by
Larry "Harris" Taylor, Ph.D.
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Long ago it was said, "If you can't measure it, it ain't science!" Science may be thought of as a method of quantifying our environment. Numbers and units are the prime means of communicating scientific information. A MEASUREMENT IS INCOMPLETE WITHOUT ITS UNITS! For example, if someone asks the distance to the nearest dive store and they are told, "It is two," there is information missing, i.e., does the inquisitive diver have to walk two yards, two meters, two blocks, or two kilometers to reach the nearest dive store? The point is do not separate measurements from their units.
In the numeric solutions to problems
in this chapter, units are always present. If you wish, you can use units to
verify the technique of solving the problem. If you are uncertain whether the
conversions you have chosen are appropriate, look at the units; if your solution
has the wrong units (i.e., the answer you get is in inches and you are
calculating a temperature), then most likely an error has been made. A numeric
solution cannot be correct without the correct units. In addition, examination
of units may provide a key to determining a numeric solution. There are many
equations in physics, and often one forgets how "numbers" or "conversion
factors" are used to calculate the solution. Examination of the units of the
quantities involved can help develop a mathematical procedure to solve the
problem.
EXAMPLE: In discussing use of lift bags, it will be necessary to calculate underwater weight of an object to be lifted. This type of calculation uses the mass and density of the object (found in reference tables) to determine the object's volume. But suppose you forgot the equation and could not remember whether volume was found by dividing or multiplying the mass by the density. For example, you notice that the units listed in the density table for salt water (64 lbs/ft3) were pounds (a unit of mass) per cubic foot (a unit of volume). Thus:
unit of mass (lbs) / unit of volume (ft3) = a unit of density (lbs/ft3)
This reminds you that to find density you divide mass by the volume. Rearranging, the volume of the object lifted is found by dividing the mass by the density.
unit of mass (lbs) / density (lbs/ft3) = unit of volume (ft3)
Although this is a trivial example, the principle of using units to aid in numeric solutions is universal. The more complex the formula, the more helpful units will be.
EXAMPLE: The "number" used to convert liters and cubic feet is 28.316. Now, if you have a choice of purchasing a 14 cubic foot pony bottle or a 400 liter pony bottle for the same price, which is the better buy? Do you multiply or divide 14 or 400 by 28.316 to determine which cylinder provides more volume for the same money?
ANSWER: Examine the units. The units of volume for the pony bottle are cubic feet (14 ft3) or liters (400 l). The conversion factor is 28.316 liters/cubic foot. Set up the units in a linear string so that units "cancel" to give the units needed in the solution:
14 ft3 x 28.316 l = 396.4 l
1 ft3
To convert from liters to cubic feet, "invert" the units:
400 l x 1 ft3 = 14.1 ft3
28.316 l
Since the 400 l cylinder contains
more available air for the same money, in this example, it is the better
bargain!
NOTE: The ability to "invert" means that only one conversion "number" with its units need be committed to memory. In this case, conversion from liters to cubic feet was accomplished by inverting the cubic feet to liter conversion factor. An alternative would be to use the 0.0353 cubic feet/l conversion factor. (Divide 1 by 28.316; the result is 0.0353). Divers must decide which conversion factor is easiest for them to use.
It has been said that the exact definition of a problem is 90% of the solution. Working the problems in dive physics (and in real life) will be easier if the same format is always used. The format that historically has made learning easier and quicker is the following:
1. Determine from reading the problem: what is known, what is not known, and if there is a relationship that will allow the determination of unknown values from known values.
2. Define the problem (write the equations or principles to be used in obtaining the solution).
3. Substitute the known values, with their units, into the equation, and finally,
4. Solve the problem (do the
arithmetic!).
BOTTOM LINE: Using units at every step increases understanding of the problem AND increases likelihood of obtaining the correct solution!
Importance Of Using Units
The procedures used to solve diving
physics problems are the same in either English or metric units. Physical
principles are independent of the units employed. However, since the numeric
values will differ (since English units are NOT exactly identical to metric
units), it is important to pay attention to the units and use the SAME
UNITS for temperature, pressure, volume, etc. during the entire calculation.
For example, if using depth in fsw, then atmospheric pressure MUST BE defined in
terms of fsw. Ignoring units can lead to problems. For example, there is a
classic formula in the United States:
Absolute Pressure = (Depth / 33 ) + 1
This formula IS ONLY VALID
when diving in sea water at sea level with depth measured in feet of sea water
to determine pressure in absolute atmospheres.. Using measurements in other
units will give rise to problems and potential unsafe environments.
BAD EXAMPLE: Assume an American trained diver on vacation rents a dive gauge calibrated in meters. The diver (used to larger numbers from a gauge calibrated in fsw) dives with a guide to 40 m. If the American diver only remembers D/33 + 1 (without units) the following diver calculation error is possible:
(Actual units the diver is using in {})
Absolute Pressure = ( 40 {m} / 33 fsw/atm) + 1 atm = 2.21 {m-atm / fsw}
SHOULD BE:
Absolute Pressure = ( 40 m / 10.1 msw/atm) + 1 atm = 4.96 ata
COMMENT: If only the "numbers" are used, there is a chance that a totally inappropriate value could be obtained as an answer. (Here, for example, a diver ignoring units would assume a calculated 2.2 ata absolute pressure for a 40 m (131 foot) depth that actually has an absolute pressure near 5 ata!) This could pose a life threat if this diver was using this type of calculation (without units) to validate the maximum depth for an oxygen enriched air mix. A diver might ASSUME a certain depth gauge measured depth was a safe area in which to play, but the greater actual pressure might pose a severe risk to oxygen toxicity because the actual partial pressure would be so much higher than calculated!
In this particular case, examination of the calculated solution shows pressure has units of meters x atm divided by feet sea water. If the units are wrong, there is very little chance that the numeric answer is correct!
Get in the habit of using units. The
solutions make more sense and they are easier to obtain!
BOTTOM LINE: If you only memorize formulas and apply formulas without understanding them, then the potential for disaster exists!
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Credit:
Portions of this article were used in my chapter on Dive Physics appearing in:
Bove and Davis' Diving Medicine (4 th Edition), published by Saunders (Elsevier)
About The
Author:
Larry "Harris" Taylor, Ph.D. is a biochemist and Diving Safety
Coordinator at the University of Michigan. He has authored more than 200 scuba
related articles. His personal dive library (See Alert Diver, Mar/Apr, 1997, p.
54) is considered one of the best recreational sources of information In North
America.
All rights reserved.
Use of these articles for personal or organizational profit is specifically denied.
These articles may be used for not-for-profit diving education