ch02

STATICS AND MECHANICS OF MATERIALS, 2nd Edition RILEY, STURGES AND MORRIS

14

Chapter 2 2-1 Use the law of sines and the law of cosines, in conjunction with sketches of the force triangles, to solve the

following problems. Determine the magnitude of the resultant R and the angle between the x-axis and the line of action of the two forces shown in Fig. P2-1.

SOLUTION From the law of cosines

2 2 290 120 2 90 120 cos90R

150.0 lbR From the law of sines

sin sin 9090 R

1 90sin 90sin 36.87150.0

150 lbR 36.87 .......................................................................................................Ans.

2-2 Use the law of sines and the law of cosines, in conjunction with sketches of the force triangles, to solve the following problems. Determine the magnitude of the resultant R and the angle between the x-axis and the line of action of the two forces shown in Fig. P2-2.


2 2 2250 200 2 250 200 cos50R

408.386 NR From the law of sines

sin sin130200 R

1 200sin130sin 22.03408.386

408 NR 36.03 .......................................................................................................Ans.



2 2 2600 800 2 600 800 cos75R

866.910 lbR From the law of sines

sin sin 75800 R

1 800sin 75sin 63.05866.910

867 lbR 86.95 .......................................................................................................Ans.


15



2 2 210 25 2 10 25 cos120R

31.225 kNR From the law of sines

sin sin12025 R

1 25sin120sin 43.9031.225

31.2 kNR 19.90 ....................................................................................................Ans.

2-5 The support ring of the traffic light shown in Fig. P2-5 is acted on by three forces – the weight of the traffic light (220 lb), a force in cable A (280 lb), and a force in cable B (FB). If the resultant of the three forces is zero, determine the magnitude and direction of FB.


2 2 212 220 280 2 220 280 cos70R

12 290.969 lbR

From the law of sines

12

12

sin sin 70220 R

112

220sin 70sin 45.28290.969

12 291 lbR 25.28

But 12 BR F 0

Therefore 291 lbBF R 25.28 ..........................................................................................Ans.

2-6 Determine the resultant of the three forces shown in Fig. P2-6. Locate the resultant with respect to the x-axis shown.

SOLUTION

2 212 3 4 5 kNR

112

3tan 36.8704

Then, using the law of cosines

2 2 25 8 2 5 8 cos113.130R

10.974 kNR From the law of sines

sin sin113.130

5 R


16

1 5sin113.130sin 24.7710.974

10.97 kNR 5.23 ....................................................................................................Ans.

2-7 A stalled automobile is being pulled by the two forces shown in Fig. P2-7. If the resultant pull is to be 120 lb in the x-direction, determine the magnitude and direction of the force P.


2 2 280 120 2 80 120 cos 20P

52.516 lbP From the law of sines

sin sin 20120 P

1 120sin 20sin 128.6052.516

52.5 lbP 31.40 ......................................................................................................Ans.

2-8 The eye bolt shown in Fig. P2-8 is subjected to a 2700 N force and an unknown force P. If the resultant pull is 2000 N in the x-direction, determine the magnitude and direction of the force P.


2 2 22000 2700 2 2000 2700 cos 60P

2426.932 kNP From the law of sines

sin sin 602000 P

1 2000sin 60sin 45.542426.932

2430 kNP 74.46 ..................................................................................................Ans.

2-9 Two forces act on the bracket shown in Fig. P2-9. Determine the angle that will make the vertical component of the resultant of these two forces zero.


2 2 2145 175 2 175 cos50R R

2 224.976 9600 0R R

167.747 lbR or 57.229 lbR From the law of sines

sin sin 50175 145

1 175sin 50sin145

67.60 or 112.40 .............................................................................................Ans.


17

167.7 lb R when 67.60

57.2 lb R when 112.40

2-10 A 270-N force and a 400-N force act at point B of the truss shown in Fig. P2-10. A third force F is to be applied at point B so that the resultant of the three forces is zero. Determine the magnitude of F and its orientation with respect to the 400-N force.

SOLUTION Using the law of cosines

2 2 212 270 400 2 270 400 cos50R

12 306.689 NR


12

12

sin sin 50270 R

112

270sin 50sin 42.41306.689

12 307 NR 42.41

But 12 3R R R 0

Therefore 3 12 307 NR R 42.41 ........................................................................................Ans.

2-11 Three forces are applied to a bracket mounted on a post as shown in Fig. P2-11. Determine (a) The magnitude and direction (angle x) of the resultant R of the three forces. (b) The magnitudes of two other forces Fx and Fy that would have the same resultant. SOLUTION (a) For the system of forces,

1 2 3 12 3R F F F R F

From the parallelogram of forces constructed using F1 and F2

2 2

12 120 140 2 120 140 cos105

206.631 lb

R

112

140sin105sin 40.878206.631

12 206.631 lbR 4.122

For the second parallelogram of forces

2 2100 206.631 2 100 206.631 cos85.878

222.993 lb

R

1 206.632sin85.878sin 67.55222.993

223 lbR 22.45 ........................................................................................................Ans. (b) From the definition of sine and cosine

223.993 cos 22.45 207.0 lb xR ......................................................................Ans.

