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5–10. Determine the components of the support reactions at the fixed support A on the cantilevered beam.
6 kN
30
SOLUTION
30
Equations of Equilibrium: From the free-body diagram of the cantilever beam, Fig. a, Ax, Ay, and MA can be obtained by writing the moment equation of equilibrium about point A. + ©F = 0; : x
1.5 m 1.5 m
1.5 m
4 kN
4 cos 30° - A x = 0 Ans.
A x = 3.46 kN + c ©Fy = 0;
A
A y - 6 - 4 sin 30° = 0 Ans.
A y = 8 kN
a+ ©MA = 0;MA - 6(1.5) - 4 cos 30° (1.5 sin 30°) - 4 sin 30°(3 + 1.5 cos 30°) = 0 MA = 20.2 kN # m
Ans.
Ans: Ax = 3.46 kN Ay = 8 kN MA = 20.2 kN # m 397
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5–11. Determine the reactions at the supports.
400 N/m 5
3 4
B A 3m
3m
Solution Equations of Equilibrium. NA and By can be determined directly by writing the moment equations of equilibrium about points B and A, respectively, by referring to the beam’s FBD shown in Fig. a. 4 1 a+ ΣMB = 0; (400)(6)(3) - NA a b(6) = 0 2 5 NA = 750 N
a+ ΣMA = 0; By(6) -
Ans.
1 (400)(6)(3) = 0 2 Ans.
By = 600 N
Using the result of NA to write the force equation of equilibrium along the x axis, + ΣFx = 0; 750 a 3 b - Bx = 0 S 5
Ans.
Bx = 450 N
Ans: NA = 750 N By = 600 N Bx = 450 N 398
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*5–12. 4 kN
Determine the horizontal and vertical components of reaction at the pin A and the reaction of the rocker B on the beam.
B
A
30
SOLUTION Equations of Equilibrium: From the free-body diagram of the beam, Fig. a, NB can be obtained by writing the moment equation of equilibrium about point A. a + ©MA = 0;
6m
2m
NB cos 30°(8) - 4(6) = 0 Ans.
NB = 3.464 kN = 3.46 kN
Using this result and writing the force equations of equilibrium along the x and y axes, we have + ©F = 0; : x
A x - 3.464 sin 30° = 0 Ans.
A x = 1.73 kN + c ©Fy = 0;
A y + 3.464 cos 30° - 4 = 0 Ans.
A y = 1.00 kN
Ans: NB = 3.46 kN Ax = 1.73 kN Ay = 1.00 kN 399
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5–13. Determine the reactions at the supports.
900 N/m 600 N/m
B
A 3m
3m
Solution Equations of Equilibrium. NA and By can be determined directly by writing the moment equations of equilibrium about points B and A, respectively, by referring to the FBD of the beam shown in Fig. a. a+ ΣMB = 0; 600(6)(3) +
1 (300)(3)(5) - NA(6) = 0 2 Ans.
NA = 2175 N = 2.175 kN a+ ΣMA = 0; By(6) -
1 (300)(3)(1) - 600(6)(3) = 0 2 Ans.
By = 1875 N = 1.875 kN
Also, Bx can be determined directly by writing the force equation of equilibrium along the x axis. + ΣFx = 0; Bx = 0 S
Ans.
Ans: NA = 2.175 kN By = 1.875 kN Bx = 0 400
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5–14. Determine the reactions at the supports. 800 N/m A
3m
B 1m
3m
Solution Equations of Equilibrium. NA can be determined directly by writing the moment equation of equilibrium about point B by referring to the FBD of the beam shown in Fig. a. a+ ΣMB = 0; 800(5)(2.5) - NA(3) = 0 Ans.
NA = 3333.33 N = 3.33 kN
Using this result to write the force equations of equilibrium along the x and y axes, + ΣFx = 0; Bx - 800(5) a 3 b = 0 S 5
Ans.
Bx = 2400 N = 2.40 kN
4 + c ΣFy = 0; 3333.33 - 800 (5)a b - By = 0 5
Ans.
By = 133.33 N = 133 N
Ans: NA = 3.33 kN Bx = 2.40 kN By = 133 N 401
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5–15. Determine the reactions at the supports.
5 kN
2m B
A 6 kN 2m
2m
8 kN 2m
Solution Equations of Equilibrium. Ay and NB can be determined by writing the moment equations of equilibrium about points B and A, respectively, by referring to the FBD of the truss shown in Fig. a. a+ ΣMB = 0; 8(2) + 6(4) - 5(2) - Ay(6) = 0 Ans.
Ay = 5.00 kN a+ ΣMA = 0; NB(6) - 8(4) - 6(2) - 5(2) = 0
Ans.
NB = 9.00 kN
Also, Ax can be determined directly by writing the force equation of equilibrium along x axis. + ΣFx = 0; 5 - Ax = 0 Ax = 5.00 kN S
Ans.
Ans: Ay = 5.00 kN NB = 9.00 kN Ax = 5.00 kN 402
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*5–16. Determine the tension in the cable and the horizontal and vertical components of reaction of the pin A. The pulley at D is frictionless and the cylinder weighs 80 lb.
D 2 1
A
SOLUTION
B
5 ft
C 5 ft
3 ft
Equations of Equilibrium: The tension force developed in the cable is the same throughout the whole cable. The force in the cable can be obtained directly by summing moments about point A. a + ©MA = 0;
+ ©F = 0; : x
T152 + T ¢
≤ 1102 - 801132 = 0 25 T = 74.583 lb = 74.6 lb 2
Ax - 74.583 ¢
1 25
Ans.
≤ = 0 Ans.
Ax = 33.4 lb + c ©Fy = 0;
74.583 + 74.583
2 25
- 80 - By = 0 Ans.
Ay = 61.3 lb
Ans: T = 74.6 lb Ax = 33.4 lb Ay = 61.3 lb 403
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5–17. The man attempts to support the load of boards having a weight W and a center of gravity at G. If he is standing on a smooth floor, determine the smallest angle u at which he can hold them up in the position shown. Neglect his weight.
4 ft
u
SOLUTION a + ©MB = 0;
G
4 ft
- NA (3.5) + W(3 - 4 cos u) = 0
As u becomes smaller, NA : 0 so that,
A
W(3 - 4 cos u) = 0
0.5 ft
B 3 ft
Ans.
u = 41.4°
Ans: u = 41.4° 404
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5–18. Determine the components of reaction at the supports A and B on the rod.
P L –– 2
SOLUTION Equations of Equilibrium: Since the roller at A offers no resistance to vertical movement, the vertical component of reaction at support A is equal to zero. From the free-body diagram, Ax, By, and MA can be obtained by writing the force equations of equilibrium along the x and y axes and the moment equation of equilibrium about point B, respectively. + ©F = 0; : x
Ax = 0
+ c ©Fy = 0;
By - P = 0
Pa
B
Ans.
Ans.
By = P a + ©MB = 0;
A
L –– 2
L b - MA = 0 2
MA =
PL 2
Ans.
Ans: Ax = 0 By = P PL MA = 2 405
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5–19. The man has a weight W and stands at the center of the plank. If the planes at A and B are smooth, determine the tension in the cord in terms of W and u.
B
SOLUTION a + ©MB = 0;
L W a cos f b - NA(L cos f ) = 0 2
+ ©F = 0; : x
T cos u -NB sin u = 0
+ c ©Fy = 0;
T sin u +NB cos u +
NA
f
W = 2
L
u
A
(1)
W - W= 0 2
(2)
Solving Eqs. (1) and (2) yields: T= NB =
W sin u 2
Ans.
W cos u 2
Ans: W T= sin u 2 406
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*5–20. A uniform glass rod having a length L is placed in the smooth hemispherical bowl having a radius r. Determine the angle of inclination u for equilibrium.
u B r A
SOLUTION By observation f = u. Equilibrium: a + ©MA = 0;
NB (2r cos u) - W a
L cos ub = 0 2
+Q ©Fx = 0;
NA cos u - W sin u = 0
+a©Fy = 0;
(W tan u) sin u +
WL 4r
NA = W tan u
WL - W cos u = 0 4r
sin2 u - cos2 u +
L cos u = 0 4r
(1 - cos2 u) - cos2 u + 2 cos2 u cos u =
NB =
L cos u = 0 4r
L cos u - 1 = 0 4r
L ; 2L2 + 128r2 16r
Take the positive root cos u =
L + 2L2 + 128r2 16r
u = cos - 1 ¢
L + 2L2 + 128r2 ≤ 16r
Ans.
Ans: u = cos - 1a 407
L + 2L2 + 12r 2 b 16r
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5–21. The uniform rod AB has a mass of 40 kg. Determine the force in the cable when the rod is in the position shown. There is a smooth collar at A.
A
3m
60 C
Solution
B
Equations of Equilibrium. TBC can be determined by writing the moment equation of equilibrium about point O by referring to the FBD of the rod shown in Fig. a. a+ ΣMO = 0; 40(9.81)(1.5 cos 600°) - TBC (3 sin 60°) = 0 Ans.
TBC = 113.28 N = 113 N
Ans: TBC = 113 N 408
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5–22. If the intensity of the distributed load acting on the beam is w = 3 kN>m, determine the reactions at the roller A and pin B.
A w 30
B
3m 4m
Solution Equations of Equilibrium. NA can be determined directly by writing the moment equation of equilibrium about point B by referring to the FBD of the beam shown in Fig. a. a+ ΣMB = 0; 3(4)(2) - NA sin 30° (3 sin 30°) - NA cos 30° (3 cos 30° + 4) = 0 Ans.
NA = 3.713 kN = 3.71 kN
Using this result to write the force equation of equilibrium along the x and y axes, + ΣFx = 0; 3.713 sin 30° - Bx = 0 S Ans.
Bx = 1.856 kN = 1.86 kN + c ΣFy = 0; By + 3.713 cos 30° - 3(4) = 0
Ans.
By = 8.7846 kN = 8.78 kN
Ans: NA = 3.71 kN Bx = 1.86 kN By = 8.78 kN 409
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5–23. If the roller at A and the pin at B can support a load up to 4 kN and 8 kN, respectively, determine the maximum intensity of the distributed load w, measured in kN>m, so that failure of the supports does not occur.
