Introduction to Fluids

Exploration 1: Density and Pressure in a Fluid

For the next set of problems, use gauge pressure, i.e. set Patm = 0.

1. A U-tube is filled with water and oil, as shown at right. Measured from the bottom of the tube, A = 25 cm, B = 18 cm, C = 31 cm.

a) Find the fluid pressure at point B.

b) What is the density of the oil (this oil does not have the same density of the generic oil listed in the text)?

2. A U-tube has 5 cm of oil with a density of 750 kg/m3 poured on top of water as shown below. Initially, the oil level is higher than the water level. A small piston is then inserted into the left side of the tube, pushing down on the oil until both levels are equal. What is the pressure exerted upon the oil by the piston?

2 3. A syringe is used to draw two fluids to h1 and h2 as shown. If the fluid on the left is water, h1 = 12 cm, and h2 = 20 cm, find the density of the fluid on the right

4. Alberto, the manager for a refrigeration plant, has drawn up a plan for the portion of a water delivery system that will be used for cooling the components of the plant's system. His diagram is shown below. In order to know what type of valves to order, he needs to know what the static fluid pressure will be at Points A and B, which are presently capped off. What is the pressure at Points A and B?

3 For the following density problems, you can express your answer in g/cm3

5. When 40 drops of water from an eyedropper are dropped into a graduated cylinder, they occupy a volume of ¾ of a cubic centimeter. What is the mass of each drop? What is the volume of each drop? (The density of water is 1.00 g/cm3.)

6. If two liters of water are mixed with three liters of alcohol, what is the density of the mixture? (The density of alcohol is 0.79 g/cm3.)

7. A liter of corn syrup with a density of 1.20 g/cm3 is dissolved into a liter of water. What is the density of the mixture?

4 Prelab for Density/Pressure Lab

These questions pertain to part 2 of the lab 1. Write the equation for the pressure of a fluid at a given depth h:

2. What units will you eventually have for the a) vertical and b) horizontal axes of your graph?

3. Would you expect Patm at Miramar college (altitude about 400 feet above sea level) to be greater than or less than Patm at sea level on a given day? Explain:

4. What mode of data-collection entry will be used for this lab?

5. Describe the graph that you will make for this lab:

5 Density and Pressure Lab

PART 1 – The Density of a Liquid

Equipment  Balance  2 graduated cylinders or beakers – 50 ml  Fluid samples (alcohol)

1. Using the balance, determine the mass of 30 ml of alcohol. You only want the mass of the fluid, so be sure to zero the balance with the empty graduated cylinder. Mass:______

2. What is the volume of this amount of alcohol in cm3?

Volume: ______

3. Determine the density of the 30 ml alcohol sample in g/cm3: - ______

4. Pour 20 ml of the sample into the other graduated cylinder so that you now have two samples: a 10 ml sample and a 20 ml sample.

5. Predict the mass of each sample in grams, without using the scale. What did you base your prediction on? ______

Predicted mass of the 10 ml sample: ______

Predicted mass of the 20 ml sample: ______

6. Now measure and record the mass of each sample (Zero the balance with the graduated cylinder first).

Measured mass of the 10 ml sample: ______

Measured mass of the 20 ml sample: ______

How did the measured masses compare to your predictions?

6 7. Do you think the density of the 10-ml sample will be greater than, less than or equal to the density of the 20-ml sample? Explain your answer.

8. Now calculate the density of each sample. Use the space below for your calculations.

Calculated density of the 10 ml sample: ______

Calculated density of the 20 ml sample: ______

Did this agree with your prediction? What conclusions can you make about the density of a particular liquid at room temperature and standard atmospheric pressure?

PART 2 – The Pressure in a Fluid

Equipment  Clear vertical tube (1.5 m tall) to hold water  Pressure Sensor  Meter stick  Clear plastic hose to fit sensor (metal weight on bottom)

1. Set up the water tube as shown and fill to the 10 cm mark. Plug the sensor into the interface. The default graph is for pressure (kPa) vs. time. Change this to pressure vs. depth (cm) graph as follows:

 Go to data collection mode (clock icon) and choose “Events With Entry.”

 Column name: depth, short name, h, units cm. When you’re ready:

7  Press the collect button

 press the blue “keep” button when ready to record a reading

 enter the depth value in cm

Read and record the fluid pressure at different depths. Begin at 10 cm (h=0 cm) and move the metal ring in about 10 cm increments until you reach the 70 cm. That’s 7 readings including h=0). Depth is measured from the top of the water level to the water level “bubble” in the plastic hose – not the bottom of the hose. Both of these values change every time. If your depths are 10,20,30 cm etc. you are probably doing something wrong. 2. After finishing the data with entry collection, change the vertical axis to read in pascals, which are Newtons per square meter, following instructions below. These are for unit conversion only: to make it easier to determine the theoretical value of the slope of your equation:  Go to Data drop-down menu and choose “New Calculated Column”.

 Name: Pressure2; short name: P2; units: Pa

 In the equation box, choose “Variables (Columns)>” and choose “Pressure” from the drop-down menu.

 Now make an equation that reads “Pressure” *1000 (you are multiplying the pressure value in Kpa by 1000).

 Choose “Done” and return to “New Calculated Column.

 Name: Depth2; short name: h; units: m

 In the equation box, choose “Variables (Columns)>” and choose “Depth” from the drop-down menu.

 Now make an equation that reads “Depth” /100 to turn centimeters into meters.

 Choose “Done” which returns to the graph.

 Bring the cursor to the word “Pressure” on the y-axis and left click for drop-down menu. Choose Pressure2. The y-axis title should change to “Pressure2” and the units should be N/m^2.

 Bring the cursor to the word “Depth” on the x-axis and left click. Choose “Depth2” . Units should now be in meters.

 Use Autoscale icon (big A on toolbar) to provide appropriate scales for data analysis.

8 2. Find the regression (“best-fit”) line for your data. If your data aren’t linear, check with an instructor. Print your graph with the information from the regression line and attach to lab. Note pressure reading is an absolute pressure – it includes atmospheric pressure.

3. What is the value of your slope (include units)? ______

4. Write down the equation for the pressure of a fluid at a depth of h:

How does

5. What quantity do you think is represented by the slope of your graph? Explain:

6. Calculate the theoretical value of the slope. Compare this value with the value from your graph. Be sure that the slope of your graph is in Pa/m . Show your work below:

7. How do the two values compare. ______Find the percent difference between the theoretical value and the value from your graph. Show your calculation in the space below.

(|Experimental value – Theoretical value|/Theoretical value) X100%.

8. Three students make the following statements:

Student 1: "To find the density of a material, it doesn't matter how much of the material you use."

Student 2: "Larger amounts of the same material have larger densities because the mass goes up."

9 Student 3: "The only way to get consistent density measurements is to always use the same volume for each sample."

Which student (if any) do you agree with? Explain your reasoning.

9. Three students are asked to describe what is meant by a density value of 2.50 g/cm3.

Student 1: "The density tells you how heavy it is relative to water. It's 2.5 times as heavy as water."

Student 2: "The density tells you how much mass will occupy one cubic centimeter."

Student 3: "The density tells you how much volume 1 gram of the material occupies."

Which of these statements gives the best explanation of density? Explain your reasoning.

10 Exploration 2 : Buoyancy Problems

8. A cubical block of wood, 0.50 m on each edge, is held below the surface of a swimming pool by a rope tied to the block and attached to the bottom of the pool. The density of the wood is 650 kg/m3. Find:

a) the buoyant force acting on the block.

b) the tension in the rope.

9. A cubical block of aluminum, 0.50m on each edge, is at rest on the bottom of a swimming pool filled with water. The density of aluminum is 2700 kg/m3. Find:

a) the buoyant force acting on the block.

b) the normal force acting on the block from the floor of the pool.

11 10. Some students are discussing the forces acting on a 1kg copper brick and a 1kg lead brick resting at the bottom of the ocean.

Student one: “The buoyant force is greater on the copper brick because it's larger.”

Student two: “The buoyant force is greater on the lead brick because lead is denser than copper.”

Student three: “The buoyant force is the same on both bricks because they have the same mass.”

Which student do you agree with and why?

11. This is an old classic, but still worth considering: If an ice cube melts in a glass of water, does the water level go down, up, or stay the same? Explain.

12. A cereal maker once included a toy called "Diving Tony" as a prize in boxes of cereal. The plastic figure is placed in a plastic 2-liter bottle filled with water, and the lid of the bottle is screwed on tightly. When the bottle is squeezed, Tony sinks. When the sides of the bottle are released, the Tony floats to the top. If just the right amount of squeeze is applied, the toy will stay at rest under water. Note that the toy is hollow, as illustrated at right, and

12 when the toy is placed in the water, some water partially fills this inner hollow volume. Explain how the toy works. (Physics teachers frequently use a similar device, called a Cartesian diver, as a demonstration.)

13. A tree trunk (density = 700 kg/m3) is floating, partially submerged, in a lake. What percentage of the tree is underwater?

14. When an object is floating in a liquid, part of the object is below the liquid surface. Write the mathematical relationship for the percentage of an object that is below the liquid surface. Express your answer in terms of the fluid density and the density of the floating object.

