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Chapter 8: Problem 41

The viscous force on a small sphere of radius \(R\) moving in a fluid varies as \((\mathrm{A}) \propto R^{2}\) (B) \(\propto R\) (C) \(\propto(1 / R)\) \((\mathrm{D}) \propto(1 / R)^{2}\)

Short Answer

Expert verified
The viscous force acting on a small sphere moving through a fluid follows Stoke's Law, given by the equation \(F = 6 \pi \eta Rv\). From this equation, we can see that the viscous force (F) is directly proportional to the radius of the sphere (R), represented as \(F \propto R\). Therefore, the correct answer is (B).

Step by step solution

01

1. Recall Stoke's Law

Stoke's law states that the viscous force (F) acting on a sphere moving through a fluid can be described by the equation: \[F = 6 \pi \eta R v\] where: - F is the viscous force - \(\eta\) is the dynamic viscosity of the fluid - R is the radius of the sphere - v is the velocity of the sphere
02

2. Identify the proportional relationship

According to the equation, the viscous force on the sphere is directly proportional to its radius (R). Therefore, we can express the viscous force as: \[F \propto R\]
03

3. Match the relationship with the given options

Now we can compare the relationship we found in step 2 with the given options. We see that option (B) correctly represents the relationship between viscous force and the radius of the sphere: \[F \propto R\] Hence, the correct answer is (B).

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