Experiment 6:  Continuity Equation and Bernoulli's Principle

Experiment 7:  Terminal Velocity of an Object in a Fluid

Experiment 8: Bottle Engine I

From the range of a horizontally directed flow out of a container, we can determine the horizontal velocity of the water as it flows from the container by considering each water particle as a projectile. We can correlate this velocity of flow with the depth of the water as measured from the point which the water flows from the container.

Set up the container above a level surface as shown on the video clip, and note the vertical distance from the outflow hole to the floor or the ground.

• Fill the container to maximum depth. Let a brief stream of water flow from the hole and take the data necessary to determine the maximum range of the stream.
• Repeat for 6 depths equally distributed between depth 0 and maximum depth.

Analyze your results.

• Make a table of maximum stream range vs. water depth.
• For each maximum stream range, determine the velocity of the water, assuming that each water particle is a projectile falling freely to the floor.
• Make a table of the velocity of the outflowing water vs. its depth relative to the hole, and construct a graph of outflow velocity v vs. depth y.
• Linearize your table using an appropriate power function transformation of the velocity column. Determine the power of the power function that gives you the best linearization.
• From the power that gives the best linearization, determine the function v = k y ^ p that models outflow velocity v vs. depth y.
• Linearize your table in another way by constructing a table of log(v) vs. log(y). Construct a graph of log(v) vs. log(y) and determine the slope m and the vertical-axis intercept b of your graph. The resulting model will be v = 10^b * y ^ m. Compare this model with your previous model.

Make sense of your results.

• Suppose that water depth is 10 cm. Then in some short time interval, if water is flowing out of the hole, 1 gram of water at the 10-cm position above the hole will be replaced by 1 gram of water flowing out of the hole.
• What is the potential energy, relative to the hole, of the 1 gram of water at the 10-cm position above the hole?
• What is the potential energy, relative to the hole, of the 1 gram of outflowing water that replaces the 1 gram at the 10 cm position?
• What therefore is the corresponding change in the potential energy of the system?
• By conservation of energy, what should be the kinetic energy change of the system?
• If the kinetic energy resides in the 1 gram of water that escapes the hole, what must be the velocity of that bit of water?
• Repeat this analysis for an unspecified small mass `dm at vertical position y relative to the hole, as is replaced over short time interval by an equal small mass flowing out of the hole. From energy conservation determine the velocity of that small mass of water as it exits the hole.
• Do you expect that any energy is dissipated in the form of thermal energy as the water flows from the hole?
• To your table of outflow velocity vs. water depth, add a column for the ideal outflow velocity at each depth, as predicted by your analysis of energy exchanges.

Analyze errors and uncertainties.

• According to your table of actual vs. ideal outflow velocities, what percent of the expected kinetic energy is actually present at each depth?
• Energy must be conserved. The potential energy changes must be manifested somewhere in the system. Clearly the lost PE  is not all going into the kinetic energy of the outflowing water. What has happened to the potential energy loss not accounted for by outflowing kinetic energy?
• How might you design an experiment, given sufficiently accurate instruments, to verify your conjecture about where the unaccounted-for energy goes?

Conservation of energy tells us that a fast-moving fluid tends to exert less pressure than a slow-moving fluid.  Using a series of tubes of different diameters we observe pressure changes in a fluid as it changes velocity.  Velocities are determined from the projectile behavior of the exiting fluid and continuity.

  Using the tube apparatus described in the video clip, we will observe the velocity and corresponding pressure changes as water flows through a series of tubes.

Begin by holding the apparatus under a faucet, as shown, and keep the top end full. Notice whether the water in the clear tubes rises or falls as the faucet fills the apparatus.

• Is the water in the tubes higher before or after the passageway narrows? Do you therefore conclude that the water pressure in the tube is greater before or after the passageway narrows?
• Does the water level in the tubes increase or decrease as water flows toward the open end of the lower tube? Do you therefore conclude that water pressure is increasing or decreasing as water flows down the tube?

