Civil Engineering Chapter 9 Mass And Massrelated Variables Engineering The Density Standard Air Function Temperature

73

Chapter 9: Mass and Mass-Related Variables in Engineering

9.2 The density of standard air is a function of temperature and may be approximated

using the ideal gas law according to

RT

P





where



= density (kg/m3)

P = standard atmospheric pressure (101.3 kPa)

R  gas constant, and its value for air is 286.9 ( Kkg

J

)

T  air temperature in Kelvin

Create a table that shows the density of air as a function of temperature in the

range of 0 C (273.15 K) to 50 C (323.15 K) in increments of 5 C. Also, create

a graph showing the value of density as a function of temperature.

SOLUTION

Temperature (C) Density (kg/m3)

20 1.20

25 1.18

50 1.09

9.3 Determine the specific gravity of the following materials: gold ( = 1208 lb/ft3),

platinum ( = 1340 lb/ft3), silver ( = 654 lb/ft3), sand ( = 94.6 lb/ft3), freshly

fallen snow ( = 31 lb/ft3), tar ( = 75 lb/ft3), and hard rubber ( = 74.4 lb/ft3).

1.20

Temperature (C)

Density of Air as a Function of

Temperature

74

SOLUTION

We will use Equation (9.3) to answer this question.

9.4 Show that the specific weight and density are related according to:

gravity) todueon accelerati(density)( weight specific



75

in whole or in part.

SOLUTION

We will use the definition given by Equation (9.2) to show this relationship.

gravity) todue onaccelerati(density)(

volume

gravity) todue ion(acceleratvolume)(density)(

volume

gravity) todue eleration(mass)(acc

volume

weight

weight specific

mass

weight



  

  

9.5 Compute the values of momentum for the following situations: (a) a 90-kg

football player running at 6 m/s; (b) a 1500-kg car moving at a rate of 100 km/h;

(c) a 200,000 kg Boeing 777 moving at a speed of 500 km/h; (d) a bullet with a

mass of 15 g traveling at a speed of 500 m/s; (e) a 140-g baseball traveling at

120 km/h; (f) a 80 kg stunt performer falling off a ten story building reaching a

speed of 30 m/s.

76

in whole or in part.

(e) a 140 g baseball traveling at 120 km/hr

















kg.m/s 67.4km m/1 1000s h/3600 1km/h 120g kg/1000 1g) 140(  VmL



(f) a 80 kg stunt performer falling off a 10 story building reaching a speed of 30 m/s.

kg.m/s 2400m/s) kg)(30 80(  VmL



9.9 Determine the mass flow rate of fuel from the gasoline tank to the car’s fuel

injection system. Assume that the gasoline consumption of the car is 20 mpg

when the car is moving at 60 mph. Use the specific gravity value of 0.72 for the

gasoline.

SOLUTION

ft 3.28

gallons 7.48

h

m

77

9.10 Determine the mass moment of inertia for the following objects: a thin disk,

circular cylinder, and a sphere. Refer to Equations (9.10) through (9.13) for

appropriate relationships. If you were to place these objects alongside of each

other on an inclined surface, which one of the object would get to the bottom first,

provided that they all have the same mass and diameter?

9.12 Determine the mass moment of inertia of a steel shaft that is 1-m long and has a

diameter of 5 cm. Determine the mass of the shaft using the density information

provided in Table 9.1.

SOLUTION

78

in whole or in part.

222 kg.m 0.005)kg)(0.025m (15.43

2

1

2

1

mRI zz

9.13 Determine the mass moment of inertia of steel balls used in ball bearings. Use a

diameter of 2 cm.

SOLUTION

9.14 Determine the mass moment of inertia of the earth about its axis of rotation, going

through the poles. Assume the shape of the earth to be spherical. Look up

information such as the mass of the earth and the radius of the earth at the

equatorial plane.

SOLUTION

79

in whole or in part.

2372

3

242 kg.m 109.7)

2

m1012751

kg)( 10(5.97

5

2

5

2





mRI zz

9.22 Referring to Figure 9.5, how much water is stored after 20 minutes in each of the

tanks. How long will it take to fill the tanks completely provided that the volume

of tank (a) is 24 m3 and tank (b) has a volume of 36 m3? Assume density of water

is 1000 kg/m3.

80

in whole or in part.

And the time that is required to fill the tank (b)

 

minutes 600s 36000

kg/s) (1–kg/s) (2

kg 36000

tank thefill torequired time 

9.24 Calculate the mass moment of inertia of the thin ring shown in the accompanying

diagram. Express your answer in lbm·ft2, lbm·in2, and slugs·ft2.

m

m.inlb 64.37)

ft

1

9.25 Determine the mass moment of inertia of a steel shaft that is 4 ft long and has

diameter of 2.5 in. Express your answer in lbm·ft2, lbm·in2, and slugs·ft2.

SOLUTION

12

81

in whole or in part.

To calculate the mass moment of inertia of the shaft about its longitudinal axis,

we use Equation (9.11).

2

m

2

m

222 .inlb 4.50.ftlb 0.35slugs.ft 0.011ft) 04slugs)(0.1 (2.07

2

1

2

1

mRI zz

9.26 Determine the mass moment of inertial of a steel ball with a diameter of 2 in.

Express your answer in lbm·ft2, lbm·in2, and slugs·ft2.

SOLUTION

12

5

9.27 Determine the mass moment of inertia of a 4-in-square steel plate. Use Equation

9.13. Express your answer in lbm·ft2, lbm·in2, and slugs·ft2.

SOLUTION

slugs 141.0)

ft

2.15)(ft 00926.0( mass 3

82

in whole or in part.

2

m

2

m

222 .inlb 05.6.ftlb 0.042slugs.ft 0.0013)

12

4

slugs)( (0.141

12

1

12

1

mWI zz

9.28 Determine the specific gravity of the following gasses by comparing their

densities to air at 1.23 kg/m3. Helium: 0.166 kg/m3, Oxygen: 1.33 kg/m3,

Nitrogen: 1.16 kg/m3, Natural gas: 0.667 kg/m3, and Hydrogen: 0.0838 kg/m3.

SOLUTION

068.0

m

kg

1.23

m

(Hydrogen) SG

3

9.29 Calculate the change in the momentum of a car, whose speed changes from 100

km/h to 20 km/h. The car has a mass of 1000 kg and is moving along a straight

line. Express your answer in SI, BG, and U.S. Customary units.

m

SOLUTION













kg.m/s 778,27km m/1 1000s h/3600 1km/h 100kg) 1000(

1 VmL















kg.m/s 555,5km m/1 1000s h/3600 1km/h 20kg) 1000(

2 VmL



slugs.ft/s -4,970.ft/slb -160,000kg.m/s 22223– 27,778 – 555,5 m12  LL



9.30 Calculate the change in the momentum of a 200,000 kg Boeing 777, whose speed

changes from 450 mph to 180 mph. Express your answer in SI, BG, and U.S.

Customary units.

