Before You Begin

Question #42121

Difficulty: 8/10 | 🟢 Free

The Question

A solid sphere of initial radius $r_0$ and density $\rho_s$ is falling in a viscous liquid of density $\rho_l$ (where $\rho_s > \rho_l$ ). The system is slowly heated such that the sphere's radius expands as $r(t) = r_0(1 + \alpha t)$ and the liquid's viscosity decreases as $\eta(t) = \eta_0(1 - \beta t)$ , where $\alpha$ and $\beta$ are small positive constants. Assuming the sphere continuously maintains its quasi-steady terminal velocity at each instant, what is the acceleration of the sphere at $t = 0$ ?

Options

A.
$\frac{2g r_0^2}{9 \eta_0} \left[ \beta (\rho_s - \rho_l) - \alpha (\rho_s + 2\rho_l) \right]$
B.
$\frac{2g r_0^2}{9 \eta_0} (\rho_s - \rho_l) (2\alpha + \beta)$
C.
$\frac{2g r_0^2}{9 \eta_0} \left[ \beta (\rho_s - \rho_l) - \alpha (\rho_s - 2\rho_l) \right]$
D.
$\frac{2g r_0^2}{9 \eta_0} \left[ \beta (\rho_s - \rho_l) - 3\alpha \rho_s \right]$

Check Your Answer

Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Variable_Constraint

Develop pure symbolic logic skills by mastering algebraic constraints without relying on numerical data.

Concepts Behind This Question

Viscosity and Stokes' Law|Thermal Expansion of Area and Volume|Differential Calculus

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to apply dynamic equilibrium to a system with time-varying parameters.

The lethal trap lies in blindly differentiating the standard terminal velocity formula $v_T = \frac{2r^2 g}{9\eta}(\rho_s - \rho_l)$ .

Students who do this implicitly assume the sphere's density remains constant while its volume expands, which violates the conservation of mass and leads directly to Trap Option B.

By forcing the mass to be constant ( $M = \frac{4}{3}\pi r_0^3 \rho_s$ ) while the displaced fluid volume changes ( $V = \frac{4}{3}\pi r(t)^3$ ), the buoyancy force becomes a dynamic function of time, leading to a completely different transient acceleration.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Mechanical Properties of Fluids - Class 11 Part 2
Concept	Viscosity and Stokes' Law
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free