Question

A thin rod, length `L_(0)` at `0^(@)C` and coefficient of linear expansion `alpha` has its two ends mintained at temperatures `theta_(1)` and `theta_(2)` respectively Find its new length .

Answer 1

Consider the diagram

Let temperature varies linearly in the rod from its one end to other end. Let `theta` be the temperature of the mid-point of the rod. At steady state,
Rate of flow of heat,
`((dQ)/(dt)) = (KA(theta_(1)-theta))/((L_(0)//2)) = (KA(theta-theta_(2)))/((L_(0)//2))`
where, K is coefficient of thermal conductivity of the rod.
or `rArr " " theta_(1)-theta = theta - theta_(2)`
or `rArr " " theta =(theta_(1)+theta_(2))/(2)`
Using relation, `L = L_(0) (1+ alpha theta)`
or `L = L_(0) [1+alpha ((theta_(1)+theta_(2))/(2))]`