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)

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 Eq. (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)