Chapter 3: Technical ProblemSolving and Communication Skills
39
: An automobile engine is advertised as producing a peak power of 118 hp (at an
engine speed of 4000 rpm) and a peak torque of 186 ft
⋅
lb (at 2500 rpm). Express those
performance ratings in the SI units of kW and N
⋅
m.
Solution:
For the torque, apply USCS to SI conversion factors from Table 3.6:
m
N
Chapter 3: Technical ProblemSolving and Communication Skills
40
From Example 3.6, express the sideways deflection of the tip in the units of mils
(defined in Table 3.5) when the various quantities are instead known in the USCS.
Use the values F = 75 lb, L = 3 in., d = 3/16 in., and E = 30 × 10
6
psi.
Solution:
Chapter 3: Technical ProblemSolving and Communication Skills
41
: Heat Q, which has the SI unit of joule (J), is the quantity in mechanical engineering
that describes the transit of energy from one location to another. The equation for the
flow of heat during the time interval ∆t through an insulated wall is
( )
lh
L
TT
L
tA
Q−=
κ
where κ is the thermal conductivity of the material from which the wall is made, A and
L are the wall’s area and thickness, and T
h
T
l
is the difference (in degrees Celsius)
between the high and lowtemperature sides of the wall. By using the principle of
dimensional consistency, what is the correct dimension for thermal conductivity in the
SI? The lowercase Greek character kappa (κ) is a conventional mathematical symbol
used for thermal conductivity. Appendix A summarizes the names and symbols of
Greek letters.
Solution:
C
m
°⋅
Chapter 3: Technical ProblemSolving and Communication Skills
42
Convection is the process by which warm air rises and cooler air falls. The Prandtl
number (Pr) is used when mechanical engineers analyze certain heat transfer and
convection processes. It is defined by the equation
κ
p
c
Pr =
where c
p
is a property of the fluid called the specific heat having the SI unit kJ /(kg
⋅
°C); & is the viscosity as discussed in Problem P3.16; and κ is the thermal conductivity
as discussed in Problem P3.22. Show that Pr is a dimensionless number. The lowercase
Greek characters mu (&) and kappa (κ) are conventional mathematical symbols used for
viscosity and thermal conductivity. Appendix A summarizes the names and symbols of
Greek letters.
Solution:
2
Chapter 3: Technical ProblemSolving and Communication Skills
43
When fluid flows over a surface, the Reynolds number will output whether the flow
is laminar (smooth), transitional, or turbulent. Verify that the Reynolds number is
dimensionless using the SI. The Reynolds number is expressed as
ρ
VD
R=
where ρ is the density of the fluid, V is the free stream fluid velocity, D is the
characteristic length of the surface, and & is the fluid viscosity. The units of fluid
viscosity are kg/(m
⋅
s).
Solution:
3
Chapter 3: Technical ProblemSolving and Communication Skills
44
Determine which one of the following equations is dimensionally consistent.
2
xmF =
2
1
,
2
xmVF =
2
1
,
2
VmxF =
2
1
,
VtF
=
,
2
tmVF = 2
where F is force, m is mass, x is distance, V is velocity, and t is time.
Solution:
2
s
2
2
2
s
s
Chapter 3: Technical ProblemSolving and Communication Skills
Referring to Problem P3.23 and Table 3.5, if the units for c
p
and & are Btu/(slug
⋅
°F)
and slug/(ft
⋅
h), respectively, what must be the USCS units of thermal conductivity in
the definition of Pr?
Chapter 3: Technical ProblemSolving and Communication Skills
46
Some scientists believe that the collision of one or more large asteroids with the
Earth was responsible for the extinction of the dinosaurs. The unit of kiloton is used to
describe the energy released during large explosions. It was originally defined as the
explosive capability of 1000 tons of trinitrotoluene (TNT) high explosive. Because that
expression can be imprecise depending on the explosive’s exact chemical composition,
the kiloton subsequently has been redefined as the equivalent of 4.186 × 10
12
J. In the
units of kiloton, calculate the kinetic energy of an asteroid that has the size
(box–shaped, 13 × 13 × 33 km) and composition (density, 2.4 g/cm
3
) of our solar
system‘s asteroid Eros. Kinetic energy is defined by
2
k
mvU
2
1
=
where m is the object’s mass and v is its speed. Objects passing through the inner solar
system generally have speeds in the range of 20 km/s.
