General Physics Ques 1
- Consider an expanding sphere of instantaneous radius $R$ whose total mass remains constant. The expansion is such that the instantaneous density $\rho$ remains uniform throughout the volume. The rate of fractional change in density $\left(\frac{1}{p} \frac{d \rho}{d t}\right)$ is constant. The velocity $v$ of any point of the surface of the expanding sphere is proportional to
(2017 Adv.)
(a) $R$
(b) $\frac{1}{R}$
(c) $R^3$
(d) $R^{\frac{2}{3}}$
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Answer:
Correct Answer: 1.( a )
Solution:
- $m=\frac{4 \pi R^3}{3} \times \rho$
On taking log both sides, we have
$ \ln (m)=\ln \left(\frac{4 \pi}{3}\right)+\ln (\rho)+3 \ln (R) $
On differentiating with respect to time,
$ \begin{aligned} & & & =0+\frac{1}{\rho} \frac{d \rho}{d t}+\frac{3}{R} \frac{d R}{d t} \\ \Rightarrow & & \left(\frac{d R}{d t}\right) & =-R \times \frac{1}{\rho}\left(\frac{d \rho}{d t}\right) \\ \because & & \frac{d R}{d t} & =v \\ \therefore & & v & =-R \propto \frac{1}{\rho}\left(\frac{d \rho}{d t}\right) \\ \therefore & & v & \propto R \end{aligned} $
BETA
