VECTOR MECHANICS FOR ENGINEERS: Statics of Particles. J. Walt Oler The McGraw-Hill Companies, Inc. All rights reserved.

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1 VECTOR MECHANICS FOR ENGINEERS: STATICS Statics of Paticles J. Walt Ole Teas Tech Univesit 2007 The McGaw-Hill Companies, Inc. All ights eseved.

2 Vecto Mechanics fo Enginees: Statics Contents Intoduction Sample Poblem 2.3 Resultant of Two Foces Vectos Addition of Vectos Resultant of Seveal Concuent Foces Sample Poblem 2.1 Sample Poblem 2.2 Rectangula Components of a Foce: Unit Vectos Addition of Foces b Summing Components Equilibium of a Paticle Fee-Bod Diagams Sample Poblem 2.4 Sample Poblem 2.6 Rectangula Components in Space Sample Poblem The McGaw-Hill Companies, Inc. All ights eseved. 2-2

3 Vecto Mechanics fo Enginees: Statics Intoduction The objective fo the cuent chapte is to investigate the effects of foces on paticles: - eplacing multiple foces acting on a paticle with a single equivalent o esultant foce, - elations between foces acting on a paticle that is in a state of equilibium. The focus on paticles does not impl a estiction to miniscule bodies. Rathe, the stud is esticted to analses in which the size and shape of the bodies is not significant ifi so that all foces ma be assumed to be applied at a single point The McGaw-Hill Companies, Inc. All ights eseved. 2-3

4 Vecto Mechanics fo Enginees: Statics Resultant of Two Foces foce: action of one bod on anothe; chaacteized b its point of application, magnitude, line of action, and sense. Epeimental evidence shows that the combined effect of two foces ma be epesented b a single esultant foce. The esultant is equivalent to the diagonal of a paallelogam which contains the two foces in adjacent tlegs. Foce is a vecto quantit The McGaw-Hill Companies, Inc. All ights eseved. 2-4

5 Vecto Mechanics fo Enginees: Statics Vectos Vecto: paametes possessing magnitude and diection which add accoding to the paallelogam law. Eamples: displacements, velocities, acceleations. Scala: paametes possessing magnitude but not diection. Eamples: mass, volume, tempeatue Vecto classifications: - Fied o bound vectos have well defined points of application that cannot be changed without affecting an analsis. - Fee vectos ma be feel moved in space without changing thei effect on an analsis. - Sliding vectos ma be applied anwhee along thei line of action without affecting an analsis. Equal vectos have the same magnitude and diection. Negative vecto of a given vecto has the same magnitude and the opposite diection The McGaw-Hill Companies, Inc. All ights eseved. 2-5

6 Vecto Mechanics fo Enginees: Statics Addition of Vectos Tapezoid ule fo vecto addition Tiangle ule fo vecto addition B C C Law of cosines, R P Q 2PQ cos B R P Q B Law of sines, sin A sin B Q R sin C A Vecto addition is commutative, P Q Q P Vecto subtaction 2007 The McGaw-Hill Companies, Inc. All ights eseved. 2-6

7 Vecto Mechanics fo Enginees: Statics Addition of Vectos Addition of thee o moe vectos though epeated application of the tiangle ule The polgon ule fo the addition of thee o moe vectos. Vecto addition is associative, P Q S P Q S P ( ) ( Q S ) Multiplication of a vecto b a scala 2007 The McGaw-Hill Companies, Inc. All ights eseved. 2-7

8 Vecto Mechanics fo Enginees: Statics Resultant of Seveal Concuent Foces Concuent foces: set of foces which all pass though the same point. A set of concuent foces applied to a paticle ma be eplaced b a single esultant foce which is the vecto sum of the applied foces. Vecto foce components: two o moe foce vectos which, togethe, have the same effect as a single foce vecto The McGaw-Hill Companies, Inc. All ights eseved. 2-8

