Mechanical Engineering Chapter 5 Problem Active Example Suppose That The Beam Subjected Knm Counterclockwise Couplea The

Problem 5.1 In Active Example 5.1, suppose that the

beam is subjected to a 6kN-m counterclockwise couple

at the right end in addition to the 4-kN downward force.

Draw a sketch of the beam showing its new loading.

Draw the free-body diagram of the beam and apply the

equilibrium equations to determine the reactions at A.

4 kN

2 m

A

Solution: The equilibrium equations are

Problem 5.2 The beam has a ﬁxed support at Aand is

loaded by two forces and a couple. Draw the free-body

diagram of the beam and apply equilibrium to determine

the reactions at A.

2 kN

4 kN

60⬚

A6 kN-m

1.5 m

1 m

Solution: The free-body diagram is drawn.

268

Problem 5.3 The beam is subjected to a load FD

400 N and is supported by the rope and the smooth

surfaces at Aand B.

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

(b) What are the magnitudes of the reactions at A

and B?

F

30°

45°

AB

1.2 m 1.5 m 1 m

Solution:

CMAD0: 1.2T2.7⊲400⊳C3.7Bcos 30°D0

Solving, we get

AD271 N

BD383 N

TD124 N

y

F

A

Problem 5.4 (a) Draw the free-body diagram of the

beam. (b) Determine the tension in the rope and the

reactions at B.

5 ft 9 ft

30⬚

B

A

600 lb

Problem 5.5 (a) Draw the free-body diagram of the

60-lb drill press, assuming that the surfaces at Aand B

are smooth.

Solution: The system is in equilibrium.

(a) The free body diagram is shown.

270

Problem 5.6 The masses of the person and the diving

board are 54 kg and 36 kg, respectively. Assume that

they are in equilibrium.

1.2 m

Solution:

(a)

AYD1.85 kN

AY

AX

BYWP

WD

1.2 m

2.4 m

Problem 5.7 The ironing board has supports at Aand

Bthat can be modeled as roller supports.

(a) Draw the free-body diagram of the ironing board.

(b) Determine the reactions at Aand B.

y

x

A

10 lb 3 lb

B

12 in 10 in 20 in

Problem 5.8 The distance xD9m.

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

(b) Determine the reactions at the supports.

10 kN

B

A

x

6 m

(b) The equilibrium equations

Fx:AxD0

Fy:AyCBy10 kN D0

MA:By⊲6m⊳⊲10 kN⊳⊲9m⊳D0

Solving we ﬁnd

AxD0,A

yD5kN,B

yD15 kN

6 m

Ax

AyBy

Problem 5.9 In Example 5.2, suppose that the 200-lb

downward force and the 300 ft-lb counterclockwise

couple change places; the 200-lb downward force acts

at the right end of the horizontal bar, and the 300 ft-lb

counterclockwise couple acts on the horizontal bar 2 ft

to the right of the support A. Draw a sketch of the object

showing the new loading. Draw the free-body diagram

of the object and apply the equilibrium equations to

determine the reactions at A.2 ft

2 ft

300 ft-lb

2 ft

100 lb

200 lb

A

30

272

Problem 5.10 (a) Draw the free-body diagram of the

beam.

(b) Determine the reactions at the supports.

100 lb 400 lb

B

A

900 ft-lb

3 ft 4 ft 3 ft 4 ft

The sum of the moments about Ais

MAD3⊲100⊳C900 7⊲400⊳C11FBD0.

From which FBD2200

11 D200 lb

Substitute into the force balance equation to obtain

FAD300 FBD100 lb

3 ft

900 ft lb

4 ft 4 ft

FAFB

Problem 5.11 The person exerts 20-N forces on the

pliers. The free-body diagram of one part of the pliers

is shown. Notice that the pin at Cconnecting the two

parts of the pliers behaves like a pin support. Determine

the reactions at Cand the force Bexerted on the pliers

by the bolt.

