# 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?

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## 4 Answers

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.

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

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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 $a/2\ \mbox{lb}$.

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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.

for some reason i can only write answers here-- no comments.

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