Approach:
Calculate the volume of the asteroid and convert it to cm
3
using Table 3.3. Then calculate
the mass of the asteroid using the volume and density. Convert the mass and velocity to
appropriate units using Table 3.3. Then, calculate the kinetic energy.
Solution:
Chapter 3: Technical ProblemSolving and Communication Skills
47
Discussion:
This kind of energy is larger than the most powerful nuclear weapons in the world,
indicating just how destructive a collision with this size asteroid would be.
Chapter 3: Technical ProblemSolving and Communication Skills
48
A structure known as a cantilever beam is clamped at one end but free at the other,
analogous to a diving board that supports a swimmer standing on it. Using the
following procedure, conduct an experiment to measure how the cantilever beam
bends. In your answer, report only the significant digits that you know reliably.
(a) Make a small tabletop test stand to measure the deflection of a plastic drinking straw
(your cantilever beam) that bends as a force F is applied to the free end. Push one end
of the straw over the end of a pencil, and then clamp the pencil to a desk or table. You
can also use a ruler, chopstick, or a similar component as the cantilever beam itself.
Sketch and describe your apparatus, and measure the length L. (b) Apply weights to the
end of the cantilever beam, and measure the tip’s deflection Ly using a ruler. Repeat the
measurement with at least a half dozen different weights to fully describe the beam‘s
forcedeflection relationship. Penny coins can be used as weights; one penny weighs
approximately 30 mN. Make a table to present your data. (c) Next draw a graph of the
data. Show tip deflection on the abscissa and weight on the ordinate, and be sure to
label the axes with the units for those variables. (d) Draw a bestfit line through the data
points on your graph. In principle, the deflection of the tip should be proportional to the
applied force. Do you find this to be the case? The slope of the line is called the
stiffness. Express the stiffness of the cantilever beam either in the units lb/in. or N/m.
Solution:
Chapter 3: Technical ProblemSolving and Communication Skills
49
“#
$%&!
“#’&&!(&!&!$
Chapter 3: Technical ProblemSolving and Communication Skills
50
Perform measurements as described in P3.28 for cantilever beams of several
different lengths. Can you show experimentally that, for a given force F, the deflection
of the cantilever’s tip is proportional to the cube of its length? As in P3.28, present your
results in a table and a graph, and report only those significant digits that you know
reliably.
Solution:
A 1.6 mmdiameter steel rod was used as the cantilever beam, and it was clamped in a vise
as shown in Figure 1. Small weights were held in a plastic bag attached to the beam‘s free
end. The distance between the beam‘s tip and the surface of the table was measured for
$%&!
51
© 2017 Cengage Learning
®
. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Chapter 3: Technical ProblemSolving and Communication Skills
52
Using SI units, calculate the change in potential energy of a 150lb person riding the
15 ft long uphill portion of a water slide (as described in P3.10). The change in
potential energy is defined as mg∆h where ∆h is the change in vertical height. The
uphill portion of the slide is set at an angle of 45˚.
Approach:
First determine the change in vertical height the rider experiences from the bottom to the
top of the uphill portion of the slide. Assume a simple triangular representation of the
uphill portion of the slide. Also, the mass of the rider is determined using gravitational
Solution:
Determine the change in vertical height the rider experiences.
2
s
ft
32.2
m = 4.66 slugs
Calculate the change in potential energy
Chapter 3: Technical ProblemSolving and Communication Skills
53
© 2017 Cengage Learning
®
. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Discussion:
This represents the amount of energy to propel the rider up the slide. The actual amount of
energy needed would be higher as there would be energy losses in the system, including
friction between the rider and the slide. To get a sense for the amount of energy this is,
approximately one Joule of energy is required to lift an apple a vertical distance of one
meter.
Chapter 3: Technical ProblemSolving and Communication Skills
Using the speed given in P3.10, calculate the power required to move the rider in
P3.30 up the incline portion of the water slide, where power is the change in energy
divided by the time required to traverse the uphill portion.
