Problem 11. The ideas of buoyancy can also be applied to the Earth itself. Where there are mountains, the crustal material which forms the mountain also extends down into the

mantle. Imagine adding material to the top of the mountain. The extra weight would force the crust to sink down even further. (This is exactly the same as adding more weight to a piece of wood floating on a lake.The wood will sink slightly and float lower in the water.) In this case, the crustal material floats on the mantle. Assume that perust = 2.8 and Pmantle = 3.3 . Notice that the crust is less dense THAN than the mantle so that it will naturally float. Pascal's principle applies here. In the diagram, the mass in a column of fixed area above point A must be the same as the mass in a column of the same area above point B. Equivalently, the pressures at A and B must be the same. Note that the thickness of the crust (D) will not be needed for either part of the question. It will always cancel out of the equations. (a) If the mountain is h = 2 km high, how far does it extend down into the mantle (H)? The answer will be much larger than 2 km. This is qualitatively identical to the fact that most of an iceberg lies below the water surface. (b) If 5 km of material were eroded from the“top" of the mountain, how high would it be? Note that as the mountain erodes, it rises up into the air since it becomes more buoyant. In the end the mountain will still be above sea level! You can assume that as the crust rises there is an infinite amount of mantle which can “flow" in to replace it.

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