Thu Dec 10 18:49:41 PST 2020

# Units

If you were to ask someone in the United States: "How tall are you?" They might answer "6".

And if they were a smart alec, they might answer: "72" or "183".

In the U.S. you'd probably assume they meant "6 feet." In a country that uses the metric system, you'd assume they meant "1.83 meters". And the smart alecs are probably answering "72 inches" or "183 centimeters."

All of these numbers represent the same height. "Feet, inches, centimeters and meters" are units.

(Feet and pounds are part of a system of units called imperial units. Meters and kilograms are part of the metric system. A nice thing about the metric system is that the big units are made of tens of smaller units: 1 kilogram = 1000 grams; 1 meter = 100 centimeters.)

Units give meaning to a number, and connect it to the real world. '6' doesn't mean much unless you know '6 of what?' 6 feet is a distance. 6 eggs is a quantity. And so on.

When a number has units, you can compare it to other numbers with the same units.
If one person is six feet tall an another is 1.5 meters tall, who is taller? We need to have the heights in the same units. It turns out that 1 foot is 0.3048 meters. This is called a conversion factor. But how do we use it? Do we multiply or divide?

Our conversion factor is:

``````1 ft = 0.3048 meters
``````

We can make this into a fraction:

`````` 0.3048 meters
--------------    =   1
1 ft
``````

Let's try dividing by the conversion factor. We'll have to turn the fraction upside down and multiply:

``````6 ft  |  1ft               6  ft * ft        19.7 ft * ft
-------------------  =    ---------     =   ----------
| .3048 meters      .3048 meters        meters
``````

19.7 is a number.. it could be meters, but what do the units say? Feet squared divided by meters?

That's not what we wanted. We wanted meters.

``````6 ft   | .3048 meters
---------------------   = 6 * .3048 meters = 1.83 meters
|  1 ft
``````

Now we can compare 1.5 meters and 1.83 meter, and see who's taller.

Notice how we wrote the problem down, so we can see clearly where the units go. Now we can see that when we multiplied we can cancel out the 'ft' since there's one on top and one on the bottom, leaving only 'meters'. Since that's the unit we were trying to get, we know we used the conversion factor correctly even before we calculated the value.

Suppose you want to calculate how heavy a balsa wood wing is before you build it. You design the wing to have an area of one square foot (1 ft2), and you know that density of your balsa wood is 6 pounds per cubic foot.

Start with the unit you want to end up with, the weight of the wing. We expect it to be in the ounces (Oz) . Do we have ounces in the numbers we have? No, but we have pounds. Since we can always convert from pounds to ounces, lets get the answer in pounds first. (We'll write area = ft2 as ft * ft and volume = ft3 as ft * ft * ft)

``````      6 lb     | 1 ft * ft          6lb
--------------------------      = ------
ft * ft * ft  |                    ft
``````

The ft * ft on top only cancels out a ft * ft on the bottom, so we have one 'ft' left over.

6 lb/ft is not a weight! What did we do wrong? Density is a mass per unit volume and we multiplied times an area. What we needed to do is multiply times the volume of our wing - which, for a flat sheet of balsa, is just its area times it's thickness.

Lets say we are going to use 1/8" balsa wood. We'll have to convert that to feet, knowing that 12 inches = 1 ft.

``````      6 lb       | 1 ft * ft | 1/8 in  |  1 ft           6 * 1/8 lb
------------------------------------------------     = -------------  = 0.0625 lb
ft * ft * ft    |                     |  12 in              12
``````

Notice how we now have ft * ft * ft on the top to cancel the ft * ft * ft on the bottom. Also there's an 'in' on both the top and the bottom, leaving only lb.

Finally we can convert lb to Oz knowing that 16 Oz is 1 lb:

``````0.0625 lb  |   16 Oz
--------------------    =  1 Oz.
|    lb
``````

Now we have to confess that we pulled a fast one on you there, and we're going to explain it so that nobody can fool you ever again.

We said that density was mass per unit volume:

``````    mass
------------
ft * ft * ft
``````

What we wrote was that density was weight per unit volume.

Mass and Weight are related, and sometimes people use a "conversion factor" between the two, but it's not the same as converting ounces to pounds or centimeters to meters. (As we'll see below the confusion between weight and mass is pretty baked into imperial units.)

Mass is a fundamental quantity of matter. An object's mass is how much stuff is in an object. Weight is a force. A force is something that makes a mass accelerate, in accordance with Newton's second law of motion:

``````F = m * a
``````

Weight is the force that a mass exerts when gravity is acting on it:

``````W = m * g
``````

Let's look at the units:

In metric, mass is usually given in kilograms (kg.) Acceleration is the rate at which something is speeding up. Its units are speed divided by time, and speed is distance divided by time. In metric, distance is in meters (m) and time is in seconds (sec):

``````meter  |   1            meter
-----------------  =  ---------  =  acceleration
sec    |  sec          sec * sec
``````

So force must have the units:

``````                             kg |   meter
F = mass * acceleration =   --------------  = 1 Newton
| sec * sec
``````

(The unit of force in metric is named after Isaac Newton, the same Newton who came up with the three laws of motion.)

Since weight is a force, that means that it must have the same units as force. Which means that 'g' must have the same units as acceleration.

In metric, g on the surface of the earth is (on average):

`````` 9.8 * meter
-------------
sec * sec
``````

In imperial units, its:

``````  32.2 * ft
-------------
sec * sec
``````

Now you can see the problem. You can't convert Mass to Weight, or Weight to Mass. But you can calculate how much Weight a Mass will have when you're on earth.

Weight divided by volume has different units than mass divided by volume. Well, you say, that's not really a problem. We'll just divide the weight by 'g' and that will give us the mass and off we go.

Of course, if we do that, we'll get the mass of our wing at the end, and not the weight, so we'll have to multiply by 'g' again to get the weight... heck, we ended up with the same answer!

It's true that you can usually shortcut and use weight when you mean to use mass because you really wanted weight in the first place, or the other way around.

But remember: Weight is not Mass. Weight is a force, and Mass is an amount of stuff.

Suppose you want to push on a ball and make it speed up.
Using Newton's second law you know that:

``````F = m * a     and   a = F / m
``````

So to find a, you need F and m.

First we weigh the ball. Our bathroom scale says its "1 lb"

Then we measure how hard are going to push on it, also using the bathroom scale. It says we can push with a force of 1 lb.

We put that into our equation:

``````      1 lb
a =  ------  = 1
1 lb
``````

The answer has no units, and acceleration should have units! What have we done? We've discovered something annoying about imperial units.

When we use a bathroom scale, what we really measure is a force. Most scales measure the deflection of a spring and calibrate that in either weight (force) or mass.

For example, a metric bathroom scale tells you weigh "84kg". It's really saying "I measured a force of 824.9 N. Since W = m * g then m = W / g and if 'g' is 9.8, that means the force was due to a mass of:

``````824.9 kg * m   |   sec * sec
------------------------------  = 84 kg.
sec * sec      |   9.8 m
``````

On the other hand, a scale that tells you your weight in pounds, gives you a choice, because in imperial units, one pound of mass has a weight of one pound of force when measured on earth.

Remember that Weight, or Force is = mass * acceleration. If we are measuring weight the acceleration is given by 'g'. What our scale is telling us when it reads 1 pound is this:

``````                1 pound mass |  1 g   | pounds force
1 pound force = -------------------------------------
|        | pounds mass
``````

This is great if we want to measure acceleration in g-units (And sometimes we do - think about a fighter pilot pulling 6g in a turn.)

But it doesn't help us we want to measure acceleration in ft/sec2.

Our scale told us "1 pound". Is that force, or is that mass? (Our metric scale told us kg: mass.) In imperial units we must choose whether to read the scale as force (pounds of force = "lbf") or as mass (pounds of mass = "lbm").

If we choose to read it as force (not unreasonable, since we are weighing something), we weigh our ball, and get the force due to the ball's mass in gravity. To get the mass we have to divide by acceleration due to g:

``````            Force                1 lbf    |   sec * sec        1
mass =     --------        =   -------------------------  =  ------  slug
acceleration                     |    32.2 ft        32.2
``````

So if we had 1 slug of mass, F = m * a gives:

``````1 slug   |   32.2 ft
---------------------   =  32.2 lbf
| sec * sec
``````

On the other hand, if we read it as pounds-mass, then we know that the scale has done this:

``````          Weight |  sec * sec
1 lbm   = ---------------------
|    32.2 ft
``````

The trouble is that now the weight can't be in pounds of force. Instead, there's a unit for force called a poundal, which is like a Newton. A force of one poundal will accelerate a mass of one lbm at 1 ft/sec2

That means that the weight of our 1 lbm ball is:

``````1 lbm   |  32.2 ft
-------------------  = 32.2 poundals
| sec * sec
``````

(Incidentally since 1 lbm at 1 g = 1 lbf, that means that 1 lbf = 32.2 poundals. And since 1 slug at 1g = 32.2 lbf, that means that 1 slug = 32.2 lbm:

``````1 slug  *  32.2  = 32.2 lbf = 32.2 lbm * 1g ;  1 slug = 32.2 lbm
``````

)

What's really important is you have to pick: lbf and slugs, or lbm and poundals. Don't mix them!

You can see that units have to work together. To compare two quantities, we need to have the quantities expressed in the same units. To do physical calculations, the units must be consistent - the units of force, mass and acceleration must be related.

In imperial units we have to choose between:

• lbf-ft-sec. slugs of mass, ft of distance, seconds of time, pounds of force or
• lbm-ft-sec. pounds of mass, ft of distance, seconds of time: poundals of force

But in metric:

• kg-meter-sec. kilograms of mass, meters of distance, seconds of time: Newtons of force.
• g-centimeter-sec. grams of mass, centimeters of distance, seconds of time: Dynes of force.

The difference between the two metric system is just that the units are smaller:

• 1 kilogram = 1000 gram
• 1 meter = 100 centimeter

Mass units are mass units and force units are force units. Much more straightforward than imperial units. (Unless somebody tries to tell us something weighs 100kgf (kilograms of force) OH, that's annoying.
Tell them to go poundal sand.)

One other thing, while we're on the subject. A scale measures force, usually by means of a spring being stretched or compressed. A balance - the kind where you put weights on it until it evens out - measures mass directly because 'g' is on both sides, and cancels out automatically. That means you can accurately measure mass with a balance on any planet, or even in a spaceship that's accelerating. (Won't work in orbit though.)

So - reader, meet units. Units meet reader. We hope that you will have a long and happy acquaintance.