Problem 75 - Physics

Question

When an object moves through a fluid, the fluid exerts a viscous force \(\overrightarrow{\mathrm{F}}\) on the object that tends to slow it down. For a small sphere of radius \(R,\) moving slowly with a speed \(v\), the magnitude of the viscous force is given by Stokes' law, \(F=6 \pi \eta R v,\) where \(\eta\) is the viscosity of the fluid. (a) What is the viscous force on a sphere of radius \(R=5.0 \times 10^{-4} \mathrm{~m}\) falling through water \(\left(\eta=1.00 \times 10^{-3} \mathrm{~Pa} \cdot \mathrm{s}\right)\) when the sphere has a speed of \(3.0 \mathrm{~m} / \mathrm{s} ?\) (b) The speed of the falling sphere increases until the viscous force balances the weight of the sphere. Thereafter, no net force acts on the sphere, and it falls with a constant speed called the "terminal speed." If the sphere has a mass of \(1.0 \times 10^{-5} \mathrm{~kg},\) what is its terminal speed?

Step-by-Step Solution

Answer

(a) The viscous force is \(2.83 \times 10^{-5} \text{ N}\). (b) Terminal speed is \(1.04 \text{ m/s}\).

Step 1: Understand Stokes' Law

Stokes' Law provides the formula for the viscous force experienced by a small sphere moving through a fluid: \(F = 6\pi \eta R v\). This formula includes the viscosity of the fluid (\(\eta\)), the radius of the sphere (\(R\)), and the speed of the sphere (\(v\)).

Step 2: Calculate the Viscous Force (Part a)

Use the given values: \(R = 5.0 \times 10^{-4} \text{ m}\), \(\eta = 1.00 \times 10^{-3}\text{ Pa} \cdot \text{s}\), and \(v = 3.0 \text{ m/s}\). Substitute these into Stokes' Law formula: \[ F = 6\pi \times 1.00 \times 10^{-3} \times 5.0 \times 10^{-4} \times 3.0 \].\ "Calculate \(F\): \(F = 2.83 \times 10^{-5} \text{ N}\)."

Step 3: Determine the Weight of the Sphere

Use the formula for weight: \(W = mg\), where \(m = 1.0 \times 10^{-5} \text{ kg}\) is the mass of the sphere and \(g = 9.81 \text{ m/s}^2\) is the acceleration due to gravity. Calculate \(W\): \[ W = 1.0 \times 10^{-5} \times 9.81 = 9.81 \times 10^{-5} \text{ N} \].

Step 4: Solve for Terminal Speed (Part b)

At terminal speed, the viscous force balances the weight, \(F = W\). Thus, \(6\pi \eta R v_{\text{terminal}} = mg\). Substitute the known values: \(6\pi \times 1.00 \times 10^{-3} \times 5.0 \times 10^{-4} \times v_{\text{terminal}} = 9.81 \times 10^{-5}\). Solve for \(v_{\text{terminal}}\): \[ v_{\text{terminal}} = \frac{9.81 \times 10^{-5}}{6\pi \times 1.00 \times 10^{-3} \times 5.0 \times 10^{-4}} = 1.04 \text{ m/s} \].

Key Concepts

Viscous Force

The viscous force is a crucial concept when discussing how objects move through fluids. It is the resistance encountered by a body moving through a fluid, which can either be a liquid or a gas. In simpler terms, imagine trying to move your hand through water. The water pushes back against your hand, trying to slow it down. This resistance is what we call viscous force.

Stokes' Law provides a way to calculate this force for a small sphere traveling slowly through a fluid. The formula is given by \( F = 6 \pi \eta R v \). Here, \( F \) is the viscous force, \( \eta \) is the fluid’s viscosity, \( R \) is the sphere's radius, and \( v \) is its speed.

• The viscous force acts in the opposite direction to the sphere's movement.
• The greater the viscosity of the fluid, the larger the resistance or viscous force.

Understanding this concept helps in predicting how different objects will behave when inserted into various fluids.

Terminal Speed

Terminal speed is a concept from fluid dynamics describing the constant speed achieved by a falling object when the force from air resistance equals the gravitational pull.

As an object like a sphere falls through a fluid, it initially accelerates due to gravity. However, as its speed increases, so does the viscous force opposing its motion. Eventually, the viscous force becomes equal to the object's weight, causing it to stop accelerating.
This constant speed, at which the net force acting on the sphere is zero, is known as terminal speed.

• At terminal speed, the sphere falls through the fluid without further acceleration.
• The equation balancing forces at terminal speed is: \( mg = 6 \pi \eta R v_{\text{terminal}} \).
• Variables include the object's mass (\( m \)), gravitational acceleration (\( g \)), and known values for the sphere and fluid based on Stokes' Law.

This balance of forces keeps the speed constant as the sphere descends steadily through the fluid.

Sphere Dynamics

Sphere dynamics in fluids examines how spherical objects interact with fluid environments. In this context, a sphere is affected by multiple forces that determine its motion through the fluid.

The forces at play include:

• Gravity, pulling the sphere downward.
• Buoyant force, pushing the sphere upward based on Archimedes' principle.
• Viscous force, which we earlier discussed as acting to slow the sphere down.

In a hypothetical situation where viscous force matches gravitational force, acceleration ceases, resulting in terminal speed.

This understanding is vital when calculating the sphere's movement through a particular viscous fluid, allowing us to predict how quickly it might reach terminal speed and how long it might continue to fall at that rate.

Fluid Viscosity

Fluid viscosity is a measure of a fluid's resistance to flow or deformation by stress. It's a critical property influencing how objects move within fluids. Imagine honey compared to water; honey’s viscosity is much higher, resisting flow greatly more than water does.

When using Stokes' Law to calculate viscous force on a sphere, viscosity (\( \eta \)) plays a significant role.

• A high-viscosity fluid, like syrup, offers more resistance, resulting in a greater viscous force.
• A low-viscosity fluid, like air, offers less resistance, producing a smaller viscous force.

The viscosity of a fluid is measured in Pascal-seconds (\( \text{Pa} \cdot \text{s} \)). In practical scenarios, knowing the viscosity helps in predicting how much resistance an object faces as it moves through a fluid. Understanding viscosity is crucial for engineers and scientists when designing systems involving fluid flow.