PHY132
Experiment
10
Buoyancy
In this experiment we will determine the density of a metal object in two different ways, including one based on
the Archimedes’ Principle. Comparison of the two results will provide some insight into the uncertainties, and a
verification of the principle. The diagram below shows a metal object hanging from a spring scale used to
determine its weight in and out of the water. The arrows indicate the forces acting on the object including the
buoyant force FB associated with the displaced volume of water.
String
tension
T
Buoyant force
of displaced
volume
FB
Wair = mg
Archimedes’ Principle tells us that the size of the buoyant force FB is equal to the weight of the displaced
volume of water and acts upward to support a portion of the weight of the object. The equation that describes
this is:
FB = water · g · V
(V = Volume of displaced fluid, water = 1000 kg/m3)
In our experiment the difference between the weight in the air and in water (as measured by the spring scale)
should be equal to FB:
FB = Wair – Wwater
(where Wwater is the tension T in the string, measured by the spring scale when the
object is fully immersed)
PROCEDURE
Your sample is a small metal cylinder with a hook. Measure its weight in the air (Wair): let each person in your
group read the spring scale and perform the measurement. Be sure you decide on an appropriate  uncertainty.
Also, measure the dimensions of the sample using a ruler or capiler.
Take a graduated cylinder and carefully fill it with water. Now hang the object in the water, so that
approximately 1/4, then 1/2, then 3/4 of the volume is immersed. Record the weight the spring scale indicates in
each case, and the displaced volume, using the markings on the side of the graduated cylinder. This procedure
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should demonstrate that the buoyant force FB is proportional to the volume of water displaced. The critical
measurement here is the displaced volume: estimate the difference in the graduated cylinder as best as you can.
Now immerse the sample entirely in the water and again record the weight on the spring scale (Wwater) and the
displaced volume. Repeat the immersed measurements three times, letting everyone in the group read at least
one measurement.
Sample material: ________________
Mass: ________________ ±__________
Diameter: _____________ ±__________
Height: _____________ ±__________
Volume of the cylindrical sample Vcalc = π · (diameter/2)2 · height = _______________________±__________
Data for PARTIAL immersion
Fraction
immersed
Weight in air
(N)
Weight in water
(N)
FB= Wair – Wwater
(N)
Displaced volume
of water
measured in the
graduated
cylinder (cm3)
none
1/4
1/2
3/4
All
±
±
±
Data for FULL immersion
Trial #
Weight in air
(N)
Weight in water
(N)
FB = Wair – Wwater
(N)
1
2
3
AVERAGE:
±
±
±
2
Displaced volume
of water
measured in the
graduated
cylinder (cm3)
ANALYSIS
Part 1
Verifying Archimedes’ Principle
1. Plot a graph of the buoyant force FB vs. the displaced volume of water (columns 4 and 5 of your data
table for Partial Immersion).
2. Is the relationship linear, i.e., is the buoyant force proportional to the displaced volume of water?
3. Draw a line of best fit through the data points and calculate the slope. Show calculation in detail in
report.
4. What should be the expected value of the slope, from the equations?
5. Calculate the % difference between your slope from the graph and the expected value. Present this result
on Analysis Table 1.
Part 2
Comparing density measurements
1. Based on Archimedes’ Principle, determine the displaced volume using
𝑉𝐴𝑟𝑐ℎ =
𝐹𝐵
𝑔𝜌𝑤𝑎𝑡𝑒𝑟
Use the average FB in the table for full immersion.
2. Now using the average weight of the sample (Wair), calculate its density from the equation
𝜌𝐴𝑟𝑐ℎ =
𝑊𝑎𝑖𝑟
𝑔𝑉𝐴𝑟𝑐ℎ
3. The data used in the previous calculations were entirely based on the immersion experiment.
4. Now, independently, measure the mass of the sample using a digital scale, and measure its volume based
𝑚
on the dimensions of the sample (use a ruler or caliper), where 𝜌𝑐𝑎𝑙𝑐 = 𝑉 .
Analysis Table 2
Material of sample:
Tabulated density
(g/cm3)
% diff
Density using Archimedes’
Principle
(g/cm3)
±
Density using mass/Vcalc
(g/cm3)
±
1 ml = 1 cm3
1 cm3 = 10–6 m3
1 g = 10–3 kg
1 g/cm3 = 103 kg/m3
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REPORT DUE IN 1 WEEK
Write a lab report containing error propagations for parts 1 & 2, and answer the question:
 Which method for density determination is more reliable? Why?
Perform the calculations required in the Video Analysis section below, and attach them (in detail, organized) to
the lab report, together with a final printout.
VIDEO ANALYSIS
Another important concept in the study of fluids is Bernoulli’s equation, which stems from the law of
conservation of energy applied to fluids:
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1
𝑃1 + 𝑚𝑣12 + 𝜌𝑔𝑦1 = 𝑃2 + 𝑚𝑣22 + 𝜌𝑔𝑦2
2
2
Bernoulli’s principle states that where the velocity of a fluid is high, the pressure is low, and where the velocity
is low, the pressure is high.
Using the equation of continuity 𝐴1 𝑣1 = 𝐴2 𝑣2 and Bernoulli’s equation, we can understand the dynamics of a
fluid pouring out of a hole as in the picture. Calling ℎ = 𝑦2 − 𝑦1, assuming P1 = P2, we find that:
(A)
𝑣12 =
2𝑔ℎ
𝐴2
1 − 12
𝐴2
area A2
We leave it as an exercise for you to demonstrate equation (A).
area A1
The purpose of this analysis is to verify equation (A).
1) Insert the movie <WaterTank.mov> in Logger Pro.
2) Enlarge the movie window. Calibrate the movie.
3) In this analysis, it will be helpful to set the origin precisely at the small hole in the bottle where fluid
comes out. Select the video analysis tool Set Origin (toolbar on the right) and drag it to that position.
4) Observe how the horizontal reach of the liquid jet decreases with time. Collect data for the horizontal
position of the end of the jet: let’s call that ∆𝑥. This horizontal displacement is related to v1, and you
know enough parabolic motion to write 𝑣1 = ∆𝑥√4.9⁄|∆𝑦| (notice, ∆𝑦 ≠ ℎ in this context). Measure
∆𝑦 in the movie using the Photo Distance tool.
5) Let’s calculate a new column for v1 with the value 𝑣1 = ∆𝑥√4.9⁄|∆𝑦|. From the top menu, select Data,
and New Calculated Column. Name it v1, write the appropriate units, and in the field Expression write
“X” * sqrt(4.9/value), where the word value should be substituted by your measurement of |∆𝑦|.
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6) Now collect a second set of data points, by tracking the position of the level of the fluid inside the bottle.
This will be h, because you have already set up the origin at the level of the hole. To collect a second set
of points you must click on Set Active Point, and select Add Point Series. Go back to the first frame of
the movie and start the new collection.
7) Equation (A) describes a straight line if we call YY = 𝑣12 , and plot YY vs. h. Create a New Calculated
Column where YY = 𝑣12 .
8) In the graph, click on top of the axis label to select the quantities you want to plot: you want YY vs h.
9) Is this graph a straight line?
10) Make the appropriate fit, and check whether your values are consistent with the fit values.
11) Calculate a % difference between your value and the fit value.
12) Save a printout and attach to report.
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