Spherical balls of radius R are falling in a viscous fluid of viscosity eta with a velocity v. The r

Q: Why is the force directly proportional to velocity?

As the object moves faster, it has to push through more fluid layers per second. This rapid displacement of the fluid increases the internal friction, leading to a higher viscous drag force.

Q: Does Stokes' Law apply to all shapes?

No, Stokes' law specifically applies to small, perfectly spherical objects moving at relatively low speeds (laminar flow) through a viscous medium.

Q: What exactly is $\eta$?

It is the coefficient of viscosity. It is a property of the fluid that quantifies its internal resistance to flow. Honey has a much higher $\eta$ than water.

Q: Is the viscous force the only force acting on the falling ball?

No, there are three main forces: gravity acting downwards, buoyant force acting upwards, and the viscous drag force acting upwards (opposing the motion).

Q: What happens when the viscous force equals the net downward force?

When the upward forces (buoyancy + viscous drag) perfectly balance the downward force (gravity), the net force becomes zero. The ball stops accelerating and continues to fall at a constant speed, known as the terminal velocity.

Question

Spherical balls of radius $R$ are falling in a viscous fluid of viscosity $\eta$ with a velocity $v$. The retarding viscous force acting on the spherical ball is

Options:

(A) directly proportional to $R$ but inversely proportional to $v$

(B) directly proportional to both radius $R$ and velocity $v$

(C) inversely proportional to both radius $R$ and velocity $v$

(D) inversely proportional to $R$ but directly proportional to velocity $v$

Sigma Percentile · Accepted Answer

Have you ever wondered why a marble falls much slower in a jar of honey compared to a jar of water? The secret lies in a fascinating property of fluids called **viscosity**, which acts like internal friction.

### The Setup: A Sphere in a Fluid
Imagine a spherical ball of radius $R$ gently dropped into a viscous fluid. As gravity pulls the ball downwards, it has to push the fluid out of its way. The fluid, in turn, resists this motion. This resistance is what we call the **viscous drag force**.

### Stokes' Law: The Master Equation
In the mid-19th century, a brilliant physicist named George Gabriel Stokes derived a beautiful mathematical expression for this exact scenario. According to **Stokes' Law**, the retarding viscous force $F$ acting on a small sphere moving through a viscous fluid is given by:

$$F = 6\pi\eta R v$$

Let's break down the characters in this equation:
- $\eta$ is the **coefficient of viscosity**, representing how 'thick' or 'sticky' the fluid is.
- $R$ is the **radius** of the spherical ball.
- $v$ is the **velocity** of the ball at any given instant.

### Analyzing the Proportionalities
The question asks us to find the relationship between the viscous force $F$, the radius $R$, and the velocity $v$. By simply looking at Stokes' equation, we can see that both $R$ and $v$ are in the numerator. There are no inverse relationships (like $1/R$ or $1/v$) here.

This means:
1. $F \propto R$: The larger the ball, the more fluid it has to displace, and thus, the greater the viscous drag it experiences.
2. $F \propto v$: The faster the ball tries to move, the harder the fluid pushes back against it.

Therefore, the retarding viscous force is **directly proportional to both the radius $R$ and the velocity $v$**.

This simple yet profound relationship is why tiny dust particles can stay suspended in the air for days, while larger pebbles fall quickly to the ground!

Animated Solution for Physics - Properties of Solids and Liquids: Spherical balls of radius $R$ are falling in a viscous fluid of viscosity $η$ with a velocity $v$ . The retarding viscous force acting on the spherical ball is

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Visualized Solution

$Falling Sphere in a Viscous Fluid$

$Stokes’ Law$

$Analyzing Proportionalities$

$Conclusion$

The Sigma Insight: Viscosity and Stokes' Law

The Setup

Stokes' Law

Analyzing the Proportionalities The question asks us to find the relationship between the viscous force $F$ , the radius $R$ , and the velocity $v$

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