$A$ cylindrical metal rod of length $L_0$ is shaped into a ring with a small gap $x$ as shown. On heating the system:

The coefficient of linear expansion of a crystalline substance in one direction is $2 \times 10^{-4} /{ }^{\circ} C$ and in every direction perpendicular to it is $3 \times 10^{-4} /{ }^{\circ} C$. The coefficient of cubical expansion of the crystal is equal to ........... $\times 10^{-4} /{ }^{\circ} C$.

$A$ 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?

$A$ 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 $= 1.2 \times 10^{-5} \ ^{\circ} C^{-1}$).

$A$ silver rod of $100 \; cm$ at $0^{\circ} C$ is heated to $100^{\circ} C$. Its length increases by $0.19 \; cm$. The coefficient of cubical expansion of the silver rod is:

The area of a circular copper coin increases by $0.4 \%$ when its temperature is raised by $100^{\circ} C$. The coefficient of linear expansion of the coin is: