Mechanical Engineering Chapter 18 Problem Points And Lie The Planesolution From The Solution Problem The Acceleration

Problem 18.31 Points Band C lie in the x–yplane.

The yaxis is vertical. The center of mass of the 18-

kg arm BC is at the midpoint of the line from B

Solution: From the solution of Problem 18.30, the acceleration of

point Bis aB=−0.323i−0.149j(m/s2).IfαBC =0, the acceleration

of the center of mass Gof arm BC is

Problem 18.32 The radius of the 2-kg disk is R=

80 mm. Its moment of inertia is I=0.0064 kg-m2.It

rolls on the inclined surface. If the disk is released from

rest, what is the magnitude of the velocity of its center

two seconds later? (See Active Example 18.2).

30⬚

R

Solution: There are four unknowns (N, f, a, α), three dynamic

equations, and one constraint equation. We have

Problem 18.33 The radius of the 2-kg disk is R=

80 mm. Its moment of inertia is I=0.0064 kg-m2. What

minimum coefﬁcient of static friction is necessary for the

disk to roll, instead of slip, on the inclined surface? (See

R

Problem 18.34 A thin ring and a homogeneous circu-

lar disk, each of mass mand radius R, are released from

rest on an inclined surface. Determine the ratio vring/vdisk

of the velocities of the their centers when they ave rolled

a distance D.

D

RR

D

490

Problem 18.35 The stepped disk weighs 40 lb and its

moment of inertia is I=0.2 slug-ft2. If the disk is

released from rest, how long does it take its center to

fall 3 ft? (Assume that the string remains vertical.)

4 in

Problem 18.36 The radius of the pulley is R=

100 mm and its moment of inertia is I=0.1 kg-

m2. The mass m=5 kg. The spring constant is k=

135 N/m. The system is released from rest with the

spring unstretched. At the instant when the mass has

fallen 0.2 m, determine (a) the angular acceleration of

R

=(0.1m)([5 kg][9.81 m/s2]−[135 N/m][0.2m])

492

Problem 18.37 The radius of the pulley is R=100 mm

and its moment of inertia is I=0.1 kg-m2. The mass

m=5 kg. The spring constant is k=135 N/m. The sys-

tem is released from rest with the spring unstretched.

What maximum distance does the mass fall before re-

bounding?

R

Problem 18.38 The mass of the disk is 45 kg and its

radius is R=0.3 m. The spring constant is k=60 N/m.

The disk is rolled to the left until the spring is com-

pressed 0.5 m and released from rest.

(a) If you assume that the disk rolls, what is its angular

acceleration at the instant it is released?

(b) What is the minimum coefﬁcient of static friction

for which the disk will not slip when it is released?

kR

494

Problem 18.39 The disk weighs 12 lb and its radius

is 6 in. It is stationary on the surface when the force

F=10 lb is applied.

(a) If the disk rolls on the surface, what is the accel-

eration of its center?

(b) What minimum coefﬁcient of static friction is nec-

essary for the disk to roll instead of slipping when

the force is applied?

F

32.2 ft/s2=8.94 ft/s2.a=17.9 ft/s2.

Problem 18.40 A 42-lb sphere with radius R=4in

is placed on a horizontal surface with initial angular

velocity ω0=40 rad/s. The coefﬁcient of kinetic fric-

tion between the sphere and the surface is µk=0.06.

What maximum velocity will the center of the sphere

attain, and how long does it take to reach that velocity?

0

ω

496

Problem 18.41 A soccer player kicks the ball to a

teammate 8 m away. The ball leaves the player’s foot

moving parallel to the ground at 6 m/s with no angu-

lar velocity. The coefﬁcient of kinetic friction between

the ball and the grass is µk=0.32. How long does it

take the ball to reach his teammate? The radius of the

ball is 112 mm and its mass is 0.4 kg. Estimate the ball’s

moment of inertia by using the equation for a thin spher-

ical shell: I=2

3mR2.

(b) Rolling — Steady motion

Problem 18.42 The 100-kg cylindrical disk is at rest

when the force Fis applied to a cord wrapped around

it. The static and kinetic coefﬁcients of friction between

the disk and the surface equal 0.2. Determine the angular

acceleration of the disk if (a) F=500 N and (b) F=

1000 N.

Strategy: First solve the problem by assuming that the

disk does not slip, but rolls on the surface. Determine the

friction force, and ﬁnd out whether it exceeds the product

of the coefﬁcient of friction and the normal force. If it

does, you must rework the problem assuming that the

disk slips.

F

300 mm

From Newton’s second law: F−f=max, where axis the accelera-

(b) For F=1000 N the acceleration is

498

Problem 18.43 The ring gear is ﬁxed. The mass and

moment of inertia of the sun gear are mS=320 kg and

IS=40 kg-m2. The mass and moment of inertia of each

planet gear are mP=38 kg and IP=0.60 kg-m2.Ifa

couple M=200 N-m is applied to the sun gear, what

is the latter’s angular acceleration? 0.50 m

M

0.86 m

Ring gear

0.18 m.

Planet Gears: Mc:Gr −Fr =IPαP

Ft:F+G=mPact

From kinematics act =−rαP

2αPrP=−RαS

We have 5 eqns in 5 unknowns. Solving, αS=3.95 rad/s2(counter-

clockwise)

GC

r

3 small disks

O

F

IP

Ms

Is

et

Problem 18.44 In Problem 18.43, what is the mag-

nitude of the tangential force exerted on the sun gear

by each planet gear at their point of contact when the

200 N-m couple is applied to the sun gear?

Problem 18.45 The 18-kg ladder is released from rest

in the position shown. Model it as a slender bar and

neglect friction. At the instant of release, determine

(a) the angular acceleration of the ladder and (b) the

normal force exerted on the ladder by the ﬂoor. (See

Active Example 18.3.)

4 m

30°

500

Problem 18.46 The 18-kg ladder is released from rest

in the position shown. Model it as a slender bar and

neglect friction. Determine its angular acceleration at the

instant of release.

20⬚

30⬚

4 m

Problem 18.47 The 4-kg slender bar is released from

rest in the position shown. Determine its angular

acceleration at that instant if (a) the surface is rough

and the bar does not slip, and (b) the surface is smooth.

1 m

502

Problem 18.48 The masses of the bar and disk are

14 kg and 9 kg, respectively. The system is released

from rest with the bar horizontal. Determine the bar’s

angular acceleration at that instant if

(a) the bar and disk are welded together at A,

(b) the bar and disk are connected by a smooth pin

at A.

Strategy: In part (b), draw individual free-body

diagrams of the bar and disk.

O

0.3 m

1.2 m

A

Solution:

Problem 18.49 The 5-lb horizontal bar is connected

to the 10-lb disk by a smooth pin at A. The system is

released from rest in the position shown. What are the

angular accelerations of the bar and disk at that instant?

A

3 ft 1 ft

O

Solution: Given

Ay

504

Problem 18.50 The 0.1-kg slender bar and 0.2-kg

cylindrical disk are released from rest with the bar

horizontal. The disk rolls on the curved surface. What is

the bar’s angular acceleration at the instant it is released?

40 mm

0.1 kg, Wb=mbg=0.981 N, md=0.2kg, Wd=mdg=1.962 N,

Problem 18.51 The mass of the suspended object Ais

8 kg. The mass of the pulley is 5 kg, and its moment of

inertia is 0.036 kg-m2. If the force T=70 N, what is

the magnitude of the acceleration of A?

120 mm

T

A

Solution: Given

The dynamic equations

FyB :T2+T−mBg−By=mBaBy

MB:−T2R+TR =IBαB

Kinematic constraints

aBy =aAy ,a

By =RαB

Solving we ﬁnd aAy =0.805 m/s2

We also have

aBy =0.805 m/s2,α

B=6.70 rad/s,T

2=68.0 N,B

y=84.9 N

mAg

T

By

T2

506

c

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Problem 18.52 The suspended object Aweighs 20 lb.

The pulleys are identical, each weighing 10 lb and hav-

ing moment of inertia 0.022 slug-ft2. If the force T=

15 lb, what is the magnitude of the acceleration of A?

4 in

T

A

Solution: Given

The FBDs

Wdisk

T4

T2

Problem 18.53 The 2-kg slender bar and 5-kg block

are released from rest in the position shown. If fric-

tion is negligible, what is the block’s acceleration at that

instant? (See Example 18.5.) 1 m

508