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Step-by-step Solution
Step 1: List the known quantities
• Young’s modulus of the steel rod, Y = 2 \times 10^{11}\,\text{N/m}^2
• Coefficient of thermal expansion, \alpha = 1 \times 10^{-5}\,^\circ \mathrm{C}^{-1}
• Initial length of the rod, l = 4\,\text{m}
• Cross-sectional area, A = 10\,\text{cm}^2 = 10 \times 10^{-4}\,\text{m}^2 (since 1\,\text{cm}^2 = 10^{-4}\,\text{m}^2 )
• Temperature change, \Delta T = (400 - 0)\,^\circ \mathrm{C} = 400\,^\circ \mathrm{C}
Step 2: Recall the formula for thermal stress
When the rod is not allowed to expand, a thermal strain is developed, equal to \alpha\,\Delta T .
The stress that develops (assuming no expansion) can be found by using Hooke’s law in terms of Young’s modulus:
\text{Stress} = Y \times (\text{Thermal strain}) = Y \,\alpha\,\Delta T.
Step 3: Compute the force (tension) in the rod
Stress is force per unit area, so
\text{Stress} = \frac{F}{A}.
Hence the force F can be written as:
F = (Y \,\alpha\,\Delta T) \times A.
Substitute the values:
F = \Bigl(2 \times 10^{11}\Bigr)\times \Bigl(1 \times 10^{-5}\Bigr)\times 400 \times \Bigl(10 \times 10^{-4}\Bigr).
Let’s simplify step-by-step:
2 \times 10^{11} \times 1 \times 10^{-5} = 2 \times 10^{6}.
Multiply by 400: 2 \times 10^{6} \times 400 = 8 \times 10^{8}.
Multiply by area 10 \times 10^{-4} = 10^{-3}.
So, 8 \times 10^{8} \times 10^{-3} = 8 \times 10^{5}\,\text{N}.
Therefore, F = 8 \times 10^{5}\,\text{N}.
Step 4: Identify the value of x
We have F = x \times 10^5\,\text{N}. Since F = 8 \times 10^5\,\text{N} , we compare to find:
x = 8.
Final Answer
The value of x is 8.
