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)