Mechanical Engineering Chapter 18 Problem The Slender Bar And Block Are Released From Rest The Position

Problem 18.54 The 2-kg slender bar and 5-kg block

are released from rest in the position shown. What

minimum coefﬁcient of static friction between the block

and the horizontal surface would be necessary for the

block not to move when the system is released? (See

Example 18.5.)

Solution: This solution is very similar to that of Problem 18.53.

Problem 18.55 As a result of the constant couple M

applied to the 1-kg disk, the angular acceleration of the

0.4-kg slender bar is zero. Determine Mand the coun-

terclockwise angular acceleration of the rolling disk.

1 m

M

0.25 m

40⬚

Solution: There are seven unknowns (M,N,f,O

x,O

y,a,α), six

dynamic equations, and one constraint equation. We use the following

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Problem 18.56 The slender bar weighs 40 lb and the

crate weighs 80 lb. At the instant shown, the velocity of

the crate is zero and it has an acceleration of 14 ft/s2

toward the left. The horizontal surface is smooth. Deter-

mine the couple Mand the tension in the rope. 6 ft

3 ft 6 ft

M

Solution: There are six unknowns (M,T,N,O

x,O

y,α),ﬁve

Problem 18.57 The slender bar weighs 40 lb and the

crate weighs 80 lb. At the instant shown, the veloc-

ity of the crate is zero and it has an acceleration of

14 ft/s2toward the left. The coefﬁcient of kinetic friction

between the horizontal surface and the crate is µk=0.2.

Determine the couple Mand the tension in the rope. 6 ft

3 ft 6 ft

M

Solution: There are seven unknowns (M,T,N,O

x,O

y,α,f),

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Problem 18.58 Bar AB is rotating with a constant

clockwise angular velocity of 10 rad/s. The 8-kg slender

bar BC slides on the horizontal surface. At the instant

shown, determine the total force (including its weight)

acting on bar BC and the total moment about its center

of mass. 0.4 m

10 rad/s

B

C

y

Problem 18.59 The masses of the slender bars AB

and BC are 10 kg and 12 kg, respectively. The angular

velocities of the bars are zero at the instant shown

and the horizontal force F=150 N. The horizontal

surface is smooth. Determine the angular accelerations

of the bars.

F

0.4 m

0.2 m

AB

C

The FBDs

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Problem 18.60 Let the total moment of inertia of the

car’s two rear wheels and axle be IR, and let the total

moment of inertia of the two front wheels be IF. The

radius of the tires is R, and the total mass of the car,

including the wheels, is m. If the car’s engine exerts a

torque (couple) Ton the rear wheels and the wheels do

not slip, show that the car’s acceleration is

a=RT

R2m+IR+IF

.

Strategy: Isolate the wheels and draw three free-body

diagrams.

Solution: The free body diagrams are as shown: We shall write

mBg

Problem 18.61 The combined mass of the motorcy-

cle and rider is 160 kg. Each 9-kg wheel has a 330-

mm radius and a moment of inertia I=0.8 kg-m2. The

engine drives the rear wheel by exerting a couple on it.

If the rear wheel exerts a 400-N horizontal force on the

road and you do not neglect the horizontal force exerted

on the road by the front wheel, determine (a) the motor-

cycle’s acceleration and (b) the normal forces exerted on

the road by the rear and front wheels. (The location of

the center of mass of the motorcycle not including its

wheels, is shown.)

1500 mm

649 mm

AB

723 mm

Solution: In the free-body diagrams shown, mw=9 kg and m=

M=−fF(0.33)=Iα.(4)

Bx

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Problem 18.62 In Problem 18.61, if the front wheel

lifts slightly off the road when the rider accelerates,

determine (a) the motorcycle’s acceleration and (b) the

torque exerted by the engine on the rear wheel.

Problem 18.63 The moment of inertia of the vertical

handle about Ois 0.12 slug-ft2. The object Bweighs

15 lb and rests on a smooth surface. The weight of the

bar AB is negligible (which means that you can treat the

bar as a two-force member). If the person exerts a 0.2-lb

horizontal force on the handle 15 in above O, what is

the resulting angular acceleration of the handle?

A

Solving Equations (1)–(3) with F=0.2 lb, we obtain α=1.06 rad/s2

B

aB

Problem 18.64 The bars are each 1 m in length and

have a mass of 2 kg. They rotate in the horizontal plane.

Bar AB rotates with a constant angular velocity of

4 rad/s in the counterclockwise direction. At the instant

shown, bar BC is rotating in the counterclockwise

direction at 6 rad/s. What is the angular acceleration of

bar BC?

BA

C

aBC

6 rad/s

4 rad/s

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Problem 18.65 Bars OQ and PQ each weigh 6 lb.

The weight of the collar Pand friction between the

collar and the horizontal bar are negligible. If the system

is released from rest with θ=45◦, what are the angular

accelerations of the two bars? 2 ft

O

Q

P

θ

2 ft

Problem 18.66 In Problem 18.65, what are the angular

accelerations of the two bars if the collar Pweighs 2 lb?

M=(N −P+Qy+Qx)(cos 45◦)=1

Problem 18.67 The 4-kg slender bar is pinned to 2-

kg sliders at Aand B. If friction is negligible and the

system is released from rest in the position shown, what

is the angular acceleration of the bar at that instant?

A

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Problem 18.68 The mass of the slender bar is mand

the mass of the homogeneous disk is 4m. The system

is released form rest in the position shown. If the disk

rolls and the friction between the bar and the horizon-

tal surface is negligible, show that the disk’s angular

acceleration is α=6g/95Rcounterclockwise.

2R

R

of the disk is

(5) ND−4mg −By=0,

Problem 18.69 Bar AB rotates in the horizontal plane

with a constant angular velocity of 10 rad/s in the coun-

terclockwise direction. The masses of the slender bars

BC and CD are 3 kg and 4.5 kg, respectively. Deter-

mine the xand ycomponents of the forces exerted on

bar BC by the pins at Band Cat the instant shown.

A

B

y

x

C

D

0.2 m 0.2 m

0.2 m

10 rad/s

Solution: First let’s do the kinematics

vC=vB+ωBC ×rC/B

A

D

0.2 m

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Problem 18.70 The 2-kg bar rotates in the horizontal

plane about the smooth pin. The 6-kg collar Aslides on

the smooth bar. At the instant shown, r=1.2m,ω=

0.4 rad/s, and the collar is sliding outward at 0.5 m/s

relative to the bar. If you neglect the moment of inertia

of the collar (that is, treat the collar as a particle), what

is the bar’s angular acceleration?

Strategy: Draw individual free-body diagrams of the

bar and collar and write Newton’s second law for the

collar in terms of polar coordinates.

r

A

2 m

ω

F=ma:Neθ=md2r

dt2−rω2er+rα +2dr

dt ωeθ.

Equating eθcomponents,

N=mrα +2dr

dt ω=(6)[rα +2(0.5)(0.4)](2).

Solving Equations (1) and (2) with r=1.2 m gives α=−0.255 rad/s2

Problem 18.71 In Problem 18.70, the moment of iner-

tia of the collar about its center of mass is 0.2 kg-m2.

Determine the angular acceleration of the bar, and com-

pare your answer with the answer to Problem 18.70.

Problem 18.72 The axis L0is perpendicular to both

segments of the L-shaped slender bar. The mass of the

bar is 6 kg and the material is homogeneous. Use inte-

gration to determine the moment of inertia of the bar

about L0.

LO

1 m

For the vertical part (Fig. b),

Iv=m

r2dm =1

0

(22+y2)ρA dy

=13

3ρA =26

3kg-m2.

Therefore I0=Ih+Iv=14 kg-m2.

Problem 18.73 Two homogenous slender bars, each of

mass mand length l, are welded together to form the T-

shaped object. Use integration to determine the moment

of inertia of the object about the axis through point O

that is perpendicular to the bars.

l

Ol

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Problem 18.74 The slender bar lies in the x−yplane.

Its mass is 6 kg and the material is homogeneous. Use

integration to determine its moment of inertia about the

zaxis.

x

y

1 m

2 m

50⬚

x

1 m

Problem 18.75 The slender bar lies in the x−yplane.

Its mass is 6 kg and the material is homogeneous. Use

integration to determine its moment of inertia about the

0

Problem 18.76 The homogeneous thin plate has mass

m=12 kg and dimensions b=1 m and h=2m.

Determine the mass moments of inertia of the plate about

the x,y, and zaxes.

Strategy: The mass moments of inertia of a thin plate

of arbitrary shape are given by Eqs. (18.37)–(18.39) in

terms of the moments of inertia of the cross-sectional

area of the plate. You can obtain the moments of inertia

of the triangular area from Appendix B.

x

y

h

b

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Problem 18.77 The brass washer is of uniform thick-

ness and mass m.

(a) Determine its moments of inertia about the xand

zaxes.

(b) Let Ri=0, and compare your results with the val-

ues given in Appendix C for a thin circular plate.

Ro

Ri

x

y

Solution:

Ri

Ro

Problem 18.78 The homogenous thin plate is of uni-

form thickness and weighs 20 lb. Determine its moment

of inertia about the yaxis.

y

x

y = 4 – x2 ft

1

–

4

Problem 18.79 Determine the moment of inertia of the

plate in Problem 18.78 about the xaxis.

Solution: The differential mass is dm =W

gA dy dx. The distance

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