1. Tardigrade
  2. Question
  3. Physics
  4. Consider an expanding sphere of instantaneous radius R whose total mass remains constant. The expansion is such that the instantaneous density ρ remains uniform throughout the volume. The rate of fractional change in density ((1/ρ ) (d ρ /d t)) is constant. The velocity v of any point on the surface of the expanding sphere is proportional to

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Q. 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 $\left(\frac{1}{\rho } \frac{d \rho }{d t}\right)$ is constant. The velocity $v$ of any point on the surface of the expanding sphere is proportional to

NTA AbhyasNTA Abhyas 2022

Solution:

Density of sphere is $\rho=\frac{m}{v}=\frac{3 m}{4 \pi R^3}$
$\Rightarrow \frac{1}{\rho} \frac{d \rho}{d t}=-\frac{3}{R} \frac{d R}{d t}$
Since $\Rightarrow \frac{1}{\rho} \frac{d \rho}{d t}$ is constant
$\therefore \frac{d R}{d t} \propto R$