Mechanical Engineering Chapter 5 Problem The Force Acting The Boom Abcy Points The Direction The Unit

Problem 5.87 The force Facting on the boom ABC

at Cpoints in the direction of the unit vector 0.512i

0.384jC0.768kand its magnitude is 8 kN. The boom

is supported by a ball and socket at Aand the cables BD

and BE. The collar at Bis ﬁxed to the boom.

(a) Draw the free-body diagram of the boom.

(b) Determine the tensions in the cables and the reac-

tions at A.

y

z

x

A

C

D

E

B

2 m

1 m

1.5 m 2 m

2 m

Solution:

In component form we have

Moment equilibrium requires:

In components:

p10.25 TBD⊲2m⊳

F

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Problem 5.88 The cables BD and BE in Problem 5.87

will each safely support a tension of 25 kN. Based on

this criterion, what is the largest acceptable magnitude

of the force F?

Solution: We have the force and distances:

The moment equations are

Problem 5.89 The suspended load exerts a force FD

600 lb at A, and the weight of the bar OA is negligible.

Determine the tensions in the cables and the reactions at

the ball and socket support O.

(0, 10, 4) ft

B

(0, 6, –10) ft

C

A(8, 6, 0) ft

x

y

–Fj

O

z

Solution: From the diagram, the important points in this problem

are A(8, 6, 0), B(0, 10, 4), C(0, 6, 10), and the origin O(0, 0, 0)

with all dimensions in ft. We need unit vectors in the directions Ato

Band Ato C. Both vectors are of the form

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Problem 5.90 In Problem 5.89, suppose that the

suspended load exerts a force FD600 lb at Aand bar

OA weighs 200 lb. Assume that the bar’s weight acts at

its midpoint. Determine the tensions in the cables and

the reactions at the ball and socket support O.

Solution: Point Gis located at (4, 3, 0) and the position vector of

Gwith respect to the origin is

The mathematical representation for all other forces and moments from

Problem 5.89 remain the same (the numbers change!). Each equation

of equilibrium has a new term reﬂecting the addition of the weight of

the bar. The new force equilibrium equation is

The new moment equilibrium equation is

Problem 5.91 The 158,000-kg airplane is at rest on

the ground (zD0 is ground level). The landing gear

carriages are at A,B, and C. The coordinates of the point

Gat which the weight of the plane acts are (3, 0.5, 5) m.

What are the magnitudes of the normal reactions exerted

on the landing gear by the ground?

y

x

6 m

21 m

A

B

C

G

6 m

Solution:

FYDNRCNLCNFWD0

mg

W0.5 m

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Problem 5.92 The horizontal triangular plate is

suspended by the three vertical cables A, B, and C.

The tension in each cable is 80 N. Determine the x

and zcoordinates of the point where the plate’s weight

effectively acts.

x

z

0.3 m

0.4 m

C

A

B

(x, 0, z)

y

Solution:

Solving

xD0.1m,zD0.1333 m

Z

80 N

240 N

Problem 5.93 The 800-kg horizontal wall section is

supported by the three vertical cables, A,B, and C. What

are the tensions in the cables?

7 m

A

B

C

Solution: All dimensions are in mand all forces are in N. Forces

A,B,C, and Wact on the wall at (0, 0, 0), (5, 14, 0), (12, 7, 0),

Problem 5.94 The bar AC is supported by the cable

BD and a bearing at Athat can rotate about the zaxis.

The person exerts a force FD10j(lb) at C. Determine

the tension in the cable and the reactions at A.

C

14 in8 in

B

A

z

x

y

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Problem 5.95 The L-shaped bar is supported by a

bearing at Aand rests on a smooth horizontal surface

at B. The vertical force FD4 kN and the distance

bD0.15 m. Determine the reactions at Aand B.

y

A

Fb

x

z:MAZC0.2F0.2BD0

AZ

B

z

b = 0.15 m

F = 4 kN

Problem 5.96 In Problem 5.95, the vertical force FD

4 kN and the distance bD0.15 m. If you represent the

reactions at Aand Bby an equivalent system consisting

of a single force, what is the force and where does its

line of action intersect the xzplane?

MAD0.4k(kN-m)

We want a single equivalent force, Rthat has the same resultant force

and moment about Aas does the set A,B, and MA.

xRD0.1m

y

F

b

R

Problem 5.97 In Problem 5.95, the vertical force FD

4 kN. The bearing at Awill safely support a force of

2.5-kN magnitude and a couple of 0.5 kN-m magnitude.

Based on these criteria, what is the allowable range of

the distance b?

MAZC0.2F0.2BD0

MAYDC0.5 (kN-m), and solve we get AYD2.5atbD0.4875 m

However, 0.3 is the physical limit of the device.