• Adjust the faucet so that the tube just barely remains full. Record the height of the water in each clear tube.
• Turn the faucet on as far as possible without creating a major mess. Record the height of water in each clear tube.

Now determine the exit velocity of the water from the lower tube:

• Position the apparatus so that the lower tube is really horizontal, and is resting at a known distance above a horizontal surface.
• Using a milk jug or some other container, fill the apparatus with water and keep it full until you have determined the range of the stream as it arcs to the ground.
• Treating the water stream as a projectile determine its horizontal velocity.

Determine the water velocities in the apparatus just before and just after the passageway narrows.

• The velocity just after the passageway narrows will equal the horizontal velocity of the water stream as it exits.
• From the relative diameters of the passageway before and after it narrows, determine the velocity of the water stream before the narrowing.

Determine the difference in pressure, and the difference in .5 `rho v^2, before and after the narrowing.

• Determine the pressure that results from the vertical column in the clear tube before narrowing, using the height of the column and the density of water.
• Determine the same quantites based on the height of water in the clear tube after narrowing.
• Using the velocities before and after narrowing determine the corresponding values of .5 `rho v^2, where `rho = 1000 kg/m^3 is the mass density of water.
• By how much does the pressure change as the water crosses the narrowing section?
• By how much does .5 `rho v^2 change as the water crosses the narrowing section?
• What do you conclude about pressure change and change in .5 `rho v^2?
• Using the heights of water in the clear tubes in the last segment of the lower section, make a graph of water pressure vs. distance along the lower tube. Describe your graph.

A marble pulled through a cylinder full of water will experience a viscous drag force which increases as marble velocity increases.  If the marble is pulled through the water by a weight suspended over a pulley we can determine the approximate velocity at which the weight of the object (corrected for pulley friction) is equal to the drag force experienced at the marble's terminal velocity.  When the experiment is conducted for a variety of suspended weights we can obtain information about drag force vs. velocity.   A graph of drag force vs. velocity demonstrates the power function relating the two quantities.

It is a common experience that when an object is dragged through the water, the faster we drag it the more force of water resistance we counter.

In this experiment we attempt to determine the drag force vs. velocity function for a marble rising in a tube of water.

We will use an Atwood machine with a marble suspended by a thread on one side of the pulley and a series of different hanging weights on the other.

• For each hanging weight total, you will use a pendulum timer to determine position vs. clock time as the weight descends from rest pulling the marble upward through the water.
• From your position vs. clock time curves you will determine the various terminal velocities of the marble as it is subjected to different forces.
• From this information you will determine the drag force on the marble vs. its velocity through the water.

Begin by setting up the Atwood machine and the water cylinder as shown on the video clip.

• As shown on the clip, find the mass that must be added to the other side of the Atwood machine, with the marble suspended in the water, to just overcome the friction in the pulley.
• Now add a mass of 2 grams to the hanging mass and observe how the mass descends, at first increasing its velocity but then maintaining a constant velocity.
• Using the pendulum timer, as shown on the video clip, determine the time required for the marble to ascend from rest through distances of 15, 30, 45 and 60 cm.
• Add an additional 2 gram mass and repeat your trial.
• Twice more, increase the hanging mass by 2 grams and repeat.

Analyze your results.

• For each of the four hanging mass totals,

• use your data to determine the time required to travel the first, second, third and fourth 15-cm. distances.
• For each 15-cm. interval determine the midpoint clock time, as measured from the instant the system was released, for that interval.
• Determine the average speed of the marble on each 15-cm interval.
• Graph the average speed of the marble on an interval vs. the midpoint clock time of that interval.

• From each graph determine whether the marble reached the terminal velocity, and determine what the terminal velocity is for that amount of hanging mass.

• For each amount of hanging mass, determine the net force exerted by the mass in excess of friction.
• Make a table showing the net force on the marble vs. the terminal velocity of the marble.

