A bakelite beaker has volume capacity of $500\, cc$ at $30^{\circ} C$. When it is partially filled with $V _{ m }$ volume (at $30^{\circ}$ ) of mercury, it is found that the unfilled volume of the beaker remains constant as temperature is varied. If $\gamma_{\text {(beaker) }}=6 \times 10^{-6}{ }^{\circ} C ^{-1}$ and $\gamma_{(\text {mercury })}=1.5 \times 10^{-4}{ }^{\circ} C ^{-1},$ where $\gamma$ is the coefficient of volume expansion, then $V _{ m }($in $cc )$ is close to

A metallic rod $1\,cm$ long with a square cross-section is heated through $1^o C$. If Young’s modulus of elasticity of the metal is $E$ and the mean coefficient of linear expansion is $\alpha$ per degree Celsius, then the compressional force required to prevent the rod from expanding along its length is :(Neglect the change of cross-sectional area)

A metallic tape gives correct value at $25^{\circ} C$. A piece of wood is being measured by this metallic tape at $10^{\circ} C$. The reading is $30 \,cm$ on the tape, the real length of wooden piece must be ………. $cm$

A steel rail of length $5\,m$ and area of cross-section $40\,cm^2$ is prevented from expanding along its length while the temperature rises by $10\,^oC$. If coefficient of linear expansion and Young's modulus of steel are $1.2\times10^{-5}\, K^{-1}$ and $2\times10^{11}\, Nm^{-2}$ respectively, the force developed in the rail is approximately

The coefficient of apparent expansion of a liquid when determined using two different vessels $A$ and  $B$ are $\gamma_1$ and $\gamma_2$ respectively. If the coefficient of linear expansion of vessel  $A$  is $\alpha$, the coefficient of linear expansion of vessel $B$ is