Properties Of Matter Ques 14
Fill in the Blank
- A piece of metal floats on mercury. The coefficients of volume expansion of the metal and mercury are $\gamma_1$ and $\gamma_2$ respectively. If the temperatures of both mercury and the metal are increased by an amount $\Delta T$, the fraction of the volume of the metal submerged in mercury changes by the factor ……..
(1991, 2M)
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Answer:
Correct Answer: 14.$\frac{1+\gamma_2 \Delta T}{1+\gamma_1 \Delta T}$
Solution:
- The condition of floating is,
$ \begin{aligned} \text { Weight } & =\text { Upthrust } \\ V \rho_1 g & =V_i \rho_2 g \\ \left(\rho_1\right. & =\text { density of metal, } \rho_2=\text { density of mercury) } \end{aligned} $
$ \begin{aligned} & \therefore \quad \frac{V_i}{V}=\frac{\rho_1}{\rho_2} \\ & =\text { fraction of volume of metal submerged in mercury } \\ & =x \text { (say) } \\ & \end{aligned} $
Now, when the temperature is increased by $\Delta T$.
$ \begin{aligned} & \rho_1^{\prime} & =\frac{\rho_1}{1+\gamma_1 \Delta T} \text { and } \rho_2^{\prime}=\frac{\rho_2}{1+\gamma_2 \Delta T} \\ \therefore & x^{\prime} & =\left(\frac{\rho_1}{1+\gamma_1 \Delta T}\right)\left(\frac{1+\gamma_2 \Delta T}{\rho_2}\right)=\frac{\rho_1}{\rho_2}\left(\frac{1+\gamma_2 \Delta T}{1+\gamma_1 \Delta T}\right) \\ \therefore & x^{\prime} & =x\left(\frac{1+\gamma_2 \Delta T}{1+\gamma_1 \Delta T}\right) \Rightarrow \frac{x^{\prime}}{x}=\frac{1+\gamma_2 \Delta T}{1+\gamma_1 \Delta T} \end{aligned} $
BETA
