Measurements Are Often Full of Lies—and That’s OK

Illustration: Rhett Allain

The symbol g represents Earth’s gravitational field, which on Earth’s surface has a pretty much constant value of 9.8 newtons/kilogram. Since that’s where almost everything we do happens, weight and mass are directly proportional. If you double the mass, you double the weight. They aren’t the same, but one maps to the other.

Of course, that all goes out the window if humans become an interplanetary species. But for now, it’s easy to see why the two quantities get conflated.

Check Your Instruments

That’s not all. As my student’s perceptive question revealed, many of the instruments we use in the lab to measure mass actually do measure weight. In fact, they don’t even measure true weight, they measure “apparent weight.” Apparent weight is essentially the force that the scale pushes UP on an object. This force can change if the floor is accelerating up or down. The apparent weight is also how you “feel” and why astronauts appear to be weightless in orbit (hint, they aren’t weightless).

Here is a quick test to see how your instrument works. Take a simple swing arm balance—you can picture the ones they use in old Western movies to measure gold nuggets—and a spring scale to measure a particular mass.

A swing arm balance works by placing a known reference mass on one side such that it exactly balances the thing you’re trying to measure. A spring scale uses a hook suspended from a calibrated spring. The mass is then determined by looking at how far the spring is stretched.

Now take these two measuring devices and put them in an elevator. When the elevator shoots upward, the spring scale will have to pull on the mass with a force MORE than the gravitational force, and thus it will give a larger reading. For the arm balance, nothing will happen. It’s not that the accelerating elevator has no effect on the mass in question. But it has exactly the same effect on the counterweight of equal mass.

So, if an instrument still gives the same value for mass in an accelerating frame of reference, it is actually measuring mass. Otherwise, it measures apparent weight (like the spring scale). So, the digital balance and the spring scale on a stationary frame actually measure weight, not mass.

Other instruments that really do measure mass are the triple beam balance and the inertial balance, shown below. In this one, the frequency of oscillation depends on the mass, so you can use that to measure mass indirectly.

Video: Rhett Allain

Measure for Measure

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