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

Answer 1

Answer: (A)

$m =\rho \frac{4}{3} \pi R ^{3}$
$0=\rho \cdot 4 \pi R ^{2} \frac{ dR }{ dt }+\frac{4}{3} \pi R ^{3} \frac{ d \rho}{ dt }$
$-\frac{1}{\rho} \frac{ d \rho}{ dt }=\frac{3}{ R } \frac{ dR }{ dt }$
$-\frac{ R }{3} \frac{1}{\rho} \frac{ d \rho}{ dt }=\frac{ dR }{ dt }$
$\frac{ dR }{ dt } \propto R$
$v \propto R$