Chapter P.1, Problem 13P

The drag force on an object moving in a liquid is quite different from that in air. Drag forces in air are largely the result of the object having to push the air out of its way as it moves. For an object moving slowly through a liquid, however, the drag force is mostly due to the viscosity of the liquid, a measure of how much resistance to flow the fluid has. Honey, which drizzles slowly out of its container, has a much higher viscosity than water, which flow's fairly freely.

The viscous drag force in a liquid depends on the shape of the object, but there is a simple result called Stokes’s law for the drag on a sphere. The drag force on a sphere of radius r moving at speed v through a fluid with viscosity η is

$\stackrel{\to }{D}$ = (6πηrv, direction opposite motion)

At small scales, viscous drag becomes very important To a paramecium (figure 1.2), a single-celled animal that can propel itself through water with fine hairs on its body, swimming through water feels like swimming through honey would to you. We can model a paramecium as a sphere of diameter $50\mu m$, with a mass of $6.5\times {10}^{-11}\text{kg}$. Water has a viscosity of $0.0010\text{N}\cdot {\text{s/m}}^2$.

Figure 1.2

You can test the viscosity of a liquid by dropping a steel sphere into it and measuring the speed at which it sinks. For viscous fluids, the sphere will rapidly reach a terminal speed. At this terminal speed, the net force on the sphere is

A. Directed downward.

B. Zero.

C. Directed upward.