A w 30
B
3m 4m
Solution Equations of Equilibrium. NA can be determined directly by writing the moment equation of equilibrium about point B by referring to the FBD of the beam shown in Fig. a. a+ ΣMB = 0; w(4)(2) - NA sin 30° (3 sin 30°) - NA cos 30° (3 cos 30° + 4) = 0 NA = 1.2376 w Using this result to write the force equation of equilibrium along x and y axes, + ΣFx = 0; 1.2376 w sin 30° - Bx = 0 S
Bx = 0.6188 w
+ c ΣFy = 0; By + 1.2376 w cos 30° - w(4) = 0
By = 2.9282 w
Thus, FB = 2Bx2 + By2 = 2(0.6188 w)2 + (2.9282 w)2 = 2.9929 w
It is required that FB 6 8 kN;
2.9929 w 6 8
w 6 2.673 kN>m
1.2376 w 6 4
w 6 3.232 kN>m
And NA 6 4 kN;
Thus, the maximum intensity of the distributed load is Ans.
w = 2.673 kN>m = 2.67 kN>m
Ans: w = 2.67 kN>m 410
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*5–24. The relay regulates voltage and current. Determine the force in the spring CD, which has a stiffness of k 120 N m, so that it will allow the armature to make contact at A in figure (a) with a vertical force of 0.4 N. Also, determine the force in the spring when the coil is energized and attracts the armature to E, figure (b), thereby breaking contact at A.
50 mm 50 mm 30 mm 10°
A
B
A
C
E
B
C k
k
D
D
SOLUTION From Fig. (a): a + ©MB = 0;
0.4(100 cos 10°) - Fs (30 cos 10°) = 0
(a)
Ans.
Fs = 1.333 N = 1.33 N Fs = kx;
(b)
1.333 = 120 x x = 0.01111 m = 11.11 mm
From Fig (b), energizing the coil requires the spring to be stretched an additional amount ¢x = 30 sin 10° = 5.209 mm. Thus x¿ = 11.11 + 5.209 = 16.32 mm Ans.
Fs = 120 (0.01632) = 1.96 N
Ans: Fs = 1.33 N Fs = 1.96 N 411
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5–25. Determine the reactions on the bent rod which is supported by a smooth surface at B and by a collar at A, which is fixed to the rod and is free to slide over the fixed inclined rod.
100 lb 3 ft
3 ft
200 lb ft
A
2 ft
3 5
4
B
13 12
5
SOLUTION 12 5 b (6) - NB a b (2) = 0 13 13
a + ©MA = 0;
MA - 100 (3) - 200 + NB a
+ ©F = 0; : x
4 5 NA a b - NB a b = 0 5 13
+ c ©Fy = 0;
3 12 NA a b + NB a b - 100 = 0 5 13
Solving, NA = 39.7 lb
Ans.
NB = 82.5 lb
Ans.
MA = 106 lb # ft
Ans.
Ans: NA = 39.7 lb NB = 82.5 lb MA = 106 lb # ft 412
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5–26. The mobile crane is symmetrically supported by two outriggers at A and two at B in order to relieve the suspension of the truck upon which it rests and to provide greater stability. If the crane and truck have a mass of 18 Mg and center of mass at G1, and the boom has a mass of 1.8 Mg and a center of mass at G2, determine the vertical reactions at each of the four outriggers as a function of the boom angle u when the boom is supporting a load having a mass of 1.2 Mg. Plot the results measured from u = 0° to the critical angle where tipping starts to occur.
6.25 m
G2 6m
SOLUTION + ©MB = 0;
θ
- NA (4) + 18 A 103 B (9.81)(1) + 1.8 A 103 B (9.81) (2 - 6 sin u)
G1
+ 1.2 A 103 B (9.81) (2 - 12.25 sin u) = 0 NA = 58 860 - 62 539 sin u
A 2m
Tipping occurs when NA = 0, or
1m 1m
Ans.
u = 70.3° + c ©Fy = 0;
B
NB + 58 860 - 62 539 sin u - (18 + 1.8 + 1.2) A 103 B (9.81) = 0 NB = 147 150 + 62 539 sin u
Since there are two outriggers on each side of the crane, NA ¿ = (29.4 - 31.3 sin u) kN = NA 2 NB¿ =
Ans.
NB = (73.6 + 31.3 sin u) kN 2
Ans.
Ans: u = 70.3° = NA = (29.4 - 31.3 sin u) kN NB= = (73.6 + 31.3 sin u) kN 413
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5–27. Determine the reactions acting on the smooth uniform bar, which has a mass of 20 kg.
B
4m
A
30º
60º
Solution Equations of Equilibrium. NB can be determined directly by writing the moment equation of equilibrium about point A by referring to the FBD of the bar shown in Fig. a. a+ ΣMA = 0; NB cos 30°(4) - 20(9.81) cos 30°(2) = 0
Ans.
NB = 98.1 N
Using this result to write the force equation of equilibrium along the x and y axes, + ΣFx = 0; Ax - 98.1 sin 60° = 0 S
Ax = 84.96 N = 85.0 N
Ans.
+ c ΣFy = 0; Ay + 98.1 cos 60° - 20(9.81) = 0 Ans.
Ay = 147.15 N = 147 N
Ans: NB = 98.1 N Ax = 85.0 N Ay = 147 N 414
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*5–28. A linear torsional spring deforms such that an applied couple moment M is related to the spring’s rotation u in radians by the equation M = (20 u) N # m. If such a spring is attached to the end of a pin-connected uniform 10-kg rod, determine the angle u for equilibrium. The spring is undeformed when u = 0°.
A
u
M (20 u) N m 0.5 m
Solution a+ ΣMA = 0; - 98.1 (0.25 cos u) + 20(u) = 0 Solving for u, Ans.
u = 47.5°
Ans: u = 47.5° 415
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5–29. Determine the force P needed to pull the 50-kg roller over the smooth step. Take u = 30°.
P
u A
50 mm
300 mm
B
Solution Equations of Equilibrium. P can be determined directly by writing the moment equation of Equilibrium about point B, by referring to the FBD of the roller shown in Fig. a. a+ ΣMB = 0; P cos 30°(0.25) + P sin 30° ( 20.32 - 0.252 2 - 50(9.81) 20.32 - 0.252 = 0 Ans.
P = 271.66 N = 272 N
Ans: P = 272 N 416
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5–30. Determine the magnitude and direction u of the minimum force P needed to pull the 50-kg roller over the smooth step.
P
u A
50 mm
300 mm
B
Solution Equations of Equilibrium. P will be minimum if its orientation produces the greatest moment about point B. This happens when it acts perpendicular to AB as shown in Fig. a. Thus u = f = cos-1 a
0.25 b = 33.56° = 33.6° 0.3
Ans.
Pmin can be determined by writing the moment equation of equilibrium about point B by referring to the FBD of the roller shown in Fig. b. a+ ΣMB = 0; Pmin (0.3) - 50(9.81)(0.3 sin 33.56°) = 0 Ans.
Pmin = 271.13 N = 271 N
Ans: Pmin = 271 N 417
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5–31. The operation of the fuel pump for an automobile depends on the reciprocating action of the rocker arm ABC, which is pinned at B and is spring loaded at A and D. When the smooth cam C is in the position shown, determine the horizontal and vertical components of force at the pin and the force along the spring DF for equilibrium. The vertical force acting on the rocker arm at A is FA = 60 N, and at C it is FC = 125 N.
E
30° F FC = 125 N
FA = 60 N
B A
C
D
SOLUTION a + ©MB = 0;
- 60(50) - FB cos 30°(10) + 125(30) = 0
50 mm
- Bx + 86.6025 sin 30° = 0 Ans.
Bx = 43.3 N + c ©Fy = 0;
20 mm
Ans.
FB = 86.6025 = 86.6 N + ©F = 0; : x
10 mm
60 - By - 86.6025 cos 30° + 125 = 0 Ans.
By = 110 N
Ans: FB = 86.6 N Bx = 43.3 N By = 110 N 418
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*5–32. Determine the magnitude of force at the pin A and in the cable BC needed to support the 500-lb load. Neglect the weight of the boom AB.
B
8 ft C
22
A
35
SOLUTION Equations of Equilibrium: The force in cable BC can be obtained directly by summing moments about point A. a + ©MA = 0;
FBC sin 13°(8) - 500 cos 35°(8) = 0 Ans.
FBC = 1820.7 lb = 1.82 kip + Q ©Fx = 0;
A x - 1820.7 cos 13° - 500 sin 35° = 0 A x = 2060.9 lb
a + ©Fy = 0;
A y + 1820.7 sin 13° - 500 cos 35° = 0 Ay = 0
Thus,
Ans.
FA = A x = 2060.9 lb = 2.06 kip
Ans: FBC = 1.82 kip FA = 2.06 kip 419
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5–33. The dimensions of a jib crane, which is manufactured by the Basick Co., are given in the figure. If the crane has a mass of 800 kg and a center of mass at G, and the maximum rated force at its end is F 15 kN, determine the reactions at its bearings. The bearing at A is a journal bearing and supports only a horizontal force, whereas the bearing at B is a thrust bearing that supports both horizontal and vertical components.
3m A 0.75 m 2m
G F
SOLUTION a + ©MB = 0;
Ans.
Ax = 25.4 kN + c ©Fy = 0;
By - 800 (9.81) - 15 000 = 0 Ans.
By = 22.8 kN + ©F = 0; : x
B
Ax (2) - 800 (9.81) (0.75) - 15 000(3) = 0
Bx - 25.4 = 0 Ans.
Bx = 25.4 kN
Ans: Ax = 25.4 kN By = 22.8 kN Bx = 25.4 kN 420
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5–34. The dimensions of a jib crane, which is manufactured by the Basick Co., are given in the figure. The crane has a mass of 800 kg and a center of mass at G. The bearing at A is a journal bearing and can support a horizontal force, whereas the bearing at B is a thrust bearing that supports both horizontal and vertical components. Determine the maximum load F that can be suspended from its end if the selected bearings at A and B can sustain a maximum resultant load of 24 kN and 34 kN, respectively.
3m A 0.75 m 2m
F
SOLUTION a + ©MB = 0;
G
B
Ax (2) - 800 (9.81) (0.75) - F (3) = 0
+ c ©Fy = 0;
By - 800 (9.81) - F = 0
+ ©F = 0; : x
Bx - Ax = 0
Assume Ax = 24 000 N. Solving, Bx = 24 kN By = 21.9 kN Ans.
F = 14.0 kN FB =
(24)2 + (21.9)2 = 32.5 kN 6 34 kN
OK
Ans: F = 14.0 kN 421
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5–35. The smooth pipe rests against the opening at the points of contact A, B, and C. Determine the reactions at these points needed to support the force of 300 N. Neglect the pipe’s thickness in the calculation.
A C
30 300 N
0.5 m
30 0.5 m
0.26 m
B
0.15 m
Solution Equations of Equilibrium. NA can be determined directly by writing the force equation of equilibrium along the x axis by referring to the FBD of the pipe shown in Fig. a. + ΣFx = 0; NA cos 30° - 300 sin 30° = 0 NA = 173.21 N = 173 N S
Ans.
Using this result to write the moment equations of equilibrium about points B and C, a+ ΣMB = 0; 300 cos 30°(1) - 173.21 cos 30°(0.26) - 173.21 sin 30°(0.15) - NC (0.5) = 0 Ans.
NC = 415.63 N = 416 N
a+ ΣMC = 0; 300 cos 30°(0.5) - 173.21 cos 30°(0.26) - 173.21 sin 30°(0.65) - NB(0.5) = 0 Ans.
