FM1 THE LEAKING RESERVOIRTo investigate the relation between water height in a leaky reservoir and the exit velocity at the leak. 2. APPARATUS: Clear plastic or glass tube 1 meter long, with an exit hole at the bottom, 2 meter sticks (one of them metal or plastic, stop watch, catch bucket. 3. REFERENCES: Any standard textbook treatment of fluid mechanics. Review Pascal's principle and Bernoulli's law. 4. GENERAL METHOD:
A vertical tube (the water reservoir) is filled with water. It has a hole at the very bottom allowing water to leak out. While leaking, the height of the water in the reservoir is measured as a function of time. From this data, graphed, you can verify the pressurevelocity relation at the exit hole. The reservoir tube should be clear so the water level may be measured. Glass or clear plastic is fine. The glass tubes from the standard "resonance tube" for measuring the velocity of sound may be repurposed for this experiment. See S1 Speed of Sound, Fig. 4. 5. THEORY: The pressure at the bottom of a liquid column is P = P_{o} + ρgh where h is the height of the column, g is the acceleration due to gravity and ρ is the density of the liquid. P_{o} is the pressure at the surface of the liquid. The pressure difference, P, at the exit orifice is related to the exit velocity of the liquid, v, is given by P = ½ρv^{2} . So the exit velocity of liquid at the bottom of a reservoir of height h is given by:
The output stream is nearly parabolic, and behaves like a horizontally fired projectile. Its horizontal range is given by:
Derive this. 6. PROCEDURE:
Mount the reservoir tube securely at the edge of a sink so the output stream of water will empty cleanly into the sink. Fasten a meter stick vertically alongside the reservoir. Cover the exit hole with a finger, and fill the reservoir with water. Uncover the output hole and record the height of the water as a function of time.
Mount the reservoir tube and meter stick as in Method A. But this time the output stream will fall all the way to the floor. To make this less messy, a bucket is placed to catch the stream and the bucket is moved as the stream becomes weaker. It is still good to have a mop handy. A plastic or metal meter stick is mounted horizontally just above the bucket. It is convenient to have its "zero" end directly below the reservoir output hole. Use a plumb line. Now record the height of water in the reservoir as a function of time AND the distance of the stream measured horizontally on the meter stick. One person can use the stopwatch, another monitors the reservoir height, and another measures the horizontal distance of the output stream. 7. ANALYSIS: Method A. The exit velocity is proportional to the velocity of the reservoir water surface as it moves downward. Why? Water is nearly incompressible. So the velocity of the water surface of the reservoir, v, is related to the exit velocity, V, by V/v = a/A where a is the radius of the exit hole and A is the radius of the reservoir tube. Derive this. From your data for reservoir height vs. time, you can generate a table of reservoir surface velocity vs time. Then you can generate a table of the exit velocity vs. time. Now use Eq. 1 to plot the square of exit velocity vs. the reservoir height. This should be a straight line of slope 2ρg. Is it? Method B.Theory suggests that the horizontal range, H, is related to the reservoir height, h, by:
Derive this. Plot these Δh vs. 1/ΔH to see whether this is satisfied by your data. 8. QUESTIONS: (1) Do your results confirm that the pressure at the bottom of a column of liquid is proportional to the height of the column? (2) Discuss the analysis of this experiment using Bernoulli's law. (3) What idealizations have we made in the theory? Are these significant? (4) We arranged the apparatus so that the water exited the reservoir horizontally. How critical is this initial condition to the conclusions of this experiment?
Text and drawings © 2014 by Donald E. Simanek.
