Question Video: Comparing Changes in Altitude Indicated on an Atmospheric Pressure Gauge Physics

Video Transcript

The apparatus shown in the diagram is used to measure atmospheric pressure. The apparatus is taken to a higher altitude above sea level, and the height of the mercury column is reduced by the length 𝑎. The apparatus is taken further above sea level, and the column height is reduced by the length 𝑏, where 𝑏 equals 𝑎. The arrow labeled Δℎ represents the increase in altitude of the apparatus that produces the initial reduction in the column height. Which of the three arrows shown most correctly represents the increase in the altitude of the apparatus that produces the second reduction in column height? The temperature of the apparatus and of the air are kept constant at all the altitudes at which the column height is measured. (A) Arrow I, (B) arrow II, (C) arrow III.

At the bottom of our diagram, we see these three arrows alongside a fourth arrow labeled Δℎ. The length of the arrow labeled Δℎ represents a change in altitude of our measurement apparatus. This measurement apparatus is filled with mercury, and it’s used to measure atmospheric pressure. The height of mercury in the column indicates that pressure, with the higher the column, the greater the atmospheric pressure. If we start out with this apparatus at sea level, the height of the mercury column reaches here, as shown in this first sketch. But then, if we take the apparatus and elevate it some height Δℎ, as a result of that change in elevation, the atmospheric pressure decreases as indicated by the decrease in the height of the mercury column.

We’re told that this change in elevation of our measurement apparatus happens twice. The first time it happens, the mercury in the column decreases by a height 𝑎. From there, the apparatus is then elevated to a yet higher elevation. That change in elevation causes the mercury in the column to go down by a distance 𝑏, where 𝑏 we’re told is equal to 𝑎. The three arrows representing our answer options, arrow I, arrow II, and arrow III, show us different possibilities for the change in elevation that our apparatus goes through for this second reduction in height of the mercury column. We can see that arrow I is the same length as Δℎ. On the other hand, arrow II is shorter than Δℎ, and arrow III is longer.

What we want to figure out then is when we elevate the apparatus the second time, do we have to elevate it just as much, less than, or more than the first elevation, where, as we’ve seen, the change in height of mercury in the column is the same for each of these two elevations? The answer to this question depends on how pressure in the atmosphere varies with altitude. Clearing some space on screen, let’s say that we plot pressure in the atmosphere against elevation above sea level. Pressure at some point in the atmosphere is due to the weight of the air column above that point, bearing down on it. We expect then that at zero elevation, sea level, our pressure value will be a maximum because there the height of the air column above that point is the greatest it can be.

One way that pressure could vary with elevation then is like this. According to this line, atmospheric pressure smoothly decreases for increases in elevation. This line would accurately represent atmospheric pressure versus elevation if the density of air in the atmosphere varied smoothly. If that was the case, then we would expect that we would have to elevate our apparatus an equal amount to Δℎ to cause the same decrease in pressure as evidenced by our apparatus. So that’s one way that pressure could vary with elevation.

Another way, though, is if our pressure versus elevation curve looked like this. According to this curve, for a given amount of change in elevation in the lower atmosphere, we would see a relatively small change in atmospheric pressure, while if we were to change that much elevation but in the upper atmosphere, at a higher elevation, then the corresponding change in pressure would be relatively large. According to this model, we wouldn’t need to increase our elevation as much the second time as we did the first time in order to generate the same change in atmospheric pressure.

So those are two models for how pressure in the atmosphere may vary with elevation. If we look though at our sketch here, we see that it looks as though the density in the atmosphere increases rapidly as we get closer to sea level. On our graph, we could indicate this relationship by this pink curve. According to this model, as we move through the top of the atmosphere, pressure doesn’t change very much. But then if we move an equal distance through the lower atmosphere, pressure changes a lot. This happens, as we’ve seen, because air density increases so rapidly as we approach ground level.

If this third model were correct, and it turns out that it is, we would need to move our apparatus through a greater change in elevation, once it’s already been elevated by some distance, in order to affect the same change in atmospheric pressure. Because arrow III is relatively longer than the arrow Δℎ, it best represents the change in elevation that will be needed for the second elevation of our apparatus. For our answer, we choose option (C).