NB = 69.22 N = 69.2 N
Ans: NA = 173 N NC = 416 N NB = 69.2 N 422
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*5–36. The beam of negligible weight is supported horizontally by two springs. If the beam is horizontal and the springs are unstretched when the load is removed, determine the angle of tilt of the beam when the load is applied.
B
A 600 N/m
kA = 1 kN/m
kB = 1.5 kN/m D
C 3m
3m
Solution Equations of Equilibrium. FA and FB can be determined directly by writing the moment equations of equilibrium about points B and A, respectively, by referring to the FBD of the beam shown in Fig. a. Assuming that the angle of tilt is small, a+ ΣMA = 0; FB(6) -
1 (600)(3)(2) = 0 FB = 300 N 2
1 a+ ΣMB = 0; (600)(3)(4) - FA(6) = 0 FA = 600 N 2 Thus, the stretches of springs A and B can be determined from FA = kAxA;
600 = 1000 xA
xA = 0.6 m
FB = kB xB;
300 = 1500 xB
xB = 0.2 m
From the geometry shown in Fig. b u = sin-1 a
0.4 b = 3.82° 6
Ans.
The assumption of small u is confirmed.
Ans: u = 3.82° 423
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5–37. The cantilevered jib crane is used to support the load of 780 lb. If x = 5 ft, determine the reactions at the supports. Note that the supports are collars that allow the crane to rotate freely about the vertical axis. The collar at B supports a force in the vertical direction, whereas the one at A does not.
8 ft B T 4 ft
x
SOLUTION
780 lb
Equations of Equilibrium: Referring to the FBD of the jib crane shown in Fig. a, we notice that NA and By can be obtained directly by writing the moment equation of equilibrium about point B and force equation of equilibrium along the y axis, respectively. a+ ©MB = 0;
NA(4) - 780(5) = 0
NA = 975 lb
Ans.
+ c ©Fy = 0;
By - 780 = 0
By = 780
Ans.
A
Using the result of NA to write the force equation of equilibrium along x axis, + : ©Fx = 0;
975 - Bx = 0
Ans.
Bx = 975 lb
Ans: NA = 975 lb Bx = 975 lb By = 780 lb 424
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5–38. The cantilevered jib crane is used to support the load of 780 lb. If the trolley T can be placed anywhere between 1.5 ft … x … 7.5 ft, determine the maximum magnitude of reaction at the supports A and B. Note that the supports are collars that allow the crane to rotate freely about the vertical axis. The collar at B supports a force in the vertical direction, whereas the one at A does not.
8 ft B T 4 ft
x 780 lb
SOLUTION
A
Require x = 7.5 ft a + ©MA = 0;
- 780(7.5) + Bx (4) = 0 Bx = 1462.5 lb
+ ©F = 0; : x
Ax - 1462.5 = 0 Ans.
Ax = 1462.5 = 1462 lb + c ©Fy = 0;
By - 780 = 0 By = 780 lb FB = 2(1462.5)2 + (780)2 Ans.
= 1657.5 lb = 1.66 kip
Ans: Ax = 1.46 kip FB = 1.66 kip 425
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5–39. The bar of negligible weight is supported by two springs, each having a stiffness k = 100 N>m. If the springs are originally unstretched, and the force is vertical as shown, determine the angle u the bar makes with the horizontal, when the 30-N force is applied to the bar.
k 2m
1m C 30 N
B k
Solution Equations of Equilibrium. FA and FB can be determined directly by writing the moment equation of equilibrium about points B and A respectively by referring to the FBD of the bar shown in Fig. a. a+ ΣMB = 0; 30(1) - FA(2) = 0 FA = 15 N
A
a+ ΣMA = 0; 30(3) - FB(2) = 0 FB = 45 N Thus, the stretches of springs A and B can be determined from FA = kxA;
15 = 100 xA
xA = 0.15 m
FB = k xB;
45 = 100 xB
xB = 0.45 m
From the geometry shown in Fig. b, d 2 - d = ; 0.45 0.15
d = 1.5 m
Thus
u = sin-1 a
0.45 b = 17.46° = 17.5° 1.5
Ans.
Note: The moment equations are set up assuming small u, but even with non-small u the reactions come out with the same FA, FB, and then the rest of the solution goes through as before.
Ans: u = 17.5° 426
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*5–40. Determine the stiffness k of each spring so that the 30-N force causes the bar to tip u = 15° when the force is applied. Originally the bar is horizontal and the springs are unstretched. Neglect the weight of the bar.
k 2m
1m C 30 N
B k
Solution Equations of Equilibrium. FA and FB can be determined directly by writing the moment equation of equilibrium about points B and A respectively by referring to the FBD of the bar shown in Fig. a. a+ ΣMB = 0; 30(1) - FA(2) = 0 FA = 15 N
A
a+ ΣMA = 0; 30(3) - FB(2) = 0 FB = 45 N Thus, the stretches of springs A and B can be determined from FA = kxA;
15 = kxA
xA =
15 k
FB = k xB;
45 = kxB
xB =
45 k
From the geometry shown in Fig. b d 2 - d = ; 45>k 15>k
d = 1.5 m
Thus, sin 15° =
45>k 1.5
Ans.
k = 115.91 N>m = 116 N>m
Note: The moment equations are set up assuming small u, but even with non-small u the reactions come out with the same FA, FB, and then the rest of the solution goes through as before.
Ans: k = 116 N>m 427
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5–41. The bulk head AD is subjected to both water and soilbackfill pressures. Assuming AD is “pinned” to the ground at A, determine the horizontal and vertical reactions there and also the required tension in the ground anchor BC necessary for equilibrium. The bulk head has a mass of 800 kg.
D 0.5 m
B
C
F
6m 4m
118 kN/m A
SOLUTION
310 kN/m
Equations of Equilibrium: The force in ground anchor BC can be obtained directly by summing moments about point A. a + ©MA = 0;
1007.512.1672 - 23611.3332 - F162 = 0 Ans.
F = 311.375 kN = 311 kN + ©F = 0; : x
Ax + 311.375 + 236 - 1007.5 = 0 Ans.
Ax = 460 kN + c ©Fy = 0;
Ay - 7.848 = 0
Ans.
Ay = 7.85 kN
Ans: F = 311 kN Ax = 460 kN Ay = 7.85 kN 428
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5–42. The boom supports the two vertical loads. Neglect the size of the collars at D and B and the thickness of the boom, and compute the horizontal and vertical components of force at the pin A and the force in cable CB. Set F1 = 800 N and F2 = 350 N.
C
3
5 4
SOLUTION a + ©MA = 0;
1m
- 800(1.5 cos 30°) - 350(2.5 cos 30°) +
3 4 F (2.5 sin 30°) + FCB(2.5 cos 30°) = 0 5 CB 5
4 Ax - (781.6) = 0 5
Ay - 800 - 350 +
30 A
F1
Ans.
Ax = 625 N + c ©Fy = 0;
D F2
Ans.
FCB = 781.6 = 782 N + ©F = 0; : x
1.5 m
B
3 (781.6) = 0 5 Ans.
Ay = 681 N
Ans: FCB = 782 N Ax = 625 N Ay = 681 N 429
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5–43. The boom is intended to support two vertical loads, F1 and F2. If the cable CB can sustain a maximum load of 1500 N before it fails, determine the critical loads if F1 = 2F2. Also, what is the magnitude of the maximum reaction at pin A?
C
3
5 4
SOLUTION a + ©MA = 0;
1m
- 2F2(1.5 cos 30°) - F2(2.5 cos 30°) +
4 3 (1500)(2.5 sin 30°) + (1500)(2.5 cos 30°) = 0 5 5
1.5 m
F1 = 2F2 = 1448 N
Ax -
30 A
F1
Ans.
F1 = 1.45 kN + ©F = 0; : x
D F2
Ans.
F2 = 724 N
B
4 (1500) = 0 5
Ax = 1200 N + c ©Fy = 0;
Ay - 724 - 1448 +
3 (1500) = 0 5
Ay = 1272 N FA = 2(1200)2 + (1272)2 = 1749 N = 1.75 kN
Ans.
Ans: F2 = 724 N F1 = 1.45 kN FA = 1.75 kN 430
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*5–44. The 10-kg uniform rod is pinned at end A. If it is also subjected to a couple moment of 50 N # m, determine the smallest angle u for equilibrium. The spring is unstretched when u = 0, and has a stiffness of k = 60 N>m.
B k 60 N/m
u
2m
0.5 m
A
Solution
50 N m
Equations of Equilibrium. Here the spring stretches x = 2 sin u. The force in the spring is Fsp = kx = 60 (2 sin u) = 120 sin u. Write the moment equation of equilibrium about point A by referring to the FBD of the rod shown in Fig. a, a+ ΣMA = 0; 120 sin u cos u (2) - 10(9.81) sin u (1) - 50 = 0
240 sin u cos u - 98.1 sin u - 50 = 0
Solve numerically
Ans.
u = 24.598° = 24.6°
Ans: u = 24.6° 431
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5–45. The man uses the hand truck to move material up the step. If the truck and its contents have a mass of 50 kg with center of gravity at G, determine the normal reaction on both wheels and the magnitude and direction of the minimum force required at the grip B needed to lift the load.
0.4 m B 0.5 m 0.2 m
G
0.4 m
60 0.4 m
Solution
A
0.1 m
Equations of Equilibriums. Py can be determined directly by writing the force equation of equilibrium along y axis by referring to the FBD of the hand truck shown in Fig. a. + c ΣFy = 0; Py - 50(9.81) = 0
Py = 490.5 N
Using this result to write the moment equation of equilibrium about point A, a+ ΣMA = 0; Px sin 60°(1.3) - Px cos 60°(0.1) - 490.5 cos 30°(0.1)
- 490.5 sin 30°(1.3) - 50(9.81) sin 60°(0.5)
+ 50(9.81) cos 60°(0.4) = 0
Px = 442.07 N
Thus, the magnitude of minimum force P, Fig. b, is P = 2Px2 + Py2 = 2442.072 + 490.52 = 660.32 N = 660 N
Ans.
and the angle is u = tan-1 a
490.5 b = 47.97° = 48.0° b 442.07
Ans.
Write the force equation of equilibrium along x axis, + ΣFx = 0; NA - 442.07 = 0 NA = 442.07 N = 442 N S
Ans.
Ans: P = 660 N NA = 442 N u = 48.0° b 432
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5–46. Three uniform books, each having a weight W and length a, are stacked as shown. Determine the maximum distance d that the top book can extend out from the bottom one so the stack does not topple over.
a
d
SOLUTION Equilibrium: For top two books, the upper book will topple when the center of gravity of this book is to the right of point A. Therefore, the maximum distance from the right edge of this book to point A is a/2. Equation of Equilibrium: For the entire three books, the top two books will topple about point B. a + ©MB = 0;
a W(a-d) -W ad- b = 0 2 d =
3a 4
Ans.
Ans: d = 433
3a 4
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5–47. Determine the reactions at the pin A and the tension in cord BC. Set F = 40 kN. Neglect the thickness of the beam.
F
26 kN
C 13
12
5
5
3
4
A
B
2m
4m
Solution a+ ΣMA = 0; - 26 a
12 3 b(2) - 40(6) + FBC(6) = 0 13 5
Ans.
FBC = 80 kN
+ ΣFx = 0; 80 a 4 b - Ax - 26 a 5 b = 0 S 5 13
+ c ΣFy = 0; Ay - 26 a
Ans.
Ax = 54 kN
12 3 b - 40 + 80 a b = 0 13 5
Ans.
Ay = 16 kN
Ans: FBC = 80 kN Ax = 54 kN Ay = 16 kN 434
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*5–48. If rope BC will fail when the tension becomes 50 kN, determine the greatest vertical load F that can be applied to the beam at B. What is the magnitude of the reaction at A for this loading? Neglect the thickness of the beam.
F
26 kN
C 13
12
5
5
3
4
A
B
2m
4m
Solution a+ ΣMA = 0; -26 a
12 3 b(2) - F(6) + (50)(6) = 0 13 5
Ans.
F = 22 kN
+ ΣFx = 0; 50 a 4 b - Ax - 26 a 5 b = 0 S 5 13
+ c ΣFy = 0; Ay - 26 a
Ans.
Ax = 30 kN
12 3 b - 22 + 50 a b = 0 13 5
Ans.
Ay = 16 kN
Ans: F = 22 kN Ax = 30 kN Ay = 16 kN 435
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5–49. The rigid metal strip of negligible weight is used as part of an electromagnetic switch. If the stiffness of the springs at A and B is k = 5 N>m, and the strip is originally horizontal when the springs are unstretched, determine the smallest force needed to close the contact gap at C.
50 mm
50 mm
k
B C
A k
10 mm
SOLUTION ©MB = 0;
FA = FC = F
©Fy = 0;
FB = 2F
x 50 - x = yB yA
(1)
kyB 2F = F kyA (2)
2yA = yB Substituting into Eq. (1): 50 - x x = yA 2yA 2x = 50 - x x =
50 = 16.67 mm 3
x 100 - x = yA 10 Set x = 16.67, then yA = 2 mm From Eq. (2), yB = 4 mm Ans.
FC = FA = kyA = (5)(0.002) = 10 mN
Ans: FC = 10 mN 436
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5–50. The rigid metal strip of negligible weight is used as part of an electromagnetic switch. Determine the maximum stiffness k of the springs at A and B so that the contact at C closes when the vertical force developed there is 0.5 N. Originally the strip is horizontal as shown.
50 mm
50 mm
k
B C
A k
10 mm
SOLUTION ©MB = 0;
FA = FC = F
©Fy = 0;
FB = 2F
x 50 - x = yA yB
(1)
kyB 2F = F kyA (2)
2yA = yB Substituting into Eq. (1): 50 - x x = yA 2yA 2x = 50 - x x =
50 = 16.67 mm 3
x 100 - x = yA 10 Set x = 16.67, then yA = 2 mm From Eq. (2), yB = 4 mm FC = FA = kyA 0.5 = k(0.002) Ans.
k = 250 N/m
Ans: k = 250 N>m 437
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5–51. The cantilever footing is used to support a wall near its edge A so that it causes a uniform soil pressure under the footing. Determine the uniform distribution loads wA and wB, measured in lb ft at pads A and B, necessary to support the wall forces of 8 000 lb and 20 000 lb.
20 000 lb 8000 lb
0.25 ft
1.5 ft B
A
SOLUTION a + ©MA = 0;
- 8000 (10.5) + wB (3)(10.5) + 20 000 (0.75) = 0
8 ft
wB 3 ft
Ans.
wB = 2190.5 lb/ft = 2.19 kip/ft + c ©Fy = 0;
wA 2 ft
2190.5 (3) - 28 000 + wA (2) = 0 Ans.
wA = 10.7 kip/ft
Ans: wB = 2.19 kip>ft wA = 10.7 kip>ft 438
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*5–52. The uniform beam has a weight W and length l and is supported by a pin at A and a cable BC. Determine the horizontal and vertical components of reaction at A and the tension in the cable necessary to hold the beam in the position shown.
C l B A
SOLUTION Equations of Equilibrium: The tension in the cable can be obtained directly by summing moments about point A. a + ©MA = 0;
l T sin 1f - u2l - W cos u a b = 0 2 T =
W cos u 2 sin 1f - u2
Using the result T =
W cos u 2 sin 1f - u2
+ ©F = 0; : x
W cos u b cos f - Ax = 0 2 sin 1f - u2
a
Ax = + c ©Fy = 0;
Ay + a Ay =
Ans.
W cos f cos u 2 sin 1f - u2
Ans.
W cos u b sin f - W = 0 2 sin 1f - u2
W1sin f cos u - 2 cos f sin u2
Ans.
2 sin f - u
Ans: W cos u 2 sin(f - u) Wcos f cos u Ax = 2 sin(f - u) W(sin f cos u - 2 cos f sin u) Ay = 2 sin (f - u) T =
439
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5–53. A boy stands out at the end of the diving board, which is supported by two springs A and B, each having a stiffness of k = 15kN>m. In the position shown the board is horizontal. If the boy has a mass of 40 kg, determine the angle of tilt which the board makes with the horizontal after he jumps off. Neglect the weight of the board and assume it is rigid.
1m A
SOLUTION
3m B
Equations of Equilibrium: The spring force at A and B can be obtained directly by summing moments about points B and A, respectively. a + ©MB = 0;
FA (1) - 392.4(3) = 0
FA = 1177.2 N
a + ©MA = 0;
FB (1) - 392.4(4) = 0
FB = 1569.6 N
Spring Formula: Applying ¢ = ¢A =
F , we have k
1177.2 = 0.07848 m 15(103)
¢B =
1569.6 = 0.10464 m 15(103)
Geometry: The angle of tilt a is a = tan - 1 a
0.10464 + 0.07848 b = 10.4° 1
Ans.
Ans: a = 10.4° 440
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5–54. The 30-N uniform rod has a length of l = 1 m. If s = 1.5 m, determine the distance h of placement at the end A along the smooth wall for equilibrium.
C
SOLUTION
h
Equations of Equilibrium: Referring to the FBD of the rod shown in Fig. a, write the moment equation of equilibrium about point A. a + ©MA = 0;
A
T sin f(1) - 3 sin u(0.5) = 0 T =
s
1.5 sin u sin f
l B
Using this result to write the force equation of equilibrium along y axis, + c ©Fy = 0;
a
15 sin u b cos (u - f) - 3 = 0 sin f (1)
sin u cos (u - f) - 2 sin f = 0
Geometry: Applying the sine law with sin (180° - u) = sin u by referring to Fig. b, sin f sin u = ; h 1.5
sin u = a
h b sin u 1.5
(2)
Substituting Eq. (2) into (1) yields sin u[cos (u - f) -
4 h] = 0 3
since sin u Z 0, then cos (u - f) - (4>3)h
(3)
cos (u - f) = (4>3)h
Again, applying law of cosine by referring to Fig. b, l2 = h2 + 1.52 - 2(h)(1.5) cos (u - f) cos (u - f) =
h2 + 1.25 3h
(4)
Equating Eqs. (3) and (4) yields h2 + 1.25 4 h = 3 3h 3h2 = 1.25 Ans.
h = 0.645 m
Ans: h = 0.645 m 441
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5–55. The uniform rod has a length l and weight W. It is supported at one end A by a smooth wall and the other end by a cord of length s which is attached to the wall as shown. Determine the placement h for equilibrium.
C
h s A
SOLUTION Equations of Equilibrium: The tension in the cable can be obtained directly by summing moments about point A.
l B
a + ©MA = 0;
l T sin f(l) - W sin u a b = 0 2 T =
Using the result T = + c ©Fy = 0;
W sin u 2 sin f
W sin u , 2 sin f
W sin u cos (u - f) - W = 0 2 sin f (1)
sin u cos (u - f) - 2 sin f = 0 Geometry: Applying the sine law with sin (180° - u) = sin u, we have sin f sin u = s h
sin f =
h sin u s
(2)
Substituting Eq. (2) into (1) yields cos (u - f) =
2h s
(3)
Using the cosine law, l2 = h2 + s2 - 2hs cos (u - f) cos (u - f) =
h2 + s2 - l2 2hs
(4)
Equating Eqs. (3) and (4) yields h2 + s 2 - l2 2h = s 2hs h =
A
s2 - l2 3
Ans.
Ans: h = 442
s2 - l 2 A 3
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*5–56. The uniform rod of length L and weight W is supported on the smooth planes. Determine its position u for equilibrium. Neglect the thickness of the rod.
L u f
c
SOLUTION a + ©MB = 0;
- Wa
L cos ub + NA cos f (L cos u) + NA sin f (L sin u) = 0 2 NA =
W cos u 2 cos (f - u)
+ ©F = 0; : x
NB sin c - NA sin f = 0
+ c ©Fy = 0;
NB cos c + NA cos f - W = 0 NB =
(1) (2)
W - NA cos f cos c
(3)
Substituting Eqs. (1) and (3) into Eq. (2): aW -
W cos u sin f W cos u cos f b tan c = 0 2 cos (f - u) 2 cos (f - u)
2 cos (f - u) tan c - cos u tan c cos f - cos u sin f = 0 sin u (2 sin f tan c) - cos u (sin f - cos f tan c) = 0 tan u =
sin f - cos f tan c 2 sin f tan c
u = tan - 1 a
1 1 cot c - cot f b 2 2
Ans.
Ans: 1
u = tan - 1 a 2 cot c 443
1 cot f b 2
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5–57. The beam is subjected to the two concentrated loads. Assuming that the foundation exerts a linearly varying load distribution on its bottom, determine the load intensities w1 and w2 for equilibrium if P = 500 lb and L = 12 ft.
P L –– 3
2P L –– 3
L –– 3
w1 w2
SOLUTION Equations of Equilibrium: Referring to the FBD of the beam shown in Fig. a, we notice that W1 can be obtained directly by writing moment equations of equilibrium about point A. a + ©MA = 0;
500(4) - W1(12)(2) = 0 Ans.
W1 = 83.33 lb>ft = 83.3 lb>ft Using this result to write the force equation of equilibrium along y axis, + c ©Fy = 0;
83.33(12) +
1 (W2 - 83.33)(12) - 500 - 1000 = 0 2 Ans.
W2 = 166.67 lb>ft = 167 lb>ft
Ans: w1 = 83.3 lb>ft w2 = 167 lb>ft 444
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5–58. The beam is subjected to the two concentrated loads. Assuming that the foundation exerts a linearly varying load distribution on its bottom, determine the load intensities w1 and w2 for equilibrium in terms of the parameters shown.
