Three rods of equal length $l$ are joined to form an equilateral triangle $PQR.$ $O$ is the midpoint of $PQ.$ The distance $OR$ remains the same for a small change in temperature. The coefficient of linear expansion for $PR$ and $RQ$ is the same,i.e.,$\alpha_2,$ but that for $PQ$ is $\alpha_1.$ Then:

An iron bar of length $10\, m$ is heated from $0^{\circ}C$ to $100^{\circ}C$. If the coefficient of linear thermal expansion of iron is $10 \times 10^{-6} {^{\circ}C^{-1}}$,the increase in the length of the bar is .......... $cm$.

$A$ 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 length of a steel rod is $5 \ cm$ longer than a copper rod at all temperatures. What are the lengths of the steel and copper rods? (The coefficients of linear expansion for steel and copper are $1.1 \times 10^{-5} /{ }^{\circ} C$ and $1.7 \times 10^{-5} /{ }^{\circ} C$,respectively.)

$A$ rail track made of steel having length $10\,m$ is clamped on a railway line at its two ends as shown in the figure. On a summer day,due to a rise in temperature by $20\,^oC$,it is deformed as shown in the figure. Find $x$ (displacement of the centre) if $\alpha_{steel} = 1.2 \times 10^{-5} \,^oC^{-1}$.

Two straight metallic strips each of thickness $t$ and length $\ell$ are riveted together. Their coefficients of linear expansion are $\alpha_1$ and $\alpha_2$. If they are heated through a temperature change $\Delta T$,the bimetallic strip will bend to form an arc of radius: