lack f tendency t tate as line f actin passes thugh O In Geneal Cnside the fce F and the pint O which lies in the shaded plane The mment O abut pint O, abut an axis passing thugh O and pependicula t the plane, is a vect quantity ment O has its specified magnitude and diectin

3 agnitude F magnitude f O, O = Fd whee d = mment am pependicula distance fm the axis at pint O t its line f actin f the fce Diectin Diectin f O is specified by using ight hand ule - finges f the ight hand ae culed t fllw the sense f tatin when fce tates abut pint O Units f mment is N.m Diectin ment acts abut an axis pependicula t the plane cntaining F and d ment axis intesects the plane at pint O Resultant ment f a System f Cplana Fces Resultant mment, R = additin f the mments f all the fces algebaically since all mment fces ae cllinea R = Fd taking clckwise t be psitive 3

4 Resultant ment f a System f Cplana Fces clckwise cul is witten alng the equatin t indicate that a psitive mment if diected alng the + z axis and negative alng the z axis ment f a fce des nt always cause tatin Fce F tends t tate the beam clckwise abut with mment = Fd Fce F tends t tate the beam cunteclckwise abut B with mment B = Fd B Hence suppt at pevents the tatin Test 4.1 F each case, detemine the mment f the fce abut pint O Slutin Line f actin is extended as a dashed line t establish mment am d Tendency t tate is indicated and the bit is shwn as a cled cul ( a) ( b) = (100 N)(m) = 00N. CW ) = (50N)(0.75m) = 37.5N. CW ) 4

5 Slutin ( c) ( d) ( e) = (40N)(4m + cs30 m) = 9N. CW ) = (60N)(1sin 45 m) = 4.4N. CCW ) = (7kN)(4m 1m) = 1.0kN. CCW ) Example 4. Detemine the mments f the 800N fce acting n the fame abut pints, B, C and D. Slutin Scala nalysis = (800N )(.5m) = 000N. CW ) = (800N )(1.5m) = 100N. CW ) = (800N )(0m) = 0kN. m Line f actin f F passes thugh C B C D = (800N )(0.5m) = 400N. CCW ) 5. ment f Fce Vect Fmulatin ment f fce F abut pint O can be expessed using css pduct O = X F whee epesents psitin vect fm O t any pint lying n the line f actin f F 5

6 5. ment f Fce - Vect Fmulatin agnitude F magnitude f css pduct, O = F sinθ whee θ is the angle measued between tails f andf Teat as a sliding vect. Since d = sinθ, O = F sinθ = F (sinθ) = Fd 5. ment f Fce - Vect Fmulatin Diectin Diectin and sense f O ae detemined by ight-hand ule - Extend t the dashed psitin - Cul finges fmtwads F - Diectin f O is the same as the diectin f the thumb 5. ment f Fce - Vect Fmulatin Diectin *Nte: - cul f the finges indicates the sense f tatin - aintain ppe de f and F since css pduct is nt cmmutative 5. ment f Fce - Vect Fmulatin Pinciple f Tansmissibility F fce F applied at any pint, mment ceated abut O is O = x F F has the ppeties f a sliding vect and theefe act at any pint alng its line f actin and still ceate the same mment abut O 6

7 5. ment f Fce - Vect Fmulatin Catesian Vect Fmulatin F fce expessed in Catesian fm, O i = XF = whee x, y, z epesent the x, y, z cmpnents f the psitin vect and F x, F y, F z epesent that f the fce vect x F x j y F y k z F z 5. ment f Fce - Vect Fmulatin Catesian Vect Fmulatin With the deteminant expended, O = ( y F z z F y )i ( x F z - z F x )j + ( x F y y F x )k O is always pependicula t the plane cntaining and F Cmputatin f mment by css pduct is bette than scala f 3D pblems 5. ment f Fce - Vect Fmulatin Catesian Vect Fmulatin Resultant mment f fces abut pint O can be detemined by vect additin R = ( x F) 5. ment f Fce - Vect Fmulatin ment f fce F abut pint, pulling n cable BC at any pint alng its line f actin, will emain cnstant Given the pependicula distance fm t cable is d = d F In 3D pblems, = BC xf 7

