Question

The area of cross - section of rod is given by $A=a_0(1+\alpha x)$ where $A_0\&\alpha$ are constant and x is the distance from one end If thermal conductivity of material is K. What is the thermal resistance of the rod if its length is

• A. $K{A}_0\alpha to(1+a{\ell }_0)$
• B. $\frac{1}{K{a}_0\alpha }\ell n(1+\alpha {\ell }_0)$
• C. $\frac{\alpha }{K{A}_0}\ell n(1+a{\ell }_0)$
• D. $\frac{K{A}_0}{\alpha }\ell n(1+\alpha {\ell }_0)$

Answer 1

The correct option is B $\frac{1}{K{a}_0\alpha }\ell n(1+\alpha {\ell }_0)$

$\begin{array}{c}A=a_0\left(1+\alpha x\right)\\ dR=\frac{dx}{KA}\\ =\frac{dx}{K{\alpha }_0\left(1+\alpha x\right)}\\ =R=\frac{1}{K{a}_0}{\int }_0^{l_0}\frac{dx}{1+\alpha x}\\ =\frac{1}{\alpha K{a}_0}\ln \left(1+\alpha l_0\right)\\ =\frac{1}{K{a}_0\alpha }\ln \left(1+\alpha l_0\right)\\ Hence,\\ optionBiscorrectanswer.\end{array}$