Consider an expanding sphere of instantaneous radius R whose tota… - Vidyadip

MCQ

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

✓
$R$
B
$\mathrm{R}^3$
C
$\frac{1}{R}$
D
$\mathrm{R}^{2 / 3}$

✓

Answer

Correct option: A.

$R$

a
$A$

$\mathrm{m}=\rho \frac{4}{3} \pi \mathrm{R}^3$

$0=\rho \cdot 4 \pi \mathrm{R}^2 \frac{\mathrm{dR}}{\mathrm{dt}}+\frac{4}{3} \pi \mathrm{R}^3 \frac{\mathrm{d} \rho}{\mathrm{dt}}$

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

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

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

$\mathrm{v} \propto R$

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