1
1.9
UNITS*
Several different systems of units are used in engineering. The most common in the United States are the U.S. foot-pound-second system ( fps), the U.S. inch-pound-sec-
ond system ( ips), and the Systeme Internationale ( SI). The metric centimeter, gram,
second ( cgs) system is being used more frequently in the U.S., particularly in international companies, e.g., the automotive industry. All systems are created from the choice of three of the quantities in the general expression of Newton’s second law mL
F =
(1.2 a)
t 2
where F is force, m is mass, L is length, and t is time. The units for any three of these variables can be chosen and the other is then derived in terms of the chosen units. The three chosen units are called base units, and the remaining one is a derived unit.
Most of the confusion that surrounds the conversion of computations between either one of the U.S. systems and the SI system is due to the fact that the SI system uses a different set of base units than the U.S. systems. Both U.S. systems choose force, length, and time as the base units. Mass is then a derived unit in the U.S. systems, which are referred to as gravitational systems because the value of mass is dependent on the local gravitational constant. The SI system chooses mass, length, and time as the base units, and force is the derived unit. SI is then referred to as an absolute system, since the mass is a base unit whose value is not dependent on local gravity.
The U.S. foot-pound-second ( fps) system requires that all lengths be measured in feet (ft), forces in pounds (lb), and time in seconds (sec). Mass is then derived from Newton’s law as
Ft 2
m =
(1.2 b)
L
and its units are pounds seconds squared per foot (lb sec2/ft) = slugs.
The U.S. inch-pound-second ( ips) system requires that all lengths be measured in
* Excerpted from Norton, Design
inches (in), forces in pounds (lb), and time in seconds (sec). Mass is still derived from of Machinery, 3ed, 2004,
McGraw-Hill, New York, with the
Newton’s law, equation 1.2 b, but the units are now
publisher’s permission.
pounds seconds squared per inch (lb sec2/in) = blobs†
† It is unfortunate that the mass
unit in the ips system has never
This mass unit is not slugs! It is worth twelve slugs or one “blob”!
officially been given a name such
as the term slug used for mass in
the fps system. The author boldly
Weight is defined as the force exerted on an object by gravity. Probably the most suggests (with tongue only slightly
common units error that students make is to mix up these two unit systems ( fps and ips) in cheek) that this unit of mass in
the ips system be called a blob (bl)
when converting weight units (which are pounds force) to mass units. Note that the to distinguish it more clearly from
gravitational acceleration constant ( g or g
the slug (sl), and to help the
c) on earth at sea level is approximately 32.17
student avoid some of the
feet per second squared, which is equivalent to 386 inches per second squared. The re-common errors listed below.
lationship between mass and weight is
Twelve slugs = one blob. Blob
does not sound any sillier than
slug, is easy to remember, implies
mass = weight / gravitational acceleration
mass, and has a convenient
abbreviation (bl) which is an
W
anagram for the abbreviation for
m =
(1.3)
pound (lb). Besides, if you have
g
ever seen a garden slug, you know
c
it looks just like a “little blob.”
22
MACHINE DESIGN -
An Integrated Approach
1
It should be obvious that if you measure all your lengths in inches and then use g = gc = 32.17 feet/sec2 to compute mass, you will have an error of a factor of 12 in your results. This is a serious error, large enough to crash the airplane you designed. Even worse off is the student who neglects to convert weight to mass at all. The results of this calculation will have an error of either 32 or 386, which is enough to sink the ship!*
The value of mass is needed in Newton’s second-law equation to determine forces due to accelerations:
F = ma
(1.4 a)
The units of mass in this equation are either g, kg, slugs, or blobs depending on the units system used. Thus, in either English system, the weight W (lbf) must be divided by the acceleration due to gravity gc as indicated in equation 1.3 to get the proper mass quantity for equation 1.4 a.
Adding further to the confusion is the common use of the unit of pounds mass (lbm).
This unit, often used in fluid dynamics and thermodynamics, comes about through the use of a slightly different form of Newton’s equation:
ma
F =
(1.4 b)
gc
where m = mass in lbm, a = acceleration, and gc = the gravitational constant. On earth, the value of the mass of an object measured in pounds mass (lbm) is numerically equal to its weight in pounds force (lbf). However, the student must remember to divide the value of m in lbm by gc when using this form of Newton’s equation. Thus the lbm will be divided either by 32.17 or by 386 when calculating the dynamic force. The result will be the same as when the mass is expressed in either slugs or blobs in the F = ma form of the equation. Remember that in round numbers at sea level on earth
* A 125-million-dollar space
1 lbm = 1 lbf
1 slug = 32.17 lbf
1 blob = 386 lbf
probe was lost because NASA
failed to convert data that had
The SI system requires that lengths be measured in meters (m), mass in kilograms been supplied in ips units by its
contractor, Lockheed Aerospace,
(kg), and time in seconds (sec). This is sometimes also referred to as the mks system.
into the metric units used in the
Force is derived from Newton’s law and the units are:
NASA computer programs that
controlled the spacecraft. It was
kg m/sec2 = newtons
supposed to orbit the planet Mars,
but instead either burned up in the
Martian atmosphere or crashed
In the SI system there are distinct names for mass and force, which helps alleviate into the planet because of this
confusion.† When converting between SI and U.S. systems, be alert to the fact that mass units error. Source: The Boston
Globe, October 1, 1999, p. 1.
converts from kilograms (kg) to either slugs (sl) or blobs (bl), and force converts from
newtons (N) to pounds (lb). The gravitational constant ( g
c) in the SI system is approxi-
† A valuable resource for
mately 9.81 m/sec2.
information on the proper use of
SI units can be found at the U. S.
Government NIST site at http://
The cgs system requires that lengths be measured in centimeters (cm), mass in physics.nist.gov/cuu/Units/
grams (g), and time in seconds (sec). Force is measured in dynes. The SI system is units.html
generally preferred over the cgs system.
Another excellent resource on the
proper use of metric units in
machine design can be found in a
The systems of units used in this textbook are the U.S. ips system and the SI sys-pamphlet “Metric Is Simple,”
tem. Much of machine design in the United States is still done in the ips system, though published and distributed by the
fastener company Bossard
the SI system is becoming more common.† Table 1-4 shows some of the variables used International Inc., 235 Heritage
in this text and their units. Table 1-5 shows a number of conversion factors between Avenue, Portsmouth, NH 03801
commonly used units. The student is cautioned always to check the units in any equa-
Chapter 1