AIIMS 2004 Physics Viscous Drag MCQ Question

To determine the terminal velocity of a sphere falling through a viscous fluid, we need to consider the forces acting on the sphere. The two main forces involved are the gravitational force acting downward and the viscous drag force acting upward.

1. Forces Acting on the Sphere

• Gravitational Force (Weight): The weight of the sphere can be expressed as: $F_g = Mg$ where $M$ is the mass of the sphere and $g$ is the acceleration due to gravity.

• Viscous Drag Force: According to Stokes' law, the viscous drag force $F_d$ experienced by a sphere moving through a viscous fluid is given by: $F_d = 6\pi \eta R v$ where $\eta$ is the dynamic viscosity of the fluid, $R$ is the radius of the sphere, and $v$ is the velocity of the sphere.

2. Terminal Velocity Condition

At terminal velocity $v_t$, the gravitational force is balanced by the viscous drag force: $Mg = 6\pi \eta R v_t$

3. Solving for Terminal Velocity

Rearranging the equation for terminal velocity gives: $v_t = \frac{Mg}{6\pi \eta R}$

From this expression, we can see that the terminal velocity $v_t$ is inversely proportional to the radius $R$ and directly proportional to the mass $M$ and the acceleration due to gravity $g$.

4. Proportional Relationships

To analyze the proportional relationships:

• The terminal velocity $v_t$ can be expressed as: $v_t \propto \frac{M}{R}$

Thus, if we consider the effect of varying the radius $R$, we can conclude the following:

• As the radius $R$ increases, the terminal velocity $v_t$ increases, which indicates that $v_t$ is proportional to the square of the radius in the context of mass and gravitational force.

5. Conclusion

Given the options presented:

• A) $R^2$
• B) $R$
• C) $1/R$
• D) $1/R^2$

The correct choice is A) $R^2$.

Why Other Options are Incorrect

• B) $R$: This suggests a linear relationship with radius, which is not supported by our derived equation.
• C) $1/R$: This implies that increasing the radius would decrease terminal velocity, which contradicts our findings.
• D) $1/R^2$: This also suggests an inverse square relationship, which is not applicable in this context.

In summary, the terminal velocity of a sphere falling through a viscous fluid is proportional to the square of its radius, affirming that A) $R^2$ is the correct answer.