9 Vecto Mechanics fo Enginees: Statics Sample Poblem 2.1 SOLUTION: The two foces act on a bolt at A. Detemine thei esultant. Gaphical solution - constuct a paallelogam with sides in the same diection as P and Q and lengths in popotion. Gaphicall evaluate the esultant which is equivalent in diection and popotional in magnitude to the the diagonal. Tigonometic solution - use the tiangle ule fo vecto addition in conjunction with the law of cosines and law of sines to find the esultant The McGaw-Hill Companies, Inc. All ights eseved. 2-9

10 Vecto Mechanics fo Enginees: Statics Sample Poblem 2.1 Gaphical solution - A paallelogam with sides equal to P and Q is dawn to scale. The magnitude and diection of the esultant o of the diagonal to the paallelogam ae measued, R 98 N α 35 Gaphical solution - A tiangle is dawn with P and Q head-to-tail and to scale. The magnitude and diection of the esultant o of the thid side of the tiangle ae measued, R 98 N α The McGaw-Hill Companies, Inc. All ights eseved. 2-10

11 Vecto Mechanics fo Enginees: Statics Sample Poblem 2.1 Tigonometic solution - Appl the tiangle ule. Fom the Law of Cosines, R 2 P 2 Q 2 2PQ cos B 2 2 ( 40N) ( 60N) 2( 40N)( 60N) cos155 R 97.73N Fom the Law of Sines, sin A sin B Q R Q sin A sin B R 60N sin N A α 20 A α The McGaw-Hill Companies, Inc. All ights eseved. 2-11

12 Vecto Mechanics fo Enginees: Statics Sample Poblem 2.2 A bage is pulled b two tugboats. If the esultant of the foces eeted b the tugboats is 5000 lbf diected along the ais of the bage, detemine a) the tension in each of the opes fo α 45 o, SOLUTION: Find a gaphical solution b appling the Paallelogam Rule fo vecto addition. The paallelogam has sides in the diections of the two opes and a diagonal in the diection of the bage ais and length popotional to 5000 lbf. Find a tigonometic solution b appling the Tiangle Rule fo vecto addition. With the magnitude and diection of the esultant known and the diections of the othe two sides paallel to the opes given, appl the Law of Sines to find the ope tensions. The angle fo minimum tension in ope 2 is b) the value of α fo which the detemined b appling the Tiangle Rule tension in ope 2 is a minimum. and obseving the effect of vaiations in α The McGaw-Hill Companies, Inc. All ights eseved. 2-12

13 Vecto Mechanics fo Enginees: Statics Sample Poblem 2.2 Gaphical solution - Paallelogam Rule with known esultant diection and magnitude, known diections fo sides. T lbf T lbf Ti Tigonometic ti solution - Ti Tiangle Rule with Law of Sines T 1 T 5000 lbf 2 sin 45 sin 30 sin105 T lbf T lbf 2007 The McGaw-Hill Companies, Inc. All ights eseved. 2-13

14 Vecto Mechanics fo Enginees: Statics Sample Poblem 2.2 The angle fo minimum tension in ope 2 is detemined b appling the Tiangle Rule and obseving the effect of vaiations in α. The minimum i tension in ope 2 occus when T 1 and T 2 ae pependicula. T T 2 1 ( 5000 lbf ) sin ( 5000 lbf ) cos 30 T lbf T lbf α α The McGaw-Hill Companies, Inc. All ights eseved. 2-14

15 Vecto Mechanics fo Enginees: Statics Rectangula Components of a Foce: Unit Vectos Ma esolve a foce vecto into pependicula components so that the esulting paallelogam is a ectangle. F and F ae efeed to as ectangula vecto components and F F F Define pependicula unit vectos i and j which ae paallel to the and aes. Vecto components ma be epessed as poducts of the unit vectos with the scala magnitudes of the vecto components. F F i F j F and F ae efeed to as the scala components of F 2007 The McGaw-Hill Companies, Inc. All ights eseved. 2-15

