# AM-1 MOTION IN VISCOUS MEDIA

1. Purpose:

To study viscous drag on a body falling in a viscous medium. To seek confirmation of Stoke's law for such motion.

2. Apparatus:

Balance scale.
Stopwatch.
Metal balls of several sizes 1" and less, drilled with hole for a string.
Glass or plastic container, preferably a long graduated cylinder.

Viscous liquid such as Calgon liquid, concentrated shampo (dilute if necessary).

3. Preliminary observations.

1. Put the container of liquid on the pan of a scale. Suspend the ball from a string, immersed in the liquid, holding the string with your hand. The ball should not touch the container.
2. Balance the scales. Record the weight reading.
3. Lower the ball to the bottom, so it sits on the bottom. Record the weight reading of the scale.
4. Lift the ball up again, so it doesn't touch the container, but it is surrounded by the liquid. Record the weight reading again.
5. Now quickly drop and raise the ball in the liquid, being careful that it never touches the sides. The buoyant force on the ball is independent of velocity, so it is constant in size, but changes direction. But as the ball drops, the pan on that side drops. When the ball rises, the pan rises. Something must be exerting a changing force on the liquid.

4. What's going on?

Explanation: The viscous force opposes the velocity. When the ball moves down, the liquid exerts an upward viscous force on it. By Newton's third law, the ball exerts and equal and opposite force down on the liquid. The scale therefore exerts an upward force on the liquid to maintain the liquid equilibrium. Therefore the scale reading increases. That's equivalent to saying the pan supporting the liqid moves down.

3. Now make an experiment of it.

In all cases you will need to measure the time of fall of the ball over a measured distance, to obtain its terminal speed. Each ball should be carefully weighed, and its diameter determined. Finally, you will directly determine the drag force from the scale imbalance as the ball falls with constant speed. Each measurement of drag force should be repeated several times, since that is very likely the measurement with the most uncertainty. This is easily and quickly done, since you have a string attached to the ball to retrieve it from the bottom of the liquid. Don't let the entire string fall into the liquid.

(1) Suspend the ball in the liquid and balance the scales. The ball should not touch the beaker. The liquid exerts an upward buoyant force on the ball, so the ball exerts and equal size and oppositely directed force downward on the water. The diagram shows that in this case the force on the pan of the balance is just the weight of beaker and water (W) plus the buoyant force.

(2) Release the string, allowing the ball to fall in the viscous liquid. The net force on the ball is zero. The liquid exerts and upward viscous drag force on the ball, and the ball exerts an equal size and oppositely directed force on the water. This causes an additional force on the pan of the balance, equal to the size of the viwscous drag force. The scales will register unbalance during the fall, by an amount equal to the viscous drag. Note the scale's balance pointer position during the fall. Time the fall over a measured distance.

(3) Now raise the ball and suspend it by the string again. As you suspend it there, add weights to one pan to duplicate the amount of unbalance you observed above. The amount of weight you add is equal to the viscous drag.

(4) Now that you know how much unbalance to expect, and in which direction, you may adjust the initial balance to allow the pointer to be to one side of the scale, allowing for larger deflections. Just be sure that the scales are not "bottoming out" initially and finally.

(5) Of course you will want to repeat this several times to see how reproducible it is. If the unbalance is small, you may wish to create a new scale and extend the pointer length to get more sensitivity. An even better arrangement is to use a mirror firmly attached to the pointer and perpendicular to the plane of its motion, reflecting a beam from a solidly fixed laser pointer to a distant wall. This is called an optical lever, because the angle of beam deflection is twice the angle of mirror deflection, and is therefore very sensitive.

(6) Now do a series of measurements using balls of different masses and different radii. Of course you must directly weigh each ball, and measure its diameter with micrometer calipers.

## Alternate apparatus.

The figure shows a specialized balance for this experiment, assembled from steel construction set parts. A horizontal beam is pivoted near one end (the pivot point determined by the weight of the suspended plastic graduate at the left end, filled with liquid). At the right end are additional balancing weights, and a laser pointer. A clothespin serves to hold the "on" button of the laser pointer, and also to keep the ponter from rolling. The pivot axis is located about 2 cm above the beam, to reduce the balance sensitivity and eliminate "jiggle" of the laser beam.

(1) The system is carefully balanced, and the laser beam position is noted on a scale on the wall. This is the reference (zero) position for the system.

(2) This is a good time to make a calibrated scale on the far wall where the laser pointer beam hits. Add weight to the top of the graduated clinder in increments of 1 gram, marking the laser pointer position for each.

(3) A metal sphere is suspended in the liquid by a string, and the new laser beam position is recorded. This represents the unbalance caused by the buoyant force on the sphere.

(4) The sphere is allowed to fall at constant speed. The time to fall a measured distance is recorded. The laser beam position is noted when it stabilizes, but, of course, while the sphere is falling. This pointer position represents unbalance caused by both buoyancy and viscous drag.

## Analysis.

Stoke's law is Fdrag = 6πμmrv where μ is the viscosity, m is the mass of the ball, r is its radius and v is its terminal velocity. You can determine the terminal velocity from d = vt by timing the fall over a measured distance.

If you take enough data you can plot Fdrag/mv vs. r. The plot should be a straight line with slope 6πμ. From this you can determine μ.

© 1999, 2004, by Donald E. Simanek.