I gave the experiment a try, and it worked. The laws of physics still apply in my house!
I have two scales. One is a person scale, that does not read any weight until 11 pounds. The other is a kitchen scale with a maximum weight of 10 pounds. As the test subject I used a 20 pound set of hand weights plus their two-footed carrier which adds 14 ounces. The test subject has a weight of 20.875 pounds.
Placing the hand weights and their carrier on the people scale shows a weight of 21 pounds.
The people scale and kitchen scale were placed on a hard floor. A book (Nature’s Deadliest Creatures) was placed under the kitchen scale to make it level with the people scale. The weights in their carrier were placed so that one foot was on the people scale, and the other foot on the kitchen scale. The people scale showed “0.0” and the kitchen scale showed “Errrr”. So 10.5 pounds each!
After the initial setback, a followup test was performed. Nature’s Deadliest Creatures was swapped for The Food Lab, Agave, and Succulents, so that the top of the kitchen scale was elevated 3.25 inches above the top of the people scale. The weights in their carrier were then placed with one foot on each scale, and a lean toward the people scale.
This people scale then showed 12 pounds, and the kitchen scale showed 8 pounds 14.5 ounces. So 12 + 8.90625 = 20.90625 pounds. So using the two scale measuring system the known weight of the test subject was recovered.
If they use the Archimedean method to weigh the table, them having a barometer (and thermometer) handy to correct for local conditions is essential to delivering an accurate answer.
The downside is just how many lab-quality barometers and thermometers can you afford to give away on the way to a reliable, repeatable answer from your physics students
I just have to share a joke from the world of aerosol science (the science of airborne particles and their behaviors):
The governor of Kentucky wants the state’s horses to win a bigger share of races, so he assembles his three top scientists. They are, of course, a geneticist, an aerodynamicist, and an aerosol scientist.
Geneticist says “This is obviously a breeding problem. We must begin a major program to breed stronger horses.”
Aerodynamicist says “Strength isn’t everything, though. We also have to consider aerodynamic drag.”
Aerosol scientist says “Yes! Let’s begin by assuming the horse is a sphere…”
I learned this knee-slapper at a course in Aerosol Science at the University of Minnesota in about 1984. Ben Liu told it. It’s the only aerosol science joke I know. If you hadn’t guessed, aerosol scientists often assume particles are spheres (and a surprisingly large number of them actually are).
Get David Letterman to verify it floats. If it does, grab a nearby duck (or verified witch) and weigh that. Per the corollary to Belvedere’s law, they will weigh the same.
I think if it needed to be done right, you could use a couple of long 2x4’s as a framework, with the table on top and put a single scale in the middle.
I bet you could find a calculation of the measuring the four legs thing that would get you closer. Just adding them up doesn’t seem right to me.