• Sketch a graph of terminal velocity vs. net force.

When air in a sealed soft-drink container is heated, water in the container can be forced upward through a thin vertical tube inserted through a stopper in the neck of the container.  The heating first occurs at nearly constant volume until water reaches the top of the tube, then continues at constant pressure until the air reaches its maximum temperature.  The potential energy of a significant amount of water is thus increased by a measurable amount.  This potential increase can be compared to the thermal energy required to heat the air in the container and the efficiency of the process can be estimated.

SAFETY NOTE:  This experiment involves working with hot water. Take the care you usually do when working with hot water in order to avoid injury to yourself or damage to your property. If you cannot perform this experiment safely, contact the instructor for an alternative.

As demonstrated on the video clip, you will

• Use the tendency for a heated gas to increase its pressure and/or to expand in order to perform work by increasing the potential energy of a measurable quantity of water.
• Measure the quantities needed to determine the change in the water's potential energy, then you will calculate the amount of thermal energy required to increase the temperature of the gas.  
• Observe the state of the system (its temperature, pressure and volume) at various stages, and you will make comparisons between what you observe and the predictions of Bernoulli's Equation and the Ideal Gas Law.
• Construct a P vs. V graph of the process and interpret the graph.

Begin by assembling the apparatus:

• Pour 16 fluid ounces plus three tablespoonfuls of water (1.5 additional fluid ounces, for a total of 17.5 fluid ounces) into a 3-liter bottle. Note that 17.5 ounces is very close to 500 ml, to that this this will leave 3000 ml - 500 ml = 2500 ml of air in the bottle.
• Calibrate the 'thermometer on a stick' using ice water and boiling water, and insert the thermometer  into the bottle.
• As shown on the video clip, remove the stoppers from the ends of the thin tubes, except for the one required to seal the end of the tube to be used as a pressure gauge. Being careful not to spill the alcohol solution in the pressure-gauge tube, insert the stopper into the top of the  bottle.
• Attach the pressure gauge tube to a convenient measuring device (e.g., the pendulum stand). As shown on the video clip, you may carefully adjust the level of the alcohol column so as to maximize the length of the air column in its end.
• Place the bottle in the deep pot or crock to which you will be adding hot water in order to heat the system.
• Be sure that the lower end of the long tube is lying on the bottom of the bottle, covered by water. Raise the other end of the long tube to a height of 50 cm above the level of the water in the bottle, and be sure that no part of the tube is higher than this level.

While the system is at room temperature take the readings you will need to determine the length of the air column in the pressure gauge and the temperature of the air in the bottle as indicated by the thermometer.

Now raise the temperature of the system just enough to raise the water to the end of the long tube, at a height of 50 cm relative to the water level in the bottle.

• Add hot but not boiling water to the pot. The temperature of the water should be approximately 60 C - 80 C, hot but significantly below boiling.
• While holding down the bottle slowly add hot water to the pot, watching the elevation of the water in the tube. The water should rise slowly as you add more and more water. When the water is nearly at the 50 cm height, stop adding hot water and do whatever is necessary to keep the liquid at that level for two minutes (e.g., if the liquid begins to rise further, threatening to overflow, you can raise the bottle slightly so that less hot water is in contact with the bottle).
• With the water at the 50 cm height, read the thermometer to determine the temperature of the gas, and take the data you will need to determine the pressure (i.e., to determine the length of the air column in the pressure gauge).

Now position a large cup or a bottle to catch the water that flows out of the high end of the tube.

• Still holding the bottle down, slowly add more hot water until the pot is full. Observe the pressure gauge and the thermometer every 30 seconds, until water stops flowing from the end of the tube.
• Remove the bottle from the water and set it aside.

Measure the mass of water you have collected at the 50 cm height. You may use the graduated cylinder, or perhaps more conveniently you can measure the number of fluid ounces of water in the cup and multiply by the approximately 28 ml per ounce, which corresponds to 28 grams per ounce. Note that a tablespoon contains 14 ml and a teaspoon contains 1/3 that amount.

