Flow Experiment

You will use a similar graduated cylinder, which is included in your lab kit, in this experiment.  If you do not yet have the kit, submit a note to the instructor asking for an alternative procedure.

• In this experiment we will observe how the depth of water changes with clock time.

In the three pictures below the stream is shown at approximately equal time intervals.

Based on your knowledge of physics, answer the following, and do your best to justify your answers with the physical reasoning and insight: 

• As water flows from the cylinder, would you expect the rate of flow to increase, decrease or remain the same as water flows from the cylinder?

the decreasing horizontal distance traveled by the outflowing stream as it drops to the pavement is evidence of decreasing flow rate

• As water flows out of the cylinder, an imaginary buoy floating on the water surface in the cylinder would descend.

• Would you expect the velocity of the water surface and hence of the buoy to increase, decrease or remain the same?

Good student response: 

Acceleration by definition is the rate of change of velocity, i.e., a change in velocity divided by a time interval. 

Newton’s second law of motion states that the acceleration of a mass is directly proportional to the net force acting on it.

• How would the velocity of the water surface, the velocity of the exiting water, the diameter of the cylinder and the diameter of the hole be interrelated?  More specifically how could you determine the velocity of the water surface from the values of the other quantities?

STUDENT RESPONSE  The nature of the force accelerating the water is gravity. As the mass of the liquid in the cylinder escapes, there is less of a mass for gravity to act upon, so as the mass decreases, the total acceleration due to gravity upon the mass will decrease.

INSTRUCTOR COMMENT:  Your argument is well-conceived, and is a very good response at this point of the course. 

However the application of Newton's Second Law in this case is more subtle than it might appear.

Newton's Second Law applies only to the net force, as you stated previously. The water in the container is subject to forces in addition to that of gravity (e.g., the container holds the water up (and in)), so the net force on the water in the container is actually quite small.

Only the water that exits the hole speeds up, and the net force on that water turns out to be due to pressure.

We'll see the details later.

• The water exiting the hole has been accelerated, since its exit velocity is clearly different than the velocity it had in the cylinder.  

• Explain how we know that a change in velocity implies the action of a force?

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• What do you think is the nature of the force that accelerates the water from inside the cylinder to the outside of the outflow hole?

From the pictures, answer the following and justify your answers, or explain in detail how you might answer the questions if the pictures were clearer:

• Does the depth seem to be changing at a regular rate, at a faster and faster rate, or at a slower and slower rate?

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• What do you think a graph of depth vs. time would look like?

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• Does the horizontal distance (the distance to the right, ignoring the up and down distance) traveled by the stream increase or decrease as time goes on?

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• Does this distance change at an increasing, decreasing or steady rate?

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• What do you think a graph of this horizontal distance vs. time would look like?  Describe in the language of the Describing Graphs exercise.

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You can easily perform this experiment in a few minutes using the graduated cylinder that came with your kit. 

Setup of the experiment is easy.  You will need to set it up near your computer, so you can use a timing program that runs on the computer.  The cylinder will be set on the edge of a desk or tabletop, and you will need a container (e.g., a bucket or trash can) to catch the water that flows out of the cylinder.  You might also want to use a couple of towels to prevent damage to furniture, because the cylinder will leak a little bit around the holes into which the tubes are inserted.

• Your kit included pieces of 1/4-inch and 1/8-inch tubing.  The 1/8-inch tubing fits inside the 1/4-inch tubing, which in turn fits inside the two holes drilled into the sides of the graduated cylinder.
• Fit a short piece of 1/8-inch tubing inside a short piece of 1/4-inch tubing, and insert this combination into the lower of the two holes in the cylinder.  If the only pieces of 1/4-inch tubing you have available are sealed, you can cut off a short section of the unsealed part and use it; however don't cut off more than about half of the unsealed part--be sure the sealed piece that remains has enough unsealed length left to insert and securely 'cap off' a piece of 1/4-inch tubing.
• Your kit also includes two pieces of 1/8-inch tubing inside pieces of 1/4-inch tubing, with one end of the 1/8-inch tubing sealed.  Place one of these pieces inside the upper hole in the side of the cylinder, to seal it.
• While holding a finger against the lower tube to prevent water from flowing out, fill the cylinder to the top mark (this will be the 250 milliliter mark).
• Remove your thumb from the tube at the same instant you click the mouse to trigger the TIMER program.
• The cylinder is marked at small intervals of 2 milliliters, and also at larger intervals of 20 milliliters.  Each time the water surface in the cylinder passes one of the 'large-interval' marks, click the TIMER.
• When the water surface reaches the level of the outflow hole, water will start dripping rather than flowing continuously through the tube.  The first time the water drips, click the TIMER.  This will be your final clock time.
• We will use 'clock time' to refer to the time since the first click, when you released your thumb from the tube and allowed the water to begin flowing.
• The clock time at which you removed your thumb will therefore be t = 0.

