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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 (1/ρ dρ/dt) is constant. The velocity v of any point on the surface of the expanding sphere is proportional to

R

R2/3

R3

1/R

Solution:

ρ = Mass/Volume
Mass = ρ × Volume = constant
On differentiating, Vdρ/dt + ρdV/dt = 0
(4/3)πR³ × dρ/dt + ρ × d/dt((4/3)πR³) = 0
(1/ρ)dρ/dt = -3(dR/dt)/R
dR/dt = v (velocity of any point on the surface)
Therefore, v ∝ R