Determine the Kelvin temperature, the volume of the air in the container, and the pressure in Pa at three stages in the process:

• At the beginning, when everything was at room temperature and atmospheric pressure,
• at the point when the water was at the top of the tube and before it started flowing out, and
• at the time when the water stopped flowing.

Determine the amount of work done by the system and the thermal energy added to the system.

• The work done is the potential energy increase of the water. Recall that the potential energy increase is equal to the product of the weight of the water and the altitude to which it was raised, relative to its starting point.
• From the original 2500 ml volume, the room temperature and the assumption of atmospheric pressure (100,000 Pa at Southwest Va. altitude), determine the number of moles of air in the system.
• It requires approximately 5/2 R = 5/2 * 8.31 Joules / (mole Kelvin) to increase the temperature of air at constant volume, and 7/2 R to increase the temperature of air at constant pressure.

• Assuming that there was no significant increase in volume as the water rose in the thin tube (i.e., that the volume of the tube was very small in proportion to that of the container, which is pretty much the case) and before water started flowing out of the tube, and that as the water flowed out of the tube the pressure remained constant, determine how much thermal energy must have been added to the system.

Determine the efficiency of the system, defined as the ratio of the useful work done by the system to the thermal energy put into the system.

Sketch and analyze a graph of the pressure of the gas vs. its volume.

• Locate three points on the graph, corresponding to the pressure and volume at the beginning, at the point when water reached the top of the tube, and at the point when water stopped flowing from the tube.
• Connect these points in the matter that seems most reasonable.
• Describe your graph and your reasoning regarding its shape.
• Determine the area under the graph and above the horizontal axis, in units of Pa * m^3.
• Show that the units of area are in fact units of energy, and explain how the area is related to an energy change observed in the experiment.

Apply Bernoulli's equation to determine the pressure required to raise the water to the 50 cm height and relate your results to the Ideal Gas Law..

• This calculation is straightforward. Fluid velocity is negligible so only changes in `rho g h and P are relevant.
• Compare a point at the top of the tube to a point at the level of the water surface in the container.
• Compare this predicted pressure with the pressure you observed.
• What Kelvin temperature would be required to build this much pressure in the system, given the starting temperature (i.e., room temperature)?

In the preceding experiment with the bottle engine, we analyzed the energy of the system as it moved through 3 states. These states were

• The original state, which we will call State 1, at room temperature, volume 2500 ml and atmospheric pressure.
• The state at which the water column just reached its maximum height, which will call State 2, in which the pressure and temperature of the system have increased without a significant increase in the 2500 ml volume.
• The state at which the system reaches its maximum temperature, which will call State 3, in which the pressure is still that required to raise the water to its maximum height, the temperature is maximized (as high as we can get it with the given water temperature), and the volume of the system has increased to its maximum with result that water has been displaced.

Begin by speculating on the answers to the following questions:

• Note that for these temperatures there is a limit to how much the system can expand, and therefore to how much water can be raised; it follows that if the water is raised only a small distance, not much work is done.
• There is also a limit to how far water can be raised given these temperatures, and that if we raise the water to this limiting height very little water will be raised so that very little work is done.
• So what would a graph of work done vs. height look like?

The actual experiment is fairly simple, though it takes awhile.

Using at least four different heights, see how much water is raised, using the same initial state, and using hot water (between 60 and 80 Celsius) at the same temperature each time.

• Start each trial with the entire system at room temperature. This includes the water in the bottle, which should be poured from a container which you are sure is at room temperature.

Analyze your results:

Design an experiment to determine the effect of maximum temperature on maximum possible efficiency.

• Describe how you would conduct an experiment to determine the relationship between the maximum temperature of the system and the maximum possible efficiency of the system.
• How would you expect maximum possible efficiency to change with maximum temperature?
• For a given maximum temperature, how would room temperature affect the maximum possible efficiency?