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Problem 5.98 The 1.1-m bar is supported by a ball

and socket support at Aand the two smooth walls. The

tension in the vertical cable CD is 1 kN.

(a) Draw the free-body diagram of the bar.

(b) Determine the reactions at Aand B.

B

C

D

A

400 mm

700 mm

x

y

z

600 mm

Solution:

(a) The ball and socket cannot support a couple reaction, but can

support a three force reaction. The smooth surface supports one-

contain only the unknown reaction at B. This will require the

position vectors of Band Drelative to A, which in turn will

1.1D33.1°.

From which,

BZD0.3819

0.6D0.6365 kN,

The reactions at Aare determined from the sums of the forces:

FZD⊲BZCAZ⊳kD0,from which AZD0.6365 kN

Problem 5.99 The 8-ft bar is supported by a ball and

socket at A, the cable BD, and a roller support at C. The

collar at Bis ﬁxed to the bar at its midpoint. The force

FD50k(lb). Determine the tension in the cable BD

y

4 ft

2 ft

D

C

The angle formed by the rod is

rDD4iC2k

and rCD⊲8 cos 22°⊳iD7.4162i.

C7.4162CYkD0,

1.192 D62.92 lb

7.4162 D17.27 lb.

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Problem 5.100 Consider the 8-ft bar in Problem 5.99.

The force FDFyj50k(lb). What is the largest value

of Fyfor which the roller support at Cwill remain on

the ﬂoor?

3.7081 1.50

0.1160 0.5960 0.7946 

C

ijk

7.4162 30

1.192 D62.92 lb.

3.708 D34.54 lb

Problem 5.101 The tower is 70 m tall. The tension in

each cable is 2 kN. Treat the base of the tower Aas a

built-in support. What are the reactions at A?

y

B

eBD D0.2265i0.7926jC0.5661k,

eBC D0.5812i0.8137jC0k.

FYD⊲AY4.7682⊳jD0,

from which AYD4.768 kN,

FZD⊲AZC0.2434⊳kD0,

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Problem 5.102 Consider the tower in Problem 5.101.

If the tension in cable BC is 2 kN, what must the ten-

sions in cables BD and BE be if you want the couple

exerted on the tower by the built-in support at Ato be

zero? What are the resulting reactions at A?

Solve:

Problem 5.103 The space truss has roller supports at

B,C, and Dand is subjected to a vertical force FD

20 kN at A. What are the reactions at the roller supports? F

A (4, 3, 4) m

y

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Problem 5.104 In Example 5.8, suppose that the cable

BD is lengthened and the attachment point Bis moved to

the end of the bar at C. The positions of the attachment

point Dand the bar are unchanged. Draw a sketch of

the bar showing cable BD in its new position. Draw the

free-body diagram of the bar and apply equilibrium to

determine the tension in the cable and the reactions at A.

x

y

z

⫺100j (lb)

A

B

C

30⬚

(2, 2, ⫺1) ftD

Solution: The sketch and free-body diagram are shown.

We must express the force exerted on the bar by cable BD in terms of

its components. The bar AC is 4 ft long. The vector from Cto Dis

Problem 5.105 The 40-lb door is supported by hinges

at Aand B. The yaxis is vertical. The hinges do not

exert couples on the door, and the hinge at Bdoes not

exert a force parallel to the hinge axis. The weight of

the door acts at its midpoint. What are the reactions at

Aand B?

z

40

4 ft

x

y

1 ft

A

B

D1.2856iC3.5jC1.532k.

The position vectors of the hinges:

rADj,rBD6j.

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Problem 5.106 The vertical cable is attached at A.

Determine the tension in the cable and the reactions at

the bearing Bdue to the force FD10i30j10k(N).

A

y

x

200 mm

100 mm

200 mm

B

Solution: The position vector of the point of application of the

force is

rBD0.1i.

0.1D60 N,

MBY DC1 N-m,

MBZ D0.1TC3D3 N-m.

Problem 5.107 In Problem 5.106, suppose that the z

component of the force Fis zero, but otherwise Fis

unknown. If the couple exerted on the shaft by the

bearing at Bis MBD6j6kN-m, what are the force

Fand the tension in the cable?

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Problem 5.108 The device in Problem 5.106 is badly

designed because of the couples that must be supported

by the bearing at B, which would cause the bearing to

“bind”. (Imagine trying to open a door supported by

only one hinge.) In this improved design, the bearings

at Band Csupport no couples, and the bearing at C

does not exert a force in the xdirection. If the force

FD10i30j10k(N), what are the tension in the

vertical cable and the reactions at the bearings Band C?

A

y

z

C

x

200 mm

50 mm

100 mm

200 mm

F

B