The units for working with mass and force tend to be confusing for many students. The key to keeping them straight is to remember Newton’s Law (the 2nd law of motion):

Historically, there have been two main approaches to systematizing the relationship between mass and force:

• in “gravitational systems”, length, force, and time are basic units, and mass is a derived unit, and
• in “absolute systems”, length, mass, and time are basic units, and force is the derived unit.

The SI system of units is absolute with respect to force, so we will follow that convention in everything we do.

Think about the dimensions on Newton’s law:

This relationship suggests that the natural units of force are mass units times accleration units:

For convenience, we often prefer to use a single “defined” unit to describe force rather than keeping track of three separate ones, hence

There are at least three different ways of handling things in English units. In most Ch.E. classes, the derived unit is the “pound-force” (lbf), defined as the product of the mass unit and the acceleration of gravity at sea level and 45 degrees latitude.

BE CAREFUL!!! 1 lbm does NOT equal 1 lbf

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Aside: Other Approaches

Others have tried to set up the English system so that “pound” was used for only one unit — either mass or force, but not both. You should be aware that these exist, although we won’t use them (Physics and CE/ME texts may).

Gravitational formulations of this type set the unit of mass as the “slug” and the unit of force as the “pound”. Absolute formulations use “pound” as the mass unit, but “poundal” as the force unit.

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g_c

The conversion between the defined unit of force (N, dyne, lbf) and natural units is so commonly used that we give it a special name and symbol, g_c.

so that

g_c is a conversion factor, so it’s magnitude is 1.0. This means that when we multiply or divide by g_c we change the units but not the magnitude of a number.

Tip: If you’re working a problem and have mass units, but want force, divide by g_c (or vice versa).

Weight

Although common usage allows the word “weight” to be used as a synonym for “mass”, in engineering and science you must be aware that weight is defined to be the force exerted on an object by gravity, so an object of mass m subjected to the gravitational acceleration g, will have weight

The acceleration of gravity is not a constant. It varies with the mass of the attracting body (earth, moon, etc.), with the distance between objects, and with position. A “typical” value is used for calculations when the exact value is uncertain. This value (standard gravity) is:

REMEMBER:g is the acceleration of gravity and varies with position, although a standard value can be used if the precise value is unknown. g_c is a constant conversion factor, and does not change.

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EXAMPLE: Himmelblau, Example 1.6, p. 10

What is the difference in the weight in newtons of a 100 kg rocket at 10 km above the earth (g=9.76 m/s²) as opposed to its weight on the surface (g=9.80 m/s²)?