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

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

Answer 1

The correct option is B

$R$

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

$Mass=\rho \times Volume=constant$

Here total mass is given as constant

On differentiating,

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

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

$\frac{1}{\rho }\frac{d\rho }{dt}=\frac{-3}{R}\frac{dR}{dt}$,
and it is given that $\left(\frac{1}{\rho }\frac{d\rho }{dt}\right)$ is constant
So, the velocity $\frac{dR}{dt}$ is propotional to $R$