# Weight, Density and Gravity

1. Oct 24, 2004

### kaboo

Hello,

I'm struggling to understand the relationship between SI units and Imperial units when it comes to weight, density and gravity. Using water as an example:

I know that F = m g

I also know that gravity is g = 9.81 m/s^2 or 32.174 ft/s^2 and that
density of water is Pw = 1,000Kg/m^3 and that
the weight of water is 62.43 pounds per cubic foot, or 62.43 lbf/ft^3.

So with this knowledge what is the formula to get
Weight of water (in SI units) == weight of water (in Imperial units)

When starting form F = m g?

Thanks!

2. Oct 26, 2004

### ramcg1

3. Oct 27, 2004

### kaboo

Does water = water?

Ok, that's what I thought! So you would think that the SI water (water density * gravity) should equal the Imperial water (water weight * a conversion factor), right?
Well, let's see...

First, the Imperial side of the equation:
If $$1 m = 3.2808 ft$$ then $$1m^3 = 35.31 ft^3$$ and if the weight of water is $$\frac{62.43 lb_{f}} {ft^3}}$$ then $$\frac{62.43 lb_{f}} {ft^3} \times \frac{35.31 ft^3} {m^3} = \frac{2,204.6 lb_{f}} {m^3}$$, right?

Now for the SI side:
If density of water is $$\frac{1,000Kg} {m^3}$$ and if density is another way of saying fluid mass, then the mass of water is also $$\frac{1,000Kg} {m^3}$$, right?

And if $$1Kg = 2.2046 lb_{m}$$ then $$\frac{1,000Kg} {m^3}\times \frac{2.2046 lb_{m}}{1Kg} = \frac{2,204.6lb_{m}}{m^3}$$.

Wow, water = water, Right? Wrong!
Even though (the Imperial water) $$\frac{2,204.6 lb_{f}} {m^3} = \frac{2,204.6lb_{m}}{m^3}$$ (the SI water) looks the same, they're not, look again!

$$lb_{f} \neq lb_{m}$$, $$lb_{f}$$ is "pounds of force" whereas, $$lb_{m}$$ is "pounds of mass".

We know that $$f=m \times g$$, thus, $$lb_{f} = lb_{m} \times g$$!

Conclusion
As I see it, there are at lest three possible outcomes:
1. The Imperial weight of water is truly the mass of water, or
2. The SI mass of water is really the weight of water, or (my personal favorite)
3. I messed up some where!

So, can someone help me out? What did I do wrong?

Yes, I can...
In writing this response, as carefully as I knew how, I discovered (just a moment ago) that I was wrong! Here's what I discovered...

In (some) physics books, applications and web based conversion utilities they state that...

$$1Kg = 2.2046 lb$$

However, it isn't clear that they are converting 1 unit of mass to 1 unit of weight. Therefore I think it should read:

$$1Kg = 2.2046 lb\ (of\ weight)$$ or
$$1Kg = 2.2046 lb_{f}$$ or better yet
$$1Kg = 0.0685 lb\ (of\ mass)$$

I've spent some serious time and energy trying to figure out why it appeared that water did not equal water. I hope that sharing my frustrations and findings will help you as much as writing this has helped me. Thanks for reading.

One final thought, explaining your problem (in writing or verbally) to someone will normally show you where you went wrong.

-kaboo

4. Oct 27, 2004

### cepheid

Staff Emeritus
Much of this makes no sense whatsoever:

Well, water density*gravitational field strength will give you weight per unit volume. As for "water weight"...that depends on how much water you have!!! I don't see how you can assign to it a single value. And "the SI water should equal the Imperial water" sounds like nonsense to me. Water does not have units! Measurable physical quanitities such as mass, volume, density, weight, etc...these have units

What do you mean by the "weight of water?" Where did you get this number from? Does it not depend on how much water you have? Besides, this number is in units of weight per unit volume. Wheter it is accurate, I have no idea.

Density is another way of saying mass per unit volume, not just mass!!! To simply state...the mass of water is 'x' is meaningless. We can only measure the mass of a certain amount of water, not just water in general, whatever the hell that means. To state, the density of water is 'x' is fine, and that is what you have done here.

Again, this is totally meaningless! You persist in speaking as though water is a measure of something! It is not! In fact, it is a compound! Water is the something we are measuring.

Knowing the distinction between mass and weight, and between mass and density would be helpful. Maybe you do, but your discussion indicated that you did not. Also learn to distinguish between quantities that indicate properties of a substance independent of amount (e.g. density) and quantities that indicate the amount of a substance (mass, or volume).

5. Oct 30, 2004

### kaboo

Frustrated and angry!

I came to the physics forums to ask, what I thought, was a very straight forward question - obviously it wasn't.

Like most people, I've been raised to think in Imperial (British) system, namely feet, pounds and seconds (where a pound is a measurement of weight). I am trying to 'think' in SI. However, I can't tell if my answer makes sense because meters and grams don’t have any 'real-world' meaning to me yet. Therefore, I'm trying to figure out how to convert between SI and the old IM way of doing thing so I can see if my numbers and formulas 'looks' right.

Also, I thought that the physics forums was to be a “safe” place to learn and to ask questions regarding physics, regardless if you are an expert on the subject or not.

So, Cepheid, instead of “griping” about “how” I didn't setup the problem correctly or that I used thing incorrectly. Why don’t you first try answering the original question or seeking to understand what it is that I am trying to ask, and I am sorry if I didn't express things correctly.

Let’s try this again. All I want to know is the force, mass and gravity (of water) in both SI and IM units, where the two sides of the equations are equal. Six values along with their proper units, that's all. If you want to show the formula on how they relate that would be a nice bonus.

6. Oct 31, 2004

### Staff: Mentor

To compare apples to apples, use the imperial unit for mass, the slug. 1 kg = 0.06852 slugs (approx). The weight of one slug can be found by w = mg --> (1 slug) (32.174 ft/s^2) = 32.174 pounds.

In the imperial system, mass is properly measured in slugs. Sure, in common usage, it is also measured in pounds. To convert, use w = mg.

The density of water (mass/volume) is 1000 kg/m^3. Let's convert:
$$1000 {kg}/m^3 (\frac{0.0685 {slug}}{1 {kg}}) (\frac{1 m^3}{35.315 {ft}^3}) = 1.940{slugs}/{ft}^3$$.
If you want to get pounds per cubic foot, use w = mg to convert from slugs to pounds; you'll get 62.4 lbs/ft^3.

Does this help?

7. Nov 13, 2004

Thank you!