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

Verified step by step guidance

1. Identify the properties of the ball bearing and the fluid. Here, the diameter of the ball bearing is 3.0 mm and the temperature of the water is 20°C, which affects the properties of the water such as density and viscosity.
2. Understand the concept of quadratic drag, which occurs when the drag force on an object moving through a fluid is proportional to the square of its velocity. This is typically significant at higher velocities.
3. Calculate the Reynolds number, which helps determine the flow regime around the ball bearing. The Reynolds number (Re) is given by the formula: \(Re = \frac{\rho v D}{\mu}\), where \(\rho\) is the density of the fluid, \(v\) is the velocity of the object, \(D\) is the diameter of the object, and \(\mu\) is the dynamic viscosity of the fluid.
4. Find the critical Reynolds number at which the flow transitions from laminar to turbulent, typically around 1000 for flow around a sphere. This transition point indicates when quadratic drag becomes significant.
5. Solve for the velocity \(v\) at which the Reynolds number equals the critical value. Use the known values of \(\rho\), \(D\), and \(\mu\) for water at 20°C to find the velocity at which the ball bearing experiences quadratic drag.

Key Concepts

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 types: linear drag at low speeds and quadratic drag at higher speeds, where the drag force increases with the square of the velocity.

Reynolds Number

The Reynolds number is a dimensionless quantity used to 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 (linear drag), while a high Reynolds number suggests turbulent flow (quadratic drag), which is crucial for determining the transition speed for drag types.

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 rapid increase in resistance as speed increases. Understanding the transition from linear to quadratic drag is essential for analyzing the motion of small objects like ball bearings in fluids.