The purpose of this lab is to explore properties of fluids in static and dynamic situations.

In Part 1, you will employ Archimede's Principle to determine the density of a known material. Then, using the same technique, you will seek to identify an unknown material.

In Part 2, you will observe that flow through small capillaries will be non-ideal. The viscosity of the fluid leads to a behavior where the pressure difference across the capillary is proportional to the flow rate through it. ^{Hoveroverthese}

Equipment

• Part 1: set-up with hanging scale, graduated cylinder, Al masses, unknown mass
• Part 2: Apparatus with connected graduated cylinders, valves and capillary tube. Timer (your watch or phone?) and a thermometer for the room.

In studying introductory physics, we typically start by treating objects as single points, which can translate in one or more dimensions. Next, we allow objects to have an extent which can rotate (as well as translate). Up to that point, all materials we consider are solids, which do not change shape when moving.

To prepare for the next level of complexity, we need the definition of pressure: \(P\) is the Force per unit Area exerted on one part or the whole of an object \( P= F/A \). Most textbooks then consider the deformation (compression, extension, shear) of solids and the pressure required to cause the material to exceed its elastic limit and fail.

Liquids and gases are examples of fluids¹, which flow under a pressure or, if confined, transmit that pressure throughout the fluid volume. In this experiment we will investigate a liquid (water) which we will treat as an incompressible fluid (also non-viscous, until Part II).¹.

In the motion of an incompressible, non-viscous fluid, we have conservation of energy. Combining this with the conservation of matter (ie. the fluid does not evaporate or react chemically) we arrive at the Bernoulli equation and Continuity condition relating conditions at points 1 and 2 in the system. \(P\) is the measured pressure, \(\rho\) is the density¹, \(v\) the fluid velocity and \(A\) the area of the flow (ie. pipe inner area)².

$$P_1 + \frac{1}{2} \rho v_1^2 + \rho gy_1 = P_2 + \frac{1}{2}\rho v_2^2 + \rho gy_2 $$ $$A_1 \rho_1 v_1 = A_2 \rho_2 v_{2} $$

In Part I we will deal with a special case: hydrostatics. In this situation the velocity terms in Bernoulli's Equation are zero. The hydrostatic term, \(\rho gh\) and Pascal's Principle (any outside pressures act throughout the fluid) combine to give Archimede's principle: if \(V_{displaced}\) is the volume of fluid displaced by an object then the buoyant force is given by

$$ F_{buoyant} = \rho_{displacedfluid} \, g V_{object} $$

In Part II we look at a deviation from Bernoulli for non-ideal fluids which exhibit viscosity but do not undergo turbulence. An important example of this is blood flow in the human arterial system². A simple model for viscous fluids is to assume laminar flow; this works for a regime of flow where the Reynold's Number is low. Internal friction in the fluid leads to energy loss, which can be seen as a pressure difference across a small pipe (or capillary) which depends on the flow rate. The Hagen-Poiseuille's Law states that the volume flow rate \( Q = \frac{dV}{dt}\) through a small pipe depends on length \(L\), pipe radius (\(r\)), the pressure change \(\Delta P\) and the dynamic viscosity of the fluid \(\eta \)³ in the following way:³:

$$ Q = \frac{\pi r^4 }{8 \eta L} \Delta P $$

Part 1: Procedure

In this part of the lab, you will use buoyancy and Arichimede’s Principle to determine the density of a prepared salt-water solution². You will hang aluminum blocks of different sizes from a force sensor and measure the tension in the supporting string, first with the block hanging in air and then with the block fully submerged in the salt water. You will then graph your results to determine the density of the salt water, and finally use that measured density, together with a new measurement on a mystery object, to determine the density of the mystery object.

1. Choose an aluminum block. Measure its weight using the force sensor. Calculate the block’s mass, and, using the density of aluminum (\(\rho_{Al} = 2700 \) kg/m³) calculate its volume.
2. Without letting the block touch the sides or bottom of the beaker, lower the block until it is fully submerged in the salt water. Keep it still and record the tension.
3. Calculate the buoyant force from the block’s weight and its tension when submerged.
4. Repeat steps 1 through 3 for each of the remaining aluminum blocks. Dry each block before hanging it in air again.
5. Hang the mystery object from the force sensor and record its tension in air and when fully submerged in the salt water.

