The coefficients of linear expansion of brass and steel are ${\alpha _1}$ and ${\alpha _2}$ respectively. If we take a brass rod of length ${l_1}$ and a steel rod of length ${l_2}$ at $0^{\circ}C$,their difference in length $({l_2} - {l_1})$ will remain the same at any temperature if:

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 .......... $cm$.

$A$ student records the initial length $l$,change in temperature $\Delta T$ and change in length $\Delta l$ of a rod as follows:

$S$.No.	$l (m)$	$\Delta T (^{\circ}C)$	$\Delta l (m)$
$(1)$	$2$	$10$	$4 \times 10^{-4}$
$(2)$	$1$	$10$	$4 \times 10^{-4}$
$(3)$	$2$	$20$	$2 \times 10^{-4}$
$(4)$	$3$	$10$	$6 \times 10^{-4}$

If the first observation is correct,what can you say about observations $(2)$,$(3)$ and $(4)$?

$A$ large steel wheel is to be fitted onto a shaft of the same material. At $27^{\circ}C$,the outer diameter of the shaft is $8.70\;cm$ and the diameter of the central hole in the wheel is $8.69\;cm$. The shaft is cooled using 'dry ice'. At what temperature (in $^{\circ}C$) of the shaft does the wheel slip on the shaft? Assume the coefficient of linear expansion of the steel to be constant over the required temperature range: $\alpha_{steel} = 1.20 \times 10^{-5}\;K^{-1}$.

$A$ 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 $\alpha_{steel} = 1.2 \times 10^{-5} \,^oC^{-1}$.

$A$ blacksmith fixes a circular iron frame on the wooden wheel of a bullock cart. The diameters of the wooden wheel and the circular iron frame are $5.012 \ m$ and $5 \ m$ respectively at $27^{\circ} C$. The temperature (in $^{\circ} C$) to which the iron ring must be heated so as to fit the wooden wheel is (Coefficient of linear expansion of iron $= 1.2 \times 10^{-5} \ ^{\circ} C^{-1}$).