# Experiment X: The Simple Pendulum ```Fluid Mechanics Laboratory
Peter Love, Physics Department, Haverford College &copy; 2011
Introduction
Fluids – the generic term for substances that can flow (i.e., gases and liquids) – are
everywhere in the world around us: from the atmosphere that sustains us, to the
plumbing systems in our homes and workplaces, to the blood that flows through our
circulatory systems. Analyzing fluid behavior with high accuracy is a great challenge,
but a good general understanding of the behavior of fluids is possible using relatively
simple theoretical constructs and experimental apparatuses. In this lab, we’ll explore
the basic properties of fluids at rest (the field of hydrostatics) and in motion
(hydrodynamics).
Experiment 1: Equations of State
In fluids, pressure, temperature and volume are related to each other by an equation of
state. If you have studied basic chemistry, you may be familiar with the equation of
state of an “ideal gas” – a gas whose molecules interact only by (rare) direct collisions
rather than through long-range forces. The equation of state for such a gas is known as
the ideal gas law, which appears as
PV = nRT = nkT.
(1)
Here P is pressure. The unit of pressure is Newtons per square meter (N/m2), and one
Newton per square meter is called a Pascal, and 1000 Pascals is 1 Kilopascal, or kPa.
These are the units measured by the pressure sensor. V is volume (in m3), n is the
number of moles of atoms or molecules in the volume (where 1 mole = 6.02 &times; 1023
molecules = Na, which is Avogadro’s number), N is the actual number of atoms or
molecules, T is the temperature (in Kelvin, K), k is “Boltzmann’s constant” (k = 1.38 &times; 10-
23 Joules/K)
and R is the “gas constant.” The gas constant is related to Boltzmann’s
constant by R = Na k. [The Joule (J) is a unit of energy, related to force by 1 J = 1 N &middot; 1 m
= 1 N&middot; m.]
The ideal gas law shows, for example, that if a fixed amount of gas is compressed at
constant temperature then PV is a constant. Processes at constant temperature are
called isothermal. When you compress a gas you perform work on it and its temperature
will tend to rise (which is why bicycle pumps heat up when you use them). An
isothermal expansion or compression must be performed slowly, so that the heat added
to the gas can leak away and the temperature of the gas remains the same as its
surroundings.
On your desk you should find the “Gas Law Apparatus” that will allow you to explore
the equation of state of air. This Apparatus is shown above. We don’t know the number
of moles or molecules of air in the piston, so n and N are unknowns. But they should
remain constant during the compression process.
Start with the valve open and raise the plunger to near the top of its range; then close the
valve. Slowly turn the screw to move the plunger down, and record the pressure and
volume at convenient points. If you change things gradually enough, the temperature
should remain essentially constant (at the ambient air temperature; variation by a few
tenths of a degree isn’t significant).
You can determine whether the air is accurately modeled by the ideal gas law by
confirming that the product PV is a constant. Plot P as a function of 1/V and perform a
power law fit (an Allometric fit in Origin under non-linear curve fitting). Print out this
plot showing the fit. Complete the section in the report form for Experiment 1.
Experiment 2: Hydrostatic pressure in Fluids
The hydrostatic pressure at the bottom of a column of fluid of height h is
p = ρgh
where p is the pressure, g is the gravitational acceleration on the surface of the earth
and ρ is the fluid density. In this experiment you will verify this relationship by
measuring the pressure at varying heights in a narrow column of fluid.
The apparatus to do this is shown at the left. Connect this apparatus to
the fluid resevoir – first making sure that all the valves are closed. The
valves in their open and closed configuration are shown below:
Valve open
Valve Closed
Open the valve at the very bottom of the tube to fill the column with water, being
careful that water that flows out goes into a graduated cylinder, then close the valve
again.
Next take six pressure measurements using the pressure sensor – attach the pressure
sensor to the valve, open the valve, take the pressure reading from Logger Pro (you can
just read it off, no need to collect data), then close the valve and remove the pressure
sensor. Be careful not to get water in the pressure sensors.
Measure the height from the surface of the water to the T junction in the column. Plot P
against h, attach the plot and complete the section in the report form for Experiment 2.
Experiment 3: Measuring viscosity in Poiseuille Flow
As you have seen in the case of terminal velocity and various experiments where the
effects of air resistance are evident, fluids resist motion through them. This tendency to
resist motion varies from fluid to fluid. Oil and honey are harder to move through than
water. Viscosity is a material property that is a measure of the internal friction of fluids,
how much they resist flow.
For flow in a pipe, a pressure difference is required between the ends of the pipe in
order to produce a flow. The fluid flows fastest in the center of the pipe, the flow
velocity is zero next to the walls and the flow profile is a parabola, as shown below.
