First slide

Viscosity

Question

A small steel sphere is allowed to fall in a long vertical viscous liquid column. When the sphere has achieved 50% of its terminal velocity, its acceleration is [take g = 10 m/s²; ignore buoyant force]

Moderate

A

2.5 m/s²
B

5 m/s²
C

4 m/s²
D

8 m/s²

Solution

At terminal velocity, $mg = 6 {πηrV}_{T}$
When $V = \frac{V_{T}}{2}$ , viscous drag, $F_{D} = 6 πηr (\frac{V_{T}}{2})$ ,
$∴$ net force $F_{net} = mg - (6 π η {rV}_{T}) \frac{1}{2}$
$= mg - mg \times \frac{1}{2} = \frac{1}{2} mg$
$∴$ Acceleration $= \frac{mg / 2}{m} = \frac{g}{2} = 5 m / s^{2}$

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