Question

The maximum possible height of a mountain on Earth is approximately.

• A. \( 9 \, \text{km} \)
• B. \( 10 \, \text{km} \)
• C. \( 12 \, \text{km} \)
• D. \( 8.8 \, \text{km} \)

Hint:

When calculating the height based on the elastic limit, ensure you correctly substitute the given values into the formula for material strength.

Answer 1

The Correct Option is B

To calculate the maximum height of the mountain, we can use the formula for the elastic limit of a material: \[ \sigma = \rho g h \] where: - \( \sigma \) is the elastic limit, - \( \rho \) is the density of the material, - \( g \) is the acceleration due to gravity, - \( h \) is the height. We are given: - \( \sigma = 30 \times 10^7 \, \text{N/m}^2 \), - \( \rho = 3 \times 10^3 \, \text{kg/m}^3 \), - \( g = 10 \, \text{m/s}^2 \). Substitute these values into the formula: \[ 30 \times 10^7 = 3 \times 10^3 \times 10 \times h \] Solving for \( h \): \[ h = \frac{30 \times 10^7}{3 \times 10^3 \times 10} = \frac{30 \times 10^7}{3 \times 10^4} = 10 \, \text{km} \] Therefore, the maximum possible height of a mountain is \( 10 \, \text{km} \).

Answer 2

Step 1: Concept behind maximum mountain height.
The maximum height of a mountain is limited by the strength of the rocks it is made of. At some point, if a mountain becomes too tall, the pressure at its base due to the weight of the rock column exceeds the rock’s ability to withstand stress, causing it to deform or collapse.

Step 2: Balance of forces.
At the base of a mountain, the pressure exerted by the column of rock is given by:
\[ P = \rho g h \] Where:
- \( \rho \) is the density of the rock (~\( 3000 \, \text{kg/m}^3 \))
- \( g \) is the acceleration due to gravity (~\( 9.8 \, \text{m/s}^2 \))
- \( h \) is the height of the mountain

The maximum stress that rock can bear is approximately \( 3 \times 10^8 \, \text{Pa} \).

Solving:
\[ h = \frac{P}{\rho g} = \frac{3 \times 10^8}{3000 \times 9.8} \approx 10,204 \, \text{m} \approx 10 \, \text{km} \]
Final Answer: \( \boxed{10 \, \text{km}} \)