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# Archimede Buoyant Force and the Density of Liquids (DOC) by mikesanye

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```									               Archimedes’ Buoyant Force and the Density of Liquids

Purpose:
The purpose of this experiment is to use Archimedes’ Principle to explore how gravity, density of
fluid, and volume of fluid displaced by an object affect the Buoyant Force on an object placed in a
fluid. In addition, Archimedes’ Principle is used to determine the density of a fluid.

Materials/Equipment: (See Figure 3 for Assembly)
GLX                                         Hooked Brass Metal Cylinder with String
High Resolution Force Sensor                Lab Jack
100 mL Graduated Cylinder                   Water or Other Fluid (more than one optional)
Ring Stand (or Table Clamp and Rod) and     Printer
Right Angle Clamp

Background/Theory:
According to Archimedes’ Principle, there exists a force on an object wholly or partially
submerged in a fluid. This force (FB) is equal to the weight of the fluid (WF) displaced by the object
and has a direction opposite that of gravity. In other words, if an object is placed in a fluid, then part of
that object is occupying a volume that was once occupied by the fluid. The fluid then attempts to push
the object back out of that volume, and its push is as strong as the weight of the fluid that once
occupied that volume.
In this lab, you will be hanging a rod from a force sensor into a graduated cylinder full of a
fluid, as depicted in Figure 1. When the rod is positioned like that, there are three forces acting on it:
its own weight due to gravity (which we know is its mass m times the gravitational acceleration g,
WOBJ=mg), tension in the string from which it is hanging (T), and the buoyant force due to it
displacing fluid in the cylinder (FB). Gravity is pulling down on the rod, the string tension is pulling up
on it, and the buoyant force is pushing it up. These forces are indicated in Figure 2.
Now that we know how the forces are acting on the rod, we can write down an equation
describing the balance of these forces. The total combination of the forces must be equal to 0 since the
rod isn’t going to be moving (it will be in equilibrium). Taking up as the positive direction, we have:

T + FB – mg = 0     (1)

Archimedes’ Principle states that the buoyant force is equal to the weight of the fluid displaced, or:

FB = WF (2)
Tension (T)
Force Sensor

Rod

Buoyant Force (FB)         Weight (WOBJ)

Figure 1: Diagram of                                Figure 2: Balance
Experiment Setup                                  of forces on the rod
Substituting that into equation (1), we have:

T + WF – mg = 0      (3)

We also know that the weight of an amount of fluid is equal to the fluid’s mass times g, and that the
fluid’s mass is equal to the fluid density (ρF) times the volume of fluid, or ρFVF. In this case the volume
of fluid is the same as the volume displaced by the rod (VD). All of that is given by:

WF = mFg = ρFVDg (4)

And substituting into equation (3):

T + ρFVDg – mg = 0 (5)
or
T = mg – ρFVDg (6)

So, if tension in the string is plotted as a function of volume of fluid displaced (string tension (T) is y,
and volume of fluid displaced (VD) is x), the result should be a straight line with slope -ρFg and y-
intercept mg. That is because the general equation of a straight line is y = mx + b where m is the slope
and b is the y-intercept. So:

T = mg – ρFVDg
becomes
y = mg – ρFgx (7)

Where the slope of the line is -ρFg and the y-intercept is mg.

Procedure/Analysis:

Setup:
1) Connect the high resolution force sensor to the
GLX, and turn the GLX on using the power
button on the bottom right of the GLX.
2) Select the Meter option on the home screen of
GLX using the arrow and check mark keys.
3) Making sure nothing is attached to the
measuring end of the force sensor, press the
zero button on the force sensor. The GLX
should now be reading 0 Newtons, or
something very close to it.
4) Assemble the lab stand, high resolution force
sensor, rod, graduated cylinder of fluid, and
lab jack as shown in Figure 3. Have enough
so that the cylinder can be completely
submerged but without spilling fluid when it
is fully submerged. Record the amount of
fluid in the cylinder on the Student Data                   Figure 3: Detailed Experiment Setup
Sheet.
Measurement:
1) Making sure the rod is not in contact with the fluid, note the force being displayed on the GLX. This
is the tension in the string connecting the rod to the force sensor, and should currently be equal to the
weight of the rod. It is negative due to the way the force sensor is constructed. Record this value for
the force (go ahead and make it positive) and the fluid level in the graduated cylinder in Table 1 (this
fluid level should be the same as the value in Setup step 4).
2) Use the lab jack to raise the graduated cylinder until the fluid reaches the second mark on the rod.
Again, record the force from the GLX and the fluid level in the graduated cylinder in Table 1.
Repeat this process, raising the graduated cylinder such that the fluid level moves up two marks on
the rod each time and recording the new force and fluid level until Table 1 is full.

Data Manipulation:
1) Recall that the data we needed was the tension in the string and the volume of fluid displaced. You
have the tension (the force from the GLX), but you don’t quite have the volume of fluid displaced.
You do have the initial fluid level, and the fluid level for each force measurement. As you lowered
the rod into the graduated cylinder, the fluid level rose. This is because the rod was displacing more
and more fluid. So, the difference between each fluid level you recorded and the initial fluid level
should be equal to the volume of fluid displace by the rod for each tension measurement. So, copy
the tensions from Table 1 into the first column of Table 2, and then calculate the difference between
each fluid level and the initial fluid level. Record that difference (which is the volume of fluid
displaced, VD) in the second column of Table 2.
2) The only problem left with the data is that it is not all in SI units – you need to convert VD from mL
to m3. One mL is one cm3, and there are 106 cm3 in one m3 (100cm*100cm*100cm). Therefore, to
convert mL to m3, you need to divide each measurement in mL by 106 (or 1,000,000). Record these
converted values in the third column of Table 2.

Plotting and Analysis:
1) Now it is time to record and plot the data using the GLX. Begin by turning the GLX off and
disconnecting the force sensor. Then turn the GLX back on and select the Table option on the GLX
home screen. Press the F3 button, check that the New Data Column option is selected, and then press
the check mark button. Then create another new data column by pressing F3 and selecting the New
Data Column option again. Press the ESC button until you see a dash mark selection box on the
table. Using the arrow keys, place this box on the first row of the first column.
2) With the dash mark box on the first row of the first column, press F2. You should see a blinking
cursor. Type the first tension value from Table 2 into the box, then press the check mark button. The
cursor should move horizontally into the first row of the second column. Here type the displaced
volume from the third column of Table 2 that corresponds to the tension you just entered, then press
the check mark button. Now the cursor will jump to the second row of the first column. Continue
entering data into the GLX until the first and third columns of Table 2 (tension and displaced volume
in m3) are recorded in the first and second columns of the GLX, respectively.
3) After you’ve entered the last value, press the check mark button so that the cursor is in a blank box,
then press the ESC button. Make sure you still see all of your data on the screen, and then press the
home button. Now select the graph option on the GLX home screen. Part of your data will now have
been graphed, but not all of it. To fix this, press the check mark button. The label for the y-axis of
the graph should now be highlighted. Use the left and right arrow keys a few times to see how to
navigate between the x and y axis labels (there should be an upper and lower choice on the y-axis
and just one choice on the x-axis). Make sure the x-axis option is highlighted, and then press the
check mark button. A list of options will pop up, including Data 1 and Data 2, which were the two
columns of data you entered in the table. Data 1 is the tension force, which should be on the y-axis
label, and Data 2 is the one that should be on the x-axis. Select Data 2, and then press the check
mark button. Now you have a plot of string tension versus volume of fluid displaced in units of
Newtons and m3, respectively.
4) With the graph now properly constructed (it should be a diagonal line which falls going from left to
right). Now you need to do a linear fit to this graph. Press F3 to access the Tools menu, and then
select the linear fit option. You’ll then be able to determine which points the fit uses by moving the
circular cursor on the graph with the left and right arrow keys. Press the right arrow key until all the
points are being used. The slope, y-intercept, and various other properties of the fit are displayed
below the graph. Record the slope on the Student Data Sheet in Table 3. Then press F4 and select the
Print option to print your graph.
5) Given that the slope of the line should be equal to -ρFg, calculate ρF (the density of the fluid) using g
= 9.8 m/s2. Record this value in Table 3, and then calculate the percent error with respect to the
known density of the fluid. For example, if your fluid was water, its density is 1000kg/m3. The
percent error is given by the formula:

Experiment al  Known
ErrorPercent                          * 100   (8)
Known

For the water example, Experimental would be the density you just calculated by the dividing the
slope by –9.8m/s2, and Known would be 1000kg/m3. Once you have calculated the error, record it in
Table 3 as well. This error represents how far your measured value for density was from the correct
value. For example, if you got a percent error of 100%, then your value was exactly twice what it
should have been. Percent error is an easy way to describe how your value compares to the expected
values.

Time permitting or according to teacher instructions: Repeat the experiment using a different fluid.
Then answer the questions on the Student Data Sheet.

When you have completed the experiment, turn off the GLX. Make sure you don’t need any more data
from it, because your data will be erased when you turn it off. When prompted, Do Not Save the file.
Student Data Sheet
Name: ____________________                     Partner’s Name(s): ________________________

Period: _____________                          Date: ________________

Initial Fluid Level/Volume (mL) _______

Table 1                                                    Table 2

Tension (N) Fluid Level/Volume                       Tension (N)    Volume Displaced         Volume
(mL)                                            (mL) (Fluid Level –      Displaced
Initial Level)          (m3)

Table 3
Slope of Linear Fit (kg/m2s2)     Calculated Fluid Density (kg/m3) Percent Error in Density

Questions:
1. Explain in your own words what the graph indicates. What are possible sources of error in the
data?

2. When floating while swimming, would you expect to float higher out of the water in fresh water or
salt water? Explain.

3. Given a ship of weight W which is capable of displacing a volume V of water, write the
mathematical expression that describes the maximum load L the ship can carry before sinking.
Explain what other factors might influence this in actual practice as the ship sails to various ocean
and river ports.

4. Icebergs float even though they are made of water - explain why this happens. Calculate the
fraction of an iceberg that is above ocean level using the density values: Sea Ice 910 kg/m3, Sea
Water 1030 kg/m3.

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