The linear expansion coefficients of brass and steel wires are $\alpha_1$ and $\alpha_2$ respectively,and their lengths at $0^\circ C$ are $L_1$ and $L_2$. If the difference $(L_2 - L_1)$ remains constant at any temperature,then:

$A$ piece of metal has a weight of $49 \ gm$ in air and $39 \ gm$ in a liquid of density $1.2 \times 10^3 \ kg/m^3$ kept at $32^{\circ}C$. When the temperature of the liquid is raised to $42^{\circ}C$, the metal piece has a weight of $40 \ gm$. If the density of the liquid at $42^{\circ}C$ is $1.0 \times 10^3 \ kg/m^3$, then the coefficient of linear expansion of the metal is:

$A$ solid metallic cube having a total surface area of $24 \; m^{2}$ is uniformly heated. If its temperature is increased by $10 \; ^{\circ}C$,calculate the increase in volume of the cube. (Given: $\alpha = 5.0 \times 10^{-4} \; ^{\circ}C^{-1}$)

Two rods are joined between fixed supports as shown in the figure. The condition for no change in the total length of the system with an increase in temperature is given by:
($\alpha_1, \alpha_2$ = linear expansion coefficients,$A_1, A_2$ = cross-sectional areas of rods,$Y_1, Y_2$ = Young's moduli)

$A$ sphere floats on a liquid whose volume does not change with temperature. At temperatures $t_1$ and $t_2$,the fractions $f_1$ and $f_2$ of the sphere are submerged in the liquid,respectively. What is the coefficient of volume expansion of the sphere?

We would like to prepare a scale whose length does not change with temperature. It is proposed to prepare a unit scale of this type whose length remains,say $10 \ cm$. We can use a bimetallic strip made of brass and iron each of different length whose length (both components) would change in such a way that the difference between their lengths remains constant. If $\alpha_{\text{iron}} = 1.2 \times 10^{-5} \ K^{-1}$ and $\alpha_{\text{brass}} = 1.8 \times 10^{-5} \ K^{-1}$,what should be the length of each strip?