P L –– 3
SOLUTION
2P L –– 3
L –– 3
w1
Equations of Equilibrium: The load intensity w1 can be determined directly by summing moments about point A. a + ©MA = 0;
Pa
L L b - w1L a b = 0 3 6 w1 =
+ c ©Fy = 0;
w2
2P L
Ans.
2P 2P 1 a w2 bL + 1L2 - 3P = 0 2 L L w2 =
4P L
Ans.
Ans: w1 = 445
2P 4P ,w = L 2 L
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5–59. The rod supports a weight of 200 lb and is pinned at its end A. If it is also subjected to a couple moment of 100 lb # ft, determine the angle u for equilibrium. The spring has an unstretched length of 2 ft and a stiffness of k = 50 lb/ft.
100 lb ft A
2 ft k
50 lb/ft
u 3 ft
3 ft
B
SOLUTION a + ©MA = 0;
100 + 200 (3 cos u) - Fs (6 cos u) = 0
Fs = kx;
Fs = 50 (6 sin u) 100 + 600 cos u - 1800 sin u cos u = 0 cos u - 1.5 sin 2u + 0.1667 = 0
Solving by trial and error, Ans.
u = 23.2° and u = 85.2°
Ans: u = 23.2° 85.2° 446
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*5–60. Determine the distance d for placement of the load P for equilibrium of the smooth bar in the position u as shown. Neglect the weight of the bar.
d P
u a
SOLUTION + c ©Fy = 0;
R cos u - P = 0
a + ©MA = 0;
- P(d cos u) + R a Rd cos2 u = R a d =
a b = 0 cos u
a b cos u
a cos3 u
Ans.
Also; Require forces to be concurrent at point O. AO = d cos u =
a>cos u cos u
Thus, d =
a cos3 u
Ans.
Ans: d = 447
a cos3 u
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5–61. If d = 1 m, and u = 30°, determine the normal reaction at the smooth supports and the required distance a for the placement of the roller if P = 600 N. Neglect the weight of the bar.
d
SOLUTION
u
Equations of Equilibrium: Referring to the FBD of the rod shown in Fig. a, a+ ©MA = 0;
a b - 600 cos 30°(1) = 0 cos 30° 450 NB = a
P
a
NB = a
a+ ©Fy¿ = 0;
NB - NA sin 30° - 600 cos 30° = 0 NB - 0.5NA = 600 cos 30°
+Q ©F = 0; x¿
NA cos 30° - 600 sin 30° = 0 NA = 346.41 N = 346 N
(1) (2) Ans.
.
Ans.
.
Ans.
.
Substitute this result into Eq (2), NB - 0.5(346.41) = 600 cos 30° NB = 692.82 N = 693 N Substitute this result into Eq (1), 450 a a = 0.6495 m 692.82 =
a = 0.650 m
Ans: NA = 346 N NB = 693 N a = 0.650 m 448
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5–62. The uniform load has a mass of 600 kg and is lifted using a uniform 30-kg strongback beam BAC and four wire ropes as shown. Determine the tension in each segment of rope and the force that must be applied to the sling at A.
F 1.25 m B
SOLUTION
A C
2m
Equations of Equilibrium: Due to symmetry, all wires are subjected to the same tension. This condition statisfies moment equilibrium about the x and y axes and force equilibrium along y axis. ©Fz = 0;
1.25 m
1.5 m 1.5 m
4 4T a b - 5886 = 0 5 Ans.
T = 1839.375 N = 1.84 kN
The force F applied to the sling A must support the weight of the load and strongback beam. Hence ©Fz = 0;
F - 60019.812 - 3019.812 = 0 Ans.
F = 6180.3 N = 6.18 kN
Ans: T = 1.84 kN F = 6.18 kN 449
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5–63. Due to an unequal distribution of fuel in the wing tanks, the centers of gravity for the airplane fuselage A and wings B and C are located as shown. If these components have weights WA = 45 000 lb, WB = 8000 lb, and WC = 6000 lb, determine the normal reactions of the wheels D, E, and F on the ground.
z
D
B
A C E
SOLUTION ©Mx = 0;
8000(6) - RD (14) - 6000(8) + RE (14) = 0
©My = 0;
8000(4) + 45 000(7) + 6000(4) - RF (27) = 0
©Fz = 0;
8 ft
6 ft x
F 8 ft
20 ft 6 ft
4 ft 3 ft
y
RD + RE + RF - 8000 - 6000 - 45 000 = 0
Solving, RD = 22.6 kip
Ans.
RE = 22.6 kip
Ans.
RF = 13.7 kip
Ans.
Ans: RD = 22.6 kip RE = 22.6 kip RF = 13.7 kip 450
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*5–64. z
Determine the components of reaction at the fixed support A. The 400 N, 500 N, and 600 N forces are parallel to the x, y, and z axes, respectively.
600 N
1m 400 N
0.5 m
0.75 m
0.75 m
Solution
A
500 N
x
Equations of Equilibrium. Referring to the FBD of the rod shown in Fig. a ΣFx = 0; Ax - 400 = 0 Ax = 400 N
Ans.
ΣFy = 0; 500 - Ay = 0 Ay = 500 N
Ans.
ΣFz = 0; Az - 600 = 0 Az = 600 N
Ans.
y
ΣMx = 0; (MA)x - 500(1.25) - 600(1) = 0 (MA)x = 1225 N # m = 1.225 kN # m
Ans.
ΣMy = 0; (MA)y - 400(0.75) - 600(0.75) = 0 (MA)y = 750 N # m
Ans.
ΣMz = 0; (MA)z = 0
Ans.
Ans: Ax = 400 N Ay = 500 N Az = 600 N (MA)x = 1.225 kN # m (MA)y = 750 N # m (MA)z = 0 451
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5–65. z
The 50-lb mulching machine has a center of gravity at G. Determine the vertical reactions at the wheels C and B and the smooth contact point A. G
4 ft
SOLUTION Equations of Equilibrium: From the free-body diagram of the mulching machine, Fig. a, NA can be obtained by writing the moment equation of equilibrium about the y axis.
1.25 ft 1.25 ft
C x
A
B 1.5 ft
©My = 0; 50(2) - NA(1.5 + 2) = 0
2 ft y
Ans.
NA = 28.57 lb = 28.6 lb
Using the above result and writing the moment equation of equilibrium about the x axis and the force equation of equilibrium along the z axis, we have ©Mx = 0; NB(1.25) - NC(1.25) = 0
(1)
©Fz = 0;
(2)
NB + NC + 28.57 - 50 = 0
Solving Eqs. (1) and (2) yields Ans.
NB = NC = 10.71 lb = 10.7 lb
Note: If we write the force equation of equilibrium ©Fx = 0 and ©Fy = 0 and the moment equation of equilibrium ©Mz = 0. This indicates that equilibrium is satisfied.
Ans: NA = 28.6 lb NB = 10.7 lb, NC = 10.7 lb 452
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5–66. z
The smooth uniform rod AB is supported by a ball-and-socket joint at A, the wall at B, and cable BC. Determine the components of reaction at A, the tension in the cable, and the normal reaction at B if the rod has a mass of 20 kg.
C 0.5 m B
2m
Solution A
Force And Position Vectors. The coordinates of points A, B and G are A(1.5, 0, 0) m, B(0, 1, 2) m, C(0, 0, 2.5) m and G(0.75, 0.5, 1) m x
FA = - Axi + Ay j + Azk TBC = TBC a NB = NBi
1m
(0 - 1)j + (2.5 - 2)k rBC 1 0.5 b = TBC c = TBC j + TBC k 2 2 rBC 11.25 11.25 2(0 - 1) + (2.5 - 2)
W = { -20(9.81)k} N rAG = (0.75 - 1.5)i + (0.5 - 0)j + (1 - 0)k = { - 0.75i + 0.5j + k} m rAB = (0 - 1.5)i + (1 - 0)j + (2 - 0)k = { - 1.5i + j + 2k} m Equations of Equilibrium. Referring to the FBD of the rod shown in Fig. a, the force equation of equilibrium gives ΣF = 0; FA + TBC + NB + W = 0 ( - Ax + NB)i + aAy -
1 0.5 TBC bj + c Az + TBC - 20 (9.81)d k = 0 11.25 11.25
Equating i, j and k components,
(1)
- Ax + NB = 0 Ay Az +
1 TBC = 0 11.25
(2)
0.5 TBC - 20(9.81) = 0 11.25
(3)
The moment equation of equilibrium gives ΣMA = 0; rAG * W + rAB * (TBC + NB) = 0 i † - 0.75 0 a
j 0.5 0
k i 1 † + † - 1.5 - 20(9.81) NB
j 1 -
1 11.25
k 2 TBC
0.5 11.25
TBC
† =0
0.5 2 0.75 1.5 TBC + TBC - 98.1bi + a TBC + 2NB - 147.15bj + a TBC - NB bk = 0 11.25 11.25 11.25 11.25
453
1.5 m
y
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5–66. Continued
Equating i, j and k Components
0.5 2 TBC + TBC - 98.1 = 0 11.25 11.25
(4)
0.75 TBC + 2NB - 147.15 = 0 11.25
(5)
1.5 TBC - NB = 0 11.25
(6)
Solving Eqs. (1) to (6)
TBC = 43.87 N = 43.9 N
Ans.
NB = 58.86 N = 58.9 N
Ans.
Ax = 58.86 N = 58.9 N
Ans.
Ay = 39.24 N = 39.2 N
Ans.
Az = 176.58 N = 177 N
Ans.
Note: One of the equations (4), (5) and (6) is redundant that will be satisfied automatically.
Ans: TBC = NB = Ax = Ay = Az = 454
43.9 N 58.9 N 58.9 N 39.2 N 177 N
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5–67. z
The uniform concrete slab has a mass of 2400 kg. Determine the tension in each of the three parallel supporting cables when the slab is held in the horizontal plane as shown.
TA
TB 15 kN
x
0.5 m
TC A
B
2m C
1m
1m
2m
y
Solution Equations of Equilibrium. Referring to the FBD of the slab shown in Fig. a, we notice that TC can be obtained directly by writing the moment equation of equilibrium about the x axis. ΣMx = 0; TC (2.5) - 2400(9.81)(1.25) - 15 ( 103 ) (0.5) = 0
Ans.
TC = 14,772 N = 14.8 kN
Using this result to write moment equation of equilibrium about y axis and force equation of equilibrium along z axis, ΣMy = 0; TB (2) + 14,772(4) - 2400(9.81)(2) - 15 ( 103 ) (3) = 0 Ans.
TB = 16,500 N = 16.5 kN ΣFz = 0; TA + 16,500 + 14,772 - 2400(9.81) - 15 ( 103 ) = 0
Ans.
TA = 7272 N = 7.27 kN
Ans: TC = 14.8 kN TB = 16.5 kN TA = 7.27 kN 455
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*5–68. z
The 100-lb door has its center of gravity at G. Determine the components of reaction at hinges A and B if hinge B resists only forces in the x and y directions and A resists forces in the x, y, z directions.
