Assuming that shear stress at the base of a mountain is equal to the force per unit area due to its weight. Calculate the maximum possible height of a mountain on the earth if breaking stress of a typical rock is `3xx10^8Nm^-3` and its density `3xx10^-3kgm^-3`. (Take `g=10ms^-2`)

Options

A. 4 Km
B. 8 Km
C. 10 Km
D. 16 Km

Explanation

To calculate the maximum possible height of a mountain on Earth, we will use the relationship between shear stress, force, area, density, height, and gravitational acceleration.

### Step-by-Step Solution:

1. **Understand the relationship between shear stress and weight**:
Shear stress (τ) at the base of the mountain is defined as the force (F) per unit area (A). The force due to the weight of the mountain can be expressed as:
\[ F = \text{Weight} = \text{Density} \times \text{Volume} \times g \]
The volume of the mountain can be expressed as:
\[ \text{Volume} = \text{Area} \times \text{Height} = A \times h \]
Therefore, the weight of the mountain becomes:
\[ F = \rho \times (A \times h) \times g \]

2. **Substituting into the shear stress equation**:
The shear stress can be expressed as:
\[ \tau = \frac{F}{A} = \frac{\rho \times (A \times h) \times g}{A} \]
Simplifying this gives:
\[ \tau = \rho \times h \times g \]

3. **Setting the shear stress equal to breaking stress**:
According to the problem, the maximum shear stress should not exceed the breaking stress of the rock. Therefore, we have:
\[ \rho \times h \times g \leq \text{Breaking Stress} \]
Substituting the values given:
\[ \rho = 3 \times 10^3 \, \text{kg/m}^3, \quad g = 10 \, \text{m/s}^2, \quad \text{Breaking Stress} = 3 \times 10^8 \, \text{N/m}^2 \]
This leads to:
\[ 3 \times 10^3 \times h \times 10 \leq 3 \times 10^8 \]

4. **Solving for height (h)**:
Rearranging the inequality gives:
\[ 30,000h \leq 3 \times 10^8 \]
Dividing both sides by 30,000:
\[ h \leq \frac{3 \times 10^8}{30,000} \]
Simplifying this:
\[ h \leq 10,000 \, \text{m} \]

5. **Final result**:
Thus, the maximum possible height of the mountain is:
\[ h \leq 10^4 \, \text{m} = 10 \, \text{km} \]

### Conclusion:
The maximum possible height of a mountain on Earth, given the breaking stress of the rock and its density, is **10 kilometers**.