# Part I Archimedes Principle

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```							GOL 106: Historical Geology
Northern Virginia Community College
Dr. Victor P. Zabielski

Isostasy

Objective:
The objective of this lab is to illustrate the concept of isostasy. This will be
accomplished by conducting lab experiments using objects and materials composed of
different densities. The results will later be used to predict the effects of isostatic forces
on continental material.

Introduction:
Isostatic forces were first illustrated by Archimedes who showed that an object
floating in water will displace a weight and volume of water equal to the object’s weight.
Modern “real world” examples of this force can be seen throughout the oceanographic
field, from icebergs floating on the ocean to continents sitting upon the earth’s mantle. A
thorough understanding of this force will help you better understand many common
phenomena ranging from the shape of the ocean basins, the rise and erosion of mountains
and the subsurface effects of ocean surface topography.

Part I: Archimedes Principle

Archimedes was a Greek Mathematician and Scientist who lived during the 3rd century
B.C. in Syracuse, Sicily. He is attributed with many inventions including the water
screw, the planetarium, and a compound pulley system as well as numerous writings in
the fields of calculus and mathematical physics. The theory that is most closely
associated with him and that is most applicable to the field of oceanography is his theory
on floating objects and the buoyancy force, known as Archimedes Principle. Archimedes
Principle states that an object floating in water will displace a weight of water equal to
the weight of that object. This can be written as follows:

Weight of floating object = weight of displaced water
For the first part of this lab you are going to use a wooden block in a pan of water to
prove that this principle is in fact valid.

Equipment needed:
1 wooden pine block
1 pan filled with water to the brim (pan A)
1 large pan that can fit the water-filled pan in it entirely (pan B)
1 scale
1 squirt bottle
1 funnel
Some paper towels

Directions:
1) Place the smaller pan A into the larger pan B.
2) Fill pan A with tap water until it is completely full and just about to overflow (you
may want to use a squirt bottle to add the last bit of water). If any water spills into
pan B, clean it out using the squirt bottle or some paper towels.
3) Weigh the wooden block that you are going to use in this experiment. Record the
weight in row 1 of Table 1 below.

Weight of pine block = ____________________                          (1)

4) Place the block into pan A (the water-filled pan) and allow any displaced water to
spill out into pan B.
5) Carefully remove pan A from pan B, making sure not to spill any more water into pan
B (you may want to remove the block first and take some water from pan A before
you attempt to lift it out of pan B)
6) Weigh the graduated cylinder before you put any water into it. Record this weight in
row 2 of Table 1.

Weight of graduated cylinder = ____________________                        (2)

7) Using the funnel, pour the water from pan B into the graduated cylinder making sure
not to spill any. Weigh the graduated cylinder with the water in it. Record this
weight in row 3 of Table 1. Also record the volume of the water in the graduated
cylinder in row 5 of Table 1 for use in the next part of the lab (we will use the weight
and volume to calculate the density of the water)

Weight of graduated cylinder with water = ____________________                   (3)

8) Subtract the weight of the graduated cylinder from the weight of the water-filled
graduated cylinder to give you the weight of the water. Enter this weight into row 4
of Table 1

(3) – (2) = weight of water               (4)
Table 1:

1   Weight of wooden block (grams)

2   Weight of graduated cylinder (grams)

3   Weight of graduated cylinder and water (grams)

4   Weight of water (grams)
(3) – (2)
5   Volume of displaced water (ml)

Questions:

How does the weight of the wooden block compare with the weight of the displaced
water? What part of the experiment could have introduced some error in this
calculation?

Determine the percentage error between the actual weight of the block and the weight
of the displaced water.

[(Weight of block – weight of displaced water) / weight of block] x 100
Part II: Density Equations

We will now take the isostasy equation one step further and use it to determine
the relative position of a floating block based on the densities and density difference
between the block and the liquid (water) in which it is floating. Since it is not always
convenient to estimate the total weight of an object floating on a liquid surface (for
example icebergs or continents), we can determine the equilibrium position of a floating
object based solely on the densities or density difference between the object and the
liquid in which it is floating.

Density:
Density (ρ, pronounced rhō) is defined as the weight per unit volume of any
substance and can be calculated using the following equation:

Density(ρ) = Weight / Volume                        (5)

The typical units for density are g/cm3.

Determine the densities of the block and water that you used in the previous part of the
lab.

Equipment needed:
1 wooden pine block
1 wooden oak block
1 scale

Directions:
1) Measure the dimensions of the block that you used in the previous part of the lab and
calculate its volume. Enter this volume in the appropriate box in Table 2. Don’t
forget to keep track of the units!

