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.

Explanation

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.