PROPRIETARY AND CONFIDENTIAL
This Manual is the proprietary property of The McGraw-Hill
Companies, Inc. (“McGraw-Hill”) and protected by copyright and other
state and federal laws. By opening and using this Manual the user
agrees to the following restrictions, and if the recipient does not agree
to these restrictions, the Manual should be promptly returned unopened
to McGraw-Hill: This Manual is being provided only to authorized
professors and instructors for use in preparing for the classes using
the affiliated textbook. No other use or distribution of this Manual
is permitted. This Manual may not be sold and may not be
distributed to or used by any student or other third party. No part
of this Manual may be reproduced, displayed or distributed in any
form or by any means, electronic or otherwise, without the prior
written permission of McGraw-Hill.
Solution Manual, Chapter 7 – Dimensional Analysis
Chapter 7
Dimensional
Analysis
Dimensions and Units, Primary Dimensions
7-1C
Solution We are to explain the difference between a dimension and a unit, and
give examples.
Analysis A dimension is a measure of a physical quantity (without
numerical values), while a unit is a way to assign a number to that dimension.
Examples are numerous – length and meter, temperature and oC, weight and lbf, mass
and kg, time and second, power and watt,…
Discussion When performing dimensional analysis, it is important to recognize
the difference between dimensions and units.
7-2C
Solution We are to append Table P7-2 with other parameters and their primary
dimensions.
Analysis Students’ tables will differ, but they should add entries such as
angular velocity, kinematic viscosity, work, energy, power, specific heat, thermal
conductivity, torque or moment, stress, etc.
Discussion This problem should be assigned as an ongoing homework problem
throughout the semester, and then collected towards the end. Individual instructors
can determine how many entries to be required in the table.
7-1
Solution Manual, Chapter 7 – Dimensional Analysis
7-3C
Solution We are to list the seven primary dimensions and explain their
significance.
Analysis The seven primary dimensions are mass, length, time, temperature,
electrical current, amount of light, and amount of matter. Their significance is
that all other dimensions can be formed by combinations of these seven primary
dimensions.
Discussion One of the first steps in a dimensional analysis is to write down the
primary dimensions of every variable or parameter that is important in the problem.
7-4
Solution We are to write the primary dimensions of the universal ideal gas
constant.
Analysis From the given equation,
Primary dimensions of the universal ideal gas constant:
{}
3
2
mL
pressure volume tL
mol temperature N T
u
R
×
×
=
==
××
2
2
mL
tTN
Or, in exponent form, {Ru} = {m1 L2 t-2 T-1 N-1}.
Discussion The standard value of Ru is 8314.3 J/kmol⋅K. You can verify that
these units agree with the dimensions of the result.
7-5
Solution We are to write the primary dimensions of atomic weight.
Analysis By definition, atomic weight is mass per mol,
Primary dimensions of atomic weight:
{}
mass
mol
M
==
m
N (1)
Or, in exponent form, {M} = {m1 N-1}.
Discussion In terms of primary dimensions, atomic mass is not dimensionless,
although many authors treat it as such. Note that mass and amount of matter are
defined as two separate primary dimensions.
7-2
Solution Manual, Chapter 7 – Dimensional Analysis
7-6
Solution We are to write the primary dimensions of the universal ideal gas
constant in the alternate system where force replaces mass as a primary dimension.
Analysis From Newton’s second law, force equals mass times acceleration.
Thus, mass is written in terms of force as
Primary dimensions of mass in the alternate system:
{}
2
2
force F Ft
mass acceleration LL/t
===
(1)
We substitute Eq. 1 into the results of Problem 7-4,
Primary dimensions of the universal ideal gas constant:
{}
2
2
2
22
Ft L
mL L
Nt T Nt T
u
R
=
==
FL
TN (2)
Or, in exponent form, {Ru} = {F1 L1 T-1 N-1}.
Discussion The standard value of Ru is 8314.3 J/kmol⋅K. You can verify that
these units agree with the dimensions of Eq. 2.
7-3
Solution Manual, Chapter 7 – Dimensional Analysis
7-7
Solution We are to write the primary dimensions of the specific ideal gas
constant, and verify are result by comparing to the standard SI units of Rair.
Analysis We can approach this problem two ways. If we have already worked
through Problem 7-4, we can use our results. Namely,
Primary dimensions of specific ideal gas constant:
{}
2
2
gas
mL
Nt T
m
N
u
R
RM
=
==
2
2
L
tT (1)
Or, in exponent form, {Rgas} = {L2 t-2 T-1}. Alternatively, we can use either form of
the ideal gas law,
Primary dimensions of specific ideal gas constant:
{}
3
2
gas
mL
pressure volume tL
mass temperature m T
R
×
×
=
==
××
2
2
L
tT (2)
For air, Rair = 287.0 J/kg⋅K. We transform these units into primary dimensions,
Primary dimensions of the specific ideal gas constant for air:
{}
2
2
air
mL
Jt
287.0 kg×K m T
R
=
==
×
2
2
L
tT (3)
Equation 3 agrees with Eq. 1 and Eq. 2, increasing our confidence that we have
performed the algebra correctly.
Discussion Notice that numbers, like the value 287.0 in Eq. 3 have no influence
on the dimensions.
7-4