18 in.
B 24 in.
SOLUTION Equations of Equilibrium: From the free-body diagram of the door, Fig. a, By, Bx, and Az can be obtained by writing the moment equation of equilibrium about the x¿ and y¿ axes and the force equation of equilibrium along the z axis.
G
A 18 in.
-By(48) - 100(18) = 0
©Mx¿ = 0;
24 in.
By = - 37.5 lb
Ans.
©My¿ = 0;
Bx = 0
Ans.
©Fz = 0;
- 100 + A z = 0;
A z = 100 lb
30 x
y
Ans.
Using the above result and writing the force equations of equilibrium along the x and y axes, we have Ans.
©Fx = 0;
Ax = 0
©Fy = 0;
A y + ( -37.5) = 0
A y = 37.5 lb
Ans.
The negative sign indicates that By acts in the opposite sense to that shown on the free-body diagram. If we write the moment equation of equilibrium ©Mz = 0, it shows that equilibrium is satisfied.
Ans: By = Bx = Az = Ax = Ay = 456
- 37.5 lb 0 100 lb 0 37.5 lb
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5–69. Determine the tension in each cable and the components of reaction at D needed to support the load.
z B 3m
6m
C 2m D
x
A y
Solution
30
Force And Position Vectors. The coordinates of points A, B, and C are A(6, 0, 0) m, B(0, -3, 2) m and C(0, 0, 2) m respectively. FAB
(0 - 6)i + ( - 3 - 0)j + (2 - 0)k rAB 6 3 2 = FAB a b = FAB c d = - FABi - FAB j + FABk rAB 7 7 7 1(0 - 6)2 + ( - 3 - 0)2 + (2 - 0)2
FAC = FAC a
(0 - 6)i + (2 - 0)k rAC 6 2 b = FAC c d = FAC i + FAC k 2 2 rAC 140 140 1(0 - 6) + (2 - 0)
F = 400 (sin 30°j - cos 30°k) = {200j - 346.41k}N FD = Dxi + Dy j + Dzk rDA = {6i} m
Referring to the FBD of the rod shown in Fig. a, the force equation of equilibrium gives ΣF = 0; FAB + FAC + F + FD = 0 6 6 3 a- FAB FAC + Dx bi + a- FAB + Dy + 200bj 7 7 140
2 2 FAC + Dz - 346.41bk = 0 + a FAB + 7 140
457
400 N
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5–69. Continued
Equating i, j and k components, 6 6 - FAB FAC + Dx = 0 7 140
(1)
3 - FAB + Dy + 200 = 0 7
(2)
2 2 F + FAC + Dz - 346.41 = 0 7 AB 140
(3)
Moment equation of equilibrium gives
ΣMD = 0; rDA * (FAB + FAC + F) = 0
5
i 6
j 0
k 0
6 6 a- FAB F b 140 AC 7
3 a- FAB + 200b 7
2 2 FAC - 346.41b a FAB + 140 7
5 = 0
2 2 3 -6 a FAB + FAC - 346.41bj + 6 a- FAB + 200b k = 0 7 7 140
Equating j and k Components,
2 2 - 6 a FAB + FAC - 346.41b = 0 7 140
(4)
3 6 a - FAB + 200b = 0 7
(5)
Solving Eqs. (1) to (5)
FAB = 466.67 N = 467 N
Ans.
FAC = 673.81 N = 674 N
Ans.
Dx = 1039.23 N = 1.04 kN
Ans.
Dy = 0
Ans.
Dz = 0
Ans.
Ans: FAB = FAC = Dx = Dy = Dz = 458
467 N 674 N 1.04 kN 0 0
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5–70. z
The stiff-leg derrick used on ships is supported by a ball-andsocket joint at D and two cables BA and BC. The cables are attached to a smooth collar ring at B, which allows rotation of the derrick about z axis. If the derrick supports a crate having a mass of 200 kg, determine the tension in the cables and the x, y, z components of reaction at D.
B
6m
7.5 m
C 6m D
Solution 2 6 T - TBC 7 BA 9 3 3 ΣFy = 0; Dy - TBA - TBC 7 9 6 6 ΣFz = 0; Dz - TBA - TBC 7 9 3 3 ΣMx = 0; TBA(6) + TBC(6) 7 9 2 6 ΣMy = 0; TBA(6) - TBC(6) 7 9 ΣFx = 0; Dx +
1m
2m
y
A
= 0
3m
4m
x
= 0 - 200(9.81) = 0 - 200(9.81)(4) = 0 + 200(9.81)(1) = 0
TBA = 2.00 kN
Ans.
TBC = 1.35 kN
Ans.
Dx = 0.327 kN
Ans.
Dy = 1.31 kN
Ans.
Dz = 4.58 kN
Ans.
Ans: TBA = 2.00 kN TBC = 1.35 kN Dx = 0.327 kN Dy = 1.31 kN Dz = 4.58 kN 459
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5–71. z
Determine the components of reaction at the ball-and-socket joint A and the tension in each cable necessary for equilibrium of the rod.
C
2m 2m D A
x
Solution
B
3m
Force And Position Vectors. The coordinates of points A, B, C, D and E are A(0, 0, 0), B(6, 0, 0), C(0, -2, 3) m, D(0, 2, 3) m and E(3, 0, 0) m respectively. FBC = FBC a
(0 - 6)i + ( - 2 - 0)j + (3 - 0)k rBC 6 2 3 b = FBC £ § = - FBCi - FBC j + FBCk 2 2 2 rBC 7 7 7 2(0 - 6) + ( - 2 - 0) + (3 - 0)
FBD = FBD a
(0 - 6)i + (2 - 0)j + (3 - 0)k rBD 6 2 3 b = FBD £ § = - FBDi + FBD j + FBDk 2 2 2 rBD 7 7 7 2(0 - 6) + (2 - 0) + (3 - 0)
FA = Axi + Ay j + Azk F = { - 600k} N rAB = {6i} m
3m E
rAE = {3i} m
Equations of Equilibrium. Referring to the FBD of the rod shown in Fig. a, the force equation of equilibrium gives ΣF = 0; FBC + FBD + FA + F = 0 6 6 2 2 3 3 a - FBC - FBD + Ax bi + a FBD - FBC + Ay b j + a FBC + FBD + Az - 600bk = 0 7 7 7 7 7 7
460
600 N
3m y
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5–71. Continued
Equating i, j and k components, 6 6 - FBC - FBD + Ax = 0 7 7
(1)
2 2 F - FBC + Ay = 0 7 BD 7
(2)
3 3 F + FBD + Az - 600 = 0 7 BC 7
(3)
The moment equation of equilibrium gives ΣMA = 0; rAE * F + rAB * (FBC + FBD ) = 0 i †3 0
j 0 0
k 0 † + 5 -600
c 1800 -
i 6
j 0
k 0
6 - (FBC + FBD) 7
2 (F - FBC) 7 BD
3 (F + FBD) 7 BC
5 = 0
18 12 (FBC + FBD) d j + (F - FBC)k = 0 7 7 BD
Equating j and k components, 1800 -
18 (F + FBD) = 0 7 BC
(4)
12 (F - FBC) = 0 7 BD
(5)
Solving Eqs. (1) to (5), FBD = FBC = 350 N
Ans.
Ax = 600 N
Ans.
Ay = 0
Ans.
Az = 300 N
Ans.
Ans: FBD = Ax = Ay = Az = 461
FBC = 350 N 600 N 0 300 N
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*5–72. z
Determine the components of reaction at the ball-andsocket joint A and the tension in the supporting cables DB and DC.
1.5 m B 1.5 m C D
Solution
1.5 m
Force And Position Vectors. The coordinates of points A, B, C, and D are A(0, 0, 0), B(0, - 1.5, 3) m, C(0, 1.5, 3) m and D(1, 0, 1) m, respectively. x FDC
(0 - 1)i + (1.5 - 0)j + (3 - 1)k rDC = FDC a b = FDC £ § rDC 2(0 - 1)2 + (1.5 - 0)2 + (3 - 1)2 = -
FDB = FDB a
1 1.5 2 FCDi + FDC j + FDC k 17.25 17.25 17.25
(0 - 1)i + ( - 1.5 - 0)j + (3 - 1)k rDB b = FDB £ § rDB 2(0 - 1)2 + ( - 1.5 - 0)2 + (3 - 1)2 = -
1 1.5 2 FDBi + FDB j + FDBk 17.25 17.25 17.25
FA = Axi + Ay j + Azk F = { - 2400k} N
rAD = (1 - 0)i + (1 - 0)k = {i + k} m rF = {4i} m
462
800 N/m
3m
1m
1.5 m
A
3m
1m y
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*5–72. Continued
Equations of Equilibrium. Referring to the FBD of the assembly shown in Fig. a. Force equation of equilibrium gives ΣF = 0; FDC + FDB + FA + F = 0 a-
1.5 1 1 1.5 FDC FDB + Ax bi + a FDC FDB + Ay bj 17.25 17.25 17.25 17.25 + a
2 2 FDC + FDB + Az - 2400bk = 0 17.25 17.25
Equating i, j and k components, -
1 1 FDC FDB + Ax = 0 17.25 17.25
(1)
1.5 1.5 FDC F + Ay = 0 7.25 DB 17.25
(2)
2 2 FDC + FDB + Az - 2400 = 0 17.25 17.25 Moment equation of equilibrium gives
(3)
ΣMA = 0; rF * F + rAD * (FDB + FDC) = 0 i †4 0
-
j 0 0
k 0 † + 5 -2400
i 1 -
1 (FDB + FDC) 17.25
j 0
k 1
1.5 (FDC - FDB) 17.25
2 (FDC + FDB) 17.25
5 = 0
3 1.5 1.5 (FDC - FDB)i + c 9600 (FDC + FDB) d j + (FDC + FDB)k = 0 17.25 17.25 17.25
Equating i, j and k Components -
1.5 (FDC - FDB) = 0 17.25
9600 -
(4)
3 (FDC + FDB) = 0 17.25
(5)
1.5 (FDC - FDB) = 0 17.25
(6)
Solving Eqs. (1) to (6)
FDC = FDB = 4308.13 N = 4.31 kN
Ans.
Ax = 3200 N = 3.20 kN
Ans.
Ay = 0
Ans.
Az = - 4000 N = -4 kN
Ans.
Negative sign indicates that Az directed in the sense opposite to that shown in FBD.
463
Ans: FDC = Ax = Ay = Az =
FDB = 4.31 kN 3.20 kN 0 -4 kN
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5–73. z
The bent rod is supported at A, B, and C by smooth journal bearings. Determine the components of reaction at the bearings if the rod is subjected to the force F = 800 N. The bearings are in proper alignment and exert only force reactions on the rod.
