Question

A unit scale is to be prepared whose length does not change with temperature and remains $20\, cm$, using a bimetallic strip made of brass and iron each of different length. The length of both components would change in such a way that difference between their lengths remains constant. If length of brass is $40 \,cm$ and length of iron will be ________$cm$.
$\left(\alpha_{\text {iron }}=1.2 \times 10^{-5} K ^{-1}\right.$ and $\left.\alpha_{\text {brass }}=1.8 \times 10^{-5} K ^{-1}\right)$.

Answer 1

$ \ell_{ B }\left(1+\alpha_{ B } \Delta T \right)-\ell_{ i }\left(1+\alpha_{ i } \Delta T \right)=\ell_{ B }-\ell_{ i }$
$ \alpha_{ B } \ell_{ B }=\ell_{ i } \alpha_{ i } $
$ 1.8 \times 10^{-5} \times 40=\ell_{ i } \times 1.2 \times 10^{-5}$
$\ell_{ i }=\frac{1.8 \times 10^{-5} \times 40}{1.2 \times 10^{-5}}=\frac{3 \times 40}{2}=60 $
$ \ell_{ i }=60\, cm$