Question

The terminal velocity of a small steel ball falling through a viscous medium is:

• A. Directly proportional to the radius of the ball
• B. Inversely proportional to the radius of the ball
• C. Directly proportional to the square of the radius of the ball
• D. Directly proportional to the square root of the radius of the ball
• E. Inversely proportional to the square of the radius of the ball

Hint:

For small objects moving through a viscous medium, the terminal velocity is directly proportional to the square of the radius of the object, according to Stokes' law.

Answer 1

The Correct Option is C

The terminal velocity \( v_t \) of a small sphere falling through a viscous medium is given by Stokes' law for low Reynolds numbers: \[ v_t = \frac{2r^2(\rho - \rho_0)g}{9\eta}, \] where: - \( r \) is the radius of the sphere,
- \( \rho \) is the density of the sphere,
- \( \rho_0 \) is the density of the fluid,
- \( g \) is the acceleration due to gravity, and
- \( \eta \) is the dynamic viscosity of the fluid.
From this equation, we can observe that the terminal velocity is directly proportional to the square of the radius of the ball. 
Thus, if the radius of the ball increases, the terminal velocity increases with the square of the radius.
Thus, the correct answer is option (C), directly proportional to the square of the radius of the ball.