# 5.2 Drag forces  (Page 4/6)

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## Stokes’ law

${F}_{\text{s}}=6\mathrm{\pi r\eta v},$

where $r$ is the radius of the object, $\eta$ is the viscosity of the fluid, and $v$ is the object’s velocity.

Good examples of this law are provided by microorganisms, pollen, and dust particles. Because each of these objects is so small, we find that many of these objects travel unaided only at a constant (terminal) velocity. Terminal velocities for bacteria (size about $\text{1 μm}$ ) can be about $\text{2 μm/s}$ . To move at a greater speed, many bacteria swim using flagella (organelles shaped like little tails) that are powered by little motors embedded in the cell. Sediment in a lake can move at a greater terminal velocity (about $\text{5 μm/s}$ ), so it can take days to reach the bottom of the lake after being deposited on the surface.

If we compare animals living on land with those in water, you can see how drag has influenced evolution. Fishes, dolphins, and even massive whales are streamlined in shape to reduce drag forces. Birds are streamlined and migratory species that fly large distances often have particular features such as long necks. Flocks of birds fly in the shape of a spear head as the flock forms a streamlined pattern (see [link] ). In humans, one important example of streamlining is the shape of sperm, which need to be efficient in their use of energy.

## Galileo’s experiment

Galileo is said to have dropped two objects of different masses from the Tower of Pisa. He measured how long it took each to reach the ground. Since stopwatches weren’t readily available, how do you think he measured their fall time? If the objects were the same size, but with different masses, what do you think he should have observed? Would this result be different if done on the Moon?

## Phet explorations: masses&Springs

A realistic mass and spring laboratory. Hang masses from springs and adjust the spring stiffness and damping. You can even slow time. Transport the lab to different planets. A chart shows the kinetic, potential, and thermal energy for each spring.

## Section summary

• Drag forces acting on an object moving in a fluid oppose the motion. For larger objects (such as a baseball) moving at a velocity $v$ in air, the drag force is given by
${F}_{\text{D}}=\frac{1}{2}{\mathrm{C\rho Av}}^{2},$
where $C$ is the drag coefficient (typical values are given in [link] ), $A$ is the area of the object facing the fluid, and $\rho$ is the fluid density.
• For small objects (such as a bacterium) moving in a denser medium (such as water), the drag force is given by Stokes’ law,
${F}_{\text{s}}=6\text{πηrv},$
where $r$ is the radius of the object, $\eta$ is the fluid viscosity, and $v$ is the object’s velocity.

## Conceptual questions

Athletes such as swimmers and bicyclists wear body suits in competition. Formulate a list of pros and cons of such suits.

Two expressions were used for the drag force experienced by a moving object in a liquid. One depended upon the speed, while the other was proportional to the square of the speed. In which types of motion would each of these expressions be more applicable than the other one?

As cars travel, oil and gasoline leaks onto the road surface. If a light rain falls, what does this do to the control of the car? Does a heavy rain make any difference?

Why can a squirrel jump from a tree branch to the ground and run away undamaged, while a human could break a bone in such a fall?

## Problems&Exercise

The terminal velocity of a person falling in air depends upon the weight and the area of the person facing the fluid. Find the terminal velocity (in meters per second and kilometers per hour) of an 80.0-kg skydiver falling in a pike (headfirst) position with a surface area of $0\text{.}\text{140}\phantom{\rule{0.25em}{0ex}}{\text{m}}^{2}$ .

$\text{115}\phantom{\rule{0.25em}{0ex}}\text{m/s;}\phantom{\rule{0.25em}{0ex}}\text{414}\phantom{\rule{0.25em}{0ex}}\text{km/hr}$

A 60-kg and a 90-kg skydiver jump from an airplane at an altitude of 6000 m, both falling in the pike position. Make some assumption on their frontal areas and calculate their terminal velocities. How long will it take for each skydiver to reach the ground (assuming the time to reach terminal velocity is small)? Assume all values are accurate to three significant digits.

A 560-g squirrel with a surface area of $\text{930}\phantom{\rule{0.25em}{0ex}}{\text{cm}}^{2}$ falls from a 5.0-m tree to the ground. Estimate its terminal velocity. (Use a drag coefficient for a horizontal skydiver.) What will be the velocity of a 56-kg person hitting the ground, assuming no drag contribution in such a short distance?

$\text{25 m/s; 9.9 m/s}$

To maintain a constant speed, the force provided by a car’s engine must equal the drag force plus the force of friction of the road (the rolling resistance). (a) What are the magnitudes of drag forces at 70 km/h and 100 km/h for a Toyota Camry? (Drag area is ${\text{0.70 m}}^{2}$ ) (b) What is the magnitude of drag force at 70 km/h and 100 km/h for a Hummer H2? (Drag area is $2\text{.}{\text{44 m}}^{2}$ ) Assume all values are accurate to three significant digits.

By what factor does the drag force on a car increase as it goes from 65 to 110 km/h?

$2\text{.}9$

Calculate the speed a spherical rain drop would achieve falling from 5.00 km (a) in the absence of air drag (b) with air drag. Take the size across of the drop to be 4 mm, the density to be $1\text{.}\text{00}×{\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}{\text{kg/m}}^{3}$ , and the surface area to be ${\mathrm{\pi r}}^{2}$ .

Using Stokes’ law, verify that the units for viscosity are kilograms per meter per second.

$\left[\eta \right]=\frac{\left[{F}_{\text{s}}\right]}{\left[r\right]\left[v\right]}=\frac{\text{kg}\cdot {\text{m/s}}^{2}}{\text{m}\cdot \text{m/s}}=\frac{\text{kg}}{\text{m}\cdot \text{s}}$

Find the terminal velocity of a spherical bacterium (diameter $2.00 \mu m$ ) falling in water. You will first need to note that the drag force is equal to the weight at terminal velocity. Take the density of the bacterium to be $1\text{.}\text{10}×{\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}{\text{kg/m}}^{3}$ .

Stokes’ law describes sedimentation of particles in liquids and can be used to measure viscosity. Particles in liquids achieve terminal velocity quickly. One can measure the time it takes for a particle to fall a certain distance and then use Stokes’ law to calculate the viscosity of the liquid. Suppose a steel ball bearing (density $7\text{.}8×{\text{10}}^{3}\phantom{\rule{0.25em}{0ex}}{\text{kg/m}}^{3}$ , diameter $3\text{.0 mm}$ ) is dropped in a container of motor oil. It takes 12 s to fall a distance of 0.60 m. Calculate the viscosity of the oil.

$0\text{.}\text{76 kg/m}\cdot s$

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