C A
2m
2m
x
Solution
B 0.75 m
1m
30 60
Equations of Equilibrium. The x, y and z components of force F are
F
Fx = 800 cos 60° cos 30° = 346.41 N
y
Fy = 800 cos 60° sin 30° = 200 N Fz = 800 sin 60° = 692.82 N Referring to the FBD of the bent rod shown in Fig. a, ΣMx = 0;
- Cy(2) + Bz(2) - 692.82 (2) = 0
(1)
ΣMy = 0;
Bz(1) + Cx(2) = 0
(2)
ΣMz = 0;
- Cy(1.75) - Cx(2) - By(1) - 346.41(2) = 0
(3)
ΣFx = 0;
Ax + Cx + 346.41 = 0
(4)
ΣFy = 0;
200 + By + Cy = 0
(5)
ΣFz = 0;
Az + Bz - 692.82 = 0
(6)
Solving Eqs. (1) to (6) Cy = 800 N Bz = - 107.18 N = 107 N By = 600 N
Ans.
Cx = 53.59 N = 53.6 N Ax = 400 N Az = 800 N
Ans.
The negative signs indicate that Cy, Bz and Az are directed in the senses opposite to those shown in FBD.
Ans: Cy = Bz = By = Cx = Ax = Az = 464
800 N 107 N 600 N 53.6 N 400 N 800 N
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5–74. The bent rod is supported at A, B, and C by smooth journal bearings. Determine the magnitude of F which will cause the positive x component of reaction at the bearing C to be Cx = 50 N. The bearings are in proper alignment and exert only force reactions on the rod.
z
C A
2m
2m
Solution
x
B 0.75 m
1m
30 60
Equations of Equilibrium. The x, y and z components of force F are
F
Fx = F cos 60° cos 30° = 0.4330 F
y
Fy = F cos 60° sin 30° = 0.25 F Fz = F sin 60° = 0.8660 F Here, it is required that Cx = 50. Thus, by referring to the FBD of the beat rod shown in Fig. a, ΣMx = 0;
(1)
-Cy(2) + Bz(2) - 0.8660 F(2) = 0
ΣMy = 0; ΣMz = 0;
(2)
Bz(1) + 50(2) = 0
(3)
-Cy(1.75) - 50(2) - By(1) - 0.4330 F(2) = 0
ΣFy = 0;
(4)
0.25 F + By + Cy = 0
Solving Eqs. (1) to (4) Ans.
F = 746.41 N = 746 N Cy = -746.41 N Bz = - 100 N By = 559.81 N
Ans: F = 746 N 465
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5–75. z
Member AB is supported by a cable BC and at A by a square rod which fits loosely through the square hole in the collar fixed to the member as shown. Determine the components of reaction at A and the tension in the cable needed to hold the rod in equilibrium.
1.5 m 400 N
3m
A y C
200 N
Solution x
Force And Position Vectors. The coordinates of points B and C are B(3, 0, - 1) m C(0, 1.5, 0) m, respectively. TBC = TBC a
B
(0 - 3)i + (1.5 - 0)j + [0 - ( - 1)]k rBC b = TBC • ¶ rBC 2(0 - 3)2 + (1.5 - 0)2 + [0 - ( -1)]2
6 3 2 = - TBC i + TBC j + T k 7 7 7 BC
F = {200j - 400k} N FA = Ax i + Ay j MA = (MA)x i + (MA)y j + (MA)z k r1{3 i} m r2 = {1.5 j} m Equations of Equilibrium. Referring to the FBD of member AB shown in Fig. a, the force equation of equilibrium gives ΣF = 0; TBC + F + FA = 0 6 3 2 a - TBC + Ax bi + a TBC + 200 + Ayb j + a TBC - 400bk = 0 7 7 7
Equating i, j and k components -
6 T + Ax = 0 7 BC
(1)
3 T + 200 + Ay = 0 7 BC
(2)
2 T - 400 = 0 7 BC
(3)
466
1m
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5–75. Continued
The moment equation of equilibrium gives ΣMA = O; MA + r1 * F + r2 * TBC = 0 i
( MA ) x i + ( MA ) y j + ( MA ) zk + † 3 0
j 0 200
k 0 † + 5 - 400
i 0 6 - TBC 7
j 1.5 3 TBC 7
k 0 2 TBC 7
3 9 T d i + 3 ( MA ) y + 1200 4 j + c ( MA ) z + TBC + 600 d k = 0 7 BC 7 Equating i, j, and k components, 3 ( MA ) x + TBC = 0 7
5 = 0
c ( MA ) x +
( MA ) y + 1200 = 0
(4) (5)
9 TBC + 600 = 0 7 Solving Eqs. (1) to (6),
( MA ) z +
(6)
Ans.
TBC = 1400 N = 1.40 kN Ay = 800 N
Ans.
Ax = 1200 N = 1.20 kN
Ans.
( MA ) x = 600 N # m
Ans.
( MA ) y = - 1200 N # m = 1.20 kN # m
Ans.
( MA ) z = - 2400 N # m = 2.40 kN # m
Ans.
The negative signs indicate that Ay, ( MA ) x, ( MA ) y and ( MA ) z are directed in sense opposite to those shown in FBD.
Ans: TBC = 1.40 kN Ay = 800 N Ax = 1.20 kN ( MA ) x = 600 N # m ( MA ) y = 1.20 kN # m ( MA ) z = 2.40 kN # m 467
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*5–76. z
The member is supported by a pin at A and cable BC. Determine the components of reaction at these supports if the cylinder has a mass of 40 kg.
0.5 m B
1m A D
Solution
y 1m
Force And Position Vectors. The coordinates of points B, C and D are B(0, -0.5, 1) m, x C(3, 1, 0) m and D(3, - 1, 0) m, respectively. FCB = FCB a
(0 - 3)i + ( -0.5 - 1)j + (1 - 0)k rCB b = FCB c d rCB 2(0 - 3)2 + ( - 0.5 - 1)2 + (1 - 0)2 6 3 2 = - FCBi - FCBj + FCBk 7 7 7
W = { - 40(9.81)k} N = { -392.4k} N. FA = Ax i + Ay j + Az k MA = ( MA ) x i + ( MA ) z k rAC = {3i + j} m rAD = {3i - j} m Equations of Equilibrium. Referring to the FBD of the assembly shown in Fig. a. the force equation of equilibrium gives ΣF = 0; FCB + W + FA = 0; 6 3 2 a - FCB + Ax bi + a - FCB + Ay bj + a FCB + Az - 392.4bk = 0 7 7 7
Equating i, j and k components
6 - FCB + Ax = 0 7 3 - FCB + Ay = 0 7 2 FCB + Az - 392.4 = 0 7 The moment equation of equilibrium gives
(1) (2) (3)
ΣMA = 0; rAC * FCB + rAD * W + MA = 0 i 3 5 6 - FCB 7
1m
j 1 3 - FCB 7
k i 0 5 + †3 2 0 FCB 7
j -1 0
k 0 † + ( MA ) x i + ( MA ) Z k = 0 - 392.4
2 6 9 6 c FCB + 392.4 + ( MA ) x d i + a - FCB + 1177.2bj + c - FCB + FCB + ( MA ) z d k = 0 7 7 7 7
468
C
3m
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*5–76. Continued
Equating i, j and k components, 2 F + 392.4 + ( MA ) x = 0 7 CB 6 - FCB + 1177.2 = 0 7 9 6 - FCB + FCB + ( MA ) z = 0 7 7 Solving Eqs (1) to (6),
(4) (5) (6)
Ans.
FCB = 1373.4 N = 1.37 kN
( MA ) x = - 784.8 N # m = 785 N # m
Ans.
( MA ) z = 588.6 N # m = 589 N # m
Ans.
Ax = 1177.2 N = 1.18 kN
Ans.
Ay = 588.6 N = 589 N
Ans.
Az = 0
Ans.
Ans: FCB ( MA ) x ( MA ) z Ax Ay Az 469
= = = = = =
1.37 kN 785 N # m 589 N # m 1.18 kN 589 N 0
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5–77. z
The member is supported by a square rod which fits loosely through the smooth square hole of the attached collar at A and by a roller at B. Determine the components of reaction at these supports when the member is subjected to the loading shown.
A
x
B
1m
2m y
2m
Solution
C
Force And Position Vectors. The coordinates of points B and C are B(2,0,0) m and C(3,0,-2) m.
300 N
FA = -Ax i - Ay j F = {300i + 500j - 400k} N NB = NB k MA = - ( MA ) x i + ( MA ) y j - ( MA ) z k rAB = {2i} m rAC = {3i - 2k} m Equations of Equilibrium. Referring to the FBD of the member shown in Fig. a, the force equation of equilibrium gives ΣF = 0; FA + F + NB = 0
( 300 - Ax ) i + ( 500 - Ay ) j + ( NB - 400 ) k = 0
470
500 N 400 N
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5–77. Continued
Equating i, j and k components, 300 - Ax = 0 Ax = 300 N
Ans.
500 - Ay = 0 Ay = 500 N
Ans.
NB - 400 = 0
Ans.
NB = 400 N
The moment equation of equilibrium gives ΣMA = 0; MA + rAB * NB + rAC * F = 0 i - ( MA ) x i + ( MA ) y j - ( MA ) z k + † 2 0
3 1000
- ( MA ) x 4 i +
3 ( MA ) y
Equating i, j and k components,
j 0 0
- 200 4 j +
k i 0 † + † 3 400 300
3 1500
j 0 500
- ( MA ) z 4 k = 0
1000 - ( MA ) x = 0 ( MA ) x = 1000 N # m = 1.00 kN # m
( MA ) y - 200 = 0 ( MA ) y = 200 N # m 1500 - ( MA ) z = 0 ( MA ) z = 1500 N # m =
k -2 † = 0 - 400
1.50 kN # m
Ans. Ans. Ans.
Ans: Ax = 300 N Ay = 500 N NB = 400 N ( MA ) x = 1.00 kN # m ( MA ) y = 200 N # m ( MA ) z = 1.50 kN # m 471
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5–78. The bent rod is supported at A, B, and C by smooth journal bearings. Compute the x, y, z components of reaction at the bearings if the rod is subjected to forces F1 = 300 lb and F2 = 250 lb. F1 lies in the y–z plane. The bearings are in proper alignment and exert only force reactions on the rod.
F1
z
45
1 ft A
C
4 ft
SOLUTION
5 ft
B
F1 = (- 300 cos 45°j - 300 sin 45°k)
2 ft
30
3 ft
= {- 212.1j - 212.1k} lb
y
45
F2 = (250 cos 45° sin 30°i + 250 cos 45° cos 30°j - 250 sin 45°k)
x
F2
= {88.39i + 153.1j - 176.8k} lb ©Fx = 0;
Ax + Bx + 88.39 = 0
©Fy = 0;
Ay + Cy - 212.1 + 153.1 = 0
©Fz = 0;
Bz + Cz - 212.1 - 176.8 = 0
©Mx = 0;
-Bz (3) - Ay (4) + 212.1(5) + 212.1(5) = 0
©My = 0;
Cz (5) + Ax (4) = 0
©Mz = 0;
Ax (5) + Bx (3) - Cy (5) = 0 Ax = 633 lb
Ans.
