Mechanical Engineering Chapter 3 Problem Active Example Suppose That The Angle Between The Ramp Supporting The

Problem 3.1 In Active Example 3.1, suppose that the

angle between the ramp supporting the car is increased

from 20°to 30°. Draw the free-body diagram of the car

showing the new geometry. Suppose that the cable from

Ato Bmust exert a 1900-lb horizontal force on the car

to hold it in place. Determine the car’s weight in pounds.

AB

20⬚

Solution: The free-body diagram is shown to the right.

Problem 3.2 The ring weighs 5 lb and is in equilib-

rium. The force F1D4.5 lb. Determine the force F2and

the angle ˛.

x

y

30⬚

F2

F1

a

Solution: The free-body diagram is shown below the drawing. The

equilibrium equations are

84

Problem 3.3 In Example 3.2, suppose that the attach-

ment point Cis moved to the right and cable AC is

extended so that the angle between cable AC and the

ceiling decreases from 45°to 35°. The angle between

cable AB and the ceiling remains 60°. What are the

tensions in cables AB and AC?

B

C

A

60⬚

45⬚

Solution: The free-body diagram is shown below the picture.

The equilibrium equations are:

Problem 3.4 The 200-kg engine block is suspended

by the cables AB and AC. The angle ˛D40°. The free-

body diagram obtained by isolating the part of the system

within the dashed line is shown. Determine the forces

TAB and TAC.

B

C

AAx

y

aa

TAB TAC

(200 kg) (9.81 m/s2)

Solution:

TAB TAC

Problem 3.5 A heavy rope used as a mooring line for

a cruise ship sags as shown. If the mass of the rope is

90 kg, what are the tensions in the rope at Aand B?B40⬚

55⬚A

Problem 3.6 A physiologist estimates that the

masseter muscle of a predator, Martes, is capable of

exerting a force Mas large as 900 N. Assume that

the jaw is in equilibrium and determine the necessary

force Tthat the temporalis muscle exerts and the force

Pexerted on the object being bitten.

T

22⬚

P

M

Solution: The equilibrium equations are

86

Problem 3.7 The two springs are identical, with un-

stretched lengths 250 mm and spring constants kD

1200 N/m.

(a) Draw the free-body diagram of block A.

(b) Draw the free-body diagram of block B.

(c) What are the masses of the two blocks?

B

A

300 mm

280 mm

Solution: The tension in the upper spring acts on block Ain the

positive Ydirection, Solve the spring force-deﬂection equation for

the tension in the upper spring. Apply the equilibrium conditions to

block A. Repeat the steps for block B.

Similarly, the tension in the lower spring acts on block Ain the nega-

300 mm

Problem 3.8 The two springs in Problem 3.7 are iden-

tical, with unstretched lengths of 250 mm. Suppose that

their spring constant kis unknown and the sum of the

masses of blocks Aand Bis 10 kg. Determine the value

of kand the masses of the two blocks.

Solution: All of the forces are in the vertical direction so we will

Problem 3.9 The inclined surface is smooth (Remem-

ber that “smooth” means that friction is negligble). The

two springs are identical, with unstretched lengths of

250 mm and spring constants kD1200 N/m. What are

the masses of blocks Aand B?

B

A

300 mm

280 mm

30⬚

Solution:

mAg

NB

88

c

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Problem 3.10 The mass of the crane is 20,000 kg. The

crane’s cable is attached to a caisson whose mass is

400 kg. The tension in the cable is 1 kN.

(a) Determine the magnitudes of the normal and

friction forces exerted on the crane by the

level ground.

(b) Determine the magnitudes of the normal and

friction forces exerted on the caisson by the

level ground.

Strategy: To do part (a), draw the free-body diagram

of the crane and the part of its cable within the

dashed line.

45°

Solution:

Ncrane D196.9kN,F

crane D0.707 kN

(b) Fy:Ncaisson 3.924 kN C1 kN sin 45°D0

Fx:1 kN cos 45°CFcaisson D0

Ncaisson D3.22 kN,F

caisson D0.707 kN

Ncrane

Fcrane

x

45°

1 kN

3.924 kN

Ncaisson

Fcaisson

Problem 3.11 The inclined surface is smooth. The

100-kg crate is held stationary by a force Tapplied to

Solution:

(a) The FBD

Ν

Problem 3.12 The 1200-kg car is stationary on the

sloping road.

(a) If ˛D20°, what are the magnitudes of the total

normal and friction forces exerted on the car’s tires

by the road?

(b) The car can remain stationary only if the total

friction force necessary for equilibrium is not

greater than 0.6 times the total normal force.

What is the largest angle ˛for which the car can

remain stationary?

a

Solution:

F–:F11.772 kN sin ˛D0

N

11.772 kN

Problem 3.13 The 100-lb crate is in equilibrium on the

smooth surface. The spring constant is kD400 lb/ft. Let

Sbe the stretch of the spring. Obtain an equation for S

(in feet) as a function of the angle ˛.

a

90

Problem 3.14 A 600-lb box is held in place on the

smooth bed of the dump truck by the rope AB.

(a) If ˛D25°, what is the tension in the rope?

(b) If the rope will safely support a tension of 400 lb,

what is the maximum allowable value of ˛?

α

A

B

Problem 3.15 The 80-lb box is held in place on

the smooth inclined surface by the rope AB. Determine

the tension in the rope and the normal force exerted

on the box by the inclined surface.

A

B

30⬚

50⬚

Solution: The equilibrium equations (in terms of a coordinate

Problem 3.16 The 1360-kg car and the 2100-kg tow

truck are stationary. The muddy surface on which the

car’s tires rest exerts negligible friction forces on them.

What is the tension in the tow cable?

