Step-by-step Derivation

1. Relate Mass, Density, and Volume: The mass M of the sphere is related to its density ρ and volume V by the formula: $M = \rho V$ For a sphere of radius R, the volume is $V = \frac{4}{3}\pi R^3$. Substituting this into the mass equation, we get: $M = \rho \left( \frac{4}{3}\pi R^3 \right)$

2. Use the Condition of Constant Mass: The problem states that the total mass M of the sphere remains constant. Since M, $\frac{4}{3}$, and $\pi$ are all constants, the product $\rho R^3$ must also be a constant. $\rho R^3 = \text{constant}$

3. Differentiate with Respect to Time: To find the relationship between the rates of change, we differentiate the equation $\rho R^3 = \text{constant}$ with respect to time t. Using the product rule for differentiation (uv)' = u'v + uv': \frac{d}{dt} (\rho R^3) = \frac{d}{dt} (\text{constant}) $$$$ \left( \frac{d\rho}{dt} \right) R^3 + \rho \left( \frac{d(R^3)}{dt} \right) = 0 Applying the chain rule to the $R^3$ term: $\frac{d(R^3)}{dt} = 3R^2 \frac{dR}{dt}$. $\left( \frac{d\rho}{dt} \right) R^3 + \rho \left( 3R^2 \frac{dR}{dt} \right) = 0$

4. Incorporate the Given Rate of Fractional Change in Density: The problem states that the rate of fractional change in density, $\frac{1}{\rho} \frac{d\rho}{dt}$, is a constant. Let's denote this constant by C: $\frac{1}{\rho} \frac{d\rho}{dt} = C$ From this, we can write $\frac{d\rho}{dt} = C\rho$.

5. Substitute and Solve for Velocity: Substitute $\frac{d\rho}{dt} = C\rho$ into the equation from Step 3: $(C\rho) R^3 + \rho \left( 3R^2 \frac{dR}{dt} \right) = 0$ The velocity v of a point on the surface is the rate of change of the radius, $v = \frac{dR}{dt}$. So we have: $C\rho R^3 + 3\rho R^2 v = 0$ We can divide the entire equation by $\rho R^2$ (assuming $\rho$ and R are non-zero): $CR + 3v = 0$ Solving for v: 3v = -CR $$$$ v = -\frac{C}{3} R

6. Conclusion: Since C is a constant, the term $-\frac{C}{3}$ is also a constant. Therefore, the velocity v is directly proportional to the instantaneous radius R. $v \propto R$ This corresponds to option A.

Evaluation of Options

• A: $R$: Our derivation shows v ∝ R. This is the correct option.
• B: $R^3$: Incorrect.
• C: $1/R$: Incorrect.
• D: $R^{2/3}$: Incorrect.