Ay = - 141 lb
Ans.
Bx = - 721 lb
Ans.
Bz = 895 lb
Ans.
Cy = 200 lb
Ans.
Cz = - 506 lb
Ans.
Ans: Ax = Ay = Bx = Bz = Cy = Cz = 472
633 lb - 141 lb - 721 lb 895 lb 200 lb - 506 lb
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5–79. The bent rod is supported at A, B, and C by smooth journal bearings. Determine the magnitude of F2 which will cause the reaction C y at the bearing C to be equal to zero. The bearings are in proper alignment and exert only force reactions on the rod. Set F1 = 300 lb.
F1
z
45
1 ft A
C
4 ft
SOLUTION
5 ft
B
F1 = ( - 300 cos 45°j - 300 sin 45°k)
2 ft
= { -212.1j - 212.1k} lb
45
F2 = (F2 cos 45° sin 30°i + F2 cos 45° cos 30°j - F2 sin 45°k)
x
= {0.3536F2 i + 0.6124F2 j - 0.7071F2 k} lb ©Fx = 0;
Ax + Bx + 0.3536F2 = 0
©Fy = 0;
Ay + 0.6124F2 - 212.1 = 0
©Fz = 0;
Bz + Cz - 0.7071F2 - 212.1 = 0
©Mx = 0;
- Bz (3) - Ay (4) + 212.1(5) + 212.1(5) = 0
©My = 0;
Cz (5) + Ax (4) = 0
©Mz = 0;
Ax (5) + Bx (3) = 0
30
3 ft
F2
Ax = 357 lb Ay = -200 lb Bx = -595 lb Bz = 974 lb Cz = - 286 lb Ans.
F2 = 674 lb
Ans: F2 = 674 lb 473
y
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*5–80. The bar AB is supported by two smooth collars. At A the connection is with a ball-and-socket joint and at B it is a rigid attachment. If a 50-lb load is applied to the bar, determine the x, y, z components of reaction at A and B.
D B z C
Solution
6 ft
4 ft
6 ft
Ax + Bx = 0 (1) By + 50 = 0
E
50 lb
5 ft x
By = - 50 lb Ans. Az + Bz = 0 (2)
3 ft
A F
y
MBz = 0 Ans. MBx + 50(6) = 0
MBx = - 300 lb # ft Ans. BCD = -9i + 3j BCD = -0.94868i + 0.316228j Require FB # uCD = 0
(Bxi - 50j + Bzk) # ( -0.94868i + 0.316228j) = 0 -0.94868Bx - 50(0.316228) = 0 Ans.
Bx = - 16.667 = - 16.7 lb From Eq. (1);
Ans.
Ax = 16.7 lb Require MB # uCD = 0
( -300i + MByj) # ( - 0.94868i + 0.316228j) = 0 300(0.94868) + MBy(0.316228) = 0
MBy = - 900 lb # ft Ans.
Ans: By = -50 lb MBz = 0 MBx = -300 lb # ft Bx = -16.7 lb Ax = 16.7 lb 474
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5–81. The rod has a weight of 6 lb>ft. If it is supported by a balland-socket joint at C and a journal bearing at D, determine the x, y, z components of reaction at these supports and the moment M that must be applied along the axis of the rod to hold it in the position shown.
z D
60 A 0.5 ft 45
x
Solution
y
ΣFx = 0; Cx + Dx - 15 sin 45° = 0
(1)
ΣFy = 0; Cy + Dy = 0
(2)
C
M
B
1 ft
1 ft
ΣFz = 0; Cz - 15 cos 45° = 0
Ans.
Cz = 10.6 lb
ΣMx = 0; - 3 cos 45°(0.25 sin 60°) - Dy(2) = 0 Dy = - 0.230 lb
Ans.
Cy = 0.230 lb
Ans.
ΣMy = 0;
- (12 sin 45°)(1) - (3 sin 45°)(1) + (3 cos 45°)(0.25 cos 60°)
From Eq. (2);
+ Dx(2) = 0 Dx = 5.17 lb
Ans.
Cx = 5.44 lb
Ans.
ΣMz = 0;
- M + (3 sin 45°)(0.25 sin 60°) = 0
From Eq. (1);
M = 0.459 lb # ft
Ans.
Ans: Cz = 10.6 lb Dy = - 0.230 lb Cy = 0.230 lb Dx = 5.17 lb Cx = 5.44 lb M = 0.459 lb # ft 475
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5–82. The sign has a mass of 100 kg with center of mass at G. Determine the x, y, z components of reaction at the ball-andsocket joint A and the tension in wires BC and BD.
z 1m
D
2m
C
SOLUTION
1m 2m
Equations of Equilibrium: Expressing the forces indicated on the free-body diagram, Fig. a, in Cartesian vector form, we have
A
FA = A xi + A yj + A zk
x
B
W = {- 100(9.81)k} N = {- 981k} N FBD = FBDuBD = FBD ≥
FBC = FBCuBC = FBC ≥
G
(- 2 - 0)i + (0 - 2)j + (1 - 0)k 2(- 2 - 0)2 + (0 - 2)2 + (1 - 0)2
¥ = a-
2 2 1 FBDi - FBDj + FBDkb 3 3 3
1m
(1 - 0)i + (0 - 2)j + (2 - 0)k
1 2 2 ¥ = a FBCi - FBCj + FBCkb 3 3 3 2(1 - 0) + (0 - 2) + (2 - 0) 2
2
2
Applying the forces equation of equilibrium, we have ©F = 0;
FA + FBD + FBC + W = 0
2 2 1 1 2 2 (A xi + A yj + A zk) + a - FBDi - FBDj + FBDk b + a FBCi - FBCj + FBCkb + ( - 981 k) = 0 3 3 3 3 3 3 a Ax -
2 1 2 2 1 2 FBD + FBC bi + a A y - FBD - FBC b j + aA z + FBD + FBC - 981bk = 0 3 3 3 3 3 3
Equating i, j, and k components, we have Ax -
2 1 F + FBC = 0 3 BD 3
(1)
Ay -
2 2 F - FBC = 0 3 BD 3
(2)
Az +
1 2 FBD + FBC - 981 = 0 3 3
(3)
In order to write the moment equation of equilibrium about point A, the position vectors rAG and rAB must be determined first. rAG = {1j} m rAB = {2j} m
476
1m
y
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5–82. Continued
Thus, ©M A = 0; rAB * (FBC + FBD) + (rAG * W) = 0 2 2 2 2 1 1 (2j) * c a FBC - FBD b i - a FBC + FBD b j + a FBC + FBD bk d + (1j) * ( - 981k) = 0 3 3 3 3 3 3 4 2 4 2 a FBC + FBD - 981 bi + a FBD - FBC b k = 0 3 3 3 3 Equating i, j, and k components we have 2 4 F + FBC - 981 = 0 3 BC 3
(4)
4 2 F - FBC = 0 3 BC 3
(5)
Ans: FBD = 294 N FBC = 589 N Ax = 0 Ay = 589 N Az = 490.5 N 477
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5–83. z
Both pulleys are fixed to the shaft and as the shaft turns with constant angular velocity, the power of pulley A is transmitted to pulley B. Determine the horizontal tension T in the belt on pulley B and the x, y, z components of reaction at the journal bearing C and thrust bearing D if u = 0°. The bearings are in proper alignment and exert only force reactions on the shaft.
200 mm 250 mm
u
300 mm C
SOLUTION
80 mm A
x
©Fx = 0; ©Fy = 0;
165 + 80210.452 - Cz 10.752 = 0 Ans.
150 + 58.0210.22 - Cy 10.752 = 0 Cy = 28.8 N
Ans.
Dx = 0
Ans.
Dy + 28.8 - 50 - 58.0 = 0 Ans.
Dy = 79.2 N ©Fz = 0;
80 N
Ans.
Cz = 87.0 N ©Mz = 0;
y
65 N
6510.082 - 8010.082 + T10.152 - 5010.152 = 0 T = 58.0 N
©My = 0;
150 mm B T
Equations of Equilibrium: ©Mx = 0;
50 N D
Dz + 87.0 - 80 - 65 = 0 Ans.
Dz = 58.0 N
Ans: T = 58.0 N Cz = 87.0 N Cy = 28.8 N Dx = 0 Dy = 79.2 N Dz = 58.0 N 478
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*5–84. z
Both pulleys are fixed to the shaft and as the shaft turns with constant angular velocity, the power of pulley A is transmitted to pulley B. Determine the horizontal tension T in the belt on pulley B and the x, y, z components of reaction at the journal bearing C and thrust bearing D if u = 45°. The bearings are in proper alignment and exert only force reactions on the shaft.
200 mm 250 mm
u
300 mm C
SOLUTION
80 mm A
x
©Fx = 0; ©Fy = 0;
165 + 80210.452 - 50 sin 45°10.22 - Cz 10.752 = 0 Ans.
58.010.22 + 50 cos 45°10.22 - Cy 10.752 = 0 Cy = 24.89 N = 24.9 N
Ans.
Dx = 0
Ans.
Dy + 24.89 - 50 cos 45° - 58.0 = 0 Ans.
Dy = 68.5 N ©Fz = 0;
80 N
Ans.
Cz = 77.57 N = 77.6 N ©Mz = 0;
y
65 N
6510.082 - 8010.082 + T10.152 - 5010.152 = 0 T = 58.0 N
©My = 0;
150 mm B T
Equations of Equilibrium: ©Mx = 0;
50 N D
Dz + 77.57 + 50 sin 45° - 80 - 65 = 0 Ans.
Dz = 32.1 N
Ans: T = Cz = Cy = Dy = Dz = 479
58.0 N 77.6 N 24.9 N 68.5 N 32.1 N
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5–85. Member AB is supported by a cable BC and at A by a square rod which fits loosely through the square hole at the end joint of the member as shown. Determine the components of reaction at A and the tension in the cable needed to hold the 800-lb cylinder in equilibrium.
z C
2 ft
A
x
FBC
3 ft
3 6 2 = FBC a i - j + k b 7 7 7
©Fx = 0;
B
6 ft
SOLUTION
y
3 FBC a b = 0 7 FBC = 0
Ans.
©Fy = 0;
Ay = 0
Ans.
©Fz = 0;
Az = 800 lb
Ans.
©Mx = 0;
(MA)x - 800(6) = 0 (MA)x = 4.80 kip # ft
Ans.
©My = 0;
(MA)y = 0
Ans.
©Mz = 0;
(MA)z = 0
Ans.
Ans: FBC = 0 Ay = 0 Az = 800 lb (MA)x = 4.80 kip # ft (MA)y = 0 (MA)z = 0 480