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Consider an expanding sphere of instantaneous radius {R} whose

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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 $(\frac{1}{ρ} \frac{d ρ}{d t})$ is constant. The velocity $v$ of any point on the surface of the expanding sphere is proportional to :

R^{3}

\frac{1}{R}

R

R^{2/3}

Views: 6,050 students

Updated on: May 17, 2023

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Olivia Discussed

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{d ρ}{d t})

is constant. The velocity

v

of any point on the surface of the expanding sphere is proportional to :

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10 mins ago

Text solutionVerified

ρ = \frac{M a ss}{V o l u m e}

Mass

= ρ \times

Volume

=

constant

On differentiating,

V \frac{dp}{dt} + ρ \frac{dV}{dt} = 0

\frac{4}{3} π R^{3} \times \frac{d p}{dt} + ρ \times \frac{d}{dt} (\frac{4}{3} π R^{3}) = 0

\frac{1}{ρ} \frac{d ρ}{dt} = \frac{- 3}{R} \frac{dR}{dt}

\frac{dR}{dt} \propto R

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Question Text	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 $(\frac{1}{ρ} \frac{d ρ}{d t})$ is constant. The velocity $v$ of any point on the surface of the expanding sphere is proportional to :
Updated On	May 17, 2023
Topic	Gravitation
Subject	Science
Class	Class 9
Answer Type	Text solution:1 Video solution: 7
Upvotes	783
Avg. Video Duration	7 min

Text solutionVerified

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