978-0073380308 Chapter 5 Solution Manual Part 26

1100 Solutions Manual

A test is conducted in which an

80 kg

person sitting in a

15 kg

cart is propelled by the jets emitted by

two household fire extinguishers with a combined initial mass of

18 kg

. The cross section of the exhaust

nozzles is

3cm

in diameter, and the density of the exhaust is

⇢D1:98 kg=m3

. The vehicle starts from rest,

and it is determined that the initial acceleration of the “jet cart” is

1:8 m=s2

. Recalling that the mass flow

rate out of the nozzle is given by

PmoD⇢Svo

, where

S

is the area of the nozzle cross section and

vo

is the

exhaust speed, determine voat the initial time. Ignore any resistance to the horizontal motion of the cart.

Solution

voDs2max

⇢⇡d2:(3)

Dynamics 2e 1101

Problem 5.165

Consider a rocket in space so that it can be assumed that no external forces act on the rocket. Let

vo

be the

constant speed of the exhaust gases relative to the rocket. In addition, let

mbCm

f

be the total mass of the

rocket and its fuel at the initial time, and let

mb

be the mass of the body after all the fuel is burned. If the

rocket is fired from rest, determine an expression for the maximum speed that the rocket can achieve.

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1102 Solutions Manual

A stationary

4cm

diameter nozzle emits a water jet with a speed

of

30 m=s

. The water jet impinges on a vane with a mass of

15 kg

.

Recalling that water has a mass density of

1000 kg=m3

, determine

the minimum static friction coefficient with the ground such that the

vane does not move if

D20ı

and

✓D30ı

. Neglect the weight of

the water layer in contact with the vane, as well as friction between

the water and the vane.

Solution

RxD1

4⇢⇡d2v2

w.cos cos ✓/and RyD1

4⇢⇡d2v2

w.sin ✓Csin /: (5)

Dynamics 2e 1103

Substituting Eqs. (5) into Eq. (8), we obtain an equation for swhose solution is

w.cos cos ✓/

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permission of McGraw-Hill, is prohibited.

Problem 5.167

A diffuser is attached to a structure whose rigidity in the horizontal

direction can be modeled by a linear spring with constant

k

. The

diffuser is hit by a water jet issued with a speed

vwD55 ft=s

from

a

2in:

diameter nozzle. Assume that the friction between the jet and

the diffuser is negligible and that the diffuser’s motion in the vertical

direction can be neglected. Recalling that the specific weight of

water is

D62:4 lb=ft3

, if the opening angle of the diffuser is

✓D40ı

, determine

k

such that the horizontal displacement of the

diffuser does not exceed

0:25 in:

from the diffuser’s rest position.

Assume that the water jet splits symmetrically over the diffuser.

Solution

Because of the symmetry we can analyze one of the two jets in which the main jet splits.

Referring to the figure on the right, we model the chosen half-jet as a steady flow system

whose control volume is defined by the cross sections at

A

and

B

. Then, applying the

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permission of McGraw-Hill, is prohibited.

Dynamics 2e 1105

Problem 5.168

A water jet with a mass flow rate

Pm

f

at the nozzle impinges with a

speed

vw

on a fixed flat vane inclined at an angle

✓

with respect to the

horizontal. Assuming that there is no friction between the water jet and

the vane, the jet will split into two flows with mass flow rates

Pm

f1

and

Pm

f2

. Neglecting the weight of the water, determine how

Pm

f1

and

Pm

f2

depend on

Pm

f

,

vw

, and

✓

.Hint: Due to the no-friction assumption,

there is no force that slows down the water in the direction tangent

to the vane, and this implies that the momentum in that direction is

conserved.

v1Dv2Dvw:(2)

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permission of McGraw-Hill, is prohibited.

Problem 5.169

A person wearing a jet pack lifts off from rest and ascends along a straight vertical trajectory. Let

M

denote the initial combined mass of the pilot and the equipment, including the fuel in the pack. Assume

that the mass flow rate

Pmo

and exhaust gas speed

vo

are known constants and that the pilot can take off as

soon as the rocket engine is started. If the engine exhaust is completely directed in the direction of gravity,

determine the expression of the pilot’s speed as a function of time,

M

,

Pmo

,

vo

, and

g

(the acceleration due

to gravity) while the pack is providing a thrust. Neglect air resistance and assume that gravity is constant.

Photo credit: Courtesy of JetPack International, LLC

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permission of McGraw-Hill, is prohibited.

Dynamics 2e 1107

Problem 5.170

A

28;000 lb

A-10 Thunderbolt is flying at a constant speed of

375 mph

when it fires a

4

s burst from its forward-facing seven-

is

3250 ft=s

. Assume that each of the plane’s two jet engines

maintains a constant thrust of

9000 lb

, that the plane is subject

the

4

s burst, modeling the airplane’s change of mass due to firing

as a continuous mass loss.

Hence, using the (full precision value of the) above expressions, we have that the (continuous) mass outflow

Pmois given by

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1108 Solutions Manual

A faucet is letting out water at a rate of

15 L=min

. Assume that the

internal diameter

d

of the faucet is uniform and equal to

1:5 cm

, the

distance

`D20 cm

, and the static water pressure at the wall is

0:30 MPa

.

Neglecting the weight of the water inside the faucet, as well as the weight

of the faucet itself, determine the forces and the moment that the wall

exerts on the faucet. Recall that the density of water is

⇢D1000 kg=m3

,

and neglect the atmospheric pressure at the spout. Hint: Define your

control volume using a section along the wall.

Solution

follows:

vAx DQ

Ay D0; vBx D0; vBy DQ

and Mwhose solution is

ND1

4⇡PAd2C4⇢Q2

⇡d2;VD4⇢Q2

⇡d2;MD4⇢Q2`

⇡d2;

ND53:37 N;VD0:3537 N;MD0:07074 Nm:

Dynamics 2e 1109

Problem 5.172

Consider a wind turbine with a diameter

dD110

m and the airflow

streamlines shown, which are symmetric relative to the axis of

the turbine. Since the airflow is tangent to the streamlines (by

definition), these lines can be taken to define the top and bottom

surfaces of a control volume. Suppose that pressure measurements

indicate that the flow experiences atmospheric pressure at the cross

sections

A

and

B

(as well as outside the control volume) where

the wind speed is

vAD7m=s

and

vBD2:5 m=s

, respectively.

Furthermore, assume that the average pressure along the streamlines

defining the control volume is also atmospheric. Finally, assume

that

O

is on the line of action of the overall weight of the turbine

and that the diameter of the flow cross section at

A

is

85

% of the

rotor diameter and that the rotor hub is at a distance

hD75

m

above the ground. If the density of air is constant and equal to

⇢D1:25 kg=m3

, determine the force exerted by the air on the wind

turbine and the reaction moment at the base of the support.

Substituting Eqs. (4)–(6) into Eqs. (1)–(3), we have, ND0,MRD0, and

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permission of McGraw-Hill, is prohibited.