Would I Be Taller in Denver?
Denver is known as the Mile-High City because its official elevation is exactly 5,280 feet above sea level. Photo by Acton Crawford on Unsplash.
As many of us prepare to go to Denver, Colorado, this fall for the 2025 Physics and Astronomy Congress, there are important questions that must be answered! One of the least important is this issue’s puzzler.
How much taller will l be in Denver?
Sometimes the way a question is posed contains information that you can use to help solve it. For this puzzler, I think it’s REALLY important we consider the specific wording of this question. Before you go any further, take a minute and guess what I’m getting at.
We can glean two pieces of information from these eight words. The first bit of information is that our geolocation matters in some way. Why is Denver special? Well, for those not overly familiar with the geography of the Western United States, Denver is pretty high up. It has an elevation of 1,609 m (or 5,280 ft for those that use the imperial system of measurement). So, solving the puzzler will likely require that piece of data.
The second piece of information is actually an assumption that I (the reader in this case) will get taller if I go to Denver. This assumption might not be true, but let’s keep it in mind as we proceed.
One of the very first things we learn to do in physics is calculate the acceleration due to gravity g. We use this to calculate the forces that masses experience as projectiles and from pulleys, weight, and even normal forces. (Pop quiz: When does the normal force not cancel the gravitational force?)
Calculating gravitational forces is common in first-year physics courses and then comes up again in all kinds of upperlevel physics courses. Another important fact we learn is that Earth’s gravitational pull changes with altitude—specifically, gravity gets weaker with altitude.
Mechanical engineers use the acceleration due to gravity g a lot as well—specifically, as it relates to the strain objects experience because of gravitational forces, like weight. It’s not hard to imagine that objects deform under stress (an applied force over an area). Strain is a measure of material deformation δl relative to length l. For example, if you stand on a chair, the chair gets a little shorter because the extra force (your weight) compresses the chair just a little bit. When NASA engineers make materials for space, they know that the material will be just a little bit bigger in space, because orbiting objects are in a nearly uniform gravitational field (they are weightless).
The ratio of compressive stress σ and axial strain ε is called Young’s modulus Y, and you can look up this value for all sorts of materials. It turns out that an object’s own weight can also deform it, usually making it smaller in the direction of the applied force.
In short, an object’s own weight compresses the object. So, if we (as people) didn’t have gravity pulling us down, we’d be a little taller! An object, say, a person, in this very specific case, experiences a total downward force of their mass times the local acceleration due to gravity. If they are standing still, they also experience a normal force pushing them upward.
Figure 1. Setting up the scenario. Image by Brad Conrad.
Now, let’s make a few assumptions before we work out this puzzler. If we can assume that cows are uniform spheres, I think it’s fair to assume that people are uniform cylinders (see Fig. 1). I think that it’s also fair to assume that when we stand still, our bones (with Young’s modulus Ybone of 1x107 kPa) are carrying most of our weight. There is cartilage (with a Young’s modulus Ycart of 1x103 kPa) and other squishier bits in between the bones, but the vast majority is bone.
So, pulling everything together (see Fig. 2), we can write out how the stress σ of someone’s weight across a normal human bone with area A divided by the axial strain ε, which is the ratio of the length δl and the height of the person l, is the Young’s modulus Y.
Figure 2. Young’s modulus helps solve for the change in height. Image by Brad Conrad.
I’ll let you solve the math on this one, but for someone at ocean level, since almost 40% of the United States population lives near a coast, I get a change in height of about 0.1056 mm.
I now ask, after fact checking my value above and considering (A) Newton’s law of universal gravitation and (B) the radius of the earth at Denver’s latitude (39.7392°) being 6369.44 km:
How much taller would you be in Denver? Hint: You don’t need anything else.
I hope to see you at the Physics and Astronomy Congress in the Mile-High City! Just remember, we could all be up to 100 microns taller, so plan accordingly. //