8 5. ment f Fce - Vect Fmulatin 5. ment f Fce - Vect Fmulatin Example 4.4 The ple is subjected t a 60N fce that is diected fm C t B. Detemine the magnitude f the mment ceated by this fce abut the suppt at. Slutin Eithe ne f the tw psitin vects can be used f the slutin, since = B xf = C xf Psitin vects ae epesented as B = {1i + 3j + k} m and C = {3i + 4j} m Fce F has magnitude 60N and is diected fm C t B 5. ment f Fce - Vect Fmulatin Slutin F = (60N) uf (1 3) i ) j + 9 0) k = (60N) ( ) + ( 1) + () = { 40i 0 j + 40k} Substitute int deteminant fmulatin i j k = XF = 1 3 = B N {[3(40) ( 0)] i [1(40) ( 40)] j + [1( 0) 3(40)] k} 5. ment f Fce - Vect Fmulatin Slutin O = XF = = {[4(40) 0( 0)] i [3(40) 0( 40)] j + [3( 0) 4(40)] k} Substitute int deteminant fmulatin = { 160i 10 j + 100k } N. m F magnitude, = (160) + (10) + (100) C = 4N. m i 3 40 j 4 0 k

9 5.3 Pinciples f ments ls knwn as Vaignn s Theem ment f a fce abut a pint is equal t the sum f the mments f the fces cmpnents abut the pint F F = F 1 + F, O = X F 1 + X F = X (F 1 + F ) = X F 5.3 Pinciples f ments The guy cable exets a fce F n the ple and ceates a mment abut the base at = Fd If the fce is eplaced by F x and F y at pint B whee the cable acts n the ple, the sum f mment abut pint yields the same esultant mment 5.3 Pinciples f ments 5.3 Pinciples f ments F y ceate ze mment abut = F x h pply pinciple f tansmissibility and slide the fce whee line f actin intesects the gund at C, F x ceate ze mment abut = F y b Example 4.6 The fce F acts at the end f the angle backet. Detemine the mment f the fce abut pint O. 9

10 5.3 Pinciples f ments Slutin ethd 1 O = 400sin30 N(0.m)-400cs30 N(0.4m) = -98.6N.m = 98.6N.m (CCW) s a Catesian vect, O = {-98.6k}N.m 5.3 Pinciples f ments Slutin ethd : Expess as Catesian vect = {0.4i 0.j}N F = {400sin30 i 400cs30 j}n = {00.0i 346.4j}N F mment, O = = XF = { 98.6k } N. m i j k Equivalent System fce has the effect f bth tanslating and tating a bdy The extent f the effect depends n hw and whee the fce is applied We can simplify a system f fces and mments int a single esultant and mment acting at a specified pint O system f fces and mments is then equivalent t the single esultant fce and mment acting at a specified pint O 5.4 Equivalent System Pint O is n the Line f ctin Cnside bdy subjected t fce F applied t pint pply fce t pint O withut alteing extenal effects n bdy - pply equal but ppsite fces F and F at O 10

11 5.4 Equivalent System 5.4 Equivalent System Pint O is n the Line f ctin - Tw fces indicated by the slash acss them can be cancelled, leaving fce at pint O - n equivalent system has be maintained between each f the diagams, shwn by the equal signs Pint O is n the Line f ctin - Fce has been simply tansmitted alng its line f actin fm pint t pint O - Extenal effects emain unchanged afte fce is mved - Intenal effects depend n lcatin f F 5.4 Equivalent System Pint O is Nt n the Line f ctin F is t be mved t pint ) withut alteing the extenal effects n the bdy pply equal and ppsite fces at pint O The tw fces indicated by a slash acss them, fm a cuple that has a mment pependicula t F 5.4 Equivalent System Pint O is Nt n the Line f ctin The mment is defined by css pduct = X F Cuple mment is fee vect and can be applied t any pint P n the bdy 11

12 5.5 Resultants f a Fce and Cuple System Example 4.14 Replace the fces acting n the bace by an equivalent esultant fce and cuple mment acting at pint. 5.5 Resultants f a Fce and Cuple System Slutin Fce Summatin F x and y cmpnents f esultant fce, + F F = 100N 400cs 45 N = 38.8N = 38.8N + F F Rx Ry Rx Ry = ΣF ; = ΣF ; = 600N 400sin 45 N = 88.8N = 88.8N x y 4.8 Resultants f a Fce and Cuple System Slutin F magnitude f esultant fce F = ( F ) + ( F ) R = 96N F diectin f esultant fce 1 F θ = tan F = 66.6 Rx Ry Rx Ry = (38.8) + (88.8) = tan Resultants f a Fce and Cuple System Slutin ment Summatin Summatin f mments abut pint, R R = Σ ; = 100N(0) 600N(0.4m) (400sin 45 N)(0.8m) (400cs 45 N)(0.3m) = 551N. m = 551N. CW ) When R and F R act n pint, they will pduce the same extenal effect eactins at the suppt 1