18⬚

10⬚

26⬚

Solution: FBD of the car being towed

13.34 kN

N

26°

92

c

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Problem 3.17 Each box weighs 40 lb. The angles are

measured relative to the horizontal. The surfaces are

smooth. Determine the tension in the rope Aand the

normal force exerted on box Bby the inclined surface.

D

C

B

A

70⬚

20⬚

45⬚

Solution: The free-body diagrams are shown.

The equilibrium equations for box Dare

Problem 3.18 A 10-kg painting is hung with a wire

supported by a nail. The length of the wire is 1.3 m.

(a) What is the tension in the wire?

(b) What is the magnitude of the force exerted on the

nail by the wire?

1.2 m

Solution:

(a) Fy:98.1N25

98.1 N

Problem 3.19 A 10-kg painting is hung with a wire

supported by two nails. The length of the wire is 1.3 m.

(a) What is the tension in the wire?

(b) What is the magnitude of the force exerted on each

nail by the wire? (Assume that the tension is the

same in each part of the wire.)

Compare your answers to the answers to Problem 3.18.

0.4 m0.4 m 0.4 m

Solution:

Fy:Tsin 27.3°49.05 N D0

TD107 N

(b) Examine one of the nails

Fx:RxTcos 27.3°CTD0

Fy:RyTsin 27.3°D0

RDRx2CRy2

RD50.5N

T

49.05 N

Rx

T

Problem 3.20 Assume that the 150-lb climber is in

equilibrium. What are the tensions in the rope on the

left and right sides?

14⬚15⬚

Solution:

y

94

Problem 3.21 If the mass of the climber shown in

Problem 3.20 is 80 kg, what are the tensions in the rope

on the left and right sides?







FxDTRcos⊲15°⊳TLcos⊲14°⊳D0

FyDTRsin⊲15°⊳CTLsin⊲14°⊳mg D0

Solving, we get

TLD1.56 kN,T

RD1.57 kN

14°15°

TL

TR

x

Problem 3.22 The construction worker exerts a 20-lb

force on the rope to hold the crate in equilibrium in the

position shown. What is the weight of the crate?

5⬚

30⬚

Solution: The free-body diagram is shown.

Problem 3.23 A construction worker on the moon,

where the acceleration due to gravity is 1.62 m/s2, holds

the same crate described in Problem 3.22 in the position

shown. What force must she exert on the cable to hold

the crate ub equilibrium (a) in newtons; (b) in pounds?

5⬚

30⬚

Solution: The free-body diagram is shown.

From Problem 3.22 we know that the weight is WD188 lb. Therefore

its mass is

The equilibrium equations for the part of the rope system where the

96

Problem 3.24 The person wants to cause the 200-lb

crate to start sliding toward the right. To achieve this,

the horizontal component of the force exerted on the

crate by the rope must equal 0.35 times the normal force

exerted on the crate by the ﬂoor. In Fig.a, the person

pulls on the rope in the direction shown. In Fig.b, the

person attaches the rope to a support as shown and pulls

upward on the rope. What is the magnitude of the force

he must exert on the rope in each case?

20⬚

(a)

Ffr D0.35N

(a) For equilibrium we have

(b) The person exerts the force F. Using the free-body diagram of

the crate and of the point on the rope where the person grabs the

rope, we ﬁnd

Problem 3.25 A trafﬁc engineer wants to suspend a

200-lb trafﬁc light above the center of the two right

lanes of a four-lane thoroughfare as shown. Determine

the tensions in the cables AB and BC.

Solution:

Fx:6

p37 TAB C2

p5TBC D0

Problem 3.26 Cable AB is 3 m long and cable BC is

4 m long. The mass of the suspended object is 350 kg.

Determine the tensions in cables AB and BC.C

B

A

5m

Solution:

TAB

98

Problem 3.27 In Problem 3.26, the length of cable AB

is adjustable. If you don’t want the tension in either cable

AB or cable BC to exceed 3 kN, what is the minimum

acceptable length of cable AB?

2.11 kN <3kN

3.43 kN

Problem 3.28 What are the tensions in the upper and

lower cables? (Your answers will be in terms of W.

Neglect the weight of the pulley.)

45°฀ 30°฀

W

Solution: Isolate the weight. The frictionless pulley changes the

direction but not the magnitude of the tension. The angle between the

right hand upper cable and the xaxis is ˛, hence

TU

βα

100

c

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Problem 3.29 Two tow trucks lift a 660-lb motorcycle

out of a ravine following an accident. If the motorcycle

is in equilibrium in the position shown, what are the

tensions in cables AB and AC?B

A

C

(12, 32) ft

(36, 36) ft

(26, 16) ft

x

y

Solution: The angles are

Problem 3.30 An astronaut candidate conducts exper-

iments on an airbearing platform. While she carries out

calibrations, the platform is held in place by the hori-

zontal tethers AB,AC, and AD. The forces exerted by

the tethers are the only horizontal forces acting on the

platform. If the tension in tether AC is 2 N, what are the

tensions in the other two tethers?

3.0 m 1.5 m

BC

TOP VIEW

D

4.0 m

3.5 m

A

Solution: Isolate the platform. The angles ˛and ˇare

3.0 m

B

102

Problem 3.31 The bucket contains concrete and

weighs 5800 lb. What are the tensions in the cables AB

and AC?

BC

A

(20, 34) ft

(5, 34) ft

(12, 16) ft

y

x

Solution: The angles are

Problem 3.32 The slider Ais in equilibrium and the

bar is smooth. What is the mass of the slider? 20⬚

45⬚

200 N

A

Solution: The pulley does not change the tension in the rope that

passes over it. There is no friction between the slider and the bar.

Eqns. of Equilibrium:

y

20°

T = 200 N

mg = (9.81) g