13 Prelab for Buoyancy Lab

1. What is special about the water used in this experiment?

2. What is the second fluid used in this experiment?

3. What is your prediction for the beginning of Part 2? What is your reason (a- e)?

4. What are you going to do with the electronic balance?

14 Buoyancy Lab

You will find the buoyant force acting on an object two different ways, using two different liquids and see how the measurements compare.

Equipment  electronic balance, set up for hanging object  wooden slat (for platform)  Lab jack  Overflow can  graduated cylinder to catch overflow  Metal cylinders with hooks or tie with string  deionized water  alcohol

PART ONE – DEIONIZED WATER A. Underwater Weighing Method: Buoyant force using Newton’s Law

1. Measure the mass of the cylinder in kilograms, and calculate the weight of the cylinder in Newtons. These will be the measurements for the cylinder when it is in air.

mcyl-air: ______kg FG: ______N

2. Set up the balance for a hanging object. Set a beaker of deionized water on the wooden slat. Make sure neither the beaker nor the wooden slat touches the piece of wire at the back end of the scale.

3. Hang the cylinder so it is completely submerged in the water, up to the top of the hook on the top (not the hook from which it hangs!). Make sure the cylinder is completely under the water, and that it is not touching the container anywhere. Measure and record the mass reading of the cylinder while it is in the water.

mcyl-sub: ______kg

4. The quantity (mcyl-sub)g is “what the scale says”, i.e. the apparent weight. Since we have set up the scale as a hanging scale, the apparent weight is a tension force. Calculate this force.

T = (mcyl-sub)g = ______N

15 5. Draw a free body diagram for the cylinder when it is submerged in the water. Show all the forces acting on the cylinder – its weight, the buoyant force and the tension in the connecting string.

6. Using your free body diagram, write down the expression for the

buoyant force in terms of the true weight of the cylinder (w) and the tension in the string (T).

FB-A = ______

7. Using your calculated values of the weight and the tension in the string, calculate the buoyant force using your equation from Step 6.

FB-A = ______N

B. Displaced fluid method: Buoyant force using Archimedes’ Principle

8. Obtain an overflow can and a 25-ml graduated cylinder.

9. Fill the overflow can with deionized water. Excess water will drip from the overflow spout for a few seconds. Wait until it stops dripping. Place the overflow can on a lab jack so that the spout extends over the edge, and position the graduated cylinder under the overflow spout. Carefully lower your cylinder into the overflow can, catching the overflow water in the graduated cylinder. When the water stops dripping into the beaker, measure the volume of the water. Read to the bottom of the meniscus.

10. vol: ______m3

11. Calculate the buoyant force using Archimedes’ Principle.

12. FB-B: ______N

16 13. Compare your two values for the buoyant force using the percent difference formula, assuming FB-A is the correct theoretical value:

[|FB-A – FB-B |/ FB-A] x 100%

14. Does this experiment show that the the buoyant force is equal to the weight of displaced water? How?

PART 2 ALCOHOL A. Submerged weighing method Prediction: If you repeated this experiment with alcohol, a fluid that is less dense than water, do you think that B, the buoyant force would be greater than, less than, or equal to the buoyant force due to the water?

What is the reason for your answer above? a) the buoyant force will decrease because the volume of alcohol displaced will be less b) the buoyant force will decrease because the mass of alcohol displaced will be less c) the buoyant force will be the same because it is based on the volume of the cylinder which doesn’t change d) the buoyant force will increase because the volume of alcohol displaced will be greater e) the buoyant force will increase because the mass of alcohol displaced will be greater

1. Repeat Part A of the experiment using alcohol instead of water. Make sure you thoroughly dry off the cylinder (it has 2 parts), and recheck the mass. mcyl-air: ______kg FG: ______N

2. Measure and record the mass reading of the cylinder submerged in the alcohol: mcyl-sub: ______kg

3. Calculate the tension: T = ______N

17 4. Draw a free body diagram for the cylinder when it is submerged in the alcohol. Show all the forces acting on the cylinder – its weight, the buoyant force and the tension in the connecting string.

5. Using your free body diagram, write down the expression for the buoyant force in terms of the weight of the cylinder and the tension in the string.

FB-Alc = ______

6. Using your calculated values of the weight and the tension in the string, calculate the buoyant force using your equation from Step 6.

FB-Alc= ______N

7. Was your prediction correct? Explain

18 B. Alcohol density.

The displaced fluid method does not work for a liquid such as alcohol; it is too thin and dribbles down the side of the overflow can. However, we can still find the density of the alcohol, using Archimedes’ Principle:

FB-alc = ρalc Volalc g

Hint: How does the volume of alcohol displaced by the cylinder compare to the volume of water displaced in either of the methods used in part 1?

3 ρalcohol = ______kg/m