If γs = unit weight of the material of falling sphere in g/cm³, γt = unit weight of the liquid medium in g/cm³, μt = viscosity of the liquid medium in g sec/cm², and D = diameter of the spherical particle in cm. According to Stokes’ law, what is the formula for the terminal velocity (v) of the spherical particle ?
- Av = (1/18) [(γs − γt)/μt] D²Correct
- Bv = (1/8) [(γs + γt)/μt] D²
- Cv = (1/12) [(γs − γt)/μt] D²
- Dv = (1/2) [(γs + γt)/μt] D²
Solution
Concept
Stokes’ law of settling
Formula
v = (γs − γw) D² / (18 μ)
Step-by-step solution
1. At terminal velocity the weight minus buoyancy equals the drag 3πμDv. 2. (π/6) D³ (γs − γt) = 3π μ D v ⇒ v = (γs − γt) D² / (18 μ).
Explanation
Stokes' law is used in the hydrometer and pipette methods of sedimentation analysis. The driving force is the submerged weight, so the difference of unit weights appears.
Why the other options are wrong
The sum γs + γt and the factors 8, 12 and 2 are incorrect.
Shortcut
v = (γs − γw) D² / 18μ.
Common mistake
Adding unit weights.
Correct answer: A) v = (1/18) [(γs − γt)/μt] D²