Ch 06: Dynamics I: Motion Along a Line

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 06: Dynamics I: Motion Along a Line

Problem 33a

Chapter 6, Problem 33a

Above what speed does a 3.0-mm-diameter ball bearing in 20°C water experience quadratic drag?

Verified step by step guidance

Understand the problem: The question asks for the speed at which a 3.0-mm-diameter ball bearing in water at 20°C transitions from experiencing linear drag to quadratic drag. This transition occurs when the Reynolds number (Re) exceeds a critical value, typically around 1 for small spherical objects in a fluid.

Step 1: Recall the formula for the Reynolds number:

Re = (ρ × v × d) / μ

, where ρ is the density of the fluid, v is the velocity of the object, d is the diameter of the object, and μ is the dynamic viscosity of the fluid.

Step 2: Look up the properties of water at 20°C. The density of water (ρ) is approximately 998 kg/m³, and the dynamic viscosity (μ) is approximately 1.002 × 10⁻³ Pa·s.

Step 3: Rearrange the Reynolds number formula to solve for the velocity (v):

v = (Re × μ) / (ρ × d)

. Use the critical Reynolds number (Re = 1) for the transition to quadratic drag.

Step 4: Substitute the known values into the formula: Re = 1, μ = 1.002 × 10⁻³ Pa·s, ρ = 998 kg/m³, and d = 3.0 mm = 3.0 × 10⁻³ m. Simplify the expression to find the critical velocity.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Drag Force

Drag force is the resistance experienced by an object moving through a fluid, such as air or water. It depends on the object's speed, shape, and the properties of the fluid. In the case of small objects like a ball bearing, drag can be classified into two regimes: linear and quadratic, with the latter dominating at higher speeds.

Reynolds Number

The Reynolds number is a dimensionless quantity that helps predict flow patterns in different fluid flow situations. It is calculated using the object's velocity, characteristic length (like diameter), fluid density, and viscosity. A low Reynolds number indicates laminar flow, while a high number suggests turbulent flow, which is crucial for determining the type of drag force acting on the ball bearing.

Quadratic Drag

Quadratic drag occurs when the drag force on an object moving through a fluid is proportional to the square of its velocity. This type of drag becomes significant at higher speeds and is characterized by a nonlinear relationship between speed and drag force. Understanding when quadratic drag takes over is essential for analyzing the motion of small objects in fluids.

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