978-0073380308 Chapter 6 Solution Manual Part 5

Problem 6.28

The wheel

A

, with diameter

dD5cm

, is mounted on the shaft of the motor shown and is rotating with

a constant angular speed

!AD250 rpm

. The wheel

B

, with center at the fixed point

O

, is connected to

A

with a belt, which does not slip relative to

A

or

B

. The radius of

B

is

RD12:5 cm

. At point

C

the

wheel

B

is connected to a saw. If point

C

is at distance

`D10 cm

from

O

, determine the velocity and

acceleration of Cwhen D20ı. Express your answers using the component system shown.

Solution

Since the belt does not slip relative to Aor B, the angular speeds of Aand Bmust be such that

d

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Dynamics 2e 1191

Substituting Eqs. (6) into Eq. (5), and using the second of Eqs. (1) as well as the last of Eqs. (3), we obtain

A

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Problem 6.29

The mechanism shown is designed to move the tool at

H

while

keeping it oriented vertically. To do so, the rotor in the motor

M

is attached to the gear

A

, which drives the gear

B

. In turn,

gear

B

drives the gear

C

, which is rigidly attached to the arm

EF

. Arms

EF

and

DG

both have length

L

and are parallel to

one another. The radii of gears

A

,

B

, and

C

are

rA

,

rB

, and

rC

,

respectively.

If the motor

A

rotates with a constant angular speed

!A

(i.e.,

˛AD0

) in the direction shown, determine the velocity and

acceleration of the tool Has functions of the angle .

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Dynamics 2e 1193

where we have used Eq. (3). Because the vertical arm is in curvilinear translation, all points on it must have

the same acceleration, which means that EaHD EaF, or

A

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Problem 6.30

The mechanism shown is designed to move the tool at

H

while

keeping it oriented vertically. To do so, the rotor in the motor

M

is attached to the gear

A

, which drives the gear

B

. In turn,

gear

B

drives the gear

C

, which is rigidly attached to the arm

EF

. Arms

EF

and

DG

both have length

L

and are parallel to

one another. The radii of gears

A

,

B

, and

C

are

rA

,

rB

, and

rC

,

respectively.

If the motor

A

rotates with angular speed

!A

and angular

acceleration in the directions shown, determine the velocity and

acceleration of the tool Has functions of the angle .

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permission of McGraw-Hill, is prohibited.

Dynamics 2e 1195

A

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permission of McGraw-Hill, is prohibited.

Problem 6.31

An acrobat lands at the end

A

of a board and, at the instant shown, point

A

has a downward vertical

component of velocity

v0D5:5 m=s

. Letting

D15ı

,

`D1

m, and

dD2:5

m, determine the vertical

component of velocity of point Bat this instant if the board is modeled as a rigid body.

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Dynamics 2e 1197

Problem 6.32

Ageosynchronous equatorial orbit is a circular orbit above the Earth’s

equator that has a period of

1

day (these are sometimes called geostationary

orbits). These geostationary orbits are of great importance for telecommu-

nications satellites because a satellite orbiting with the same angular rate

as the rotation rate of the Earth will appear to hover in the same point in

the sky as seen by a person standing on the surface of the Earth. Using

this information, modeling a geosynchronous satellite as a rigid body, and

noting that the satellite has been stabilized so that the same side always

faces the Earth, determine the angular speed !sof the satellite.

Solution

The motion of the satellite is a fixed-axis rotation about the center of the Earth where the axis is perpendicular

to the plane of the satellite’s orbit. For the satellite to be geosynchronous, the satellite’s angular velocity must

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Problem 6.33

Wheels

A

and

C

are mounted on the same shaft and rotate together. Wheels

A

and

B

are connected by a belt, and so are wheels

C

and

D

. The axes

of rotation of all the wheels are fixed, and the belts do not slip relative

to the wheels they connect. If, at the instant shown, wheel

A

has an

angular velocity

!AD2rad=s

and an angular acceleration

˛AD0:5 rad=s2

,

determine the angular velocity and acceleration of wheels

B

and

D

. The

radii of the wheels are

RAD1ft

,

RBD0:25 ft

,

RCD0:6 ft

, and

RDD

0:75 ft.

RDD0:75 ft.

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Dynamics 2e 1199

Problem 6.34

At the instant shown,

A

is moving upward with a speed

v0D5ft=s

and acceleration

a0D0:65 ft=s2

. Assuming that the rope that connects the pulleys does not slip relative to

the pulleys, and letting

`D6in:

and

dD4in:

, determine the angular velocity and angular

acceleration of pulley C.

Solution

First doing the pulley kinematics, we can define the rope length using the

coordinates shown on the right as

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