978-0073380308 Chapter 6 Solution Manual Part 21

Problem 6.142

Bar

AB

is pinned to the fixed support at

A

, and the pin

P

is fixed

to the disk at radius

Ri

. The disk with outer radius

Ro

rolls without

slipping over the horizontal ﬂat surface with angular speed

!d

and

angular acceleration

˛d

in the directions shown. At the instant shown,

sD35 in:,RiD8:3 in:,RoD11 in:, and !dD15 rad=s.

If

˛d

is zero at the instant shown, determine the angular velocity

and angular acceleration of the bar AB.

sqs2.RoCRi/2D Psand .RoCRi/2!d

sDs!AB ;

s2and !AB D.RoCRi/2!d

s2

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Problem 6.143

Bar

AB

is pinned to the fixed support at

A

, and the pin

P

is fixed

to the disk at radius

Ri

. The disk with outer radius

Ro

rolls without

slipping over the horizontal ﬂat surface with angular speed

!d

and

angular acceleration

˛d

in the directions shown. At the instant shown,

sD35 in:,RiD8:3 in:,RoD11 in:, and !dD15 rad=s.

If

˛dD30 rad=s2

at the instant shown, determine the angular

velocity and angular acceleration of the bar AB.

sqs2.RoCRi/2D Psand .RoCRi/2!d

sDs!AB ;

the solution of which is

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permission of McGraw-Hill, is prohibited.

Dynamics 2e 1353

s22

Noting that

O

IDcos O{sin O|

and

O

JDsin O{Ccos O|

, this vector equation is equivalent to the following

two scalar equations in the unknowns ˛AB and Rs:

d

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Problem 6.144

A vertical shaft has a base

B

that is stationary relative to an inertial

reference frame with vertical axis

Z

. Arm

OA

is attached to the

vertical shaft and rotates about the

Z

axis with an angular velocity

!OA D5rad=s

and an angular acceleration

˛OA D1:5 rad=s2

. The

´

axis is coincident with the

Z

axis, but is part of a reference frame

that rotates with the arm

OA

. The

x

axis of the rotating reference

frame coincides with the axis of the arm

OA

. At the instant shown,

the

y

axis is perpendicular to the page and directed into the page. At

this instant, the collar

C

is sliding along

OA

with a constant speed

vCD5ft=s

and is at a distance

dD1:2 ft

from the

´

axis. Compute

the inertial acceleration of the collar, and express it relative to the

rotating coordinate system.

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Dynamics 2e 1355

Problem 6.145

Let frames

A

and

B

be the frames with origins at points

A

and

B

, respectively. Point

B

does not move

relative to point A. The velocity and acceleration of point Prelative to frame Bare

EvPrel D.6:14 O{BC23:7 O|B/ft=s and EaPrel D.3:97 O{BC4:79 O|B/ft=s2:

Knowing that, at the instant shown, frame

B

rotates relative to frame

A

at a constant angular velocity

!BD1:2 rad=s

, that the position of point

P

relative to frame

B

is

ErP =B D.8 O{BC4:5 O|B/ft

, and that

frame

A

is fixed, determine the velocity and acceleration of

P

at the instant shown and express the results

using the frame Acomponent system.

Solution

Relating the velocity of Pto that of Bvia the rotating xByB´Breference frame, we obtain

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permission of McGraw-Hill, is prohibited.

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permission of McGraw-Hill, is prohibited.

Dynamics 2e 1357

Problem 6.146

Repeat Prob. 6.145, but express the results using the component

system of frame B.

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permission of McGraw-Hill, is prohibited.

Problem 6.147

A vertical shaft has a base

B

that is stationary relative to an inertial

reference frame with vertical axis

Z

. Arm

OA

is attached to the

vertical shaft and rotates about the

Z

axis with an angular velocity

!OA D5rad=s

and an angular acceleration

˛OA D1:5 rad=s2

. The

´

axis is coincident with the

Z

axis but is part of a reference frame

that rotates with the arm

OA

. The

x

axis of the rotating reference

frame coincides with the axis of the arm

OA

. At the instant shown,

the

y

axis is perpendicular to the page and directed into the page.

At this instant, the collar

C

is sliding along

OA

with a constant

speed

vCD3:32 m=s

and is rotating with a constant angular velocity

!CD2:3 rad=s

relative to the arm

OA

. At the instant shown, point

D

happens to be in the

x´

plane and is at a distance

`D0:05

m from

the

x

axis and at a distance

dD0:75

m from the

´

axis. Compute the

inertial acceleration of point

D

, and express it relative to the rotating

reference frame.

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Dynamics 2e 1359

Problem 6.148

Bar

AB

rotates about point

A

with angular speed

!1

and angular

acceleration

˛1

, both in the directions shown. The curved bar

BC

is pinned to bar

AB

at

B

and rotates with angular speed

!2

and

angular acceleration

˛2

about the pin at

B

in the directions shown,

relative to bar

AB

. At this instant, collar

D

is at the top of the

curved bar

BC

. The speed and acceleration of

D

relative to bar

BC

are given by

v0

and

a0

, respectively. Express your answers in

the

xy´

frame that is attached to bar

BC

with its origin at

B

.Hint:

As we will see in Eq. (10.14),

E!BC D E!AB C E!BC =AB D E!1C E!2

.

Determine the velocity of the collar

D

in terms of

d

,

!1

,

!2

,

R, and v0.

Solution

Using the given rotating reference frame, we can determine the velocity of Dusing

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permission of McGraw-Hill, is prohibited.