Moment and couple. In 3-D, because the determination of the distance can be tedious, a vector approach becomes advantageous. r r

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1 Moment and couple In 3-D, because the detemination of the distance can be tedious, a vecto appoach becomes advantageous. o k j i M k j i M o ) ( ) ( ) ( + + M o M M M + O A

2 Moment about an abita ais O M O n λ λ M 1. Calculate moment M o ind moment about λ ais 2. Calculate pojection of moment on λ ais n n n n M M O ) ( ) ( λ ).( ) ( k j i k j i n γ β α + + γ β α γ β α k j i γ β α + +

3 Vaignon s Theoem O 1 A Sum of the moments of a sstem of concuent foces about a given point equals the moment of thei sum about the same point M o ( ) ) M o ( ) R (

4 Couples(1) B B O M A A d -Couple is a moment poduced b two foce of equal magnitude but opposite in diection. M M + ( ) ( ) A B A B - vecto fom an point on the line of action of to an point on the line of action of - Moment of a couple is the same about all point Couple ma be epesented as a fee vecto. - Diection: nomal to the plane of the two foces (ight hand ule) - Recall: Moment of foce about a point is a sliding vecto.

5 Couples(2) 1 M 1 M 1 M M 2 M [Couple fom 1 ]+[Couple fom 2 ] [Couple fom ] couples ae fee vecto. the line of action o point of action ae not needed!!!

6 oce couple sstems B A B A B M A No changes in the net etenal effects Couple - M Moment of about point B - is a vecto stat fom point B to an point on the line of action of

7 Sample 1 A Tension T of magnitude 10 kn is applied to the cable attached to the top A of the igid mast and secued to the gound at B. Detemine the moment M of T about the -ais passing though the base O.

8 Sample 2 Detemine the magnitude and diection of the couple M which will eplace the two given couples and still poduce the same etenal effect on the block. Specif the two foce and, applied in the two faces of the block paallel to the - plane, which ma eplace the fou given foces. The 30-N foces act paallel to the - plane.

9 Sample 3 A foce of 400 N is applied at A to the handle of the contol leve which is attached to the fied shaft OB. In detemining the effect of the foce on the shaft at a coss section such as that at O, we ma eplace the foce b an equivalent foce at O and a couple. Descibe this couple as a vecto M.

10 Sample 4 If the magnitude of the moment of about line CD is 50 Nm, detemine the magnitude of.

11 Sample 5 Tension in cable AB is N. Detemine the moment about the -ais of this tension foce acting on point A. Compae ou esult to the moment of the weight W of the 15-kg unifom plate about the -ais. What is the moment of the tension foce acting at A about line OB

12 Summa (oce-moment 3-D) oce 1. Detemine coodinate 2. Detemine unit vecto 3. oce can be calculate Angle between foce and -,-,-ais 1. oce i + j + k 2. Detemine amplitude of foce 3. cosθ /, cosθ /, cosθ / Angle between foce and abita ais 1. Detemine unit vectos (n, n) 2. cosθ n n

13 Summa (oce-moment 3-D) Moment Conside to use vecto method o scala method Vecto method Moment about an abita point O 1. Detemine and 2. Coss vecto Moment about an abita ais 1. Detemine moment about an point on the ais M O 2. Detemine unit vecto of the ais n 3. Moment about the ais M O n Angle between moment and ais Same as angle between foce and ais

14 Resultants(1) Step1 Step2 Step3 Select a point to find moment Replace foces with foces at point O + couples R M M + M + M +... ( ) Add foces and couples vectoiall to get the esultant foce and moment

15 Resultants(2) 2-D B A Md B A M v v oce + couple can be eplaced b a foce b changing the position of. 3-D M 2 M O M 1 R v M 2 v R M 2 and R can be eplaced b one foce R b changing the position of R. M v // R v 1 M 1 can not be eplaced

16 Wench esultant(1) M 2 Rd

17 Wench esultant(2) 2-D: All foce sstems can be epesented with onl one esultant foce o couple 3-D: All foce sstems can be epesented with a wench esultant Wench: esultant couple M paallel to the esultant foce R

18 Sample 6 Detemine the esultant of the sstem of paallel foces which act on the plate. Solve with a vecto appoach.

19 Sample 7 Replace the two foces and the negative wench b a single foce R applied at A and the coesponding couple M.

20 Sample 8 Detemine the wench esultant of the thee foces acting on the backet. Calculate the coodinates of the point P in the - plane though which the esultant foce of the wench acts. Also find the magnitude of the couple M of the wench.

21 Sample 9 The esultant of the two foces and couple ma be epesented b a wench. Detemine the vecto epession fo the moment M of the wench and find the coodinates of the point P in the - plane though which the esultant foce of the wench passes

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