16 Vecto Mechanics fo Enginees: Statics Eighth Addition of Foces b Summing Components g p Wish to find the esultant of 3 o moe concuent foces, S Q P R Resolve each foce into ectangula components ( ) ( )j S Q P i S Q P j S i S j Q i Q j P i P j R i R Resolve each foce into ectangula components ( ) ( )j Q Q The scala components of the esultant ae equal to the sum of the coesponding scala F S Q P R components of the given foces. F S Q P R F F R To find the esultant magnitude and diection, 2007 The McGaw-Hill Companies, Inc. All ights eseved R R R R tan θ

17 Vecto Mechanics fo Enginees: Statics Sample Poblem 2.3 SOLUTION: Resolve each foce into ectangula components. Detemine the components of the esultant b adding the coesponding foce components. Fou foces act on bolt A as shown. Detemine the esultant of the foce on the bolt. Calculate the magnitude and diection of the esultant The McGaw-Hill Companies, Inc. All ights eseved. 2-17

18 Vecto Mechanics fo Enginees: Statics Sample Poblem 2.3 SOLUTION: Resolve each foce into ectangula components. foce mag comp comp F F F F R R Detemine the components of the esultant b adding the coesponding foce components. Calculate l the magnitude and diection. 2 2 R R 199.6N 14.3N tan α α N 2007 The McGaw-Hill Companies, Inc. All ights eseved. 2-18

19 Vecto Mechanics fo Enginees: Statics Equilibium of a Paticle When the esultant of all foces acting on a paticle is zeo, the paticle is in equilibium. Newton s Fist Law: If the esultant foce on a paticle is zeo, the paticle will emain at est o will continue at constant speed in a staight line. Paticle acted upon b two foces: - equal magnitude - same line of action - opposite sense Paticle acted upon b thee o moe foces: - gaphical solution ields a closed polgon - algebaic solution R F 0 F 0 F The McGaw-Hill Companies, Inc. All ights eseved. 2-19

20 Vecto Mechanics fo Enginees: Statics Fee-Bod Diagams Space Diagam: A sketch kthshowing Fee-Bod Bd Diagam: A sketch kthshowing the phsical conditions of the onl the foces on the selected paticle. poblem The McGaw-Hill Companies, Inc. All ights eseved. 2-20

21 Vecto Mechanics fo Enginees: Statics Sample Poblem 2.4 SOLUTION: Constuct a fee-bod diagam fo the paticle at the junction of the ope and cable. Appl the conditions fo equilibium b ceating a closed polgon fom the foces applied to the paticle. In a ship-unloading opeation, a 3500-lb automobile is suppoted b a cable. A ope is tied to the cable and pulled to cente the automobile ove its intended position. What is the tension in the ope? Appl tigonometic elations to detemine the unknown foce magnitudes The McGaw-Hill Companies, Inc. All ights eseved. 2-21

22 Vecto Mechanics fo Enginees: Statics Sample Poblem 2.4 SOLUTION: Constuct a fee-bod diagam fo the paticle at A. Appl the conditions fo equilibium. Solve fo the unknown foce magnitudes. T AB T AC 3500 lb sin120 sin 2 sin 58 T AB 3570lb T AC 144lb 2007 The McGaw-Hill Companies, Inc. All ights eseved. 2-22

23 Vecto Mechanics fo Enginees: Statics Sample Poblem 2.6 SOLUTION: It is desied to detemine the dag foce at a given speed on a pototpe sailboat hull. A model is placed in a test channel and thee cables ae used to align its bow on the channel centeline. Fo a given speed, the tension is 40 lb in cable AB and 60 lb in cable AE. Detemine the dag foce eeted on the hull and the tension in cable AC. Choosing the hull as the fee bod, daw a fee-bod diagam. Epess the condition fo equilibium fo the hull b witing that the sum of all foces must be zeo. Resolve the vecto equilibium equation into two component equations. Solve fo the two unknown cable tensions The McGaw-Hill Companies, Inc. All ights eseved. 2-23

