When subjected to a force of compression, the length of a bone decreases by `2.7xx10^(-5)m`. When this same bone is subjected to a tensile force of the same magnitude, by how much does it stretch ?

Options

A. `4.6xx10^(-5)m`
B. `1.6xx10^(-5)m`
C. `3.1xx10^(-5)m`
D. `2.0xx10^(-5)m`

Answer

To solve the problem, we need to find out how much the bone stretches when subjected to a tensile force, given that it compresses by a certain amount under a compressive force. We can use the relationship between compression and tension through Young's modulus.

### Step-by-Step Solution:

1. **Identify the Given Data:**
- Compression of the bone: \( \Delta L_c = 2.7 \times 10^{-5} \, \text{m} \)
- Young's modulus of compression for the bone: \( Y_c = 9.4 \times 10^9 \, \text{N/m}^2 \)
- Young's modulus of tensile strength for the bone: \( Y_t = 1.6 \times 10^{10} \, \text{N/m}^2 \)

2. **Understand the Relationship:**
- The amount of stretch (tensile deformation) \( \Delta L_t \) when the same force is applied in tension can be calculated using the ratio of Young's moduli:
\[ \Delta L_t = \Delta L_c \cdot \frac{Y_c}{Y_t} \]

3. **Substitute the Values:**
- Plug in the values we have:
\[ \Delta L_t = (2.7 \times 10^{-5} \, \text{m}) \cdot \frac{9.4 \times 10^9 \, \text{N/m}^2}{1.6 \times 10^{10} \, \text{N/m}^2} \]

4. **Calculate the Ratio of Young's Moduli:**
- Calculate \( \frac{Y_c}{Y_t} \):
\[ \frac{Y_c}{Y_t} = \frac{9.4 \times 10^9}{1.6 \times 10^{10}} = 0.5875 \]

5. **Calculate the Stretch:**
- Now, substitute this ratio back into the equation:
\[ \Delta L_t = (2.7 \times 10^{-5} \, \text{m}) \cdot 0.5875 \]
- Perform the multiplication:
\[ \Delta L_t = 1.58775 \times 10^{-5} \, \text{m} \approx 1.6 \times 10^{-5} \, \text{m} \]

6. **Final Answer:**
- The amount by which the bone stretches when subjected to a tensile force of the same magnitude is:
\[ \Delta L_t \approx 1.6 \times 10^{-5} \, \text{m} \]

### Conclusion:
The bone stretches by approximately \( 1.6 \times 10^{-5} \, \text{m} \) when subjected to a tensile force of the same magnitude.