Length (cm ) = __________             (6)
Width (cm) = __________               (7)
Height (cm) = __________              (8)

Volume (cm3) = Length x Width x Height = __________ (9)

2) Determine the density of your block using the calculated volume (9) and the measured
weight of the block from Table 1 (1). Enter all values in Table 2

Density of pine block (ρp) = weight of block (1) / Volume (8) = __________         (10)
3) Now determine the density of the water used in your experiment. Since we already
have the weight of a known volume of the water from the previous part of the lab, the
density of the water can be determined by dividing the weight of the water in Row 4
(Table 1) by the volume in Row 5 (Table 1). First you will need to convert ml to cm3
(note: the conversion formulas for several units can be found on the rear cover of
your textbook). The conversion formula for this is:

1 ml = 1 cm3

Density of water(ρw) = weight of water / volume = __________              (11)

Enter all numbers in Table 2.

Questions:

Compare the densities of the pine block and the water. Which one is denser and which
one is less dense? Which one would you predict would float on the other?

4) Now determine the density of one of the oak blocks (darker color). Proceed exactly as
you did for the previous pine block. First measure the dimensions of the oak block
and determine its volume.

Length (cm ) = __________                (12)
Width (cm) = __________                  (13)
Height (cm) = __________                 (14)

Volume (cm3) = Length x Width x Height = __________ (15)

5) Now weigh the oak block on the scale.

Weight of oak block = ___________               (16)

6) Finally determine the density of the oak block

Density of oak block (ρo) = Weight of oak block (16) / Volume (15)
= __________ (17)

Enter all the data in Table 2.
Questions:
Compare the densities of the pine and oak blocks.

Which one is denser?

Given blocks of equal volume, which one, oak or pine, will weigh more?

Based on your answer to the previous question, which block will sink deeper in the
water? Test your hypothesis to see if you were correct.

Table 2

Material       Volume (cm3)      Weight (grams)    Density (g/cm3)
Pine Block

Water

Oak Block
Part III: Isostatic Equilibrium

Since a block will displace a volume of water equal to its own weight, the total
weight of material in an imaginary column overlying the bottom of the tank including the
floating block and the water below it has not changed at all. When no block is floating
above it, the weight of material overlying the bottom of the tank is just the weight of the
overlying water. However, when a block is floating above it you must consider the sum
of the weights of the water (which is now less than that with no floating block due to the
displacement of some water by the block) and the weight of the block (whose weight is
equal to the water it displaced). Its not as confusing as it sounds.

Look at Columns 1, 2 and 3 in Figure 1. If we assume a constant cross-sectional
area for all the columns (for example 1 cm2), the weight of material in Column 1must be
equal to the weight of the material in Columns 2 and 3. The volume is calculated by
multiplying our cross-sectional area (1 cm2) by the height of the material (X or Y in the
diagram). Since the density of all the materials is known, the weight of each substance in
the column can be calculated using the following formula:

Weight = Density x Volume               (18)

This is the same formula given previously for density with the terms rearranged. Thus
the weight of material in each column is equal to the sum (Σ) of the weights of water and
wood in that column, or;

Total weight of column (g) = Σ [Density(g/cm3) x Height(cm) x cross-sectional area (cm2)]
(19)

Or for a single floating block,

Total weight of column(g) = weight of column of block + weight of column of water

or

Total weight of Column = [(Densityx ) x (X) x (1 cm2)] + [(Densityy ) x (Y) x (1 cm2)]   (20)

Equipment needed:
3 wooden pine blocks
3 wooden oak blocks
1 scale
1 meter stick
1 transparent tank filled partially with water
Figure 1

Pine
Block                      Oak
X1                 Block     X2
Column 1                              r

Column 2

Column 3
water
Y0                        Y1                         Y2

1                            2                              3

Directions:
1) Determine the weight of material in a free standing column of water (Column 1).
Measure the height of the water column without any block floating over it and enter
you value in the “Y” column in Table 3.

2) Enter the density of water from Table 2 into the Density Column in Table 3 and
calculate the Weight of Y using equation (19) above. For this case, the weight of “Y”
will be equal to the total weight of the Column, so the last two columns will have the
same number in them. Remember, the cross-sectional area is always 1 cm2 based on
our original assumption

3) Determine the weight of material in column 2 with the pine block floating above it.
Measure X1 and Y1 in your experiment and record the values in Table 3.

4) Use the density of pine from Table 2 and Equation (20) to calculate the total weight of
the column.

5) Determine the weight of material in column 3 with the oak block floating above it.
Measure X2 and Y2 in your experiment and record the values in Table 3.