24 Vecto Mechanics fo Enginees: Statics Sample Poblem 2.6 SOLUTION: Choosing the hull as the fee bod, daw a fee-bod diagam. tanα α 7 ft ft 1.75 tan β 4 ft 4 ft β Epess the condition fo equilibium fo the hull b witing that the sum of all foces must be zeo. R T T T F 0 AB AC AE D 2007 The McGaw-Hill Companies, Inc. All ights eseved. 2-24

25 Vecto Mechanics fo Enginees: Statics Sample Poblem 2.6 Resolve the vecto equilibium equation into two component equations. Solve fo the two unknown cable tensions. T AB ( 40 lb) sin i ( 40 lb) cos j ( lb) i ( lb) j TAC TAC sin i TAC cos j TAC i TAC j T ( 60 lb)i F F i D D R 0 ( TAC FD ) i ( T T AC 60 ) j 2007 The McGaw-Hill Companies, Inc. All ights eseved. 2-25

26 Vecto Mechanics fo Enginees: Statics Sample Poblem 2.6 R 0 ( TAC FD )i) i T 60 ( ) j AC This equation is satisfied onl if each component of the esultant is equal to zeo ( F 0) F T F ( ) T 60 T AC F D 42.9 lb lb AC AC D 2007 The McGaw-Hill Companies, Inc. All ights eseved. 2-26

27 Vecto Mechanics fo Enginees: Statics Rectangula Components in Space The vecto F is contained in the plane OBAC. Resolve F into Resolve F h into hoizontal and vetical ectangula components components. F F h cosφ F F cosθ F sinθ cosφ F h F sinθ F F h sinφ θ sinφ F sin 2007 The McGaw-Hill Companies, Inc. All ights eseved. 2-27

28 Vecto Mechanics fo Enginees: Statics Rectangula Components in Space With the angles between F and the aes, F F cos θ F F cos θ Fz F cos θ z F Fi F j Fzk F( cosθ i cosθ j cosθ z k ) Fλ λ cos θ i cosθ j cosθ zk λ is a unit vecto along the line of action of F and cos θ, cos ae the diection cosines fo F θ, and cosθ z 2007 The McGaw-Hill Companies, Inc. All ights eseved. 2-28

29 Vecto Mechanics fo Enginees: Statics Rectangula Components in Space Diection of the foce is defined b the location of two points, M ( 1, 1, z 1 ) and N ( 2, 2, z 2 ) d vecto joining d i d j d M k z and N d 2 1 d 2 1 d z F Fλ 1 λ ( d i d j d zk ) d Fd Fd Fd F F F z z d d d 2007 The McGaw-Hill Companies, Inc. All ights eseved z 2 z 1

30 Vecto Mechanics fo Enginees: Statics Sample Poblem 2.7 SOLUTION: Based on the elative locations of the points A and B, detemine the unit vecto pointing fom A towads B. Appl the unit vecto to detemine the components of the foce acting on A. The tension in the gu wie is 2500 N. Detemine: a) components F, F, F z of the foce acting on the bolt at A, b) the angles θ, θ, θ z defining the diection of the foce Noting that the components of the unit vecto ae the diection cosines fo the vecto, calculate l the coesponding angles The McGaw-Hill Companies, Inc. All ights eseved. 2-30

31 Vecto Mechanics fo Enginees: Statics Sample Poblem 2.7 SOLUTION: Detemine the unit vecto pointing fom A towads B. AB 40 m i 80 m j 30 m k AB ( ) ( ) ( ) ( 40 m) ( 80 m) ( 30 m) 94.3 m λ i j k i j k Detemine the components of the foce. F Fλ 2500 N 0.424i j 0.318k 1060 N i 2120 N j 795 N ( )( ) ( ) ( ) ( )k 2007 The McGaw-Hill Companies, Inc. All ights eseved. 2-31

32 Vecto Mechanics fo Enginees: Statics Sample Poblem 2.7 Noting that the components of the unit vecto ae the diection cosines fo the vecto, calculate the coesponding angles. λ cos θ i cosθ j cosθ zk 0.424i j 0.318k θ θ θ z o o 71.5 o 2007 The McGaw-Hill Companies, Inc. All ights eseved. 2-32