Solids expand on heating because

Three uniform thin aluminium rods each of length $2 \,m$ form an equilateral triangle $PQR$ as shown in the figure. The midpoint of the rod $PQ$ is at the origin of the coordinate system. If the temperature of the system of rods increases by $50^{\circ} C$, the increase in $y$-coordinate of the centre of mass of the system of the rods is ............ $mm$. (Coefficient of volume expansion of aluminium $= 12 \sqrt{3} \times 10^{-6} \,K^{-1}$)

At a certain temperature,rod $A$ and rod $B$ of different materials have lengths $L_A$ and $L_B$ respectively. Their coefficients of linear expansion are $\alpha_A$ and $\alpha_B$ respectively. It is observed that the difference between their lengths remains constant at all temperatures. The ratio $L_A / L_B$ is given by

$A$ copper rod of length $l_1$ and an iron rod of length $l_2$ are always maintained at the same common temperature $T$. If the difference $(l_2 - l_1)$ is $15\,cm$ and is independent of the value of $T$,then $l_1$ and $l_2$ have the values (given the linear coefficients of expansion for copper and iron are $\alpha_c = 2.0 \times 10^{-6}\,^{\circ}C^{-1}$ and $\alpha_i = 1.0 \times 10^{-6}\,^{\circ}C^{-1}$ respectively).

$A$ solid rectangular sheet has two different coefficients of linear expansion $\alpha_{1}$ and $\alpha_{2}$ along its length and breadth respectively. The coefficient of surface expansion is (for $\alpha_{1} \Delta t \ll 1, \alpha_{2} \Delta t \ll 1$):

Two different wires having lengths $L_{1}$ and $L_{2}$ and respective temperature coefficients of linear expansion $\alpha_{1}$ and $\alpha_{2}$ are joined end-to-end. Then the effective temperature coefficient of linear expansion is