The Fugue Counterpoint by Hans Fugal

20Aug/077

Measure Your Body Density

When you try to lose weight, what you are really trying to do is lose fat.
Weighing yourself is a first approximation of your progress, but a better
indicator is your body fat percentage (%BF). Unfortunately, measuring %BF can
be expensive and/or difficult. It doesn't need to be so. All you need is a body
of water (e.g. a swimming pool), a gallon jug, and your bathroom scale.

One of the most accurate ways to measure body fat is hydrostatic weighing. You are weighed underwater and on land, and your body's density is determined. Then body fat is estimated from the measured density. This is the same basic technique that we will use, but we don't require an underwater scale or special tank.

First the how, then I'll give you the physics. Get in the pool and exhale all
the air you can, and allow yourself to sink. You will sink unless you're
particularly obese. Take note of the sensation of sinking. Then do the same
thing but with your lungs full. Take note of the sensation of floating. Now, we
want to reach the point of neutral buoyancy when your lungs are empty, where
you are neither sinking nor floating. You will be weightless under the water.
Take the gallon jug and hold it under the water, then exhale completely. If the
jug is full of air, you will probably float (unless you are quite lean, in
which case you'll need two jugs). Keep adding water and repeating until you
reach neutral buoyancy. If you sink, add air (pour out some water). If you
float, add water. Once you've found the magic amount of water, use this
equation to calculate your body density (ρ):

rho=m/(m liters/kg - v_\text{air}))

where m is your mass (what the scale tells
you), v is the volume of you and your buoy combined, and vair is the
volume of air in your buoy. If you have ¾ gallon of water in your gallon jug,
then vair is ¼ gallon. ρw=1 kg/liter for all the
precision we need.

Once you have density, you may like to estimate your body fat. The equation for that is Siri's equation, which says

This equation assumes your lungs are completely empty, which they can't be, so we need to introduce a term for the residual volume of your lungs. This is about ¼ of your total lung capacity, or ⅓ of your vital lung capacity. You can measure your vital lung capacity with a balloon or a by blowing air through a straw into an inverted container filled with water. The average residual lung capacity for an american male is 1.2 liters; mine is about 1.9 liters. So we can adjust the formula for density as follows:

rho=m/(m/rho w - v a - v_r)

If you do this experiment you will probably find that your estimated %BF is not too far from your BMI, which is a statistical tool for estimating %BF. It can be wildly inaccurate for statistical outliers (e.g. people who are actually in shape), but it's easy to calculate and a decent sanity check in this case.

Here's what's going on. We're using Archimedes' principle: the buoyant force on a submerged object is equal to the weight of the fluid displaced. When the buoyant force balances the force of gravity, we have neutral buoyancy. The buoyant force is expressed as F_b = -rho_w v g. Substitute weight for the buoyant force and solve for the volume of the body (v = vbody + vair), then substitute that into the definition of density (m/v), and you get the formula I gave you above (if you consider the mass of air and the gallon jug as negligible). I glossed over that—if you'd like me to go into more detail say so in the comments.

If you're particularly obese and don't sink when you exhale completely, then all is not lost. You just need some counterbalance. The modified equation is:

rho = m<em>b / ((m</em>b + m<em>c)/rho</em>w - v<em>a - v</em>c).

You can find the volume of your counterbalance by taking a cue from Archimedes and measuring displacement.

About accuracy: the biggest variable in this process is how much air is left in your lungs. You will find with practice that you are able to exhale more air, which will lower your %BF estimation, as if by magic. However it always overestimates and once you figure out how to completely exhale will be very consistent. Siri's equation is the next place to look—it basically takes the density of fat and the density of muscle and ignores bone mass and density, what you ate for lunch, etc. It will also almost certainly overestimate %BF. The astute reader will wonder about air compression in the milk jug. I measured this and found that when the jug is held within a foot or so from the surface, it does compress. However, the amount it compresses conveniently offsets the extra capacity of the jug (they don't pack milk spilling over the brim of the jug, after all). All in all I think it's accurate within a few percentage points for %BF, gives you an upper bound (i.e. you are free to brag about the number you get, even if it may be slightly high), and is more accurate than BMI.