The length of a ceramic block is being measured by a metallic ruler, which gives readings $L_{1}$ at temperature $T_{1}$ , and $L_{2}$ at a temperature $T_{2}$ . It is known that the ruler gives a correct reading at temperature $T$ , and that the ceramic block does not significantly expand for the temperature differences in question. What is the coefficient of linear expansion of the ruler?

$(\frac{L_{2} - L_{1}}{T (L_{2} - L_{1}) + L_{1} T_{1} - L_{2} T_{2}})$

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$(\frac{L_{1} - L_{2}}{T (L_{1} - L_{2}) + L_{1} T_{1} - L_{2} T_{2}})$

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$(\frac{L_{1} - L_{2}}{T (L_{1} - L_{2}) + \frac{T_{1} T_{2}}{T} L_{1}})$

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(\frac{L_{2} - L_{1}}{T (L_{2} - L_{1}) + L_{2} T_{2}})

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Solution

The correct option is A
$(\frac{L_{2} - L_{1}}{T (L_{2} - L_{1}) + L_{1} T_{1} - L_{2} T_{2}})$

It is given in the question, that the scale gives a correct reading at temperature $T$ . Calling the correct length $L$ , we can write the following relations.

$L_{1} [1 + α (T_{1} - T)]$ = $L_{2} [1 + α (T_{2} - T)]$

$\Rightarrow$ = $α [L_{1} (T_{1} - T) - L_{2} (T_{2} - T)]$ = $L_{2} - L_{1}$

$\Rightarrow$ $α$ = $[\frac{L_{2} - L_{1}}{T (L_{2} - L_{1}) + L_{1} T_{1} - L_{2} T_{2}}]$

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The length of a ceramic block is being measured by a metallic ruler, which gives readings $L_{1}$ at temperature $T_{1}$ , and $L_{2}$ at a temperature $T_{2}$ . It is known that the ruler gives a correct reading at temperature $T$ , and that the ceramic block does not significantly expand for the temperature differences in question. What is the coefficient of linear expansion of the ruler?

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