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  • A thin rod having length Lo at 0^oC and coefficient of linear expansion α has its two ends maintained at temperatures θ _1 and θ _2, respectively. Find its new length.

A thin rod having length Lo at 0^{\circ}C and coefficient of linear expansion α has its two ends maintained at temperatures \theta _{1}and \theta _{2}, respectively. Find its new length.

Answers (1)

We know that from one end to another end the temperature of the rod varies from \theta _{1} to \theta _{2}.

The mean temperature of rod= \frac{\theta _{1} + \theta _{2}}{2 ^{\circ}C}

 

Here, the rate of flow of heat from A to C to B is equal

Thus, \theta _{1} > \theta >\theta _{2}

Thus, \frac{d\theta}{dt} = \frac{KA (\theta _{1} - \theta )}{\frac{L_{0}}{2}}

                        = \frac{KA (\theta - \theta _{2})}{ {\frac{L_{0}}{2}}}

Here K is the coefficient of thermal conductivity.

Therefore, \theta _{1} - \theta = \theta - \theta _{2}

\theta = \theta _{1} + \frac{\theta _{2}}{2}

Thus, L = L_{0} (1+\alpha \theta )

          = L_{0} \left [1 + \alpha \left (\frac{ \theta _{1} +\theta _{2}}{2} \right ) \right ]

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