Mechanical Engineering Chapter 5 Homework If determine the distance hof placement at the end

5–50.

SOLUTION

Substituting into Eq.

(1):

The rigid metal strip of negligible weight is used as part

of an electromagnetic switch. Determine the maximum

stiffness kof the springs at Aand Bso 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

10 mm

A

B

C

k

438

5–51.

The cantilever footing is used to support a wall near its

edge Aso that it causes a uniform soil pressure under the

footing. Determine the uniform distribution loads and

, measured in lb ft at pads Aand B,necessary to support

the wall forces of 8 000 lb and 20 000 lb.

wB

w

A

w

A

AB

w

B

1.5 ft

8000 lb

20 000 lb

0.25 ft

SOLUTION

Ans:

439

*5–52.

SOLUTION

The uniform beam has a weight Wand length land is

supported by a pin at Aand a cable BC.Determine the

horizontal and vertical components of reaction at Aand

the tension in the cable necessary to hold the beam in the

position shown.

C

B

A

l

Equations of Equilibrium: The tension in the cable can be obtained directly by

summing moments about point A.

Ans:

5–53.

SOLUTION

Equations of Equilibrium: The spring force at Aand Bcan be obtained directly by

summing moments about points Band A, respectively.

B

A

1m 3 m

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.

Ans:

441

5–54.

h

s

C

B

The uniform rod has a length of .If,

determine the distance hof placement at the end Aalong the

smooth wall for equilibrium.

SOLUTION

Equations of Equilibrium: Referring to the FBD of the rod shown in Fig. a, write

the moment equation of equilibrium about point A.

Using this result to write the force equation of equilibrium along yaxis,

Substituting Eq. (2) into (1) yields

Again, applying law of cosine by referring to Fig. b,

Equating Eqs. (3) and (4) yields

s=1.5 ml=1 m30-N

Ans:

442

5–55.

SOLUTION

Equations of Equilibrium: The tension in the cable can be obtained directly by

summing moments about point A.

Substituting Eq. (2) into (1) yields

Using the cosine law,

Equating Eqs. (3) and (4) yields

h

s

C

A

l

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.

443

*5–56.

SOLUTION

Substituting Eqs

. (1) and (3) into Eq. (2):

T

he uniform rod of length Land weight Wis supported on

the

smooth planes. Determine its position for equilibrium.

N

eglect the thickness of the rod.

u

L

u

f

c

Ans:

444

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

and for equilibrium if and .L=12 ftP=500 lbw2

w1

P2P

w2

w1

L

––

3

L

––

3

L

––

3

SOLUTION

Equations of Equilibrium:Referring to the FBD of the beam shown in Fig. a,we

Ans:

445

5–58.

SOLUTION

P2P

w1

L

––

3

L

––

3

L

––

3

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.

Ans:

446

5–59.

SOLUTION

3 ft

2ft

100 lb ft k50 lb/ft

B

Au

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 ufor equilibrium. The

spring has an unstretched length of 2 ft and a stiffness

of k = 50 lb/ft.

Ans:

447

*5–60.

Determine the distance

d

for placement of the load Pfor

equilibrium

of the smooth bar in the position as shown.

Neglect the weight of the bar.

u

SOLUTION

d

a

u

P

Ans:

448

5–61.

If ,and ,determine the normal reaction at

the smooth supports and the required distance afor the

placement of the roller if .Neglect the weight of

the bar.

SOLUTION

Equations of Equilibrium: Referring to the of the rod shown in Fig. a,

FBD

P=600 N

u=30°d=1

m

P

d

a

u

Ans:

449

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.

SOLUTION

Equations of Equilibrium: Due to symmetry, all wires are subjected to the same

tension. This condition statisfies moment equilibrium about the xand yaxes and

force equilibrium along yaxis.

2 m

1.5 m

1.25 m

1.5 m

1.25 m

F

A

BC

Ans:

450

5–63.

A

B

D

E

z

C

SOLUTION

Due to an unequal distribution of fuel in the wing tanks, the

c en t er s o f gr avi t y f o r t h e ai r p l an e f u s el age Aand wings B

an d Car e l o c at ed as s h o w n . I f t h es e c o m p o n en t s h ave

weights and

determine the normal reactions of the wheels D, E, an d F

on the ground.

W

C=6000 l b ,W

B=8000 l b ,W

A=45 000 l b ,

Ans:

451

*5–64.

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.

SOLUTION

Equations of Equilibrium. Referring to the FBD of the rod shown in Fig. a

400 N

600 N

500 N

1 m

0.5 m

0.75 m

z

x

A

0.75 m

452

5–65.

The

5

0-lb mulching machine has a center of gravity at G.

Determine the vertical reactions at the wheels Cand Band

the smooth contact point A.

x

z

G

1.25 ft

4 ft

C

B

A

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

yaxis.

Using the above result and writing the moment equation of equilibrium about the

xaxis and the force equation of equilibrium along the zaxis, we have

Ans:

453

SOLUTION

Force And Position Vectors. The coordinates of points A, B and G are A(1.5, 0, 0) m,

Equations of Equilibrium. Referring to the FBD of the rod shown in Fig. a, the force

equation of equilibrium gives

Equating i, j and k components,

The moment equation of equilibrium gives

5–66.

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.

z

A

B

2 m

0.5 m

C

454

Equating i, j and k Components

Solving Eqs. (1) to (6)

Note: One of the equations (4), (5) and (6) is redundant that will be satistied

automatically.

5–66. Continued

455

Ans:

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

5–67.

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.

x

A

C

B

TC

TB

TA

y

z

2 m

1 m

2 m

0.5 m

15 kN

456

*5–68.

The 100-lb door has its center of gravity at G. Determine the

components of reaction at hinges Aand Bif hinge Bresists

only forces in the xand ydirections and Aresists forces in

the x,y,zdirections.

SOLUTION

Equations of Equilibrium:From the free-body diagram of the door,Fig.a, By,B

x,

and Azcan be obtained by writing the moment equation of equilibrium about the

and axes and the force equation of equilibrium along the zaxis.

Using the above result and writing the force equations of equilibrium along the

y¿

x¿

B

G

z

18 in.

24 in.

Ans: