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 is constant. The velocity ν of any point on the surface of the expanding sphere is proportional to

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Question

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 $(\frac{1}{ρ} \frac{dρ}{dt})$ is constant. The velocity ν of any point on the surface of the expanding sphere is proportional to

R^2/3

R³

$\frac{1}{R}$

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Let partical on surface moves dR in dt time.

V = $\frac{dR}{dt}$

$\frac{1}{ρ} \frac{dρ}{dt}$ = c ( $∴$ C = constant)

$\frac{dρ}{dt} = (\frac{M}{(4 / 3) π}) (\frac{- 3}{R^{4}}) \frac{dR}{dt}$

= $\frac{- 3 ρdR}{dt}$ .....(ii)

by (i) & (ii)

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