978-0073380308 Chapter 5 Solution Manual Part 27

1110 Solutions Manual

obtain

FRD270:4 kN:

Dynamics 2e 1111

Problem 5.173

Let

pA

and

pB

be given static pressure measurements at the cross sections

A

and

B

in the air duct shown.

Assume that any cross section between

A

and

B

is circular with diameter

d

. Assume that the flow is steady

and that the mass density

⇢A

at

A

is known along with

vA

, the speed of the flow at

A

, and

vB

, the speed of

the flow at

B

. Determine the expression of mass density at

B

and the expression of the force

F

acting on

the fan.

Solution

The flow under consideration is steady and therefore we will solve this problem applying concepts pertaining

to steady flows systems. We begin with observing that since the mass flow rate

Pm

f

is constant and the cross

sectional area at Aand Bare equal, we must have

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Problem 5.174

A rope with weight per unit length of

0:1 lb=ft

is lifted at a constant

upward speed

v0D8ft=s

. Treating the rope as inextensible, deter–

mine the force applied to the top end of the rope after it is lifted

9ft

.

Assume that the top end of the rope is initially at rest and on the

floor. In addition, disregard the horizontal motion associated with

the uncoiling of the rope.

Solution

To solve this problem we model the portion of the rope that has been lifted as a variable mass

system gaining mass. Hence, referring to the FBD shown to the right, using the appropriate

force balance for variable mass systems and focusing only on the vertical motion, we have

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Dynamics 2e 1113

Problem 5.175

A rope with mass per unit length of

0:05 kg=m

is lifted at a constant

upward acceleration

a0D6m=s2

. Treating the rope as inextensible,

determine the force that must be applied at the top end of the rope

after it is lifted

3

m. Assume that the top end of the rope is initially

at rest and on the floor. In addition, disregard the horizontal motion

associated with the uncoiling of the rope.

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 5.176

A rope with mass per unit length of

0:05 kg=m

is lifted by applying a

constant vertical force

FD10

N. Treating the rope as inextensible,

plot the velocity and position of the top end of the string as a function

of time for

0t3

s. Assume that the top end of the rope is

initially at rest and

1mm

off the floor. In addition, disregard the

horizontal motion associated with the uncoiling of the rope.

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.

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 5.177

An amateur rocket with a body weight of

6:5 lb

is equipped with a rocket engine

holding

2:54 lb

of solid propellant with a burnout time (time required to burn all

the fuel) of

5:25

s (this is the typical data made available by amateur rocket engine

manufacturers). The initial thrust is

68 lb

. Assuming that the mass flow rate and the

speed of the exhaust relative to the rocket remain constant, determine the exhaust

mass flow rate

Pmo

and the speed relative to the rocket

vo

. In addition, determine

the maximum speed achieved by the rocket

vmax

if the rocket is launched from rest.

Neglect air resistance, and assume that gravity does not change with elevation.

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

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Dynamics 2e 1117

Letting

vmax

be the speed at burnout, multiplying both sides of the above equation by

dt

and integrating, we

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Problem 5.178

Continue Prob. 5.177 and determine the maximum height reached by the rocket, again

neglecting air resistance and changes of gravity with elevation. Hint: For

0<t<t

0

,

Zln✓1t

t0◆dt D.t0t/1ln✓1t

t0◆CC:

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

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to time from tD0to tDtbo as follows:

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