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

A thin rod having length L_(0) at 0^(@)C and coefficient of linear exp

1.	A thin rod having length L_(0) at 0^(@)C and coefficient of linear expansion alpha has its two ends maintained at temperature theta_(1) and theta_(2) respectively. Find its new length.
Answer» Solution :The temperature in rod changed by going linearly from its one end to another end and temperature at midpoint is `theta`. In thermal steady state heat current `=(dQ)/(dt)=` constant. `:.KA(theta_(1)-theta)/((L_(0//2)))=(KA(theta-theta_(2)))/((L_(0//2)))` where K is thermal conductivity, `:.theta_(1)-theta=theta-theta_(2)` `:.theta_(1)+theta_(2)=2THETA` `:.theta=(theta_(1)+theta_(2))/(2)` temperature of midpoint Now, its length INCREASES with increase in temperature, `:.L=L_(0)(1+alpha theta)` `:.L=L_(0)[1xxalpha((theta_(1)+theta_(2))/(2))]`which is new length.