A man grows into a giant such that his linear dimension increase by a factor of 9. Assuming that his density remains same, the stress in the leg will change by a factor of

Options

A. 81
B. `1//9`
C. 9
D. `1//81`

Answer

To solve the problem, we need to determine how the stress in the leg of a man changes when his linear dimensions increase by a factor of 9, while maintaining the same density.

### Step-by-Step Solution:

1. **Understanding Stress**:
Stress (\( \sigma \)) is defined as the force (\( F \)) acting on an area (\( A \)):
\[ \sigma = \frac{F}{A} \]

2. **Force Acting on the Body**:
The force acting on the man's leg due to gravity is the weight, which can be expressed as:
\[ F = mg \]
where \( m \) is the mass of the man and \( g \) is the acceleration due to gravity.

3. **Expressing Mass in Terms of Density**:
The mass can be expressed in terms of density (\( \rho \)) and volume (\( V \)):
\[ m = \rho V \]

4. **Volume in Terms of Linear Dimensions**:
The volume of a three-dimensional object is proportional to the cube of its linear dimension:
\[ V \propto L^3 \]
Therefore, if the linear dimension increases by a factor of 9, the new volume (\( V_2 \)) can be expressed as:
\[ V_2 = 9^3 V_1 = 729 V_1 \]

5. **Area in Terms of Linear Dimensions**:
The cross-sectional area (\( A \)) is proportional to the square of the linear dimension:
\[ A \propto L^2 \]

6. **Calculating Stress**:
Now we can express the initial and final stress:
\[ \sigma_1 = \frac{F_1}{A_1} = \frac{m_1 g}{A_1} \]
\[ \sigma_2 = \frac{F_2}{A_2} = \frac{m_2 g}{A_2} \]

7. **Relating Initial and Final States**:
Since the density remains constant, we have:
\[ m_1 = \rho V_1 \quad \text{and} \quad m_2 = \rho V_2 = \rho (729 V_1) \]
Thus, the new mass is:
\[ m_2 = 729 m_1 \]

8. **Area Change**:
The area also changes according to the square of the linear dimension:
\[ A_2 = 9^2 A_1 = 81 A_1 \]

9. **Final Stress Calculation**:
Now substituting these into the stress equations:
\[ \sigma_1 = \frac{m_1 g}{A_1} \]
\[ \sigma_2 = \frac{729 m_1 g}{81 A_1} = \frac{729}{81} \cdot \frac{m_1 g}{A_1} = 9 \sigma_1 \]

10. **Conclusion**:
The stress in the leg changes by a factor of 9.

### Final Answer:
The stress in the leg will change by a factor of **9**.