223.993 sin 22.45 85.5 lb yR ...........................................................................Ans.


18

2-12 Four forces act on a small airplane in flight, as shown in Fig. P2-12; its weight (25 kN), the thrust provided by the engine (10 kN), the lift provided by the wings (24 kN), and the drag resulting from its motion through the air (3 kN). Determine the resultant of the four forces and its line of action with respect to the axis of the plane.

SOLUTION For the system of forces,

1 2 3 4

12 3 4

123 4

R F F F FR F FR F


2 2

12 3 24 2 3 24 cos90

24.1868 kN

R

11

3sin 90sin 7.12524.1868

12 24.1868 kNR 73.875

From the parallelogram of forces constructed using R12 and F3

2 2

123 12 1210 2 10 cos82.875

25.000 kN

R R R

1 122

sin 82.875sin 73.74025.000

R

123 25.000 kNR 83.740

Finally, from the parallelogram of forces constructed using R123 and F4

2 2

123 12325 2 25 cos 6.260

2.730 kN

R R R

1 1233

sin 6.260sin 86.9022.730

R

2.730 kNR 3.088 ..................................................................................................Ans.

13.088 below the -axisx .........................................................................................Ans.

2-13 Determine the resultant force R of the three forces applied to the gusset plate shown in Fig. P2-13. SOLUTION For the system of forces,

1 2 3 12 3R F F F R F


2 2

12 11,000 4500 2 11,000 4500 cos135

14,534.565 lb

R

112

4500sin135sin 12.64614,534.565

12 14,534.565 lbR 77.354


19


2 2

12 129000 2 9000 cos137.354

22,015.69 lb

R R R

1 9000sin137.354sin 16.07822,015.69

22,020 lbR 86.57 ............................................................................Ans.

2-14 Determine the magnitude and direction of the resultant of the three forces shown in Fig. P2-14. SOLUTION For the system of forces,

1 2 3 12 3R F F F R F


2 2

12 5 7.5 2 5 7.5 cos130

11.3780 kN

R

112

7.5sin130sin 30.32811.3780

12 11.3780 kNR 20.328


2 2

12 1210 2 10 cos65.328

11.5961 kN

R R R

1 10sin 65.328sin 51.59511.5961

11.60 kNR 71.92 ..................................................................................................Ans.

2-15 Two forces A and B are applied to an eyebolt as shown in Fig. P2-15. If the magnitudes of the two forces are A = 50 lb and B = 100 lb, calculate and plot the magnitude of the resultant R as a function of the angle A (0

A 180 ). Also calculate and plot the angle R that the resultant makes with the force B as a function of the angle A. When is the resultant a maximum? When is the resultant a minimum? When is the angle R a maximum? Repeat for A = 100 lb and B = 50 lb.


2 2 2 cos AR A B AB


sin 180sin AB

A R

1 sinsin AB

AR

For A = 50 lb and B = 100 lb


20

2 250 100 2 50 100 cos

100 1.25 cos lb

A

A

R

1

1

50sinsin100 1.25 cos

sinsin2 1.25 cos

AB

A

A

A

For A = 100 lb and B = 50 lb

2 2100 50 2 100 50 cos

100 1.25 cos lb

A

A

R

1

1

100sinsin100 1.25 cos

sinsin1.25 cos

AB

A

A

A

2-16 Two forces A and B are applied to a bracket as shown in Fig. P2-16. If the magnitude of the force B is 325 N, calculate and plot the magnitude of the resultant R as a function of the magnitude of the force A (0 A 900 N).


2 2 2325 2 325 cos 75R A A

2105,625 168.2324 NR A A


21

2-17 Determine the magnitudes of the u- and v-components of the 1000-lb force shown in Fig. P2-17. SOLUTION From the law of sines

1000

sin 35 sin100 sin 45u vF F

582 lbuF ........................................................................Ans.

718 lbvF .........................................................................Ans.

2-18 Determine the components of the 3000-N force in the directions of members AB and BC of the truss shown in Fig. P2-18 when = 45 .

SOLUTION From the law of sines

3000

sin15 sin 90 sin 75BCAB FF

776 NABF ......................................................................Ans.

2900 NBCF ....................................................................Ans.

2-19 Two cables are used to support a stop light as shown in Fig. P2-19. The resultant R of the cable forces Fu and Fv has a magnitude of 300 lb and its line of action is vertical. Determine the magnitudes of the forces Fu and Fv.


300

sin 36.87 sin 98.13 sin 45u vF F

181.8 lbuF .....................................................................Ans.

214.3 lbvF .....................................................................Ans.

2-20 Two ropes are used to tow a boat upstream as shown in Fig. P2-20. The resultant R of the rope forces Fu and Fv has a magnitude of 1500 N and its line of action is directed along the axis of the boat. Determine the magnitudes of the forces Fu and Fv.


1500

sin 40 sin110 sin 30u vF F

1026 NuF .......................................................................Ans.

798 NvF .........................................................................Ans.

2-21 Determine the u- and v-components of the 5200-lb force acting on the bracket shown in Fig. P2-21. SOLUTION From the law of sines

5200

sin 22.38 sin135 sin 22.62u vF F

2800 lbuF ...............................................................Ans.

