Density_and_Buoyant_Force_6c.docx
DensityandBuoyantForce
An exploration of the relationships between weight, density, and buoyant force.
1
1.1
OBJECTIVES
EXPERIMENTAL GOAL
Students will use Archimedes' principle to predict whether various materials will sink or float in
deionized water based on their estimates of buoyant force.
1.2
PREREQUISITE SKILLS AND KNOWLEDGE
Students should have some familiarity with writing and adapting VI’s for LabVIEW, using Excel and
basic error analysis as covered in previous labs. Some prior knowledge of density and buoyancy is
helpful, but not vital.
1.3
RESEARCH SKILLS
After this lab, students will have had practice in:
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1.4
following laboratory protocols
using a laboratory notebook
using LabVIEW to control and collect data from a sensor
using a Vernier caliper to measure length
using a force sensor
organizing data
using Excel to analyze experimental data
LEARNING OBJECTIVES
After this lab, students will be able to:
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Use a force diagram to determine the net force on an object
Calculate the density of an object.
Determine whether or not an object will float in a given medium.
Determine what percentage of a floating object will be submerged.
Apply their knowledge of density and buoyancy in order to figure out how to make an object
float.
2
2.1.1
PRE-EXPERIMENT
Density
Density is an intensive property, in that it is intrinsic to the material and does not depend on how much of
the material you have. In this lab, we will be interested in mass density, the proportionality of mass to
volume of a given material.
The symbol for mass density (and for many other densities as well) is the Greek letter  (rho). The mass
density of a given amount of a substance is the ratio of its mass m to its volume V:
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 = m/V
Since density is a derived unit, it does not have its own SI unit, but is composed of the SI units kilogram
(kg) and meter (m): kg/m3. While a commonly used unit of density is g/mL, in this lab you will use SI
units.
2.1.2
The Buoyant Force
If you have ever been swimming, you will
have experienced the force that buoys you
up when you are partly or fully
submerged in the water. This buoyant
force Fb acts on a body in the opposite
direction of the gravitational force Fg.
The net force acting on the swimmer is
the sum of the two forces
(1)
Fnet = Fb + Fg
The free-body diagram shows the relative
sizes of the forces as well as their
direction. By convention, the buoyant
force is positive in sign, while the
gravitational force on an object is
negative.
Free-body diagram.
Image of fish from Wikipedia
According to Newton’s laws, if the net force acting on the swimmer is nonzero, the swimmer will
accelerate in the direction of the force with the greater magnitude. The acceleration of a swimmer of mass
m and acceleration a is Fnet = ma. Force F and acceleration a are vectors; they have both a magnitude and
a direction.
2.1.3
Measuring the Buoyant Force of an Object in a Fluid
In the laboratory, a third force is present: the tension of the bar connecting the object to the force sensor.
This tension will always be equal but opposite to the net force in equation (1) above. The force sensor
reports this tension as the net force on the object (as if the force sensor were not also exerting a force on
the object).
What is the actual net force on the object?*
The gravitational force on the object can be calculated from the mass m of the object and the free-fall
acceleration g due to the Earth’s gravity,
Fg = mg
Close to the ground the magnitude of g is 9.8 m/s2.
2.1.4
Predicting the Buoyant Force of an Object in a Fluid
The buoyant force is described by Archimedes’ principle, which says that the upward force on a partially
or completely submerged object is equal to the weight of the fluid that it displaces.
The buoyant force on an object submerged in a fluid is a result of the difference in pressure of the fluid on
the top and on the bottom of the submerged object. Pressure in a fluid increases as the depth increases.
*
Is the object accelerating?
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Density and Buoyant Force |3
Why is the pressure different at different depths of a fluid?†
Pressure is exerted by a fluid equally in all directions. Thus the pressure is directed downwards against
the top of the block and upwards against the bottom of the block.
What about the pressure on the sides of the block?1
Let us assume that the object submerged in a fluid is a
cylinder, with a cross-sectional area A and length l. Then
the buoyant force, Fb, equals the difference in the force
of pressure on the top of the cylinder, Ftop, and the force
of pressure on the bottom of the cylinder, Fbottom. The
force of the pressure on the top of the cylinder is simply
atmospheric pressure (Patm) plus the weight of the
column of fluid directly above the cube. If htop is the
difference in height between the surface of the fluid and
the top of the cylinder, fluid is the density of the fluid,
and g is the acceleration due to gravity, then
Ftop = A(Ptop) = A(Patm) + Ahtopfluid g
The same equation can be used to determine the pressure
at the bottom of the cylinder, since the pressure of a
fluid is exerted in all directions. From this relation, the
buoyant force Fb is given by:
Fb = A(Ptop – Pbottom)
= A[(Patm + htopfluid g) – (Patm + hbottomfluid g)]
= A(Patm – Patm + htopfluid g – hbottomfluid g)
= A(htopfluid g – hbottomfluid g)
= Afluid g (htop – hbottom)
= Afluid g (l)
Fb = fluid g V
where V is the volume of the cylinder (A∙l).