C

50 mm

45⬚

80

mm

25

mm

C

Cy

Cx

B

Solution: The equilibrium equations

MC:B⊲25 mm⊳20 N cos 45°⊲80 mm⊳

20 N sin 45°⊲50 mm⊳D0

Fx:Cx20 N sin 45°D0

Fy:CyB20 N cos 45°D0

Solving:

Problem 5.12 (a) Draw the free-body diagram of the

beam.

(b) Determine the reactions at the pin support A.

30⬚

AB

2 kN-m

8 kN 8 kN

600

mm

500

mm

600

mm

600

mm

(b) The equilibrium equations

MA:⊲8kN⊳⊲0.6m⊳C⊲8kN⊳⊲1.1m⊳2 kNm

Bcos 30°⊲2.3m⊳D0

Fx:AxBsin 30°D0

Fy:Ay8kNC8kNBcos 30°D0

Solving

AxD0.502 kN,A

yD0.870 kN,BD1.004 kN

Ax

Ay

B

Problem 5.13 (a) Draw the free-body diagram of the

beam.

(b) Determine the reactions at the supports.

A

y

6 m

8 m

12 m

Bx

40 kN

274

Problem 5.14 (a) Draw the free-body diagram of the

beam.

(b) If FD4 kN, what are the reactions at Aand B?

A2 kN-m

B

F0.2 m

0.4 m

0.3 m

0.2 m

Solution:

Ax

2 kN-m

Problem 5.15 In Example 5.3, suppose that the attach-

ment point for the suspended mass is moved toward

point Bsuch that the horizontal distance from Ato

the attachment point increases from 2 m to 3 m. Draw

a sketch of the beam AB showing the new geometry.

Draw the free-body diagram of the beam and apply the

equilibrium equations to determine the reactions at A

to B.

2 m

A

B

3 m

2 m

276

Problem 5.16 A person doing push-ups pauses in the

position shown. His 180-lb weight Wacts at the point

shown. The dimensions a=15in,b= 42 in, and c=16

in. Determine the normal force exerted by the ﬂoor on

each of his hands and on each of his feet. W

ab

c

We ﬁnd that

66.3 lb on each hand

Problem 5.17 The hydraulic piston AB exerts a 400-lb

force on the ladder at Bin the direction parallel to the

piston. Determine the weight of the ladder and the reac-

tions at C.

AB

C

3 ft

6 ft 3 ft

6 ft

W

Solution: The free-body diagram of the ladder is shown.

The angle between the piston AB and the horizontal is

Problem 5.18 Draw the free-body diagram of the

structure by isolating it from its supports at Aand E.

Determine the reactions at Aand E.

C

D

BA

100 lb

E

400 lb

200 ft-lb

2 ft

1 ft

2 ft 2 ft 2 ft

Solution: The free-body diagram is shown.

The equilibrium equations are

Problem 5.19 (a) Draw the free-body diagram of the

beam.

(b) Determine the tension in the cable and the reactions

at A.A

30 in 800 lb

BC

30 in 30 in

30°

Solution:

TT

278

Problem 5.20 The unstretched length of the spring CD

is 350 mm. Suppose that you want the lever ABC to

exert a 120-N normal force on the smooth surface at A.

Determine the necessary value of the spring constant k

and the resulting reactions at B.

20⬚

B

A

C

k

D450

mm

180

mm

230

mm

Solution: We have

Problem 5.21 The mobile is in equilibrium. The ﬁsh

Bweighs 27 oz. Determine the weights of the ﬁsh A,C,

and D. (The weights of the crossbars are negligible.) 12 in 3 in

2 in6 in

2 in7 in

A

C

D

B

Solution: Denote the reactions at the supports by FAB,FCD, and

FBCD as shown. Start with the crossbar supporting the weights C

and D. The sum of the forces is

CD

FCD

280

Problem 5.22 The car’s wheelbase (the distance

between the wheels) is 2.82 m. The mass of the car is

1760 kg and its weight acts at the point xD2.00 m,

yD0.68 m. If the angle ˛D15°, what is the total

normal force exerted on the two rear tires by the sloped

ramp?

y

x

α

Solution: Split Winto components:

Wcos ˛acts ?to the incline

FY:NRCNFWcos ˛D0

MR:⊲2⊳⊲W cos ˛⊳ C⊲0.68⊳W sin ˛C2.82NFD0

Solving: NRD5930 N, NFD10750 N

0.68 m

fNR

α

y

x

W

W = (1760X9.81) N

Problem 5.23 The link AB exerts a force on the bucket

of the excavator at Athat is parallel to the link. The

weight W= 1500 lb. Draw the free-body diagram of the

bucket and determine the reactions at C. (The connection

at Cis equivalent to a pin support of the bucket.)

