Chapter 6: Fundamental Dimensions and Systems of Units
6.1 Convert the information given in the accompanying table from SI units to U.S.
Customary units. Refer to conversion tables on the front and back end sheets of
this book. Show all steps of your solutions. See Examples 6.3 and 6.4.
Convert from SI units
To U.S. Customary units
120 km/h
miles/h and ft/s
1000 W
Btu/hr and horsepower
100 m
3
ft
3
80 kg
lb
m
1000 kg/m
3
lb
m
/ft
3
900 N
lb
f
100 kPa
lb
f
/in
2
9.81 m/s
2
ft/s
2
SOLUTION
miles/h 5.74
ft 5280
mile 1
m 1
ft 28.3
km 1
m 1000
h
km
120
ft/s 109.3
s 3600
h 1
m 1
ft 28.3
km 1
m 1000
h
km
120
Btu/h3412
3
3ft 3529
ft 28.3
m 100
2.2046 lbm
3 3
kg 1 m 2.2046 lbm lbm
1000 62.5
m 3.28 ft 1 kg ft
14
© 2020 Cengage Learning®. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website,
in whole or in part.
-3
224.809 10 lbf
900 N 202.3 lbf
6.2 Convert the information given in the accompanying table from U.S. Customary
units to SI units. Refer to conversion tables on the inside and back covers of this
book. Show all steps of your solutions. See Examples 6.3 and 6.4.
Convert from U.S. Customary
Units
To SI Units
65 miles/h
km/h and m/s
60,000 Btu/h
W
120 lb
m
/ft
3
k
g/m
3
30 lb/in
2
kPa
200 lb
m
kg
200 lb
f
N
SOLUTION
m 1
km 1
ft 5280
miles
m/s 29
s 3600
km 1
h
6.104
120 1920 kg/m
16
6.4 Which one of the following equations is dimensionally homogenous? Show your
proof.
a.
ma
F
b.
R
V
mF
2
c. )Vm(V)tF(t 1212
d.
mV
F
e. )(
)(
12
12
tt
VV
mF
where
F = force (N)
m = mass (kg)
a = acceleration (m/s2)
V = velocity (m/s)
R = radius (m)
t = time (s)
SOLUTION
ma
F
R
17
© 2020 Cengage Learning®. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website,
in whole or in part.
d.
mV
F
N = (kg)(m/s)
Not dimensionally homogenous.
e. )(
)(
12
12
tt
VV
mF
N = (kg)(m/s)/(s) = (kg)(m/s2)
Dimensionally homogenous.
6.5 Determine the number of significant digits for the following numbers:
Number
o
f Significant Digits
286.5
2.2 x 10
2
2200
0.0286
SOLUTION
Number
o
f Significant Digits
286.5
Four
2.2 x 10
2
Two
3
digits; If we meant to write 2200.0 then four significant digits
0.0286
Three
18
6.6 Present the results of the following operations using the proper number of
significant digits.
Your Calculator Displays
Should be
recorded as
1.2856 + 10.1 =
155
–
0.521 =
155
–
0.5
2
=
1558 x 12 =
3.585 / 12 =
SOLUTION
Your Calculator Displays
Should be recorded as
1.2856 +
10.1
11.3856
11.3
155
–
0.521 =
154.479
154
155
–
0.5
2
=
154.
48
154
1558 x 12 =
18696
1.86 x 10
4
3.585 / 12 =
0.29875
0.29
6.11 How many gallons of gasoline would be saved if someone drives a car with 35
miles per gallon versus a car with 20 miles per gallon? Assume the car is driven
12,000 miles per year for the next 10 years.
6.13 Which one of the following equation is dimensionally homogenous? Show your
proof.
a. 2
1
2
212
2
1
2
1
)( mVmVxxF
b. 2
1
2
2
2
1
2
1mVmVF
19
c. 2
1
2
212
2
1
2
1
)( mxmxVVF
d. 1212 )( mVmVttF
where
F = Force (N)
x = distance (m)
m = mass (kg)
V = velocity (m/s)
t = time (s)
s
s
6.14 A car has a mass of 1500 kg. Express the mass and the weight of the car using B.G.
and U.S. Customary units. Show the conversion steps.
SOLUTION
mass:
slug 1
20
© 2020 Cengage Learning®. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website,
in whole or in part.
m
mlb 3307
kg 0.4536
lb 1
(kg) 1500
weight:
N 47151m/s 9.81(kg) 1500 2
lb 3308
N 4.448
lb 1
(N) 14715
6.15 Express the kinetic energy
2
1
(mass)(speed)
2
of a car with a mass of 1200 kg and
moving at a speed of 100 km/h using SI, BG, and U.S. Customary units. Show the
conversion steps.