The flow rate is the total volume of fluid per unit time that comes out of one end of the
pipe. As the pressure difference increases, the flow rate increases. For Poiseuille flow,
the flow rate is related to the pressure difference by the Hagen-Poiseuille equation:
(3)
Where  is the flow rate, P is the pressure difference between the ends of the pipe, L is
the length of the pipe, r is the radius of the pipe and  is the viscosity. The units of
viscosity are Pa s, or N s /m2. The viscosity of water is 10-3 Pa s at 20o C. Notice that for
lower viscosity you get a larger flow rate for the same pressure difference.
On your lab bench there is a Poiseuille viscometer – a pipe connected to a reservoir of
fluid (the fluid is water with food coloring in it). There are two points at the end of the
pipe where you can attach the pressure sensor so that you can measure P. You will
measure flow rate by taking a video of the fluid filling a graduated cylinder and
analyzing the video in VideoPoint, just as you did for the collisions lab. Make these
measurements one after the other – don’t try and do everything at once.
Procedure – Pressure measurement
1. In Logger Pro set the data collection parameters to collect ten samples per second
for twenty seconds.
2. Connect the gas pressure sensors to the measurement points and open the valves
to connect the pressure sensors to the flow line.
3. With the end of the flow line pointing in the bucket, open the valve at the end of
the flow line so that the fluid begins to flow.
4. Collect a set of pressure data.
5. Stop the flow by closing the valve at the end of the flow line
6. Create a new calculated column that gives the pressure difference.
7. Record the mean and standard deviation of the pressure difference using the
“statistics” option under analysis in logger pro.
Procedure – Flow rate measurement
1. Make a video of the graduated cylinder filling up with fluid.
2. Edit the video in Videopoint Capture so that you have the graduated cylinder
filling over a 30mL range
3. Select one frame out of every 15 so that you have around 50 frames.
4. Save the video and open in VideoPoint
5. Track the height of the liquid in VideoPoint and use the ruler function to convert
to mL using the scale visible in the video (there is no mL unit available so just
choose meters)
6. Import your data into Origin and perform a linear fit to find the flow rate.
Once you have your flow rate and pressure difference and errors, measure the distance
the data analysis spreadsheet from blackboard – this will propagate the errors for you
and give you a viscosity measurement and standard deviation.
Complete the section of the report form for Experiment 3
Experiment 4: Low pressure experiments
Perhaps the most obvious characteristic of a fluid (besides its density) is its pressure, P,
defined as force per unit area: P = F/A. The unit of pressure is Newtons per square
meter, and one Newton per square meter is called a Pascal. The layer of atmosphere
above us creates a pressure at ground level of roughly 101000 (N/m2), or 101 kPa. That
means that each square inch of your body’s surface essentially is supporting
approximately 14.7 pounds worth of air. So why don’t we feel the pressure?
In certain situations we are aware of atmospheric pressure – for example, when we use a
suction cup to hold up some object. In expelling the air from the inside of the cup when
it is compressed, we create an imbalance between the pressure of the atmosphere
outside and the little remaining air inside. If we could remove all of the air from inside
the cup, and it had an area of one square inch, then it could support an object weighing
up to 14.7 pounds.
On your lab bench you should see the setup shown above – you can connect the
pressure sensor and use the syringe to pump air out of the chamber. You should be
able to get the pressure down to less than 10 kPa (for comparison, at an altitude of
100km, the edge of space, atmospheric pressure falls to 1 Pa).
Demo A – Inflation
When you breathe in your diaphragm lowers, expanding your chest cavity and
lowering the pressure. Air is forced into your lungs from the atmosphere. To familiarize
yourself with the vacuum chamber, perform the following demo. Place a balloon with
only a small amount of air in, and with the end tied off, in the pressure chamber. Then
evacuate air from the chamber using the syringe. Observe the balloon – what you are
seeing is exactly the same physics that causes your lungs to inflate when you breathe in.
Demo B – Suction cups don’t suck
Suction is simply a pressure differential between one place and another – its better
understood in terms of a higher pressure pressing onto something rather than a low
pressure ``sucking’’ onto something. To show this, lets remove the higher pressure and
observe what happens. Stick the suction cup to the side of the vacuum chamber – then
evacuate some of the air. What happens to the suction cup?
Demo C – Vapor pressure of water
The temperature at which a liquid boils is dependent on pressure. In the report form is
a plot showing how the boiling point of water depends on pressure. In this experiment
you will find the pressure at which water that is below 100 degrees Celcius boils. Take
some hot water in the small plastic cup and measure its temperature. Then place it in
the vacuum chamber with the pressure sensor attached and pump out the air. Record
the pressure as a function of time and mark when the water starts to boil. Complete the
section in the report form for Experiment 4.
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