Run the experiment, and copy and paste the contents of the TIMER program below:

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Now make a table of the position of the water surface vs. clock time.  The water surface positions will be the positions of the large marks on the cylinder relative to the outflow position (i.e., the distances you measured in the preceding question) and the clock times will as specified above (the clock time at the first position will be 0).  Enter 1 line for each event, and put clock time first, position second, with a comma between.

For example, if the first mark is 25.4 cm above the outflow position and the second is 22.1 cm above that position, and water reached the second mark 2.45 seconds after release, then the first two lines of your data table will be

0, 25.4

2.45, 22.1

If it took another 3.05 seconds to reach the third mark at 19.0 cm then the third line of your data table would be

5.50, 19.0

Note that it would NOT be 3.05, 19.0.  3.05 seconds is a time interval, not a clock time.  Again, be sure that you understand that clock times represent the times that would show on a running clock. 

The second column of your TIMER output gives clock times (though that clock probably doesn't read zero on your first click), the third column gives time intervals.  The clock times requested here are those for a clock which starts at 0 at the instant the water begins to flow; this requires an easy and obvious modification of your TIMER's clock times. 

For example if your TIMER reported clock times of 223, 225.45, 228.50 these would be converted to 0, 2.45 and 5.50 (just subtract the initial 223 from each), and these would be the times on a clock which reads 0 at the instant of the first event.

Do not make the common error of reporting the time intervals (third column of the TIMER output) as clock times.  Time intervals are the intervals between clicks; these are not clock times.

Despite the above warning, students still often report time intervals rather than clock times.  Another frequent error is to report clock times vs. depth (a report of depth vs. clock time would have clock time in the first column, depth in the second).

You data could be put into the following format:

clock time (in seconds, measured from first reading)	Depth of water (in centimeters, measured from the hole)
0	14
10	10
20	7
etc.	etc.

Your numbers will of course differ from those on the table.

The following questions were posed above.  Do your data support or contradict the answers you gave above?

• Is the depth changing at a regular rate, at a faster and faster rate, or at a slower and slower rate?

The data reported by students always indicates a trend toward increasing time intervals for the same distance interval.  This indicates that it takes longer and longer to achieve the same change in depth, so depth is changing at a slower and slower rate (more properly, the magnitude of the rate of change of depth with respect to clock time is decreasing; the depth is decreasing at a decreasing rate).

• Sketch a graph of depth vs. clock time (remember that the convention is y vs. x; the quantity in front of the 'vs.' goes on the vertical axis, the quantity after the 'vs.' on the horizontal axis).  You may if you wish print out and use the grid below.

Describe your graph in the language of the Describing Graphs exercise.

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caution:  Be sure you didn't make the common mistake of putting time intervals into the first column; you should put in clock times.  If you made that error you still have time to correct it.  If you aren't sure you are welcome to submit your work to this point in order to verify that you really have clock times and not time intervals

Now analyze the motion of the water surface:

• For each time interval, find the average velocity of the water surface.

Explain how you obtained your average velocities, and list them:

The average velocity for a time interval is defined as

• ave vel = average rate of change of position with respect to clock time,

which is in turn equal to

• ave vel = (change in position) / (change in clock time). 

To get the average velocity for each interval you would find the difference in the corresponding depths, and divide this by the difference in the clock times. 

