Question

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

• A. R
• B. $R^3$
• C. $\frac{1}{R}$
• D. $R^{2/3}$

Answer 1

The correct option is A R

$\rho =\frac{Volume}{Mass}$

$Mass=\rho \times Volume=cons\tan t$

On differentiating we get,

$V\frac{d\rho }{dt}+\rho \frac{dV}{dt}=0$

$\frac{4}{3}\pi R^3\times \frac{d\rho }{dt}+\rho \times \frac{d}{dt}\left(\frac{4}{3}\pi R^3\right)=0$

$\frac{1}{\rho }\frac{d\rho }{dt}=\frac{-3}{R}\frac{dR}{dt}$

$\frac{dR}{dt}\propto R$

Hence the correct option is A