Question

A solid ball having a radius of \(2\ \text{cm}\), has a terminal speed of \(32\ \text{cm/s}\) in a viscous liquid. The terminal speed of a solid ball of the same material, with radius of \(0.5\ \text{cm}\) in the same liquid under identical conditions is

• A. \(4\ \text{cm/s}\)
• B. \(8\ \text{cm/s}\)
• C. \(2\ \text{cm/s}\)
• D. \(16\ \text{cm/s}\)

Hint:

By Stokes' law, terminal speed is proportional to the square of the radius for the same material and liquid.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
By Stokes' law, the terminal speed of a ball is \(v_t=\dfrac{2r^{2}(\rho-\sigma)g}{9\eta}\), so \(v_t\propto r^{2}\) when the material and liquid are the same.

Step 2: Calculation
\(\dfrac{v_2}{v_1}=\left(\dfrac{r_2}{r_1}\right)^{2}=\left(\dfrac{0.5}{2}\right)^{2}=\dfrac{1}{16}\).

Step 3: Final value
\(v_2=\dfrac{32}{16}=2\ \text{cm/s}\).

Final Answer:
The terminal speed of the smaller ball is 2 cm/s. \[ \boxed{\text{(C) }2\ \text{cm/s}} \]