For example

if the depth changed from 8 cm to 5 cm while the clock time changed from 40 seconds to 50 seconds, the average velocity of the water surface would be

ave vel = (change in depth) / (change in clock time) = (5 cm - 8 cm) / (50 s - 40 s) = -3 cm / (10 s) = -.3 cm/s.

• Assume that this average velocity occurs at the midpoint of the corresponding time interval.

What are the clock times at the midpoints of your time intervals, and how did you obtain them?  (Give one midpoint for each time interval; note that it is midpoint clock time that is being requested, not just half of the time interval.  The midpoint clock time is what the clock would read halfway through the interval.  Again be sure you haven't confused clock times with time intervals.  Do not make the common mistake of reporting half of the time interval, i.e., half the number in the third column of the TIMER's output):

If for example the depth changed from 8 cm to 5 cm while the clock time changed from 40 seconds to 50 seconds, the midpoint clock time for this interval would be 45 seconds, halfway between 40 s and 50 s. 

The midpoint clock time would not be half of 40 seconds (i.e., 20 seconds) or half of 50 seconds (i.e., 25 seconds).

• Make a table of average velocity vs. clock time.  The clock time on your table should be the midpoint clock time calculated above.

Give your table below, giving one average velocity and one clock time in each line.  You will have a line for each time interval, with clock time first, followed by a comma, then the average velocity.

If for example the depth changed from 8 cm to 5 cm while the clock time changed from 40 seconds to 50 seconds, the midpoint clock time for this interval would be 45 seconds and the average velocity -.3 cm/s.

The corresponding point on the specified graph would therefore be (45 s, -.3 cm/s).

There would be one such point for each interval observed in your data. 

• Sketch a graph of average velocity vs. clock time.  Describe your graph, using the language of the Describing Graphs exercise.

Your graph would depict negative velocities.  The points would lie below the horizontal axis, and would be approaching the horizontal axis.  The graph could be described as 'negative and increasing, approaching zero'.  The graph might be increasing at an increasing, decreasing or constant rate. 

If you left off the negative signs on your velocities, your graph would in fact depict the speed of the water surface vs. clock time, as opposed to its velocity vs. clock time.  Your graph would then be decreasing toward zero.  It might be decreasing at a constant, increasing or decreasing rate.

• For each time interval of your average velocity vs. clock time table determine the average acceleration of the water surface.   Explain how you obtained your acceleration values.

The average acceleration for an interval is the average rate at which velocity changes with respect to clock time, which in turn is average acceleration = (change in velocity) / (change in clock time). 

This calculation would be done for every interval of your velocity vs. clock time graph.

• Make a table of average acceleration vs. clock time, using the clock time at the midpoint of each time interval with the corresponding acceleration.

Give your table in the box below, giving on each line a midpoint clock time followed by a comma followed by acceleration.

Acceleration should be very nearly constant. The reasons for this are related to the energy analysis, but the reasoning itself is beyond the scope of the Phy 122 or Phy 202 course.  Phy 232 students would be expected to have the background to understand the explanation, but at this point this is not required of students in any course.

In this experiment the accelerations are determined from the original data by taking difference quotients, which is done twice. This magnifies the inevitable uncertainties in the data and makes it difficult to tell whether the acceleration of the system is indeed constant.

• Do your data indicate that the acceleration of the water surface is constant, increasing or decreasing, or are your results inconclusive on this question?
• Do you think the acceleration of the water surface is actually constant, increasing or decreasing?

One student's response:

The results would seem to indicate that the acceleration is constant because they show no general trend in either direction, but I would say that they are inconclusive. I think that the acceleration is constant.

INSTRUCTOR RESPONSE:

I agree. The ideal behavior would in fact be a constant acceleration; there's a slight deviation due mainly to second-order frictional effects in the outflow, especially near the end, but it would require very accurate measurements to detect this.

The unavoidable 'noise' the data for this experiment introduces a lot of uncertainty into your interval-by-interval results for the acceleration, making it impossible to draw any sort of conclusion from those calculations.

The linearity of your v vs. t graph is, however, good evidence of constant acceleration.