Question

This table gives the terminal speeds of various spheres falling through the same fluid. The spheres all have the same radius. $$\begin{array}{llllllll}\hline m= & 8 & 12 & 16 & 20 & 24 & 28 & (\mathrm{g}) \\\ \hline v_{1}= & 1.0 & 1.5 & 2.0 & 2.5 & 3.0 & 3.5 & (\mathrm{cm} / \mathrm{s}) \\\\\hline\end{array}$$, Is the drag force primarily viscous or turbulent? Explain your reasoning.

Short answer

The drag force is primarily viscous, as indicated by the linear relationship between mass and terminal speed.

Step-by-step solution

Understand Terminal Speed and Drag Force

Terminal speed is the constant speed that an object reaches when the drag force due to the fluid is equal to the gravitational force on the object. The drag force can be primarily viscous or turbulent depending on the regime of flow over the object.

Analyze Drag Force Nature

The nature of the drag force is determined by the relationship between mass and terminal speed values. Viscous drag is proportional to the velocity \(v\), while turbulent drag is proportional to \(v^2\).

Examine the Data

From the table provided, as the mass (\(m\)) increases, the terminal speed (\(v_1\)) increases linearly. Plotting mass against velocity would show if it's a linear relationship. If the velocity \(v_1\) and mass \(m\) showed a quadratic relationship, that would imply a turbulent drag. The data shows a consistent increase in terminal speed per increase in mass which suggests a linear relationship rather than quadratic.

Determine the Dominant Drag Force

The linear change in terminal speed with mass suggests that the drag force is more likely viscous in nature. This is because a linear relationship suggests drag \(F_d \propto v\), which is characteristic of viscous drag.

Key concepts

Terminal Speed

When an object moves through a fluid, it experiences a resistive force known as drag. At some point, the object stops accelerating and moves at a constant speed, known as its terminal speed. Terminal speed happens when the gravitational force pulling the object downwards is balanced by the drag force pushing it upwards. This balance results in zero net force on the object, causing it to fall at a steady pace.
Terminal speed is different for every falling object and depends on several factors:

• The object's mass
• The object's shape and size
• The density of the fluid through which it falls
• The nature of the drag (viscous or turbulent)

In the exercise's table, various spheres fall through the same fluid, reaching different terminal speeds based on their mass. Understanding this concept is crucial to determine the kind of drag acting on the objects.

Viscous Drag

Viscous drag occurs when an object moves through a fluid at low speeds, where the fluid's viscosity – its stickiness, if you will – is significant. This kind of drag force is directly proportional to the speed of the object. In formula terms, we often write viscous drag as \(F_d = b \, v\), where \(b\) is a constant related to the fluid's viscosity and the object's shape, and \(v\) is the speed of the object.
Viscous drag is prevalent in scenarios where objects move slowly enough that the fluid can "sneak around" them smoothly. This drag mechanism is typical in systems like raindrops falling through air at low speeds, or tiny particles settling in water.
In the exercise example, if the relationship between the spheres' mass and their terminal speed appears linear, this would be an indicator that viscous drag is the predominant force.

Turbulent Drag

Turbulent drag takes center stage at higher velocities; it's a different beast compared to viscous drag. It arises when the speed of an object through a fluid is high enough that the fluid flow becomes chaotic rather than smooth. The drag force due to turbulence is not just proportional to the speed, but rather to the square of the speed. We can express the turbulent drag force with the formula: \(F_d = c \, v^2\), where \(c\) is a constant based on several factors, including the object's shape and the fluid's characteristics.
Turbulent drag occurs in scenarios such as when a speeding car moves through air, or when hailstones plummet from the sky. The increased velocity leads to increased drag force, causing more energy to be dissipated.
From the exercise data, if the relationship between mass and terminal speed is quadratic, that could suggest turbulent drag. However, since the table data indicates linearity, turbulent drag is likely not the major player here.

Mass-Velocity Relationship

The relationship between an object's mass and its terminal velocity offers insight into the kind of drag force at work. In fluid dynamics, the drag force's behavior is reflective of how mass and velocity interact.
For viscous drag, which is dominant at low speeds, the terminal speed is proportional to the ratio of mass to drag coefficient. Essentially, a graph of mass versus terminal speed should reveal a direct, or linear, relationship. For turbulent drag, the relationship becomes more complex. Since the drag force is proportional to the square of the velocity, the graph would have a quadratic nature, or in simpler terms, it would curve upwards more dramatically as velocity increases. Analyzing the data from the exercise shows a linear relationship between mass and terminal speed, aligning with the principles of viscous drag.
This understanding allows us to predict the behavior of objects closely based on how mass and velocity interact in different fluids.