There is same change in length when a 33000 N tensile force is applied on a steel rod of area of cross-section `10^(-3)m^(2)` . The change of temperature reuired to produce the same elongation, if the steel rod is heated , if (The modulus of elasticitay is ` 3xx10^(11) N//m^(2)` and the coefficient of linear expansion of steel is ` 1.1 xx 10^(-5) //^(@)C`).

Options

A. `20^(@)`C
B. `15^(@)C`
C. `10^(@)C`
D. `0^(@)C`

Explanation

To solve the problem, we need to find the change in temperature required to produce the same elongation in a steel rod when it is heated, given the tensile force applied, the area of cross-section, the modulus of elasticity, and the coefficient of linear expansion.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Tensile Force (F) = 33,000 N
- Area of Cross-section (A) = \(10^{-3} \, m^2\)
- Modulus of Elasticity (E) = \(3 \times 10^{11} \, N/m^2\)
- Coefficient of Linear Expansion (α) = \(1.1 \times 10^{-5} \, /^{\circ}C\)

2. **Calculate the Stress (σ):**
\[ \sigma = \frac{F}{A} \]
Substituting the values:
\[ \sigma = \frac{33,000 \, N}{10^{-3} \, m^2} = 33,000,000 \, N/m^2 = 3.3 \times 10^7 \, N/m^2 \]

3. **Relate Stress to Strain using Young's Modulus:**
Young's modulus (E) is defined as:
\[ E = \frac{\sigma}{\epsilon} \]
where ε (strain) can be expressed as:
\[ \epsilon = \frac{\Delta l}{L_0} \]
Rearranging gives:
\[ \epsilon = \frac{\sigma}{E} \]

4. **Calculate the Strain (ε):**
\[ \epsilon = \frac{3.3 \times 10^7 \, N/m^2}{3 \times 10^{11} \, N/m^2} = 1.1 \times 10^{-4} \]

5. **Relate Change in Length to Temperature Change:**
The change in length due to thermal expansion is given by:
\[ \Delta l = L_0 \alpha \Delta \theta \]
Therefore, we can express strain in terms of temperature change:
\[ \frac{\Delta l}{L_0} = \alpha \Delta \theta \]
Thus,
\[ \Delta \theta = \frac{\epsilon}{\alpha} \]

6. **Calculate the Change in Temperature (Δθ):**
Substituting the values:
\[ \Delta \theta = \frac{1.1 \times 10^{-4}}{1.1 \times 10^{-5}} = 10 \, ^{\circ}C \]

### Final Answer:
The change of temperature required to produce the same elongation is \(10 \, ^{\circ}C\).

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