Ch 06: Dynamics I: Motion Along a Line

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796당신이 사용하는 게 아니라요?교과서 변경

Knight Calc 5th Edition

Ch 06: Dynamics I: Motion Along a Line

문제 33b

6장, 문제 33b

Below what speed does a 3.0-mm-diameter ball bearing in 20°C air experience linear drag?

검증된 단계별 안내

Step 1: Understand the concept of linear drag. Linear drag occurs when the drag force on an object moving through a fluid is proportional to its velocity. This happens at low speeds, where the flow around the object is laminar, and the Reynolds number is less than approximately 1.

Step 2: Recall the formula for the Reynolds number, which determines the flow regime:

R e = \frac{ρ}{v}

. Here,

ρ

is the density of the fluid (air),

v

is the velocity of the object,

d

is the diameter of the object, and

μ

is the dynamic viscosity of the fluid.

Step 3: For linear drag to occur, the Reynolds number must be less than 1. Rearrange the Reynolds number formula to solve for the velocity:

v = \frac{μ}{ρ}

. Substitute the known values: the diameter of the ball bearing (

3.0 	imes 10^{-3} \(\text{ m}\)

), the density of air at 20°C (

1.2 \(\text{ kg/m}\)^3

), and the dynamic viscosity of air at 20°C (

1.8 	imes 10^{-5} \(\text{ Pa·s}\)

Step 4: Perform unit analysis to ensure consistency. The units of velocity should be in meters per second (

m/s

). Verify that the units of the numerator (

Pa·s

) and denominator (

kg/m^3 	imes m

) simplify correctly.

Step 5: Substitute the values into the formula and simplify the expression to find the maximum velocity below which linear drag occurs. This velocity represents the threshold speed for laminar flow and linear drag.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Linear Drag Force

Linear drag force, also known as viscous drag, is the resistance experienced by an object moving through a fluid. It is directly proportional to the object's velocity and the fluid's viscosity. The equation for linear drag is given by F_d = -b*v, where F_d is the drag force, b is the drag coefficient, and v is the velocity. Understanding this concept is crucial for determining the conditions under which the ball bearing will experience significant drag.

Reynolds Number

The Reynolds number is a dimensionless quantity that helps predict flow patterns in different fluid flow situations. It is calculated as Re = (ρ*v*D)/μ, where ρ is the fluid density, v is the velocity, D is the characteristic length (diameter of the ball bearing), and μ is the dynamic viscosity of the fluid. A low Reynolds number indicates laminar flow, where linear drag is significant, while a high Reynolds number suggests turbulent flow, where drag behaves differently.

Drag Coefficient

The drag coefficient is a dimensionless number that quantifies the drag or resistance of an object in a fluid environment. It depends on the shape of the object and the flow conditions. For a sphere, the drag coefficient can vary with the Reynolds number, affecting how the ball bearing interacts with the air. Knowing the drag coefficient is essential for calculating the speed at which the ball bearing will start to experience linear drag.