2830 lbvF ...............................................................Ans.


22

2-22 A 5000-N force acts in the vertical direction on the block shown in Fig. P2-22. Determine the components of the force perpendicular to and parallel to the line AB.


5000

sin 60 sin 90 sin 30pa FF

4330 NaF .................................................................................... Ans.

2500 NpF .................................................................................... Ans.

2-23 Two forces Fu and Fv are applied to a bracket as shown in Fig. P2-23. If the resultant R of the two forces has a magnitude of 725 lb and a direction as shown on the figure, determine the magnitudes of the forces Fu and Fv.


725

sin 47.73 sin 49.40 sin82.88u vF F

707 lbuF .................................................................Ans.

948 lbvF ..................................................................Ans.

2-24 A 20-kN force acts on the member shown in Fig. P2-24. Determine the components of the force in the horizontal and vertical directions (FAC and FBC).


20

sin 53.13 sin 90 sin 36.87AC BCF F

16.00 NACF ................................................................................. Ans.

12.00 NBCF ................................................................................. Ans.

2-25 A 500-lb force acts along the line AB shown in Fig. P2-25. Determine the magnitude of the components in the direction of lines AC and AD.


500

sin 40.60 sin108.44 sin 30.96ACAD FF

271 lbACF ........................................................Ans.

343 lbADF ........................................................Ans.


23

2-26 Two forces Fu and Fv are applied to a bracket as shown in Fig. P2-26. If the resultant R of the two forces has a magnitude of 375 N and a direction as shown on the figure, determine the magnitudes of the forces Fu and Fv.


375

sin 70.35 sin 67.38 sin 42.27u vF F

383 NuF ...........................................................Ans.

273 NvF ...........................................................Ans.

2-27 Three forces are applied to a bracket as shown in Fig. P2-27. The magnitude of the resultant R of the three forces is 50 kip. If the force F1 has a magnitude of 30 kip, determine the magnitudes of the forces F2 and F3.


1 2 3 1 23R F F F F R

From the parallelogram of forces constructed using F1 and R23

2 2

23 30 50 2 30 50 cos75

51.2205 kip

R

12

50sin 75sin 70.54651.2205

23 51.2205 kipR 27.546

For the second parallelogram of forces 12 27.546 39.546

38 27.546 10.454

180 130

3 251.2205sin sin sinF F

2 42.6 kipF ....................................................................................................................... Ans.

3 12.13 kipF ..................................................................................................................... Ans.

2-28 A homogeneous cylinder is subjected to the three forces shown in Fig. P2-28. If the resultant of the three forces is zero, determine the magnitude of N1 and the angle of the surface to which N1 is normal.

SOLUTION Since the resultant of the three forces is zero, they must form a closed figure (a triangle). Using the law of sines and the law of cosines gives

2 2

1 2220 1940 2 2220 1940 cos15

609.837 N

N

1 1940sin15sin 55.42609.837

1 610 NN ........................................................................Ans.

55.42 ..........................................................................Ans.


24

2-29 Four forces act on the machine component shown in Fig. P2-29. If the resultant of F1, F2, and F3 is horizontal to the left and has a magnitude of 400 lb, determine the magnitudes of the forces F2 and F3.


1 2 3 1 23 400 lb F F F F R

From the force triangle constructed using F1 and R23

2 223 400 800 894.427 lbR

12

400tan 26.565800

23 894.427 lbR 63.435


32 894.427sin 33.435 sin 75 sin 71.565

FF

2 510 lbF .......................................................................................................................... Ans.

3 878 lbF ........................................................................................................................... Ans.

2-30 A 140-N light is supported by two cables as shown in Fig. P2-30. If the resultant of the three forces is zero, determine the force T1 and the angle .

SOLUTION Since the resultant of the three forces is zero, they must form a closed figure (a triangle). Using the law of sines and the law of cosines gives

2 2

1 140 155 2 140 155 cos50

125.411 N

T

1

sin 40 sin 50140 T

1 140sin 5040 sin 58.777125.411

1 125.4 NT ........................................................................................................................ Ans.

18.77 ............................................................................................................................ Ans.

2-31 Two forces A and B are applied to an eye bolt using ropes as shown in Fig. P2-31. The resultant R of the two forces has a magnitude R = 4000 lb and makes an angle of 30 with the force A as shown. If both of the forces pull on the eye bolt as shown (ropes cannot push on the eye bolt), what is the range of angles ( min B

max) for which this problem has a solution? Calculate and plot the required magnitudes A and B as functions of the angle angles B ( min B max). Why is the magnitude of B a minimum when B = 90 ?


4000

sin sin sin 30A B

180 B

180 30 30B


25

4000sin 30

lbsin 180

B

B

A

4000sin 30 lb

sin 180 B

B

30B or else 0A

180B or else 0B

The magnitude of B is a minimum when B is vertical because that results in the shortest distance from the end of the 4000 lb force to the horizontal.

