Consider an expanding sphere of instantaneous radius ð 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 ð£ of any point on the surface of the expanding sphere is proportional to

Question

Consider an expanding sphere of instantaneous radius ð 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 ð£ of any point on the surface of the expanding sphere is proportional to (A) R (B) R3 (C) (1/R) (D) R2/3

Tardigrade Admin · Accepted Answer

m =ρ (4/3) π R 3 0=ρ ⋅ 4 π R 2 ( dR / dt )+(4/3) π R 3 ( d ρ/ dt ) -(1/ρ) ( d ρ/ dt )=(3/ R ) ( dR / dt ) -( R /3) (1/ρ) ( d ρ/ dt )=( dR / dt ) ( dR / dt ) propto R v propto R

$R$

$R^{3}$

$\frac{1}{R}$

$R^{2/3}$

$m = ρ \frac{4}{3} π R^{3}$
$0 = ρ \cdot 4 π R^{2} \frac{d R}{d t} + \frac{4}{3} π R^{3} \frac{d ρ}{d t}$
$- \frac{1}{ρ} \frac{d ρ}{d t} = \frac{3}{R} \frac{d R}{d t}$
$- \frac{R}{3} \frac{1}{ρ} \frac{d ρ}{d t} = \frac{d R}{d t}$
$\frac{d R}{d t} \propto R$
$v \propto R$

R

R3

R1​

R2/3

m=ρ34​πR3 0=ρ⋅4πR2dtdR​+34​πR3dtdρ​ −ρ1​dtdρ​=R3​dtdR​ −3R​ρ1​dtdρ​=dtdR​ dtdR​∝R v∝R

$R$

$R^{3}$

$\frac{1}{R}$

$R^{2/3}$

$m = ρ \frac{4}{3} π R^{3}$
$0 = ρ \cdot 4 π R^{2} \frac{d R}{d t} + \frac{4}{3} π R^{3} \frac{d ρ}{d t}$
$- \frac{1}{ρ} \frac{d ρ}{d t} = \frac{3}{R} \frac{d R}{d t}$
$- \frac{R}{3} \frac{1}{ρ} \frac{d ρ}{d t} = \frac{d R}{d t}$
$\frac{d R}{d t} \propto R$
$v \propto R$