8 in

14 in

8 in

W

CD

A

16 in

4 in

B

Solution: The free-body diagram is shown.

The angle between the link AB and the horizontal is

Problem 5.24 The 14.5-lb chain saw is subjected to

the loads at Aby the log it cuts. Determine the reactions

Solution: The sum of the forces are

FXD5CBXRcos 60°D0.

BXD11.257 lb,

Problem 5.25 The mass of the trailer is 2.2 Mg (mega-

grams). The distances aD2.5 m and bD5.5 m. The

truck is stationary, and the wheels of the trailer can turn

Solution:

(a) The free body diagram is shown.

ab

A

282

Problem 5.26 The total weight of the wheelbarrow

and its load is W= 100 lb. (a) What is the magnitude of

the upward force Fnecessary to lift the support at Aoff

the ground? (b) What is the magnitude of the downward

force necessary to raise the wheel off the ground?

B

W

A

F

Problem 5.27 The airplane’s weight is WD2400 lb.

Its brakes keep the rear wheels locked. The front (nose)

wheel can turn freely, and so the ground exerts no hori-

zontal force on it. The force Texerted by the airplane’s

propeller is horizontal.

(a) Draw the free-body diagram of the airplane. Deter-

mine the reaction exerted on the nose wheel and

the total normal reaction on the rear wheels

(b) when TD0,

(c) when TD250 lb.

W

T

B

A

5 ft

4 ft

2 ft

MAD5WC7BYD0,

from which BYD5W

284

Problem 5.28 A safety engineer establishing limits on

the load that can be carried by a forklift analyzes the

situation shown. The dimensions are a= 32 in, b=30

in, and c= 26 in. The combined weight of the forklift and

operator is WF= 1200 lb. As the weight WLsupported

by the forklift increases, the normal force exerted on the

ﬂoor by the rear wheels at Bdecreases. The forklift is

on the verge of tipping forward when the normal force

at Bis zero. Determine the value of WLthat will cause

this condition.

abc

WF

BA

WL

Solution: The equilibrium equations and the special condition for

this problem are

Problem 5.29 Paleontologists speculate that the ste-

gosaur could stand on its hind limbs for short periods

to feed. Based on the free-body diagram shown and

assuming that mD2000 kg, determine the magnitudes

of the forces Band Cexerted by the ligament –muscle

brace and vertebral column, and determine the angle ˛.

mg

B

22°

α

C

160

mm

580

mm

415

mm

790

mm

Solution: Take the origin to be at the point of application of the

force C. The position vectors of the points of application of the forces

Band Ware:

from which

444.72 D34.85 kN.

The sums of the forces:

51.93 D14.1°.

The magnitude of C,

286

Problem 5.30 The weight of the fan is WD20 lb. Its

base has four equally spaced legs of length bD12 in.

Each leg has a pad near the end that contacts the ﬂoor

and supports the fan. The height hD32 in. If the fan’s

blade exerts a thrust TD2 lb, what total normal force

is exerted on the two legs at A?

h

b

T

W

AB

T

Problem 5.31 The weight of the fan is WD20 lb. Its

base has four equally spaced legs of length bD12 in.

Each leg has a pad near the end that contacts the ﬂoor

and supports the fan. The height hD32 in. As the thrust

Tof the fan increases, the normal force supported by the

two legs at Adecreases. When the normal force at Ais

zero, the fan is on the verge of tipping over. Determine

the value of Tthat will cause this condition.

h

b

T

W

Side View

AB

Top View

T

Solution: The free-body diagram is shown.

The equilibrium equations are