A metal rod of length L and cross-sectional a | vedclass

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(C) The thermal expansion of the rod is given by $\Delta L = L \alpha T$. To prevent this expansion,we must apply a compressive force $F$ such that th

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$YA \alpha T /(1+\alpha T)$

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(C) The thermal expansion of the rod is given by $\Delta L = L \alpha T$. To prevent this expansion,we must apply a compressive force $F$ such that the compressive strain equals the thermal strain. The stress developed is $\sigma = Y 	imes 	ext{strain} = Y 	imes \frac{\Delta L}{L_{final}}$. The final length of the rod after heating is $L_{final} = L(1 + \alpha T)$. Thus,the force $F = 	ext{Stress} 	imes A = Y 	imes \frac{\Delta L}{L_{final}} 	imes A$. Substituting the values,$F = Y 	imes \frac{L \alpha T}{L(1 + \alpha T)} 	imes A = \frac{YA \alpha T}{1 + \alpha T}$.

$A$ metal rod of length $L$ and cross-sectional area $A$ is heated through $T^{\circ} C$. What is the force required to prevent the expansion of the rod lengthwise? ($Y=$ Young's modulus of the material of the rod,$\alpha=$ coefficient of linear expansion of the rod.)

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