# Units of $sec^2/ft.lb$: What does that mean?

I'm working through a text that talks about mechanical vibrations, and I found something I haven't seen before, nor understand, yet. In the example, it gives me a weight of 16 lb. It says this is actually a measure of weight and not mass. To convert to mass, I must divide by 32. Now, I understand that, if I have a measure of weight, then dividing by g will give me the mass of the thing. What I don't understand is this:

$\frac{16 lb}{32 \frac{ft}{sec^2}} = \frac{1}{2}\frac{(lb)sec^2}{ft}$

That's the example it gives and it says that the mass is $\frac{1}{2}$. Is that nonsense?

Is the unit of mass lb($sec^2$)/ft ? Why not kg?

-

You shouldn't expect a metric system unit of mass like kg to result from a calculation in English units (lb and ft). You should expect the English system unit of mass, the slug, and that's exactly what you got, considering that a pound is a slug foot per second per second.

-
Thank you. I've never heard of a slug. Seems like an odd distraction to put in the middle of a lesson. :/ – Korgan Rivera Dec 2 '12 at 4:33

More generally, it is possible to manipulate the multiplication/division of units as if they were mathematical objects.

Consider the simple example: you travel 50 feet per second for 300 seconds, how many miles have you gone? This is a straightforward calculation, but often we overlook the operations we do with the units:

$$50\ \frac{\color{#F00}{\rlap{/}}{\text{ft}}}{\color{#0FF}{\rlap{//}}\text{sec}}\cdot 300\ \color{#0FF}{\rlap{//}}\text{sec} \cdot 1 \frac{\text{mi}}{5280\ \color{#F00}{\rlap{/}}\text{ft}} \approx 2.841\ \text{mi}.$$

So from Newton's laws, what you have is $$F = ma \implies F\ \text{lbf} = m\ \text{lbm} \cdot 32\ \frac{\text{ft}}{\text{sec}^2}.$$

So what we've derived is $$1\ \text{lbm} = \frac{1}{32} \frac{F}{m} \frac{\text{lbf}\cdot\text{sec}^2}{\text{ft}}.$$

If you wanted things in terms of kilograms, you can now do some truly heinous algebra:

$$1\ \text{lbm} \approx 0.454\ \text{kg}$$ meaning $$1\ \text{kg} \approx \frac{1}{14.528} \frac{F}{m} \frac{\text{lbf}\cdot\text{sec}^2}{\text{ft}} \frac{1\ \text{ft}}{.3048\ \textrm{meters}}$$ $$1\ \text{kg} \approx \frac{1}{4.428} \frac{F}{m} \frac{\text{lbf}\cdot\text{sec}^2}{\text{meters}}$$

-

If the brackets denote your favorite units, and if we disregard constants of proportionality, then $$[\mbox{force}] = \frac{[\mbox{mass}][\mbox{distance}]}{[\mbox{time}]^2}$$ so that $$[\mbox{mass}] = \frac{[\mbox{force}][\mbox{time}]^2}{[\mbox{distance}]}.$$

Mass is a measure of inertia. To figure an object's mass, subject it to a force and measure the corresponding acceleration, then figure the ratio. That's all that's going on here.

Your object is subject to a force of $16\ \mbox{lb}$ due to gravity, and when subjected to that force, it accelerates at $32\ \mbox{ft}/\mbox{s}^2$, so its mass must be $\frac{1}{2} \frac{\mbox{lb}\cdot\mbox{s}^2}{\mbox{ft}}$. This tells you that if it's accelerating at $a\ \mbox{ft}/\mbox{s}^2$, then it must be subject to a force of $a/2\ \mbox{lb}$.

-

When you divide weight by acceleration, you get the mass. and when you manipulate the formula for the units of each, you get what you see there. nothing behind it.