Question

A solid cube is first floating in a liquid. The coefficient of linear expansion of the cube is $\alpha$ and the coefficient of volume expansion of the liquid is $\gamma$. On increasing the temperature of the (liquid + cube) system, the cube will sink if

• A. $\gamma =3\alpha$
• B. $\gamma >3\alpha$
• C. $\gamma <3\alpha$
• D. $\gamma =2\alpha$

Answer 1

The correct option is B $\gamma >3\alpha$

From thermal expansion of a liquid, we can say that,
${\gamma }_{real}={\gamma }_{app}+{\gamma }_{cube}$
Let the linear expansion of the cube be $\alpha$
Then, volume expansion coefficient of cube $\left({\gamma }_{cube}\right)=3\alpha .......\left(1\right)$
$\therefore {\gamma }_{real}={\gamma }_{app}+3\alpha ........\left(2\right)$
Given,
Initially cube is floating in the liquid.
$\Rightarrow$ Density of cube is less than the density of liquid i.e ${\rho }_{cube}<{\rho }_{liquid}$
Cube will sink if density of cube is greater than the density of liquid i.e ${\rho }_{cube}^{\prime }>{\rho }_{liquid}^{\prime }.....\left(3\right)$

From the formula of variation of density with temperature
${\rho }^{\prime }=\rho (1-\gamma \Delta T)$, we can write,
For cube: ${\rho }_{cube}^{\prime }={\rho }_{cube}(1-{\gamma }_{cube}\Delta T)........\left(4\right)$
For liquid : ${\rho }_{liquid}^{\prime }={\rho }_{liquid}(1-{\gamma }_{real}\Delta T)........\left(5\right)$
Applying condition $\left(3\right)$ and substituting $\left(4\right)$ and $\left(5\right)$, we get
${\rho }_{cube}(1-{\gamma }_{cube}\Delta T)>{\rho }_{liquid}(1-{\gamma }_{real}\Delta T)$
Since ${\rho }_{cube}<{\rho }_{liquid}$,
$1-{\gamma }_{cube}\Delta T>1-{\gamma }_{real}\Delta T$
$\Rightarrow {\gamma }_{real}>{\gamma }_{cube}$
i.e ${\gamma }_{real}>3\alpha$
[using (1)]
Hence, option (b) is the correct answer.