2-32 Three forces A, B, and C are applied to an eye bolt using ropes as shown in Fig. P2-32. Force A has a magnitude A = 50 N, and the resultant of the three forces is zero. If all of the forces pull on the eye bolt as shown (ropes cannot push on the eye bolt), what is the range of angles ( min C max) for which this problem has a solution? Calculate and plot the required magnitudes B and C as functions of the angle C ( min C max). Why is the magnitude of C a minimum when C = 90 ?

SOLUTION Since the resultant of the three forces is zero, they must form a closed figure (a triangle). Using the law of sines gives

50

sin sin sin 40B C

180 C

180 40 40C

50sin 40

kNsin 180

C

C

B

50sin 40 kN

sin 180 C

C

40C or else 0B

180C or else B C

The magnitude of C is a minimum when C is vertical because that results in the shortest distance from the end of the 50 kN force to the horizontal.

2-33 Determine the x- and y-components of the 1000-lb force shown in Fig. P2-33. SOLUTION

1000cos30 866 lbxF ...............................................................................................Ans.

1000sin 30 500 lbyF ...............................................................................................Ans.

2-34 Determine the x- and y-components of the 800-N force shown in Fig. P2-34. SOLUTION

800sin 25 338 NxF .................................................................................................Ans.

800cos 25 725 NyF .................................................................................................Ans.


26

2-35 Determine the x- and y-components of each force shown in Fig. P2-35. SOLUTION For the 600 lb force

600sin 60 520 lbxF .................................................................................................Ans.

600cos60 300 lbyF .................................................................................................Ans.

For the 800 lb force

800sin 45 566 lbxF ............................................................................................Ans.

800cos 45 566 lbyF .................................................................................................Ans.

2-36 Determine the x- and y-components of each force shown in Fig. P2-36. SOLUTION

1 3 5 950 570 NxF ................................................................................................Ans.

1 4 5 950 760 NyF ...............................................................................................Ans.

2 1 2 800 566 NxF ......................................................................................Ans.

2 1 2 800 566 NyF ............................................................................................Ans.

2-37 Two forces are applied to a post as shown in Fig. P2-37. Determine (a) The x- and y-components of each force. (b) The x’- and y’-components of each force. SOLUTION

(a) 1 500cos70 171.0 lbxF ............................................................................................Ans.

1 500sin 70 470 lbyF ................................................................................................Ans.

2 750cos30 650 lbxF ...............................................................................................Ans.

2 750sin 30 375 lbyF ..........................................................................................Ans.

(b) 1 500sin 25 211.3 lbxF .......................................................................................Ans.

1 500cos 25 453 lbyF ...............................................................................................Ans.

2 750cos15 724 lbxF ...............................................................................................Ans.

2 750sin15 194.1 lbyF .............................................................................................Ans.

2-38 For the 900-N force shown in Fig. P2-38 (a) Determine the x, y, and z scalar components of the force. (b) Express the force in Cartesian vector form. SOLUTION

(a) 900cos35 737.237 NxyF

cos60 369 Nx xyF F ..................................................................................................Ans.

sin 60 638 Ny xyF F ...................................................................................................Ans.

900sin 35 516 NzF ..................................................................................................Ans.

(b) 369 638 516 NF i j k ..........................................................................................Ans.


27

2-39 As an automobile rounds a curve, a force F of magnitude 600 lb is exerted on one of the tires, as shown in Fig. P2-39. Determine the x, y, and z scalar components of the force. The xy plane is parallel to the roadway.

SOLUTION

2 2 2

2 2 12600 97.3 97.3 584 lb2 2 12

F i j kF e i j k

97.3 lbxF ....................................................................................................................... Ans.

97.3 lbyF ......................................................................................................................... Ans.

584 lbzF .......................................................................................................................... Ans.

2-40 A 50-kN force is applied to an eye bolt as shown in Fig. P2-40. (a) Determine the direction angles x, y, and z. (b) Determine the x, y, and z scalar components of the force. (c) Express the force in Cartesian vector form. SOLUTION

(c) 2 2 2

3 2 250 36.38 24.25 24.25 kN3 2 2

F i j kF e i j k

36.4 24.3 24.3 kNF i j k ..................................................................................Ans.

(b) 36.4 kNxF ..................................................................................................................... Ans.

24.3 kNyF ..................................................................................................................... Ans.

24.3 kNzF ....................................................................................................................... Ans.

(a) 1 36.38cos 136.6950x ............................................................................................Ans.

1 24.25cos 119.0150y ............................................................................................Ans.

1 24.25cos 60.9950z ................................................................................................Ans.

2-41 Two forces are applied at a point on a body as shown in Fig. P2-41. Determine (a) The x- and y- components of each force. (b) The x’- and y’- components of each force. SOLUTION

(a) 1 800sin 40 514 lbxF ...........................................................................................Ans.

1 800cos 40 613 lbyF ...............................................................................................Ans.

2 1000cos 20 940 lbxF .............................................................................................Ans.

2 1000sin 20 342 lbyF .............................................................................................Ans.

(b) 1 800sin 70 752 lbxF ..........................................................................................Ans.

1 800cos70 274 lbyF ...............................................................................................Ans.

2 1000cos50 643 lbxF .............................................................................................Ans.

2 1000sin 50 766 lbyF .............................................................................................Ans.


28

2-42 Two forces are applied to an eye bolt as shown in Fig. P2-42. (a) Determine the x, y, and z scalar components of the 30-kN force F1. (b) Express the 30-kN force F1 in Cartesian vector form. (c) Determine the magnitude of the rectangular component of the 30-kN force F1 along the line of action of

the 50-kN force F2. (d) Determine the angle between the two forces. SOLUTION

(a) 1 1 1 2 2 2

5 330 25.355 15.213 5.071 kN5 3 1

F i j kF e i j k

1 25.4 kNxF ...................................................................................................................... Ans.

1 15.21 kNyF .................................................................................................................Ans.

1 5.07 kNzF ...................................................................................................................... Ans.

(b) 1 25.4 15.21 5.07 kNF i j k ..................................................................................Ans.

(c) 2 2 2 2

2 3 2 0.48507 0.72761 0.485072 3 2i j ke i j k

12 1 2 25.355 0.48507 15.213 0.72761 5.071 0.4850725.83 kN

F F e

12 25.8 kNF ...................................................................................................................... Ans.

(d) 1 2 125.83 kN cosFF e

1 25.83cos 30.5730

.................................................................................................Ans.

2-43 A wire is stretched between two pylons, one of which is shown in Fig. P2-43. The 250-lb force in the wire is parallel to the xy-plane and makes an angle of 30 with the y-axis. Point A lies in the xz-plane and point C lies in the yz-plane. Determine

(a) The magnitude of the rectangular component of the 250-lb force in the direction of member AD. (b) The magnitude of the rectangular component of the 250-lb force in the direction of member CD. (c) The angle between members AD and CD. SOLUTION

(a) 250sin 30 250cos30 125.00 216.51 lbFF e i j i j

2 2

12 18 0.55470 0.8320512 18

ADi ke i k

2 2

9 18 0.44721 0.894439 18

CDj ke j k

125 0.55470 216.51 0 0 0.83205AD ADF F e

69.3 lbADF ....................................................................................................................... Ans.

(b) 125 0 216.51 0.44721 0 0.89443CD CDF F e

96.8 lbCDF ....................................................................................................................... Ans.

(c) 1cos 41.91AD CDe e ..........................................................................................Ans.


29

2-44 The hot-air balloon shown in Fig. P2-44 is tethered with three mooring cables. The force TA in cable AD has a magnitude of 1860 N.

(a) Express TA in Cartesian vector form. (b) Determine the magnitude of the rectangular component of the force TA along BD. (c) Determine the angle between cables AD and BD. SOLUTION

(a) 2 2 2

20 30 501860 603 905 1509 N20 30 50

A A AT i j kT e i j k ..........................Ans.

(b) 2 2 2

16 25 50 0.27517 0.42995 0.8599016 25 50

BDi j ke i j k

/ 603 0.27517 905 0.42995 1509 0.85990A BD A BDT T e

/ 1074 NA BDT .................................................................................................................... Ans.

(c) 1 1 1 1074cos cos cos 54.721860

A BDAD BD

ATT ee e ..........................Ans.

2-45 A system of three cables supports the cylinder shown in Fig. P2-45. The magnitude of the force T1 in cable AO is 2000 lb. Determine

(a) The magnitude of the rectangular component of the force T1 along the line OB. (b) The angle between the force T1 and the line OB. SOLUTION

(a) 1 1 1 2 2 2

4 3 62000 1024.30 768.22 1536.44 lb4 3 6

T i j kT e i j k

2 2 2

4 6 4 0.48507 0.72761 0.485074 6 4

OBi j ke i j k

1 807.39 lb 807 lbOB OBT T e .................................................................................Ans.

(b) 1 807.39 lb 2000 1 cosOBT e

1 807.39cos 66.192000

..........................................................................................Ans.

2-46 A 2000-N force F acts on a machine component as shown in Fig. P2-46. (a) Determine the x, y, and z scalar components of the force. (b) Express the force in Cartesian vector form. (c) Determine the angle between the force and the line AB. SOLUTION

(a) 2 2 2

8 14 102000 843.27 1475.73 1054.09 N8 14 10

F i j kF e i j k

843 NxF ........................................................................................................................... Ans.

1476 NyF ........................................................................................................................ Ans.

1054 NzF ...................................................................................................................... Ans.

(b) 843 1476 1054 NF i j k ......................................................................................Ans.


30

(c) 2 2

150 130 0.75569 0.65493150 130

ABi je i j

1603.75 N 2000 1 cosAB ABF F e

1 1603.75cos 36.692000

.............................................................................................Ans.

2-47 Two flower pots are supported with wires as shown in Fig. P2-47. If = 6 , the tension force T1 has a magnitude of 13 lb, and the tension force T2 has a magnitude of 9 lb, determine the magnitude and orientation of the resultant of the forces T1 and T2.

SOLUTION

1 13cos 45 13sin 45 9.1924 9.1924 lbT i j i j

2 9cos6 9sin 6 8.9507 0.94076 lbT i j i j

1 2 9.1924 8.9507 9.1924 0.94076

0.2417 10.1332 lb

R T T i j

i j

10.14 lbR 88.63 .....................................................................................................Ans.

2-48 Two forces are applied to an eye bolt as shown in Fig. P2-48. Determine the magnitude of the resultant R of the two forces and the angle x between the line of action of the resultant and the x-axis.

SOLUTION

1 500cos60 500sin 60 250.00 433.01 NF i j i j

2 375cos30 375sin 30 324.76 187.50 NF i j i j

1 2 250 324.76 433.01 187.50

574.76 620.51 N

R F F i j

i j

846 NR 47.19 ........................................................................................................Ans.

2-49 An automobile stuck in a muddy field is being moved by using a cable fastened to a tree as shown in Fig. P2-49. The forces T1 and T2 in the two segments of the cable each have a magnitude of 650 lb. If the resultant of the three forces T1, T2, and P is to be zero, determine the magnitude of P.

SOLUTION

1 650cos5 650sin 5 647.53 56.65 lbT i j i j

2 650cos5 650sin 5 647.53 56.65 lbT i j i j

1 2 113.30R T T P j P 0

113.3 lbP j

113.3 lbP ......................................................................................................................... Ans.

2-50 Three forces act on the structural member shown in Fig. P2-50. Determine the resultant of the forces and express the result in Cartesian vector form.

SOLUTION

2500cos30 900 1265 NxR

2000 2500sin 30 3250 NyR

1265 3250 NR i j ....................................................................................................Ans.


31

2-51 Express the resultant of the two forces shown in Fig. P2-51 in Cartesian vector form, and determine the angles x, y, and z between the line of action of the resultant and the positive coordinate axes.

SOLUTION

1 100cos 44 71.934 lbxF

1 0 lbyF

1 100sin 44 69.466 lbzF

2 200cos53 120.363 lbxyF

2 2 cos70 41.167 lbx xyF F

2 2 sin 70 113.104 lby xyF F

2 200sin 53 159.727 lbzF

1 2 113.101 113.104 229.193 lbR F F i j k

113.1 113.1 229 lbR i j k ....................................................................................Ans.

279.49 lbR

1 113.101cos 66.13279.49x ............................................................................................Ans.

1 113.104cos 66.13279.49y ............................................................................................Ans.

1 229.193cos 34.91279.49z ............................................................................................Ans.

2-52 Determine the magnitude R of the resultant of the two forces shown in Fig. P2-52. Also determine the angles x, y, and z between the line of action of the resultant and the positive coordinate axes.

SOLUTION

1 10cos67 3.9073 kNxF

1 0 kNyF

1 10sin 67 9.2051 kNzF

2 0 kNxF

2 20cos60 10.0000 kNyF

2 20sin 60 17.3205 kNzF

1 2 3.907 10.000 26.526 kNR F F i j k

28.616 28.6 kNR .......................................................................................................Ans.

1 3.907cos 82.1528.616x ..............................................................................................Ans.

1 10cos 69.5528.616y .............................................................................................Ans.

1 26.526cos 22.0328.616z .............................................................................................Ans.


32

2-53 Four forces are applied to the block shown in Fig. P2-53. Determine the magnitude of the resultant and the angle between the resultant and the x-axis.

SOLUTION

15 700 649.93 lb29xF 1

2 700 259.97 lb29yF

23 300 154.35 lb34xF 2

5 300 257.25 lb34yF

32 600 222.83 lb29xF 3

5 600 557.09 lb29yF

42 900 804.98 lb5xF 4

1 900 402.49 lb5yF

223.54 lbxR 1476.80 lbyR

1494 lbR 81.39 ......................................................................................................Ans.

2-54 Determine the magnitude and orientation of the resultant of the two forces applied to the eye bolt shown in Fig. P2-54.

SOLUTION

1 2 2

120 100 44.7214 N120 240

xF 1 2 2

240 100 89.4427 N120 240

yF

1 0 NzF 2 0 NxF

2 2 2

90 75 33.5410 N90 180

yF 2 2 2

180 75 67.0820 N90 180

zF

1 2 44.721 122.984 67.082 NR F F i j k

147.054 147.1 NR .....................................................................................................Ans.

1 44.721cos 72.30147.054x ............................................................................................Ans.

1 122.984cos 33.25147.054y ............................................................................................Ans.

1 67.082cos 62.86147.054z ............................................................................................Ans.

2-55 The three forces acting on the machine component shown in Fig. P2-55 have magnitudes F1 = 500 lb, F2 = 300 lb, and F3 = 200 lb. Determine the resultant R of the three forces and express the resultant in Cartesian vector form. Also, determine the angles x, y, and z between the line of action of the resultant and the positive coordinate axes.

SOLUTION

1 1 1 2 2 2

2 35 40500 18.80 329.02 376.02 lb2 35 40

F i j kF e i j k

2 2 2 2 2 2

9 22 20300 86.92 212.46 193.15 lb9 22 20

F i j kF e i j k

3 200 lbF j


33

1 2 3 105.72 83.44 569.17 lbR F F F i j k

105.7 83.4 569 lbR i j k ..................................................................................Ans.

584.89 lbR

1 105.72cos 100.41584.89x .........................................................................................Ans.

1 83.44cos 98.20584.89y .............................................................................................Ans.

1 569.17cos 166.69584.89z .........................................................................................Ans.

2-56 Determine the magnitude R of the resultant of the three forces shown in Fig. P 2-56. Also determine the angles x, y, and z between the line of action of the resultant and the positive coordinate axes.

SOLUTION

1 10cos 26 cos 42 6.679 kNxF 1 10sin 26 4.384 kNzF

1 10cos 26 sin 42 6.014 kNyF

2 16cos 40 sin 35 7.030 kNxF 2 16sin 40 10.285 kNzF

2 16cos 40 cos35 10.040 kNyF

3 24cos50 cos60 7.713 kNxF 3 24sin 50 18.385 kNzF

3 24cos50 sin 60 13.360 kNyF

1 2 3 7.362 2.694 33.054 kNR F F F i j k

33.971 34.0 kNR .......................................................................................................Ans.

1 7.362cos 77.4833.971x ..............................................................................................Ans.

1 2.694cos 94.5533.971y .............................................................................................Ans.

1 33.054cos 13.3433.971z ..............................................................................................Ans.

2-57 Three forces are applied at the corner of the box shown in Fig. P2-57. Determine the magnitude R of the resultant and the angles x, y, and z between the line of action of of the resultant and the positive coordinate axes.

SOLUTION

1 1 1 2 2 2

8 15 2010 3.0478 5.7145 7.6194 kip8 15 20

F i j kF e i j k

2 2 2 2 2 2

16 15 2020 10.7811 10.1073 13.4763 kip16 15 20

F i j kF e i j k

3 3 3 2 2 2

16 15 525 17.7822 16.6708 5.5569 kip16 15 5

F i j kF e i j k


34

1 2 3 31.611 32.493 26.653 kipR F F F i j k

52.587 52.6 kipR .......................................................................................................Ans.

1 31.611cos 53.0552.587x ..............................................................................................Ans.

1 32.493cos 128.1652.587y .........................................................................................Ans.

1 26.653cos 59.5552.587z ..............................................................................................Ans.

2-58 The pin A shown in Fig. P2-58 supports a load F of magnitude 1250 N, and is held in place by a wire AD and compression members AB and AC. If the magnitudes of the forces CB and CC in the compression members are 995 N and 700 N, respectively, and the resultant of the four forces is zero, determine the magnitude of the force TD.

SOLUTION

2 2 2

2 6 2995 300.00 900.01 300.00 N2 6 2

Bi j kC i j k

2 2 2

3 6 2700 300.00 600.00 200.00 N3 6 2

Ci j kC i j k

2 2

6 3 0.89443 0.44721 N6 3

D D D DT T Tj kT j k

1250 NF k

0xR

1500.01 0.89443 0y DR T

750.00 0.44721 0z DR T

1677 NDT ........................................................................................................................ Ans.

2-59 Three forces are applied to an eye bolt as shown in Fig. P2-59. Determine the magnitude R of the resultant of the forces and the angle x between the line of action of the resultant and the x-axis.

SOLUTION

12 5000 4472.136 lb5xF 1

1 5000 2236.068 lb5yF

21 2000 894.427 lb5xF 2

2 2000 1788.854 lb5yF

32 1000 894.427 lb5xF 3

1 1000 447.214 lb5yF

2683.28 lbxR 4472.14 lbyR

5220 lbR 59.04 ......................................................................................................Ans.


35

2-60 Two forces, F1 and F2, are applied to an eye bolt as shown in Fig. P2-60. Determine (a) The magnitude and direction (angle x) of the resultant R of the two forces. (b) The magnitudes of two other forces Fu and Fv (along the axes u and v) that would have the same

resultant. SOLUTION

(a) 1 80cos15 77.2741 NxF 1 80sin15 20.7055 NyF

2 60cos 26 53.9276 NxF 2 60sin 26 26.3023 NyF

23.3465 NxR 47.0078 NyR

52.5 NR 63.59 .......................................................................................................Ans.

(b) 63.5886 45 18.5886

70 45 115

180 46.4114

52.4861

sin sin sinu vF F

41.9 NuF ......................................................................................................................... Ans.

18.46 NvF ........................................................................................................................ Ans.

2-61 Three forces are applied at a point on a body as shown in Fig. P2-61. Determine the resultant R of the three forces and the angles x, y, and z between the line of action of the resultant and the positive x-, y-, and z-coordinate axes.

SOLUTION

1 2 2

2 2500 353.553 353.553 lb2 2j kF j k

2 2 2

4 4800 565.685 565.685 lb4 4i jF i j

3 2 2

2 2700 494.975 494.975 lb2 2i kF i k

1 2 3 1060.660 919.238 848.528 lbR F F F i j k

1061 919 849 lbR i j k .........................................................................................Ans.

