978-0073380308 Chapter 8 Solution Manual Part 16

Problem 8.90

A uniform disk

W

of radius

RWD7mm

and mass

mWD0:15 kg

is

connected to point

O

via the rotating arm

OC

. Disk

W

also rolls without

slip over the stationary cylinder

S

of radius

RSD15 mm

. Assuming that

!WD25 rad=s

, determine the angular momentum of

W

about its own

center of mass C, as well as about point O.

Solution

Recalling that in the component system shown

O

kDOur⇥Ou✓

, the angular

momentum of the the wheel Wabout its own mass center is

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Dynamics 2e 1809

Problem 8.91

A rotor

B

with center of mass

G

, weight

WD3000 lb

, and radius of gyration

kGD15 ft

is spinning with an angular speed

!BD1200 rpm

when a braking

system is applied to it, providing a time-dependent torque

MDM0.1 Cct/

,

with

M0D3000 ftlb

and

cD0:01 s1

. If

G

is also the geometric center of

the rotor and is a fixed point, determine the time

ts

that it takes to stop the rotor.

Photo credit: NASA

Solution

We use subscripts

1

and

2

to denote the time at which the braking moment

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Problem 8.92

The uniform bar

AB

has length

LD4:5 ft

and weight

WAB D14 lb

. At the

instant shown,

✓D67ı

and

vAD5:8 ft=s

. Determine the magnitude of the

linear momentum of

AB

, as well as the angular momentum of

AB

about its mass

center Gat the instant shown.

Solution

The linear momentum of the bar is

EpAB DmAB EvG

where

G

is the mass

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Dynamics 2e 1811

Problem 8.93

A uniform pipe section

A

of radius

r

, mass center

G

, and mass

m

is gently placed

(i.e., with zero velocity) on a conveyor belt moving with a constant speed

v0

to the right. Friction

between the belt and pipe causes the pipe to move to the right and eventually to roll without slip. If

k

is

the coefficient of kinetic friction between the pipe and the conveyor belt, find an expression for

tr

, the time

it takes for

A

to start rolling without slip. Hint: Using the methods of Chapter 7, we can show that the

force between the pipe section and the belt is constant.

Solution

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permission of McGraw-Hill, is prohibited.

Computation. Substituting Eqs. (3)–(6) into Eqs. (1) and (2), we have

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Dynamics 2e 1813

Problem 8.94

An automobile wheel test rig consists of a uniform disk

A

, of mass

mAD5000 kg

and radius

rAD1:5

m, that can rotate freely about

its fixed center

C

and over which the wheel of an automobile is

made to roll. A wheel

B

, whose center and center of mass coincide

at D, is mounted on a shaft (not shown) that holds Dfixed while

it allows the wheel to rotate about

D

. The wheel has diameter

dD0:62

m, mass

mBD21:5 kg

, and mass moment of inertia

about its mass center

IDD44 kgm2

. Both

A

and

B

are initially

at rest when

B

is subject to a constant torque

M

that causes

B

to

roll without slip on A.

If

MD1500 Nm

, use the angular impulse-momentum prin-

ciple to determine how long it takes to reach conditions simulating

a car speed of 100 km=h.

Solution

We model the wheel and the disk as rigid bodies in fixed axis ro-

tations about

D

and

C

, respectively. We assume that wheel

B

is

Force Laws. All forces are accounted for on the FBD.

Kinematic Equations.

Since the system starts from rest, we have

!A1 D0

and

!B1 D0

. Also, in order to

simulate a car moving at a speed

vs

, the speed of point

D

relative to the point on the wheel in contact with

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Dynamics 2e 1815

Problem 8.95

An automobile wheel test rig consists of a uniform disk

A

, of mass

mAD5000 kg

and radius

rAD1:5

m, that can rotate freely about

its fixed center

C

and over which the wheel of an automobile is

made to roll. A wheel

B

, whose center and center of mass coincide

at D, is mounted on a shaft (not shown) that holds Dfixed while

it allows the wheel to rotate about

D

. The wheel has diameter

dD0:62

m, mass

mBD21:5 kg

, and mass moment of inertia

about its mass center

IDD44 kgm2

. Both

A

and

B

are initially

at rest when

B

is subject to a constant torque

M

that causes

B

to

roll without slip on A.

Use the angular impulse-momentum principle and determine

M

if it takes 15 seconds to achieve conditions simulating a car

speed of 60 km=h.

t1

where

ID

is the mass moment of inertia of

B

about

D

, and where

E!BD!BO

k

is the angular velocity of

B

. Choosing the fixed

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permission of McGraw-Hill, is prohibited.

Force Laws. All forces are accounted for on the FBD.

Kinematic Equations.

Since the system starts from rest, we have

!A1 D0

and

!B1 D0

. Also, in order to

simulate a car moving at a speed

vs

, the speed of point

D

relative to the point on the wheel in contact with

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permission of McGraw-Hill, is prohibited.

Dynamics 2e 1817

Problem 8.96

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

and the

center of mass of the spool coincide, 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. The spool rolls

without slip, and the cord unwinds from the spool without slip.

Neglecting the inertia of the pulley system, use the impulse-

momentum principles to determine the angular speed of the spool

3

s

after release.

Solution