Approach:
Using the relationship, Power = LEnergy/LTime, calculate the power required. The change
s 0.79
m 4.57
L
m
=== V
d
t
Discussion:
This represents the amount of power to propel the rider up the slide at the given speed. The
actual amount of power needed would be higher as there would be energy losses in the
system, including friction between the rider and the slide. If a higher speed is desired, more
Chapter 3: Technical ProblemSolving and Communication Skills
55
$*
For these estimation problems, students should set up the problems using the
ApproachSolutionDiscussion model as presented in the chapter. In the Approach phase,
it will be very important to make appropriate assumptions concerning the problem and
model. In the Solution phase, it will be critical to develop the correct set of equations to
match the assumptions. In the Discussion phase, it is essential to consider the
reasonableness of the solution, the relevance of the assumptions, and any conclusions one
could draw from the solution.
Chapter 3: Technical ProblemSolving and Communication Skills
56
The modulus of elasticity, modulus of rigidity, Poisson’s ratio, and the unit weight
for various materials are shown below. The data is given as Material; Modulus of
Elasticity, E (Mpsi & GPa); Modulus of Rigidity, G (Mpsi & GPa); Poisson‘s Ratio;
and Unit Weight (lb/in
3
, lb/ft
3,
kN/m
3
). Prepare a single table that captures this
technical data in a professional and effective manner.
Aluminum alloys 10.3 71.0 3.8 26.2 0.334 0.098 169 26.6
Beryllium copper 18.0 124.0 7.0 48.3 0.285 0.297 513 80.6
Brass 15.4 106.0 5.82 40.1 0.324 0.309 534 83.8
Carbon steel 30.0 207.0 11.5 79.3 0.292 0.282 487 76.5
Cast iron, grey 14.5 100.0 6.0 41.4 0.211 0.260 450 70.6
Copper 17.2 119.0 6.49 44.7 0.326 0.322 556 87.3
Glass 6.7 46.2 2.7 18.6 0.245 0.094 162 25.4
Lead 5.3 36.5 1.9 13.1 0.425 0.411 710 111.5
Magnesium 6.5 44.8 2.4 16.5 0.350 0.065 112 17.6
Molybdenum 48.0 331.0 17.0 117.0 0.307 0.368 636 100.0
Nickel silver 18.5 127.0 7.0 48.3 0.322 0.316 546 85.8
Nickel steel 30.0 207.0 11.5 79.3 0.291 0.280 484 76.0
Phosphor bronze 16.1 111.0 6.0 41.4 0.349 0.295 510 80.1
Stainless steel 27.6 190.0 10.6 73.1 0.305 0.280 484 76.0
Solution:
A possible layout of the table is shown in Table 1. The columns should be clearly labeled
with units and the data should be easy to read and understand.
$+&,%
Material Modulus of Elasticity Modulus of Rigidity Poisson’s
Ratio
Unit Weight
(Mpsi) (Gpa) (Mpsi) (Gpa) (lb/in
3
) (lb/ft
3
) (kN/m
3
)
Chapter 3: Technical ProblemSolving and Communication Skills
57
For the data in P3.41, prepare a graph that charts the relationship between the
modulus of elasticity (yaxis) and unit weight (xaxis) using the USCS unit system data.
Explain the resulting trend, including a physical explanation of the trend, noting any
deviations from the trend.
Solution:
A possible graph of the data is shown in Figure 1.
“#%&-! (./#
Trend observations could include:
The general trend of an increasing Modulus of Elasticity as Unit Weight increases.
They could explain this both from the data and a physical explanation of the materials
in the data (e.g., in general, the denser the material, the harder it will be to deform). A
trend line helps to visualize the trend. An exponential curve is shown in Figure 1.
Two significant deviations exist on the plot:
y = 4.2048e4.9437x
0
10
20
30
50
60
Modulus of Elasticity (Mpsi)
Modulus of Elasticity vs. Unit Weight for
Various Materials
Chapter 3: Technical ProblemSolving and Communication Skills
58
0*0
These estimation problems are very openended and will be driven largely by the
assumptions made by the students. The problems are all complex enough so that they could
Regardless of how the problems are administered, the students should set up the problem
using the ApproachSolutionDiscussion model as presented in the chapter. In the
Approach phase, it will be critical for the students to make appropriate assumptions
For these types of problems, it is always interesting to track the maximum and minimum
estimates for each problem from all the students/groups and report these to the class when
the problems are returned or discussed. Discussing the sets of assumptions that led to the