EXAMPLE 6 How High Can a Mountain Be?

Just like skyscrapers, the height of mountains is also limited, by gravity, their composition, and their shape. How tall can a mountain be?

We build a simple mathematical model of a mountain. Suppose that it is made of granite, a common material, with uniform density. Granite weighs and has a crushing strength of about 4 million .

In the interests of both realism and simplicity, we assume that the mountain is a solid cone whose width at the base is the same as its height. Let’s model Mount Everest, the tallest mountain on Earth, at about 6 mi high. The base, then, is a circle with a diameter (distance across) of 6 mi. The radius (half the diameter) is 3 mi (Figure 18.7). Because we took round numbers (6 mi) for the height and width, we record as significant only the first digit or two of the results of the calculations.

What does the model Everest weigh? The relevant formula is , or

We already know the density of granite , so to find the weight, we need the formula for the volume of a cone of radius and height :

Using a radius of 3 mi and a height of 6 mi, we find that the model Everest has a volume of about .

751

To find the weight of of granite, we need to convert units because the density that we know is in pounds per cubic foot . Let’s convert to units of feet as follows:

Thus,

So we have

Now that we know the mountain weighs 1.4 quadrillion pounds, we want to find the pressure on the base of the cone and compare it with the crushing strength of granite. (Everest is standing, so if our model is any good, that pressure will be below the crushing strength.)

Physics tells us that the weight of the mountain is spread evenly over the base of the cone (though we are oversimplifying the geology underlying mountains). Because

we need to calculate the area of the base of the cone. The shape is a circle, and the familiar formula

gives for a radius mi an area of .

Once again, we need to convert units to express the pressure in pounds per square foot, the units in which the crushing strength is expressed. We get

Then

This number is about half the crushing strength of granite, . For a mountain to come close to the limitation of the crushing strength of granite, it would have to be only about 10 mi high, not quite twice as high as Everest. Other physical considerations suggest a maximum height of at most 15 mi. The fact that no current mountains are that high may be a consequence of the Earth’s high amount of volcanic activity and the structural deformation of the Earth’s crust. For a mountain that is too big, the rock below it would flow and the mountain would sink.