Mechanical Engineering Chapter 18 Problem Box Subjected Horizontal Force Neglect Friction The Box Remains The Oor

Problem 18.112 A 2-kg box is subjected to a 40-N

horizontal force. Neglect friction.

(a) If the box remains on the ﬂoor, what is its accel-

eration?

(b) Determine the range of values of cfor which the

box will remain on the ﬂoor when the force is

applied.

40 N

BA

100 mm

c

100 mm

Solution:

Problem 18.113 The slender, 2-slug bar AB is 3 ft

long. It is pinned to the cart at Aand leans against it

at B.

(a) If the acceleration of the cart is a=20 ft/s2, what

normal force is exerted on the bar by the cart at B?

(b) What is the largest acceleration afor which the bar

will remain in contact with the surface at B?A

B

a

60°

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Problem 18.114 To determine a 4.5-kg tire’s moment

of inertia, an engineer lets the tire roll down an inclined

surface. If it takes the tire 3.5 s to start from rest and

roll 3 m down the surface, what is the tire’s moment of

inertia about its center of mass?

330 mm

Problem 18.115 Pulley Aweighs 4 lb, IA=0.060

slug-ft2, and IB=0.014 slug-ft2. If the system is

released from rest, what distance does the 16-lb weight

fall in 0.5 s?

12 in

8 in

8 lb

16 lb

A

B

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Problem 18.116 Model the excavator’s arm ABC as a

single rigid body. Its mass is 1200 kg, and the moment

of inertia about its center of mass is I=3600 kg-m2.If

point Ais stationary, the angular velocity of the arm is

zero, and the angular acceleration is 1.0 rad/s2counter-

clockwise, what force does the vertical hydraulic cylin-

der exert on the arm at B?

x

y

B

C

3.0 m

A

1.7 m 1.7 m

Solution: The distance from Ato the center of mass is

IA=I+d2m=28,270 kg-m2.

From the equation of angular motion: 1.7B−3.4mg =IAα.

Substitute α=1.0 rad/s2, to obtain B=40,170 N.

Problem 18.117 Model the excavator’s arm ABC as a

single rigid body. Its mass is 1200 kg, and the moment of

inertia about its center of mass is I=3600 kg-m2. The

angular velocity of the arm is 2 rad/s counterclockwise

and its angular acceleration is 1 rad/s2counterclockwise.

What are the components of the force exerted on the arm

at A?

Problem 18.118 To decrease the angle of elevation of

the stationary 200-kg ladder, the gears that raised it are

disengaged, and a fraction of a second later a second

set of gears that lower it are engaged. At the instant

the gears that raised the ladder are disengaged, what is

the ladder’s angular acceleration and what are the com-

ponents of force exerted on the ladder by its support

at O? The moment of inertia of the ladder about Ois

I0=14,000 kg-m2, and the coordinates of its center of

mass at the instant the gears are disengaged are x=3m,

y=4m.

y

x

O

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Problem 18.119 The slender bars each weigh 4 lb and

are 10 in. long. The homogenous plate weighs 10 lb. If

the system is released from rest in the position shown,

what is the angular acceleration of the bars at that

instant?

45°

8 in

since ω=0 upon release.

The acceleration of the plate:

ijk



=10

12 sin θαi−10

12 cos θαj(ft/s2).

From which aPx =10

12 sin θα,aPy =−

10

12 cos θα.

Bx=4.42 lb, By=5.68 lb. α=30.17 rad/s2.

Problem 18.120 A slender bar of mass mis released

from rest in the position shown. The static and kinetic

friction coefﬁcients of friction at the ﬂoor and the wall

have the same value µ. If the bar slips, what is its angu-

lar acceleration at the instant of release?

Solution: Choose a coordinate system with the origin at the inter-

section of wall and ﬂoor, with the xaxis parallel to the ﬂoor. Denote

the points of contact at wall and ﬂoor by P and N respectively, and

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Problem 18.121 Each of the go-cart’s front wheels

weighs 5 lb and has a moment of inertia of 0.01 slug-ft2.

The two rear wheels and rear axle form a single rigid

body weighing 40 lb and having a moment of inertia of

0.1 slug-ft2. The total weight of the go-cart and driver

is 240 lb. (The location of the center of mass of the go-

cart and driver, not including the front wheels or the rear

wheels and rear axle, is shown.) If the engine exerts a

torque of 12 ft-lb on the rear axle, what is the go-cart’s

acceleration?

15 in

AB

6 in 4 in

Problem 18.122 Bar AB rotates with a constant angu-

lar velocity of 10 rad/s in the counterclockwise direc-

tion. The masses of the slender bars BC and CDE are

2 kg and 3.6 kg, respectively. The yaxis points upward.

Determine the components of the forces exerted on bar

BC by the pins at Band Cat the instant shown.

A

B

C

400 mm

10 rad/s D

700 mm

x

y

700 mm 400

mm

E

Solution: The velocity of point Bis

ij k



0.7−0.40

=−4i+0.4ωBCi+0.7ωBCj(m/s).

By

Bx

The acceleration of the center of mass of BC is

ijk



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Problem 18.123 At the instant shown, the arms of the

robotic manipulator have the constant counterclockwise

angular velocities ωAB =−0.5 rad/s, ωBC =2 rad/s,

and ωCD =4 rad/s. The mass of arm CD is 10 kg, and

the center of mass is at its midpoint. At this instant, what

force and couple are exerted on arm CD at C?

x

D

C

y

B

30°

20°

A

250 mm

300 mm

250 mm

from which

Problem 18.124 Each bar is 1 m in length and has

a mass of 4 kg. The inclined surface is smooth. If the

system is released from rest in the position shown, what

are the angular accelerations of the bars at that instant?

A

B

45°

30°

O

the acceleration of Ato obtain the two equations:

(6)−Ax−Bsin β=maGABx ,

556

Problem 18.125 Each bar is 1 m in length and has

a mass of 4 kg. The inclined surface is smooth. If the

system is released from rest in the position shown, what

is the magnitude of the force exerted on bar OA by the

support at Oat that instant?

Problem 18.126 The ﬁxed ring gear lies in the

horizontal plane. The hub and planet gears are bonded

together. The mass and moment of inertia of the

combined hub and planet gears are mHP =130 kg and

IHP =130 kg-m2. The moment of inertia of the sun gear

is Is=60 kg-m2. The mass of the connecting rod is

5 kg, and it can be modeled as a slender bar. If a 1 kN-

m counterclockwise couple is applied to the sun gear,

what is the resulting angular acceleration of the bonded

hub and planet gears?

240 mm

720 mm

340

mm

140

mm

Planet gear

Connecting

rod

Sun gear

Hub gear

Ring gear

Solution: The moment equation for the sun gear is

Problem 18.127 The system is stationary at the instant

shown. The net force exerted on the piston by the

exploding fuel-air mixture and friction is 5 kN to the

left. A clockwise couple M=200 N-m acts on the crank

AB. The moment of inertia of the crank about Ais

0.0003 kg-m2. The mass of the connecting rod BC is

0.36 kg, and its center of mass is 40 mm from Bon

the line from Bto C. The connecting rod’s moment

of inertia about its center of mass is 0.0004 kg-m2.

The mass of the piston is 4.6 kg. What is the piston’s

acceleration? (Neglect the gravitational forces on the

crank and connecting rod.)

40°

50 mm

A

M

B

C

125 mm

Solution: From the law of sines:

from which

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Problem 18.128 If the crank AB in Problem 18.127

has a counterclockwise angular velocity of 2000 rpm at

the instant shown, what is the piston’s acceleration?

Solution: The angular velocity of AB is