Consider an expanding sphere of instantaneous radius whose total mass remains constant. The expansion is such that the instantaneous density remains uniform throughout the volume. The rate of fractional change in density is constant. The velocity of a

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 $(\frac{1}{ρ} \frac{d ρ}{d t})$ is constant. The velocity $v$ of any point on the surface of the expanding sphere is proportional to [2017]

Smart Achievers · Accepted Answer

(1)

$Given : \frac{1}{ρ} \frac{d ρ}{d t} = constant$

$∴$ $\frac{4 π R^{3}}{3 m} \frac{d}{d t} [\frac{m}{\frac{4}{3} π R^{3}}] = constant$

$\Rightarrow R^{3} \frac{d}{d t} (R^{- 3}) = constant$

$\Rightarrow R^{3} (- 3 R^{- 4}) \frac{d R}{d t} = constant$ $∴$ $| \frac{d R}{d t} |$ $\propto R$

Solution

1 $R$

2 $R^{3}$

3 $\frac{1}{R}$

4 $R^{2 / 3}$

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Solution

1 R

2 R3

3 1R

4 R2/3

Ans. (1) Given : 1ρdρdt=constant ∴ 4πR33mddt[m43πR3]=constant ⇒R3ddt(R-3)=constant ⇒R3(-3R-4)dRdt=constant ∴ |dRdt|∝R

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1 $R$

2 $R^{3}$

3 $\frac{1}{R}$

4 $R^{2 / 3}$