Question

At $0{}^\circ C$ , the densities of a cork and a liquid, in which the cork floats are ${{S}_{1}}$ and ${{S}_{2}}$ respectively. The coefficients of expansion of the material of the cork and the liquid are 100r and respectively. If the cork sinks when temperature of the liquid is t ${}^\circ C$ , then the $\left( \frac{{{P}_{2}}}{{{P}_{1}}} \right)$ is:

• A. $\frac{1+100\gamma t}{1+\gamma t}$
• B. $\frac{1+\gamma t}{1+100\,\gamma t}$
• C. $\frac{100+\gamma t}{1+\gamma t}$
• D. $\frac{1+\gamma t}{100+\gamma t}$

Answer 1

Answer: (A)

$\frac{1+100\gamma t}{1+\gamma t}$

Answer 2

Let $\rho {{}_{1}}$ is the density of liquid at $t{{\,}^{o}}C$ and $\gamma$ is the coefficient of expansion. Then, $p{{}_{2}}={{\rho }_{1}}(1-\gamma t)$ ?(i) Similarly, if $\rho {{}_{2}}$ is the density of cork having coefficient of expansion $100\gamma$ at $t{{\,}^{o}}C$ Then $\rho {{}_{2}}={{\rho }_{2}}(1-\gamma t)={{\rho }_{2}}(1-100\gamma \,t)$ ?(ii) the cork will sink k at $t{{\,}^{o}}C$ if $\rho {{}_{1}}=\rho {{}_{2}}$ or $\frac{\rho {{}_{1}}}{\rho {{}_{2}}}=1$ ?(iii) Now, putting the values from Eqs. (i) and (ii) in E (iii), we obtain $\frac{{{\rho }_{1}}(1-\gamma t)}{{{\rho }_{2}}(1-100\gamma t)}=1$ $\frac{{{\rho }_{2}}}{{{\rho }_{1}}}=\frac{1-\gamma t}{1-100\gamma t}$ or $\frac{{{\rho }_{2}}}{{{\rho }_{1}}}=\frac{{{(1-100\gamma t)}^{-1}}}{{{(1-\gamma t)}^{-1}}}$ $\frac{{{\rho }_{2}}}{{{\rho }_{1}}}=\frac{1+100\gamma t}{(1+\gamma t)}$ (using Binomial theorem)