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

When a liquid is heated in a copper vessel, its coefficient of apparent expansion is $6 \times 10^{-6} /{ }^{\circ} C$. When the same liquid is heated in a steel vessel, its coefficient of apparent expansion is $24 \times 10^{-6} /{ }^{\circ} C$. If the coefficient of linear expansion for copper is $18 \times 10^{-6} /{ }^{\circ} C$, find the coefficient of linear expansion for steel.

The apparent coefficient of expansion of a liquid when heated in a brass vessel is $X$ and when heated in a tin vessel is $Y$. If $\alpha$ is the coefficient of linear expansion for brass, the coefficient of linear expansion of tin is ..........

The apparent coefficient of expansion of a liquid,when heated in a copper vessel is $C$ and when heated in a silver vessel is $S$. If $A$ is the linear coefficient of expansion of copper,then the linear coefficient of expansion of silver is:

Give the name of a substance that contracts with an increase in temperature.

In a vertical $U-$tube containing a liquid,the two arms are maintained at different temperatures ${t_1}$ and ${t_2}$. The liquid columns in the two arms have heights ${l_1}$ and ${l_2}$ respectively. The coefficient of volume expansion of the liquid is equal to