# 2.4: Untitled Page 16

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## Chapter 2

where *f* is used to represent the magnitude of the vector **f**. Note that the force is expressed in terms of *three* of the four fundamental standards of measure, i.e., mass, length and time. There is no real need to go beyond Eq. 2‐7 in our description of force; however, our intuitive knowledge of force is rather different from our knowledge of mass, length and time. Consider for example, pushing against a wall with a “force” of 91 kg m/s2. This is simply not a satisfactory description of the event. What we want here is a unit that describes the *physical* *nature* of the event, and we obtain this unit by *defining* a unit of force as

2

1 newton =

1 kg

1 m/s

(2‐8)

When pushing against the wall with a force of 91 kg m/s2 we feel comfortable describing the event as

2

force = 91 kg m/s

2

91 kg m/s 1

(2‐9)

2

1 newton

91 kg m/s

91 newtons

2

kg m s

Here we have arranged Eq. 2‐8 in the form

newton

N

1

(2‐10)

2

2

kg m/s

kg m/s

and multiplied the quantity 91 kg m/s2 by *one* in order to affect the change in units. Note that in our definition of the unit of force we have made use of a *one-to‐one* correspondence represented by Eq. 2‐8. This is a characteristic of the SI system and it is certainly one of its attractive features. One must keep in mind that Eq. 2‐8 is *nothing more than a definition* and if one wished it could be replaced by the *alternate definition* given by (Truesdell, 1968)

2

1 euler =

17.07 kg

1 m/s

(2‐11)

However, there is nothing to be gained from this second definition, aside from honoring Euler, and it is clearly less attractive than the definition given by Eq. 2‐8.

In the British system of units the *one‐to‐one* correspondence is often lost and confusion results. In the British system we choose our standards of mass, length and time as

standard of mass = lbm

standard of length = ft

standard of time = s

19

and we define the pound‐force according to

1 lb

1 lb

2

f

m

32.17 ft/s

(2‐12)

This definition was chosen so that mass and force would be numerically equivalent when the mass was acted upon by the earth’s gravitational field.

While this may be convenient under certain circumstances, the definition of a unit of force given by Eq. 2‐12 is certainly less attractive than that given by Eq. 2‐8.

In summary, we note that there are only four standards needed to assign numerical values to all observables, and the choice of standard is arbitrary, i.e.

the standard of length could be a foot, an inch or a centimeter. Once the standard is chosen, i.e. the meter is the standard of length, other *alternate units* can be constructed such as those listed in Table 2‐2. In addition to a variety of *alternate units* for mass, length and time, we construct for our own convenience a series of *derived units*. Some of the derived units for the SI system are given in Table 2‐3. Finally, we find it convenient to tabulate *conversion factors* for the *derived units* for the various different systems of units and some of these are listed in Table 2‐4a and in Table 2‐4b. An interesting history of the SI system is available from the Bureau International des Poids et Mesures (see references).

**2.3 Dimensionally Correct and Dimensionally Incorrect Equations** In defining a unit of force on the basis of Eq. 2‐6 we made use of what is known as the *law of dimensional homogeneity*. This law states that natural phenomena proceed with no regard for man‐made units, thus the basic equations describing physical phenomena must be valid for all systems of units.

It follows that each term in an equation based on the laws of physics must have the same units. This means that the units of **f** in Eq. 2‐6 must be the same as the units of *d*( *m***v**) *dt *. It is this fact which leads us to the definition of a unit of force such as that given by Eq. 2‐8. Equations that satisfy the *law of dimensional* *homogeneity* are sometime referred to as *dimensionally correct* in order to distinguish them from equations that are *dimensionally incorrect*. While the law of dimensional homogeneity requires that all terms in an equation have the same units, this constraint is often ignored in the construction of empirical equations found in engineering practice. For example, in the sixth edition of *Perry’s* *Chemical Engineers’ Handbook* (Perry *et* *al*., 1984), we find an equation for the drop size produced by a certain type of atomizer that takes the form