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Classical Mechanics Level 5

A long rod has one end at 0 o C 0^{o} C and another at a higher temperature.The coefficient of thermal conductivity for the rod k varies with distance x x from the lower temperature end as k = k o ( 1 + α x ) k=k_{o}(1+\alpha x) ,where k o = 1 0 2 k_{o}=10^{2} SI Unit and α = 1 m − 1 \alpha =1 m^{-1} .At what distance from the first end the temperature will be 10 0 o C 100^{o} C ?

Take area of cross-section of rod to be 1 0 − 2 d m 2 10^{-2} dm^{2} and rate of heat conduction to be 1 W 1 W

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The answer is 1.71.

Md Zuhair by Md Zuhair , 20, India

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1 solution

Prabhat Rao
Jun 6, 2018

Thermal resistance is L/KA where all the variables have general meaning. Net thermal resistance upto a length L after which the temperature is 100 can be calculated by integrating dx/(1+ax)Ak which comes out to be ln(L+1)A/k Now using the anamoly of heat conduction with electricity We get Thermal resistance = temperature diff/heat current. R =100 As R was calculated from integral. On substitution we get L as e-1 which is 1.71

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