Question

Suppose a mountain at location A is in isostatic equilibrium with a column at location B, which is at sea-level, as shown in the figure. The height of the mountain is 4 km and the thickness of the crust at B is 1 km. Given that the densities of crust and mantle are 2700 kg/m\(^3\) and 3300 kg/m\(^3\), respectively, the thickness of the mountain root (r1) is km. (Answer in integer)

 

Hint:

The concept of isostasy often involves balancing the weight of elevated topography with the buoyant force provided by a deeper crustal root. The density difference between the crust and the mantle is the key factor in determining the thickness of the root.

Answer 1

Step 3: Buoyant Force by Root
Given:

• Density of mantle: \( \rho_m = 3300 \, \text{kg/m}^3 \)

The crustal root provides a buoyant force by displacing mantle material: \[ \text{Buoyant force} = r_1 \times (\rho_m - \rho_c) = r_1 \times (3300 - 2700) = r_1 \times 600 \]

 

Step 4: Equating Mass and Buoyant Force
For isostatic equilibrium: \[ h_{\text{mountain}} \times \rho_c = r_1 \times (\rho_m - \rho_c) \] \[ 4 \times 2700 = r_1 \times 600 \quad \Rightarrow \quad 10800 = r_1 \times 600 \]

 

Step 5: Solving for \( r_1 \)
\[ r_1 = \frac{10800}{600} = 18 \, \text{km} \]