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 = Vobj * Dfluid

where D = ρg is the weight density of the fluid.  B is the weight of displaced fluid.  That's what Archimedes figured out.

The Mass Density (ρ) of Water :

Historically, 1gram, was defined to be the mass of 1cm3 of pure water at 4.0oC.  This means that ρwater 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:

If you have not done so, you may need to add the following Website to your Java exception list: http://www.mhhe.com/

To do this, follow the path (Windows operating system),

Start → All Programs → Java → Configure Java → Security (use High) → Edit Site List … → Add → Type in the site URL (http://www.mhhe.com/).

To verify the "Archimedes' principle," we will use an applet the can measure the buoyancy on a barge.  For the barge safety (avoiding it from sinking), we load it such that only 3/4 of its volume goes in water.  In other words, we find the measured value of the load that causes 3/4 submersion of the barge.   To find the accepted value of the load that can cause 3/4 submersion of the barge, we will use the buoyancy formula.

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 3cm2 to 7 cm2.  This makes the volume of the barge to change from 12cm3 to 28cm3.  We will use water as the first choice for fluid.

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

1. Set the barge volume to 12cm3.  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(12cm3) = 9 cm3  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 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.
2. Set the barge volume to 16cm3. 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.
3. Repeat the above steps for each of the 20cm3, 24cm3 and 28cm3 options of the barge volume.

Case II: Alcohol

1. Since the weight density of alcohol is 0.8 gf/cm3, only 0.8 gram force of load is allowed for every cm3 of alcohol displacement.  For example, if you choose a barge volume of 12cm3, the minimum downward force that sinks the barge is 12cm3x0.8gf/cm3 = 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.
2. Repeat this experiment for all possible volumes for the barge.

Case III: Mercury

The weight density of mercury is 13.55 gf/cm3.  If you choose V = 12 cm3, the max. downward force that puts the barge on the verge of sinking is 12cm3(13.55gf/cm3) = 162.6gf rounded to 160gf.  For ¾ submersion, the maximum downward force on the barge is (3/4)(160gf) = 120gf. Since the barge itself weighs 5.0gf; therefore, the safe load is only 115gf. 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.

### Data: Given and Measured:

Table 1

 Trial Barge Volume (cm3) 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/cm3 1 2 3 4 5 Alcohol D = 0.8 gf/cm3 1 2 3 4 5 Mercury D = 13.55 gf/cm3 1 2 3 4 5

### Provide the calculations as required.

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.