To keep constant time,watches are fitted with a balance wheel made of:

$A$ steel rod at $25^{\circ} C$ is observed to be $1 \ m$ long when measured by another metal scale which is correct at $0^{\circ} C$. The exact length of the steel rod at $0^{\circ} C$ is $(\alpha_{\text{steel}} = 12 \times 10^{-6} \ ^{\circ} C^{-1}, \alpha_{\text{metal}} = 20 \times 10^{-6} \ ^{\circ} C^{-1})$. (in $m$)

Two rods,one of aluminium and the other of steel,having initial lengths $L_1$ and $L_2$ are connected together to form a single rod of length $(L_1+L_2)$. The coefficients of linear expansion of aluminium and steel are $\alpha_1$ and $\alpha_2$ respectively. If the length of each rod increases by the same amount,when their temperatures are raised by $t^{\circ}C$,then the ratio $\frac{L_1}{L_1+L_2}$ will be

Two rods of same area of cross-section have lengths $L$ and $2 \,L$ and coefficients of linear expansion $2 \alpha$ and $\alpha$ respectively. If they are welded to form a composite rod of length $3 \,L$, then the coefficient of linear expansion of the composite rod is

$A$ 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 the difference between their lengths remains constant. If the length of brass is $40\,cm$,what is the length of iron in $cm$?
($\alpha_{\text{iron}} = 1.2 \times 10^{-5} K^{-1}$ and $\alpha_{\text{brass}} = 1.8 \times 10^{-5} K^{-1}$)

The area of cross-section of a railway track is $0.01\, m^2$. The temperature variation is $10^{\circ}C$. The coefficient of linear expansion of the material of the track is $10^{-5} /^{\circ}C$. The energy stored per meter in the track is ...... $J/m$. (Young's modulus of the material of the track is $10^{11}\, Nm^{-2}$)