Part 1: Analysis

1. Graph the buoyant force (vertical axis) versus the block’s volume (horizontal axis). Fit a straight line through the data and use the slope to determine the density of the salt water.
2. Using the buoyant force on the mystery object and the density of the salt water you found, determine the volume of the mystery object. Then use its weight and volume to calculate its density.
3. Calculate the uncertainty in your results and compare the density you found for the mystery object with some standard metals. Here is a Table of Candidates .
4. Compare the density of the mystery object to the accepted value of the material you decided it must be. Is the difference consistent with your estimated uncertainty?

Part 1: Questions

1. When the block hangs in air, we treated the tension as equal to the block’s weight. Air also exerts a buoyant force. Estimate its size relative to the weight of an aluminum block (the density of air is about 1.2 kg/m³). Was neglecting it justified?
2. Suppose the block touched the bottom of the container during step 2. How would this affect the measured tension, and how would it change your calculated value of the density of the salt water (too high or too low)? Explain.
3. Suppose a small air bubble were clinging to the underside of a submerged aluminum block. How would this affect the measured tension and your result for the density of the salt water?

Part II: Procedure

In this experiment, a fluid is forced to flow through a small capillary by hydrostatic pressure. The capillary is chosen so that water will exhibit nearly laminar flow and obey the Hagen-Poiseuille law. As the experiment progresses, the change in column height (and, therefore \(\Delta P\)) is related to volume flow \(Q = \frac{dV}{dt}\) by that law.

1. Flow valve A was left open (with reservoir valve B closed) for a long time prior to beginning your experiment, allowing the water levels to equilibrate in the vertical tubes. We will close valve A, open valve B and adjust the height of the reservoir cup until the water level in the columns is 25cm, then close valve B.
2. Next we carefully pour water into column 1 until \(y_1 \approx 75\)cm. Record these two levels, \(y_1\) and \(y_2\). The difference between the two values is \(y = (y_1 -y_2\)). The initial height difference \(y_0 \approx 50\)cm.
3. There are three values of interest at each time: \(y_1\), \(y_2\) and \(t\). Simultaneously start a timer and open valve A; the water level in the higher column should start dropping and the lower column should start rising. Record \(y_1\), \(y_2\) and \(t\)⁴ every cm as the level drops until the difference in heights between the two water columns \( y \lt 5 \) cm. Record the Celsius temperature in the lab room.

Part II: Analysis

Calculate \(y/y_0\) for each time interval. Propagate uncertainty. Calculate \(ln(y/y_0)\) and assign the same relative uncertainty as for \(y/y_0\).

Make a plot of \(ln(y/y_0)\) vs. elapsed time \(t\). The slope should be ⁴ $$ Slope = - \frac{\pi r^4 ρ g}{A_{cyl} 8 \eta L} $$

Use your slope, radius of the capillary \(r_{capillary} = 0.5 \)mm, length \(L_{capillary}= 10\)cm, area of the graduated cylinder \(A_{cylinder} = 5.0 \)cm² and the density of water \(\rho\) to extract a value for \(\eta\), the viscosity of water. Assume all quantities in Equation 10 have small (negligible) uncertainty except the slope of the graph. Compare to the accepted value obtained using room temperature T (in \(^\circ\)C): \(\eta = 1.002 + .024(20- T) x 10^{-3}\)Pa-sec.

Part II: Questions

1. The temperature of the water, taken as the temperature of the room, affects the viscosity of water. If the temperature in the lab were 5 degrees warmer, would the viscosity of the water be higher or lower?
2. In the Hagen-Poiseuille law, the volume flow rate Q depends on the fourth power of the capillary radius. If the radius of the capillary were 10% larger than you assumed, by what percentage would your calculated viscosity change?
3. Suppose you replaced the water with a more viscous liquid, such as a glycerin-water mixture. How would the time for draining change? Would the laminar assumption be better or worse justified? Explain.

Guide to Uncertainty Propagation & Error Analysis (Quick Reference)

PHY IntroLabs Plotting Tool

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