As shown in the figure,an equilateral triangle $ABC$ is formed by joining three rods of equal lengths $l$. $D$ is the midpoint of $AB$. The coefficient of linear expansion of the material of rod $AB$ is $\alpha_1$ and that of rods $AC$ and $BC$ is $\alpha_2$. If the length $DC$ remains constant for small changes in temperature,then:

$A$ 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?

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 $= 12 \sqrt{3} \times 10^{-6} \,K^{-1}$)

$A$ uniform metal rod of length $L$ and mass $M$ is rotating about an axis passing through one of the ends and perpendicular to the rod with angular speed $\omega$. If the temperature increases by $t^{\circ} C$,then the change in its angular velocity is proportional to which of the following? (Coefficient of linear expansion of rod $= \alpha$)

$A$ steel tape of $300 \,cm$ length is graduated at $27^{\circ} C$. The length of a steel rod measured using the tape is found to be $110 \,cm$ at $50^{\circ} C$. The actual length of the steel rod at $50^{\circ} C$ is $(\alpha_{\text{steel}} = 1.2 \times 10^{-5} \,K^{-1})$. (in $\,cm$)

Show that the coefficient of area expansion,$(\Delta A / A) / \Delta T,$ of a rectangular sheet of a solid is twice its linear expansivity,$\alpha_{l}$.