$A$ steel tape gives correct measurement at $20^{\circ} C$. $A$ piece of wood is being measured with the steel tape at $0^{\circ} C$. The reading is $25 \ cm$ on the tape. The real length of the given piece of wood must be:

$A$ composite rod made up of two rods $AB$ and $BC$ are joined at $B$. The rods are of equal length at room temperature and have equal masses. The coefficient of linear expansion $\alpha$ of $AB$ is more than that of $BC$. The composite rod is suspended horizontally by means of a thread at $B$. When the rod is heated:

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:

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

What must be the lengths of steel and copper rods at $0^o C$ for the difference in their lengths to be $10\,cm$ at any common temperature? $(\alpha_{steel}=1.2 \times 10^{-5} \;^o C^{-1})$ and $(\alpha_{copper} = 1.8 \times 10^{-5} \;^o C^{-1})$

The coefficient of volume expansion of a material is $5 \times 10^{-4} {^{\circ}C}^{-1}$. The fractional change in its density for a $40^{\circ}C$ rise in temperature is nearly.