M-1 MOTION IN VISCOUS MEDIA

1. PURPOSE

(1) To study a body falling through a medium with non-negligible viscous drag.

(2) To determine the terminal velocity of such a body and to investigate whether the drag is proportional to the square of the velocity.

2. APPARATUS

Two-meter stick, coffee filters or paper muffin cups, dark cloth background, stroboscopic lamp, camera.

3. THEORY

Textbooks often assert that when a body moves under viscous or frictional drag, that the retarding force is velocity dependent. If the velocity is large enough, the drag force can be proportional to the square of the velocity. Seldom does the student test these assertions by experiment or direct experience.

When a body falls through a viscous medium, the net force on it is the sum of the gravitational force and the drag force:

[1]

F = mg - f

where F is the net force, m is the body's mass, and f is the drag force. We assume a coordinate system were positive is downward.

Consider a drag force proportional to the speed:

[2]

f = kv²

The net force on the body is, by Newton's law:

[3]

F = mg - kv² = ma

If the viscous drag increases with velocity, given enough time it will become as large as the gravitational force, and being opposite to mg, the net force on the body becomes zero, the acceleration is then zero, and the body continues moving with constant velocity. This condition is called "terminal velocity."

[4]

mg = kv² , or v² = mg/k

[5]

so, v = √(mg/k)

This shows that the terminal velocities of bodies falling through the same medium are proportional to the square roots of the masses. Since y = vt, then the distance two such bodies fall in the same time is:

[6]

y₂/y₁ = √(m₂/m₁)

Therefore if two such bodies were dropped simultaneously, and one were four times as heavy as the other, the heavier one would fall twice as far in the same time.

Question: If two bodies were dropped simultaneously, one being twice as heavy as the other, what distance will the heavier one fall compared to the lighter one in the same time?

4. PROCEDURE

The predictions of Eq. 5 may be easily tested with minimal equipment if we can find a body that reaches terminal speed very quickly, and falls slowly enough to time with simple equipment. Such a situation may be easily achieved!

The small paper cups used for baking muffins not only are light in weight, but their shape causes them to fall through air in a stable manner (they don't tumble). Also, they are easily stacked, to give a body of the same shape, but mass 2m, 3m, 4m, etc., where m is the mass of one paper cup.

The larger cups used as filters in automatic coffee machines also may be used.

By dropping such cups and timing their fall we can test the assumptions of Eq. 1 and Eq. 2. For much of the work we won't even need a timing device. We will use the technique of comparing two events occurring simultaneously, a method often used by Galileo when he studied the motion of moving bodies. In this experiment you will drop cups of different mass from different heights simultaneously so that they reach the floor simultaneously. You will be able to not only see, but hear, whether they hit the floor at the same time.

(1) First simultaneously drop two cups, one of mass m from a height of 2 meters, and the other of mass 2m (two nested cups) from a lower height, say 50cm. Which one hits the floor first? Try it again, adjusting the distance of fall of the heavier one until they reach the floor simultaneously.

(2) In this case, and subsequent cases, do each trial several times, so that you can determine the variation (error) in the determination of the smaller height. This gives you information about how reliable and reproducible your experiment is, including errors in the simultaneous dropping, in determining whether they land simultaneously, and all other sources of indeterminate error.

(3) Repeat with cups of m and 3m, m and 4m, m and 5m, etc. You may wish to use a larger distance of fall, say 4 meters, but then you may have a problem of starting them falling simultaneously.

(4) Set up a black cloth background, strobe lamp, and camera to photograph the fall of one of the cups. The photo should answer the question of how quickly terminal speed is attained, and also give you a direct measure of that terminal velocity.

(5) Investigate the fall of other bodies, using all of the above techniques. Suggestions: a wadded paper napkin, a feather, a sponge ball, a toy parachute.

(6) Consider the fall of a rubber ball. Using the methods of parts 1 and 2, you will find that the only way that two balls dropped at the same time will hit the floor simultaneously is if they fall from the same initial height. This will be found to be true even if the balls have different mass! Clearly something is very different here compared to the case of the falling paper cups. What does this fact tell you about the nature and/or size of the drag force? Does it tell you anything about the gravitational force?

(7) If a strobe light and suitable camera are available, photograph the fall of objects against a black background in a darkened room. The photos may be enlarged (or negatives may be enlarged by projection) to analyze the motion.

5. ANALYSIS

(1) Try to develop from your data a table of terminal speed vs. mass for the falling paper cups. Consider using a graph of v² vs. m to do this.

(2) You have been given the job of testing how well the data confirms the equation:

f = kvⁿ

Your experiment seems to confirm that the exponent n has a value of 2. But, considering the uncertainties in your measurements of time (simultaneity) and distance, what is the uncertainty in your determination of n? You do not need to know the actual times of fall, nor the actual terminal velocities to answer this question.

6. QUESTIONS

(1) Has this experiment confirmed Newton's law F = ma? If so, how? If not, why not?

(2) Has this experiment confirmed that the terminal velocity of a paper cup falling in air is proportional to the square root of its mass? If so, how? If not, why not?

(3) Has this experiment confirmed that the viscous drag force on a paper cup falling in air is proportional to the square of its speed? If so, how? If not, why not?