Recall that Archimedes’ principle says the upward force on a submerged object is equal to the weight of
the fluid it displaces. Compare this statement to the derived equation for the buoyant force.
2.1.5
Practice with Density and Buoyancy
Go to the PhET Buoyancy simulation and play:
http://phet.colorado.edu/sims/density-and-buoyancy/buoyancy_en.html
Try both the “Intro” and “Buoyancy Playground” tabs. There will be questions on the online assessment
directly related to this simulation.
Consider each object floating partially or fully submerged in a fluid. Draw yourself a free-body diagram
showing the directions of all the forces on the object.
What is the net force on the object?
†
Think of the pressure at the bottom of a stack of books, versus near the top.
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Write an equation that relates the forces on the object. Include the density of the object and the density of
the fluid.
2.1.6
Review of Vernier Calipers
You will be using calipers with a Vernier scale to measure the dimensions of the objects in this lab. The
Vernier scale provides an additional significant digit in your measurements. There are a number of on-line
tutorials on Vernier scales and calipers. Here are links to two:
http://hyperphysics.phy-astr.gsu.edu/hbase/class/phscilab/vernier.html
http://www.upscale.utoronto.ca/PVB/Harrison/Vernier/Vernier.html
In the example image above, the diameter of the acrylic sphere is read as 1.280 ± 0.002 cm. The zero line
(white arrow) points to a measurement between 1.2 cm and 1.3 cm. Estimation might get you to 1.27 or
1.28 cm, but the Vernier scale allows you one more significant figure. Look for the spot where a line from
the upper scale exactly meets a line from the lower scale. In this example, that meeting takes place at
1.280 cm (orange box). If the line just to the right of the 8 had been the place where the two lines joined
exactly, then you would record 1.282 cm. The last digit is always open to dispute, so the measurement
would be reported as 1.280 ± 0.002 cm. Why 0.002 cm?
2.2
PREPARE FOR THE EXPERIMENT
Read through the entire lab procedure and prepare your lab notebook, as well as an Excel workbook to
collect and analyze your data. Include any necessary formulae. Dust off your force measurement VI from
X-Lab 1 and adapt it to use in for this experiment. Email it to yourself.
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When you feel ready for the lab, test your preparation with the Pre-Experiment Quiz on e-Learning.
3
3.1
LABORATORY MANUAL
MATERIALS CHECK OFF LIST
Each small group of (2-3) students will have:
Laptop computer with LabVIEW
Ring stand with clamp and rod to suspend force sensor
Force sensor
SensorDAQ with USB cable
100-mL graduated cylinder
Lab jack
Threaded rod
Lab balance
Vernier calipers
Beaker with approximately 100 mL water
Each large group of 1 or 2 small groups will share:
5 cylinders of different materials
Disposables (can be disposed of in regular trash, or in bench-top waste container):
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3.2
Paper Towels and Kimwipes
SAFETY AND WASTE DISPOSAL PROTOCOLS, INCLUDING WASTE LABELS TO BE PREPARED
A lab coat, leg coverings, and closed-toe shoes must be worn. Eating, drinking or applying lotion to the
skin is not allowed in the laboratory.
Keep water away from electronics and electrical outlets. Try to wipe up spilled water immediately. Resist
the temptation to taste the cylinders.
3.3
3.3.1
EXPERIMENTAL PROCEDURE
Calculate densities and predict buoyancy
1. Use the Vernier calipers to measure the dimensions of each cylinder. Record the dimensions and
estimated error (in m) in your Excel workbook.
2. Calculate the volume in SI units (m3).
3. Use the laboratory balance to measure the mass of each cylinder. Record the mass and estimated error
in kg in your Excel workbook.
