Force, weight, mass, and all that

The weight of an object is the force exerted on it by gravity. (Well, this is not exactly a watertight definition, but if you know its flaws, then you shouldn't have to read these notes.) The weight $W$ is proportional to the mass $m$, which is expressed as $W = mg$, where $g$ is called the gravitational acceleration.

As simple as the equation $W = mg$ is, it does pose difficulties to the uninitiated when it comes down to plugging in numbers. If you work in the metric system of units, it's likely that you will express the mass $m$ in grams or kilograms, and $g$ in cm/sec$^2$ or m/sec$^2$. The value of $g$ varies slightly with the location on Earth, but it is common to take $g = 9.81$ m/sec$^2$ (or $g = 981$ cm/sec$^2$) as a representative value. If you work in the avoirdupois (av'-er-de-poiz') units, it's likely that you will express $m$ in pounds and $g$ in ft/sec$^2$. It is common to take $g = 32$ ft/sec$^2$ as a representative value.

The nitty-gritty

• A rock of mass $m=5$ kilograms has a weight of $W = mg = 5 \times 9.81 = 49.05$. But 49.05 what? 49.05 newtons, that's what!
• An earring of mass of $m=5$ grams has a weight of $W = mg = 5 \times 981 = 4905$. But 4905 what? 4905 dynes, that's what!
• You buy sugar in four-pound bags. So the mass is $m=4$ lb and the weight is $W = mg = 4 \times 32 = 128$. But 128 what? 128 poundals, that's what!

  Aside: The “lb” symbol stands for the ancient Roman weight libra. So much for notational consistency.

• A guy weighs $W = 192$ pounds…

  Oops, wait a minute! I thought a “pound” is a unit of mass, not weight. What gives?

  Well, the technically correct expression is to say “A guy weighs $W = 192$ pounds-force.” A pound-force is a unit of force (or weight) in the avoirdupois system but in the common daily usage the “-force” suffix is carelessly dropped and one ends up saying so-and-so weighs 192 pounds. You will have to infer the meaning from the context.

  OK, let's say Joe weighs $W = 192$ pounds-force. What is Joe's mass? It is $m = W/g = 192/32 = 6$. But 6 what? 6 slugs, that's what!

So both slugs and pounds are units of mass. How are they related?

A slug is defined to be 32.1740 pounds. That 32.1740 is the standardized value of the average gravitational acceleration. In solving our Math 225 homework problems, however, we take one slug to be exactly 32 pounds in order to simplify hand calculations.

Example: Joe weighs $W = 192$ pounds-force. So his mass is $m = W/g = 192/32 = 6$ slugs which is the same as $m = 32\times 6 = 192$ pounds. As we see, Joe's weight expressed in pounds-force is numerically equal to Joe's mass expressed in pounds. Therein lies the source of confusion about pound and pound-force. Don't trip over it!