The Mechanics of Measuring My Weight

In a comment below, Don Juan de Bubba asks "Where did you find a scale that worked for you?"

The short answer is that I bought a Healthometer model 402KL Physician Balance Beam Scale from an outfit called Balkowitsch Enterprises:

http://www.balkowitsch.com/products.php?catid=17

Now for the long answer. My background includes some science and engineering, so I've given this a lot more thought than...well...than anyone needs to.

The Healthometer 402KL is the same kind of scale your doctor has. You get on and the beam at the top bounces up and the nurse adjusts the sliding weights until the pointer on the right side is floating in the center of its little metal loop.

If you followed the link, you can see that a scale like this is much more expensive than an ordinary bathroom scale. The list price is about $230, but Balkowitsch has them for $170.

So what costs so much? Or more to the point, why did I think it was worth spending $170 for this scale when I can buy a bathroom scale from Target for under $50?

First of all, this scale performs better than most bathroom scales. That's because it's a balance beam scale, not a spring scale. Let me explain:

A spring scale uses a system of levers to direct the force of your weight to compress a spring. This is useful because springs generally have a linear response to an applied force: The distance the spring compresses is directly proportional to the force on the spring. If the spring compresses 1 inch when a 100 pound person steps on the scale, it will compress 2 inches when a 200 pound person steps on. The spring is connected to the dial mechanism so that the dial turns as the spring is compressed. The markings on the dial convert this compression into weight.

Spring scales are not the best way to measure a person's weight. They tend to be sensitive to your position and movement on the scale. Most spring scales will visibly change their measurement by up to 5 or 10 pounds if you change the way your weight is distributed left-to-right or heel-to-toe. Since I only lose about 2 pounds per week, the errors are larger than the change I'm trying to see.

A digital spring scale still works on the same principles, except that the spring's compression is measured with some kind of sensor that puts out a voltage in proportion to the spring's compression. The scale's display does some simple math to calculate body weight from the voltage.

This isn't much of an improvement. The spring mechanism still jumps around because the person standing on the scale is moving. This means the voltage from the sensor is also jumping around, which would make it hard to read the digital readout. For this reason, digital scales usually "lock up" the reading rather than showing the moment-to-moment changes. However, the number they lock on is somewhat random. Step off the scale and step back on and it will probably lock up a different number.

You can compensate for this variation somewhat by weighing yourself several times and taking a middle value. This reduces the effect of random errors, but it doesn't do anything about systematic errors. That is, if I change the weighing procedure from one week to the next, it could shift the whole set of weight measurements. For example, I've had digital scales that would show weights differing by 3-5 pounds depending on which foot I stepped up with first.

What it comes down to is that spring scales don't produce repeatable and reliable measurements. If this week's measurement is lower than last weeks by a pound, is it me or is it the scale?

Balance scales operate on a different principle. The traditional balance is an equal-arm pan balance like the one carried be the Goddess of Justice in all the statues. There's a pretty good picture of one at the Michelli Scales site. Operation is simple: Put the test sample to be weighed in one pan, then place various members of a set of known calibrated weights in the other. When the scale is perfectly level, the weight of the test sample is the same as the weight of the calibrated weights.

A physician's balance beam works like this, with a couple of differences. First, it uses an unequal-length beam, longer on one side than the other, to take advantage of the physics of leverage. The turning force on each side of a beam is equal to the weight times the distance from the pivot. The weight of a 200-pound man can be therefore be balanced by a 20-pound counterweight that is 10 times further from the pivot. The physician's scale uses several such levers to divide down the weight of the person until all it takes is a couple of small weights.

Because the turning force is proportional to the distance from the pivot point, the balancing force of the counterweights can be adjusted by sliding them along the beam. When the scale is in balance, you can read the positions of the counterweights to find the amount of weight they must be balancing. It's a very useful principle.

It's also inherently more stable. With a spring scale, when the user moves, the change in force makes the spring move; but when the spring moves, it changes the amount of force it's applying. The spring and the user are both moving and changing the forces on the weighing mechanism, but not at the same time, so the mechanism wobbles back and forth a lot. With a balance scale, the user manually sets the counterweights and then lets go. Even if the user moves, the counterweight stays the same. As soon as the user stops moving, the scale is back in balance.

Also, a spring scale's mechanism has to move over the whole weight range of the scale, whereas the balance scale is constrained to move only near the balanced position. If the pointer is pegged in either the direction, the user adjusts the weights until it's free again.

So the balance scale is inherently more precise and resistant to measurement "noise." I can carefully adjust the counterweights until the pointer is in the exact center of its movement range and read my weight to within about a quarter of a pound. If I step off and then back on, the balance scale will still balance out at the exact same weight.

When it says I've lost weight, I believe it.

This brings me to the second reason for buying the Healthometer balance scale, which is that it has a large enough range for my weight. Most bathroom scales only go up to 300 pounds or maybe 330 if they're based on the metric system. My scale, like the one in my doctor's office, uses two weights: A coarse weight that fits into a series of slots to indicate 50-pound increments from 0 to 300, and a fine weight that slides smoothly to get an exact balance. The the fine weight on my scale is marked with quarter-pound increments from 0 to 50, implying precision to about 4 ounces. The real level of precision, however, depends on how careful you are in balancing the pointer in the center of its range. I think I can get the full quarter-pound precision out of it.

So if one weight goes up to 300 pounds and the other goes up to 50 pounds, how am I weighing myself, since I'm over 350 pounds? The scale comes with a counterweight that can be hung off the beam to shift the weight range up by 100 pounds, for a maximum weight of 450 pounds. Thankfully, that's more than I needed.

(I imagine that doctor's offices get the scales with the counterweights too, but I assume they've lost them, since they never go get them for me. The nurse usually just makes something up. Lately I've been telling them not to bother. When I go back, probably later this month, I'll have accurate data for the doctor.)

By the way, I found the dealer, Balkowitsch, on eBay. I was satisfied with their service. They promised delivery in two weeks and got it to me in about 4 days. The scale itself is built cleanly and solidly of strong materials and was carefully packed for shipping. In its own way, it's a fine measuring instrument.