A) \[\frac{{{f}_{1}}-{{f}_{2}}}{{{f}_{2}}{{t}_{1}}-{{f}_{1}}{{t}_{2}}}\]
B) \[\frac{{{f}_{1}}-{{f}_{2}}}{{{f}_{1}}{{t}_{1}}-{{f}_{2}}{{t}_{2}}}\]
C) \[\frac{{{f}_{1}}+{{f}_{2}}}{{{f}_{2}}{{t}_{1}}-{{f}_{1}}{{t}_{2}}}\]
D) \[\frac{{{f}_{1}}+{{f}_{2}}}{{{f}_{1}}{{t}_{1}}-{{f}_{1}}{{t}_{2}}}\]
Correct Answer: A
Solution :
[a] As with the rise in temperature, the liquid undergoes volume expansion therefore the fraction of solid submerged in liquid increases. Fraction of solid submerged at \[{{t}_{1}}{}^\circ C={{f}_{1}}=\] Volume of displaced liquid \[={{V}_{0}}(1+\gamma {{t}_{1}})\] (i) and fraction of solid submerged at \[{{t}_{2}}{}^\circ C={{f}_{2}}=\] Volume of displaced liquid \[={{V}_{0}}(1+\gamma {{t}_{2}})\] (ii) From Eqs. (i) and (ii), \[\frac{{{f}_{1}}}{{{f}_{2}}}=\frac{1+\gamma {{t}_{1}}}{1+\gamma {{t}_{2}}}\Rightarrow \gamma =\frac{{{f}_{1}}-{{f}_{2}}}{{{f}_{2}}{{t}_{1}}-{{f}_{1}}{{t}_{2}}}\]
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