SOLUTION
m 000 1
h 1
km
J .3558 1
6.16 A machine shop has a rectangular floor shape with dimensions of 30 ft by 50 ft.
Express the area of the floor in ft2, m2, in2, and cm2. Show the conversion steps.
SOLUTION
ft 3.28
21
© 2020 Cengage Learning®. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website,
in whole or in part.
2
2
2in 216000
ft 1
in 12
)ft (1500 A
2 4
2
2cm10 139.4
ft 0.0328
cm 1
)ft (1500 A
6.17 A trunk of a car has a listed luggage capacity of 18 ft3. Express the capacity in in3,
m3, and cm3. Show the conversion steps.
SOLUTION
3
ft 0.0328
6.18 A car has a listed 300 horsepower, 3.5 liter engine. Express the engine size in both
kW and in3. Show the conversion steps.
SOLUTION
kW 0.7457
6.19 The density of air at standard room conditions is 1.2 kg/m3. Express the density in
BG and U.S. Customary units. Show the conversion steps.
SOLUTION
3
m
3/ftlb 0.075
kg 0.4536
ft 3.28
m
6.20 On a summer day, in Phoenix, Arizona, the inside room temperature is maintained
at 68F while the outdoor air temperature is a sizzling 110F. What is the outdoor-
indoor temperature difference in:
(a) degrees Fahrenheit,
(b) degrees Rankine,
(c) degrees Celsius, and
(d) kelvin?
Is one degree temperature difference in Celsius equal to one degree temperature
difference in kelvin, and is one degree temperature difference in Fahrenheit equal
to one degree temperature difference in Rankine? If so, why?
23
© 2020 Cengage Learning®. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website,
in whole or in part.
(c)
CFTCT outdooroutdoor
3.43)32110(
9
5
)32)((
9
5
)(
CFTCT indoorindoor
20)3268(
9
5
)32)((
9
5
)(
CCCTT indooroutdoor
3.23203.43
(d)
KCTKT outdooroutdoor 3.3162733.43273)()(
KCTKT indoorindoor 29327320273)()(
KKKTT indooroutdoor 3.232933.316
Note that the temperature difference expressed in degrees Celsius is equal to the
temperature difference expressed in Kelvin.
460)(460)(460)(460)()()( 21
)(
2
)(
121
21
FTFTFTFTRTRT
RTRT
And simplifying the above relationship leads to:
)()()()( 2121 FTFTRTRT
Also note that you could have converted the temperature difference in degree
Rankine to Kelvin directly in the following manner:
KRTKT 3.23)42)(
9
5
()(
9
5
)(
24
6.21 A cantilever beam shown in the accompanying figure is commonly used to
support a load acting on a balcony. The deflection of the centerline of the beam is
given by the following equation:
3
2 2
( 4 6 )
24
wx
y x Lx L
EI
where
y = deflection at a given x location, (m)
w = distributed load
E = modulus of elasticity (N/m2)
I = second moment of area (m4)
x = distance from the support as shown (m)
L = length of the beam (m)
A cantilever beam
What is the appropriate unit for w, if the preceding equation is to be homogenous
in units? Show all steps of your work.
SOLUTION
25
6.22 A model known as stopping sight distance is used by civil engineers to design
roadways. This simple model estimates the distance a driver needs in order to stop
his car traveling at a certain speed after detecting a hazard. The model proposed
by the American Association of State Highway and Transportation Officials
(AASHTO) is given by
TV
Gfg
V
S
)(2
2
where
S = stopping sight distance (ft)
V = initial speed (ft/s)
g = acceleration due to gravity, 32.2 ft/s2
f = coefficient of friction between tires and roadway
G = grade of road
T = driver reaction time (s)
What are the appropriate units for f and G if the preceding equation is to be
homogenous in units? Show all steps of your work.
6.23 In an annealing process—a process wherein materials such as glass and metal are
heated to high temperatures and then cooled slowly to toughen them —the
following equation may be used to determine the temperature of a thin piece of
material after some time t.
t
cL
h
TT
TT
enviromentinitial
tenvironmen
2
exp
where
T = temperature (°C)
h = heat transfer coefficient
= density (kg/m3)
c = specific heat (J/kg·K)
L = plate thickness (m)
t = time (s)
Those of you who will pursue aerospace, chemical, mechanical, or materials
engineering will learn in your heat-transfer class about the underlying concepts
that leads to the solution. What is the appropriate unit for h if the preceding
equation is to be homogenous in units? Show all steps of your work.
SOLUTION
(s)
C22
h
h
6.24 The air resistance to motion of a vehicle is something important that engineers
investigate. As you may also know, the drag force acting on a car is determined
experimentally by placing the car in a wind tunnel. For a given car, the
experimental data is generally represented by a single coefficient that is called the