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Chapter 9: Problem 66

This table gives the terminal speeds of various spheres falling through the same fluid. The spheres all have the same radius. $$\begin{array}{llllllll}\hline m= & 5.0 & 11.3 & 20.0 & 31.3 & 45.0 & 80.0 & (\mathrm{g}) \\\ \hline v_{1}= & 1.0 & 1.5 & 2.0 & 2.5 & 3.0 & 4.0 & (\mathrm{cm} / \mathrm{s}) \\\\\hline\end{array}$$, Is the drag force primarily viscous or turbulent? Explain your reasoning.

Short Answer

Expert verified
Answer: The drag force is primarily viscous.

Step by step solution

01

1. Recap drag forces and Reynolds number formula

There are two main types of drag forces, viscous and turbulent. We can use the Reynolds number formula to determine the type of flow: $$R_e = \frac{\rho v d}{\mu}$$ where \(R_e\) is the Reynolds number, \(\rho\) is the fluid density, \(v\) is the velocity, \(d\) is the diameter of the sphere, and \(\mu\) is the fluid viscosity.
02

2. Calculate the Reynolds number for each sphere

We don't have all the variables needed for the formula. Instead, we'll look for the relationship between mass (\(m\)) and terminal velocity (\(v\)), as a possible indication of viscous or turbulent flow. Let's organize the data in a new table with mass and corresponding terminal velocity: $$\begin{array}{llllllll}\hline m(g)= & 5.0 & 11.3 & 20.0 & 31.3 & 45.0 & 80.0 \\ \hline v_{1}(cm/s)= & 1.0 & 1.5 & 2.0 & 2.5 & 3.0 & 4.0 \\\hline\end{array}$$
03

3. Look for relationships between mass and terminal velocity

When comparing the mass and terminal velocity values, we can identify a possible relationship. The higher the mass, the higher the terminal velocity, increasing linearly. This indicates that there may be a direct relationship between the mass and terminal velocity.
04

4. Identify viscous or turbulent flow based on the relationship

In viscous flow, the drag force increases linearly with the particle velocity, whereas, in turbulent flow, the drag force increases nonlinearly with the particle velocity, often with a power greater than one. In this case, the relationship between the mass and terminal velocity seems to be linear, leading to the conclusion that the drag force is primarily viscous.
05

5. Conclusion

Based on the relationship observed between the mass and terminal velocity, it appears that the drag force is primarily viscous for the spheres falling through the same fluid with the same radius.

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