Mechanical Engineering Chapter 3 Homework We can express each of the forces on the free-body

201

3–41.

The single elastic cord ABC is used to support the 40-lb

load.Determine the position xand the tension in the cord

that is required for equilibrium. The cord passes through

the smooth ring at Band has an unstretched length of 6 ft

and stiffness of k=50 lb>ft.

A

C

B

x

5 ft

1 ft

SOLUTION

Equations of Equilibrium:Since elastic cord ABC passes over the smooth ring at B,

the tension in the cord is constant throughout its entire length.Applying the equation

Geometry: Referring to Fig. (b), the stretched length of cord ABC is

Spring Force Formula: Applying the spring force formula using Eq.(2), we obtain

202

3–42.

SOLUTION

Free-Body Diagram: The tension force in the cord is the same throughout the cord,

that is, 10 lb. From the geometry,

Equations of Equilibrium:

A “scale”is constructed with a 4-ft-long cord and the 10-lb

block D.The cord is fixed to a pin at Aand passes over two

small pulleys at Band C.Determine the weight of the

suspended block at B if the system is in equilibrium when

s = 1.5 ft.

s 1.5 ft

C

D

A

1 ft

Ans:

203

Ans:

SOLUTION

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

3–43.

The three cables are used to support the 40-kg owerpot.

Determine the force developed in each cable for

equilibrium.

2 m

z

1.5 m

D

y

x

A

B

204

Ans:

SOLUTION

Equations of Equilibrium. Referring to the FBD shown,

*3–44.

Determine the magnitudes of F1, F2, and F3 for equilibrium

of the particle.

y

30fi

25

24

7

4 kN 10 kN

F1

F2

F3

z

x

205

3–45.

If t

h

e

b

uc

k

et an

d

i

ts contents

h

ave a tota

l

we

i

g

h

t of 20

lb

,

determine the force in the supporting cables DA,DB,

and DC.

SOLUTION

4.5 ft

2.5 ft

3ft

A

C

z

Ans:

206

3–46.

SOLUTION

Cartesian Vector Notation:

Equations of Equilibrium:

Equating i, j, and k components, we have

Solving Eqs. (1),(2) and (3) yields

Spring Elongation: Using spring formula, Eq. 3–2, the spring elongation is .

s=F

k

Determine the stretch in each of the two springs required to

hold the 20-kg crate in the equilibrium position shown.

Each spring has an unstretched length of 2 m and a stiffness

of k=360 N>m.

y

x

z

O

C

B

A12 m

6m

4m

207

Ans:

SOLUTION

Equations of Equilibrium.

3–47.

Determine the force in each cable needed to support the

20-kg owerpot.

D

y

x

C

A

B

4 m

2 m

6 m

3 m

z

208

*3–48.

SOLUTION

Force Vectors: We can express each of the forces on the free-body diagram shown

in Fig. (a) in Cartesian vector form as

Determine the tension in the cables in order to support the

100-kg crate in the equilibrium position shown.

2.5 m 2 m

2 m

1 m

A

z

D

y

x

B

C

Equations of Equilibrium: Equilibrium requires

Equating the i, j, and kcomponents yields

Solving Eqs. (1) through (3) yields

Ans:

209

3–49.

SOLUTION

Force Vectors: We can express each of the forces on the free-body diagram shown

in Fig. (a) in Cartesian vector form as

Equations of Equilibrium: Equilibrium requires

Equating the i, j, and kcomponents yields

W

hen cable AD is subjected to maximum tension,.Thus,by

FAD =3000 N

Determine the maximum mass of the crate so that the tension

developed in any cable does not exceeded 3 kN.

2.5 m 2 m

2 m

1 m

A

z

D

y

x

B

C

Ans:

210

Ans:

SOLUTION

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

3–50.

Determine the force in each cable if F = 500 lb. z

A

B

y

D

F

1 ft

2 ft 3 ft

3 ft

6 ft

2 ft

1 ft

C

211

SOLUTION

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

Solving Eqs (1), (2) and (3)

3–51.

Determine the greatest force F that can be applied to the

ring if each cable can support a maximum force

of 800 lb.

z

A

B

y

D

F

1 ft

2 ft 3 ft

3 ft

6 ft

2 ft

1 ft

C

212

*3–52.

SOLUTION

Force Vectors: We can express each of the forces on the free-body diagram shown in

Fig. ain Cartesian vector form as

Equations of Equilibrium: Equilibrium requires

Equating the i,j,and kcomponents yields

Solving Eqs. (1) through (3) yields

gF=0;

FAB +FAC +FAD +W=0

Determine the tension developed in cables AB and AC and

the force developed along strut AD for equilibrium of the

400-lb crate.

x

z

5.5 ft

2 ft

A

B

D

C

4 ft

2.5 ft

6 ft

Ans:

213

3–53.

If the tension developed in each of the cables cannot

exceed 300 lb,determine the largest weight of the crate that

can be supported. Also,what is the force developed along

strut AD?

z

5.5 ft

2 ft

A

B

D

C

4 ft

2.5 ft

6 ft

SOLUTION

Force Vectors: We can express each of the forces on the free-body diagram

shown in Fig. ain Cartesian vector form as

Equations of Equilibrium: Equilibrium requires

Equating the i,j,and kcomponents yields

Let us assume that cable AC achieves maximum tension first. Substituting

gF=0;

FAB +FAC +FAD +W=0

214

3–54.

SOLUTION

Force Vectors: We can express each of the forces shown in Fig. ain Cartesian vector

form as

Equations of Equilibrium: Equilibrium requires

Equating the i,j,and kcomponents yields

Solving Eqs. (1) through (3) yields

y

A

B

C

D

x

z

6 ft

3 ft

2 ft 2 ft

3 ft

4 ft

Determine the tension developed in each cable for

equilibrium of the 300-lb crate.

Ans: