978-0073380308 Chapter 8 Solution Manual Part 8

1728 Solutions Manual

A spool of mass

msD150 kg

and inner and outer radii

⇢D0:8

m

and

RD1:2

m, respectively, is connected to a counterweight

A

of

mass

mAD50 kg

by a pulley system whose cord, at one end, is

wound around the inner hub of the spool. The center

G

of the spool

is also the center of mass of the spool, and the radius of gyration of

the spool is

kGD1

m. The system is at rest when the counterweight

is released, causing the spool to move to the right. Assume that the

spool rolls without slip.

Neglecting the inertia of the pulley system, determine the angular

speed of the spool after the counterweight has dropped 0:5 m.

Solution

Dynamics 2e 1729

of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the

permission of McGraw-Hill, is prohibited.

1730 Solutions Manual

A spool of mass

msD150 kg

and inner and outer radii

⇢D0:8

m

and

RD1:2

m, respectively, is connected to a counterweight

A

of

mass

mAD50 kg

by a pulley system whose cord, at one end, is

wound around the inner hub of the spool. The center

G

of the spool

is also the center of mass of the spool, and the radius of gyration of

the spool is

kGD1

m. The system is at rest when the counterweight

is released, causing the spool to move to the right. Assume that the

spool rolls without slip.

Assume that the inertia of the cord and of pulleys

B

and

D

can be

neglected, but model pulley

C

as a uniform disk of mass

mCD15 kg

and radius

rCD0:3

m. If the cord does not slip relative to pulley

C

,

determine the angular speed of the spool after Adrops 0:5 m.

Solution

Dynamics 2e 1731

Kinematic Equations. Denoting by Lthe length of the cord, we have

of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the

permission of McGraw-Hill, is prohibited.

Problem 8.48

The uniform slender bar

AB

has length

LD1:45 ft

and weight

WAB D20 lb

. Rollers

D

and

E

, which are pinned at

A

and

B

,

respectively, can be modeled as two identical uniform disks, each with

radius

rD1:5 in:

and weight

WrD0:35 lb

. Rollers

D

and

E

roll

without slip on the surface of a cylindrical bowl with center at

O

and

radius

RD1ft

. Determine the system’s kinetic energy when

G

(the

center of mass of bar AB) moves with a speed vD7ft=s.

Solution

of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the

permission of McGraw-Hill, is prohibited.

Dynamics 2e 1733

Problem 8.49

The Charpy impact test is one test that measures the resistance of a material

to fracture. In this test, the fracture toughness is assessed by measuring the

energy required to break a specimen of a given geometry. This is done by

releasing a heavy pendulum from rest at an angle

✓i

and then measuring

the maximum swing angle

✓

f

reached by the pendulum after the specimen

is broken.

Consider a test rig in which the striker

S

(the pendulum’s bob) can

be modeled as a uniform disk of mass

mSD19:5 kg

and radius

rSD

150 mm

, and the arm can be modeled as a thin rod of mass

mAD2:5 kg

and length

LAD0:8

m. Neglecting friction and noting that the striker

and the arm are rigidly connected, determine the fracture energy (i.e.,

the kinetic energy lost in breaking the specimen) in an experiment where

✓iD158ıand ✓

fD43ı.

Solution

Computation. Substituting Eqs. (2)–(6) into Eq. (1), we have

of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the

permission of McGraw-Hill, is prohibited.

Dynamics 2e 1735

Problem 8.50

The Charpy impact test is one test that measures the resistance of a material

to fracture. In this test, the fracture toughness is assessed by measuring the

energy required to break a specimen of a given geometry. This is done by

releasing a heavy pendulum from rest at an angle

✓i

and then measuring

the maximum swing angle

✓

f

reached by the pendulum after the specimen

is broken.

Consider a test rig in which the striker

S

(the pendulum’s bob) can be

modeled as a uniform disk of weight

WSD40 lb

and radius

rSD6in:

,

if the striker impacts the specimen when the pendulum’s arm is vertical,

determine the speed of the point

Q

on the striker immediately before the

striker impacts with the specimen. Neglect friction, and observe that the

striker and the arm are rigidly connected.

of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the

permission of McGraw-Hill, is prohibited.

Computation. Substituting Eqs. (2), and (4)–(5) into Eq. (1), we have

v2

Q2

of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the

permission of McGraw-Hill, is prohibited.

Dynamics 2e 1737

Problem 8.51

A crate, with weight

WD155 lb

and mass center

G

, is placed on

a slide and released from rest as shown. The lower part of the slide

is circular, with radius

RD6ft

. Model the crate as a uniform body

with

bD3:6 ft

and

hD2ft

. Take into account the gap between

the crate and the slide when the crate is in its lowest position, and

assume that when the crate is in its lowest position on the slide, the

crate’s center of mass is moving to the left with a speed

vGD12 ft=s

.

Determine the work done by friction on the crate as the crate moves

from the release point to the lowest point on the slide.

Solution

Referring to the

FBD

on the left in the figure below, we model the crate as a rigid body subject to its own