Ncert Exercises & Solutions

A thin rod having a length $L_{o}$ at 0°C and coefficient of linear expansion $α$ has its two ends maintained at temperatures $θ_{1} a n d θ_{2}$ respectively. Find its new length.

Hint: The length of the rod will increase due to the temperature difference at its two ends.

Consider the diagram,

Step 1: Find the median temperature of the rod.

Let the temperature varies linearly in the rod from one end to another end. Let

$θ$ be the
the temperature of the mid-point of the rod. At a steady-state,

Rate of flow of heat,

$(\frac{d Q}{d t}) = \frac{K A (θ_{1} - θ)}{(L_{0} / 2)} = \frac{K A (θ - θ_{2})}{(L_{0} / 2)}$

where K is the coefficient of thermal conductivity of the rod.

$\Rightarrow θ_{1} - θ = θ - θ_{2}$

$\Rightarrow θ = \frac{θ_{1} + θ_{2}}{2}$

Step 2: Find the final length of the rod.

Using relation,

$L = L_{0} (1 + α θ)$

$L = L_{0} [1 + α (\frac{θ_{1} + θ_{2}}{2})]$

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