1640.121 lbR

1 1060.660cos 49.711640.121x ..........................................................................................Ans.

1 919.238cos 55.911640.121y ..........................................................................................Ans.

1 848.528cos 58.841640.121z ..........................................................................................Ans.


36

2-62 Three forces are applied at a point on a body as shown in Fig. P2-62. Determine the resultant R of the three forces and the angles x, y, and z between the line of action of the resultant and the positive x-, y-, and z-coordinate axes.

SOLUTION

1 2 2 2

3 2 330 19.1881 12.7920 19.1881 kN3 2 3i j kF i j k

2 2 2 2

3 4 1.520 11.4939 15.3252 5.7470 kN3 4 1.5i j kF i j k

3 2 2

1.5 425 8.7781 23.4082 kN1.5 4

i jF i j

1 2 3 39.4601 51.5254 24.9351 kNR F F F i j k

39.5 51.5 24.9 kNR i j k .................................................................................Ans.

69.5250 kNR

1 39.4601cos 124.5869.5250x .......................................................................................Ans.

1 51.5254cos 42.1769.5250y ...........................................................................................Ans.

1 24.9351cos 68.9869.5250z ............................................................................................Ans.

2-63 Three forces are applied to a stalled automobile as shown in Fig. P2-63. Determine the magnitude of the force F3 and the magnitude of the resultant R if the line of action of the resultant is along the x-axis.

SOLUTION

1 50cos 48 33.4565 lbxF 1 50sin 48 37.1572 lbyF

2 30cos 23 27.6152 lbxF 2 30sin 23 11.7219 lbyF

3 3 3cos37 0.79864 lbxF F F 3 3 3sin 37 0.60182 lbyF F F

333.4565 27.6152 0.79864 lbxR F R

337.1572 11.7219 0.60182 0 lbyR F

3 81.2 lbF .......................................................................................................................... Ans.

125.9 lbR ......................................................................................................................... Ans. 2-64 Three cables are used to drag a heavy crate along a horizontal surface as shown in Fig. P2-64. The resultant

R of the forces has a magnitude of 2800 N and its line of action is directed along the x-axis. Determine the magnitudes of the forces F1 and F3.

SOLUTION

1 1 1cos30 0.86603 NxF F F 1 1 1sin 30 0.50000 NyF F F

2 1600cos10 1575.6924 NxF 2 1600sin10 277.8371 NyF

3 3 3cos 45 0.70711 NxF F F 3 3 3sin 45 0.70711 NyF F F

1 30.86603 1575.6924 0.70711 2800 NxR F F


37

1 30.50000 277.8371 0.70711 0 NyR F F

1 1100 NF ......................................................................................................................... Ans.

3 385 NF ........................................................................................................................... Ans.

2-65 Two forces are applied at a point in a body as shown in Fig. P2-65. Determine (a) The magnitude and direction (angles x, y, and z) of the resultant R of the two forces. (b) The magnitude of the rectangular component of the force F1 along the line of action of the force F2. (c) The angle between forces F1 and F2. SOLUTION

(a) 1 1 1 2 2 2

1.5 6 4.5150 29.4174 117.6697 88.2523 lb1.5 6 4.5

F i j kF e i j k

21.5120cos60 36.0000 lb2.5xF

22120cos60 48.0000 lb

2.5yF

2 120sin 60 103.9231 lbzF

65.4174 69.6697 192.1754 lbR i j k

214.6269 lb 215 lbR ................................................................................................Ans.

1 65.4174cos 72.25214.6269x .........................................................................................Ans.

1 69.6697cos 71.06214.6269y .........................................................................................Ans.

1 192.1754cos 26.44214.6269z .........................................................................................Ans.

(b) 1 21/ 2

2

4582.333 38.2 lb120

FF

F F................................................................................Ans.

(c) 1 11 2

2

4582.333cos cos 75.25150 120F

F F.............................................................Ans.

2-66 Three forces are applied with cables to the anchor block shown in Fig. P2-66. Determine (a) The magnitude and direction (angles x, y, and z) of the resultant R of the three forces. (b) The magnitude of the rectangular component of the force F1 along the line of action of the force F2. (c) The angle between forces F1 and F3. SOLUTION

(a) 1 2 2 2

2.4 2.7 3.6136 64.000 72.000 96.000 N2.4 2.7 3.6

i j kF i j k

2 2 2 2

0.6 1.8 2.7250 45.455 136.364 204.545 N0.6 1.8 2.7

i j kF i j k

3 2 2 2

3.6 1.2 0.9325 300.000 100.000 75.000 N3.6 1.2 0.9

i j kF i j k

409.455 164.364 375.545 NR i j k


38

579.399 N 579 NR ...................................................................................................Ans.

1 409.455cos 45.03579.399x ...........................................................................................Ans.

1 164.364cos 106.48579.399y .......................................................................................Ans.

1 375.545cos 49.60579.399z ............................................................................................Ans.

(b) 1 21/ 2

2

12,727.232 50.9 N250

FF

F F............................................................................Ans.

(c) 1 11 3

1 3

19, 200cos cos 64.25136 325FF

F F............................................................Ans.

ch02

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