Question

In the given diagram, a 6 km high plateau is supported by a crustal root of thickness \(r\). The system is in isostatic equilibrium as per Airy’s hypothesis of isostasy. Densities of the crust and the mantle are 2800 kg/m\(^3\) and 3100 kg/m\(^3\), respectively. Assuming the acceleration due to gravity to be same throughout the region, the thickness of the root (\(r\)) is _________ km. 

Hint:

In isostatic equilibrium, the thickness of the crustal root can be calculated using Airy’s hypothesis, which ensures that the weight of the crust is balanced by the buoyant force from the mantle.

Answer 1

Correct Answer: 56

Step 1: Understanding Airy’s Hypothesis of Isostasy.
Airy’s hypothesis of isostasy states that the crust “floats” on the denser mantle, and the height of the crust is in equilibrium with the thickness of the root. The balance of forces is maintained such that the weight of the crust and the root is balanced by the buoyant force exerted by the mantle. Mathematically, this is expressed as: \[ \text{Weight of the crust} = \text{Buoyant force from the mantle}. \] Step 2: Applying the Isostatic Equilibrium Condition.
The weight of the plateau (crust) and the crustal root must balance with the weight of the mantle displaced by the root. Using the equation: \[ (\text{Crust density}) \times (\text{Height of plateau}) = (\text{Mantle density}) \times (\text{Depth of compensation}). \] Substituting the given values: \[ 2800 \times 6 = 3100 \times r, \] \[ 16800 = 3100 \times r, \] \[ r = \frac{16800}{3100} = 5.42 \, \text{km}. \] Step 3: Conclusion.
The thickness of the root (\(r\)) is approximately 15 km.