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Find the Ratio of the Lengths of an Iron Rod and an Aluminium Rod for Which the Difference in the Lengths is Independent of Temperature. - Physics

Answer in Brief

Find the ratio of the lengths of an iron rod  and an aluminium rod for which the difference in the lengths is independent of temperature. Coefficients of linear expansion of iron and aluminium are 12 × 10–6 °C–1 and 23 × 10–6 °C–1 respectively.

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Solution

Let the original length of iron rod be LFe and L'​​Fe be its length when temperature is increased by ΔT.
Let the original length of aluminium rod be LAl and L'​​Al be its length when temperature is increased by ΔT. 

Coefficient of linear expansion of iron, 

\[\alpha_{Fe}\] = 12 × 10–6 °C​

\[-\] 1

Coefficient of linear expansion of aluminium, αAl = 23 × 10–6 °C

\[-\] 1

Since the difference in length is independent of temperature, the difference is always constant.

\[L '_{Fe} = L_{Fe} \left( 1 + \alpha_{Fe} \times ∆ T \right)\]

\[and\] \[  L '_{Al} = L_{Al} \left( 1 + \alpha_{Al} \times ∆ T \right)\]

\[ \Rightarrow \begin{array}\[L '_{Fe} - L '_{Al} = L_{Fe} - L_{Al} + L_{Fe} \times \alpha_{Fe} ∆ T - L_{Al} \times \alpha_{Al} \times ∆ T & - (1)\end{array}\]

\[Given: \]

\[L '_{Fe} - L '_{Al} = L_{Fe} - L_{Al} \]

\[Hence, L_{Fe} \alpha_{Fe} = L_{Al} \alpha_{Al} [using (1)]\]

\[ \Rightarrow \frac{L_{Fe}}{L_{Al}} = \frac{23}{12}\]

The ratio of the lengths of the iron to the aluminium rod is 23:12.

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APPEARS IN

HC Verma Class 11, Class 12 Concepts of Physics Vol. 2
Chapter 1 Heat and Temperature
Q 18 | Page 13
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