__Experiment 9__

Archimedes' Principle

__Objective:__

To verify Archimedes' principle by designing a barge

__Equipment:__

A computer with internet connection, a calculator (The built-in calculator of the computer may be used.), a ruler, a few sheets of paper, and a pencil

__Theory:__

Archimedes' principle states that when a non-dissolving object is submerged (or partially inserted) into a fluid, the fluid exerts an upward force onto the object called the buoyancy force, B, that is equal to the weight of displaced fluid. We may write:

**B = (V _{object}
)( D_{fluid})**

Where **D = ρ g** is the
weight
density of the fluid

__The Mass
Density ( ρ) of Water:__

Historically,
1gram, was defined to be the
mass of 1cm^{3} of pure water at 4.0^{o}C.
This means that *ρ*_{water}
can be written as follows:

__
The Weight Density (D) of Water: __

If we name the weight of 1gram to be 1gram force (gf); therefore,

__
Procedure:__

To verify Archimedes' principle, the buoyancy force on a barge will be measured. For barge's safety measures, let's load a barge such that only 3/4 of it is to go under water in each experiment. In other words, find the measured value of the load for 1/4 of the barge to be above water surface. Also, by calculation, find the accepted value of the load for 1/4 of the barge to be above water surface.

Click on the following applet:
http://www.mhhe.com/physsci/physical/giambattista/fluids/fluids.html .
A barge will appear. You can change the volume of the barge by clicking on
the top slider in the applet. The height of the barge remains constant at
4cm. The base area of the barge can be changed from 3cm^{2} to 7
cm^{2}. This makes the volume of the barge to change from 12cm^{3}
to 28cm^{3}. Our first choice of fluid is ,of course, water.

Before starting this experiment, make sure that you have understood the mass density and weight density of water as explained under "Theory", above.

__
Case I: Water__

- Set the barge volume to 12cm
^{3}. This means that this barge can provide a maximum buoyancy force of 12 gram force (12gf). If its total weight becomes 12 gram force, it will be on the verge of sinking. Since we want it to be only 3/4 in water (or 1/4 out of water), only 3/4 of its volume must be used to provide buoyancy. (3/4)x(12cm^{3}) = 9 cm^{3}. Therefore, the maximum downward force must be 9 gram force. Out of this 9 gram force load, 5 gram force of it is its own weight. This means that only 4 gram force of load it can carry if only 3/4 of it in in water. The accepted value for the allowable load in this case is 4gf. To find the measured value of the allowable load, click on the "set" button. The magnet in the applet puts the barge in water. Under "cargo," select 1 gram force increments and click "Add Cargo." Add enough cargo until you can judge that 1/4 of the height of the barge is out of water. If with "eye-balling" you are happy with your estimate of the 1/4 of height being out of water, count the number of gram forces of load and record it as your measured value in Table 1. Calculate a %error. As you load the barge, the barge goes down. Water level goes up and overflows into the measuring tube on the left. The tube measures the volume of displaced liquid. - Set the barge volume to 16cm
^{3}. Estimate the maximum buoyancy that brings the barge onto the verge of sinking. Next, calculate 3/4 of it. Then, calculate the accepted value of the load you can place in it considering the barge's own weight. Finally run the experiment by loading the barge until you are pleased with your eye-estimation of 1/4 of the barge being out of water. Count the number of gram forces you put in the barge. Record this as your measured value, and calculate a %error. - Repeat the above steps for each of the 20cm
^{3}, 24cm^{3}and 28cm^{3}options of the barge volume.

__Case II: Alcohol__

- Since the weight density of alcohol is 0
**.**8 gf /cm^{3}, only 0**.**8 gram force of load is allowed for every cm^{3}of alcohol displacement. For example, if you choose a barge volume of 12cm^{3}, the downward force that brings the barge onto the verge of sinking is (12 cm^{3})(0**.**8gf/cm^{3}) = 9**.**6 gf**.**For 3/4 submersion, the max**.**downward force on the barge is (3/4)(9**.**6gf) = 7**.**2gf**.**Since the barge itself weighs 5**.**0gf**;**therefore, the safe load is only 2**.**2gf. This is your accepted value of safe load for 3/4 submersion. The measured value for 3/4 submersion can be done by running the applet for the alcohol option. Repeat the steps as you did for water and record the values in the Table. - Repeat this experiment for all possible volumes for the barge.

__Case III: Mercury__

The weight density of mercury is 13**.**55 gf /cm^{3}. If
you choose V = 12 cm^{3}, the max. downward force that puts the barge on
the verge of sinking is (12cm^{3})(12.55gf/cm^{3})=162**.**6gf
rounded to 160gf. Again, this is the calculated, or
expected, or accepted value. Find the downward force for 3/4
submersion as well as the allowable load considering the barge's weight itself.
Repeat all possible cases as you did for water and alcohol above with finding
the corresponding %errors.

Trial | Barge Volume (cm |
On-The-Verge-of-Submersion Weight (gf) |
3/4 Submersion Downward Force (gf) |
Accepted
(Adjusted) Allowable Load (gf) |
Measured Allowable Load (gf) |
% Error |

Water
D = 1 gf/cm |
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1 | ||||||

2 | ||||||

3 | ||||||

4 | ||||||

5 | ||||||

Alcohol
D = 0 |
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1 | ||||||

2 | ||||||

3 | ||||||

4 | ||||||

5 | ||||||

Mercury
D = 13 |
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1 | ||||||

2 | ||||||

3 | ||||||

4 | ||||||

5 |

__
Comparison of the results:__

Provide the percent error formula used as well as the calculation of percent errors.

__
Conclusion:__

State your conclusions of the experiment.

__
Discussion:__

Provide a discussion if necessary.