4. Calculate the density of each cylinder.
Q1. Assuming the density of water is 1000 kg/m3, predict whether each cylinder will sink or float.2
Q2. Predict the buoyant force (in N) the water will exert on each cylinder.3
Q3. For any cylinder that will float, predict the percent of the cylinder that will be submerged.4
3.3.2
Set up apparatus
1. Remove the hook from the force sensor and replace it with the threaded rod. Insert the rod until the
nut just touches the force sensor. Do not insert further! Place the hook someplace you will be able to
find it at the end of the lab.
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2. Set up the force sensor so that it hangs centered
above the lab jack. There should be enough
space for a cylinder attached to the rod to dangle
just above the 100-mL graduated cylinder
resting on the lab jack in its lowest position.
3. Raise the lab jack so that the weight hangs
inside the graduated cylinder below the 100 mL
mark.
4. Lower the lab jack so the weight hangs just above the cylinder, as before.
5. Unscrew the weight from the threaded rod and set it aside.
3.3.3
Set up VI
1. Connect the force sensor to your computer through the SensorDAQ and load the VI.
2. Run the VI.
3. Record the force from the weight of the threaded rod. Unless you incorporated a “tare” function in
your VI, you will need to subtract this value from each measured force.
4. Gently pull down and push up on the threaded rod.
Q4. What is the sign of a force pulling toward the center of the earth? Pushing toward the ceiling?
6. Adapt your VI so that a force pointing in the direction of the earth’s center is negative.
3.3.4
Record data for a metal weight
1. Attach the metal weight to the threaded rod on the force sensor.
2. Record your observations of the weight.
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3. Record the force reading. Remember to subtract the weight of the threaded rod.
Q5. What does this force reading represent?5
Q6. How is this force reading related to gravity? Write the equation.6
Q7. What is the net force on this object? (Consider whether the object is accelerating). Write the
equation describing the forces on this object.7
4. Raise the lab jack so the weight hangs inside the graduated cylinder, well below the 100 mL mark.
Pour enough water into the cylinder to completely cover the weight, but no higher than the 100 mL
mark. Tap firmly, but lightly to dislodge any air bubbles.
5. Record the new force reading and the water level.
Q8. What does this new force reading represent?8
Q9. Write the equation describing the forces on this object. Include the density of the object and the
density of the fluid in your equation.9
6. Lower the lab jack so that the weight is once again hanging above the graduated cylinder. You may
need to tap the cylinder a few times to loosen water drops as you go.
7. Record the force reading and the water level.
8. Raise the lab jack so the weight is again submerged. Tap to remove bubbles.
9. Record the force reading and water level.
10. Repeat until you have at least three (3) force and water level readings with the weight in air and in the
water.
11. Remove the weight and empty the water from the graduated cylinder back into the beaker.
3.3.5
Record data for the remaining weights
Use the same procedure to measure the buoyant force on the remaining weights. As above, collect at least
three values for each measurement. Record your data in the workbook and your observations, thoughts,
and questions in your notebook.
3.3.6
For cylinders predicted to float
Because the threaded rod is not flexible, it may be hard to tell when your cylinder is actually floating,
unless you are looking at the force sensor data.
Q10. How will you know from the force sensor that your cylinder is freely floating?10
Stop raising the platform as soon as your cylinder is floating so you can record the water level at the float
point.
Q11. What is the net force on the floating cylinder?11
Q12. Write the equation describing the forces on the floating weight.12
Continue raising the platform so that the cylinder is completely submerged and record the water level as
for the other cylinders.
Q13. What is the sign of the force on the cylinder when it is completely submerged? How can you
explain this phenomenon?13
3.3.7
Clean Up
Empty the water into the sink and dry off all the cylinders. Gently remove the threaded rod from the force
sensor and replace it with the hook. Disconnect the sensor and power supply so they are as they were at
the start of lab.
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3.3.8
Analysis
Average the data and calculate the buoyant force for each cylinder in water.
Use your water level data to calculate the submerged fraction of the “floating” cylinder.
Q14. Compare your experimental results to your original predictions:
a. Discuss whether (and, if so, how) your results validated your predictions about which
cylinder would sink or float.
b. How well did your experimental results match your predicted buoyant force for each
cylinder?
c. How well did your experimental results match your predicted percent of submersion for
each cylinder?
Q15. Note any surprising results.
3.4
POST-LAB ASSIGNMENT
Each small group will submit a one-page abstract in class, describing the experiment you just completed.
Include your figure in your abstract.
For details, refer to the Abstract Writing Guidelines posted in the Student Resources folder on e-Learning.
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