4) Use the density of oak from Table 2 and Equation (20) to calculate the total weight of
the column.
Table 3
Wooden Block                           Water
Total
X (cm)               3     mb (g)     Y (cm)              3    mw (g)    weight
ρ b(g/cm )                         ρw (g/cm )
Experiment       thickness      Density     Weight      height    Density of    Weight        of
of block      of block)    of block   of water     water       of water   column
A             B        AxB=C         D          E          DxE=F        (g)
C+F
Column 1
water (alone)

Column 2
water &
pine block
Column 3
water &
oak block

Questions:
Compare the total weights of Columns 1, 2 and 3. How similar are they?

Calculate the percentage error for the pine and oak block experiments using the
following formula:

Percentage error = weight of column w/block – weight of column w/o block   x 100
Weight of column w/o block

How does this error compare to the error you calculated in Part I of this lab?

What might account for this error?

5) Now add another block of similar composition to each of the floating blocks and
determine the weight of Columns 2 and 3 again with the added blocks. Record your
data and calculations in Table 4.
6) Repeat the experiment with 3 blocks of similar composition on top of each other.
Record your data and calculations in Table 4

Table 4
Wooden Blocks                            Water
Total
X (cm)     ρ b(g/cm3)    mb (g)     Y (cm)    ρw (g/cm3)    mw (g)    weight
Experiment      thickness    Density     Weight      height    Density of   Weight        of
of block    of block)    of block   of water     water      of water   column
A)          B        AxB=C         D          E         DxE=F        (g)
C+F
2 Blocks
Column 2
water &
2 pine blocks
Column 3
water &
2 oak blocks
3 Blocks
Column 2
water &
3 pine blocks
Column 3
water &
3 oak blocks

Questions:
Compare the weight of all the columns. Are the values close?

Calculate the error for each column using the same equation you used previously.
What happens to the error as you add more blocks?

What might account for this error?

6) What will happen to the elevation of the blocks if you took a block off of the stack?
Will the absolute height of the blocks (the height from the bottom of the tank to the
top of the block stack) decrease by the thickness of the block that you removed?
Calculate the absolute height of the blocks for the 3 block stacked scenario. Measure
the distance from the bottom of the tank to the top of the block for both the pine and
oak blocks. Record these numbers in Table 5.
7) Now remove one block from each stack, leaving two blocks in each stack and
determine the absolute height of the new columns. Record these numbers in Table 5.

8) Calculate the difference in height between the 2-block and 3-block scenarios for each
the pine and oak scenarios by subtracting the 2-block absolute height from the 3-
block absolute height and record the difference in Table 5

9) Now measure the thickness of the blocks that you removed from the stack and record
that value in the last column of Table 5

Table 5

Experiment         Height of         Height of        Difference in     Thickness of
3-block stack     2-block stack         height        block removed
(cm)              (cm)               (cm)              (cm)
Pine Blocks

Oak Blocks

Questions:
Did the height of the stack of blocks decrease less or more (difference in height) than
the thickness that was removed? Why?

This phenomenon is termed isostatic compensation, or isostatic rebound. You
experience this phenomenon every time you get out of a bed or a couch and the surface
rebounds after your weight is removed. Does the crust of the earth experience isostatic
rebound when weight is removed from it? How is weight commonly removed from the
surface of the earth?
Now you are going to determine how much each block will sink based on the relative
densities of the block and water. The height to which a block will float or sink is
determined by the difference between the densities of the block and the material in which
it is floating. A block that is half the density of water will sink half way into the water.
A block that is one-quarter the density water will only sink one-quarter of its depth into
the water. The actual depth to which an object will sink is thus related to the ratio if its
density to that of the density of the liquid in which it is floating. This concept is
illustrated in Figure 2. The equation describing this phenomenon is as follows:

R = density of wooden block                   (21)
H     density of liquid

Figure 2

H1

Pine                             Oak
Block       R1                   Block       R2       H2
Density= ρp                      Density= ρo

Water
Density= ρw

10) Place one pine block and one oak block in the tank and measure the respective
values for H and R for each block. Record the values in Table 6.

11) Calculate the R/H ratio for each block and record that value in Table 6.

12) Place the values for the densities of each of the blocks and the water (already
determined in table 2) into Table 6.

13) Calculate the ratio of densities (density of wooden block(ρb) / ratio of water(ρw)) for
each block and enter the values in Table 6.
Table 6

Density     Density
Material      R (cm)        H (cm)        R/H      of block    of water     ρb/ ρw
ρb          ρw
(A)       (g/cm3)    (g/cm3)       (B)
Pine Block

Oak Block

Questions:
Compare R/H (A) and ρb/ ρw (B) for each of the block types. Are they close?
Part IV: Tectonic Isostasy

Now it is time to bring all this new knowledge into a real life example. Although we
used examples of wood floating on water, the same isostatic laws can be applied to any
material floating on top of other material. One example where this is particularly
relevant is the case of the crustal material (granite, limestone and basalt) floating on top
of the mantle (peridotite) and the implications that isostasy has for the height of a land
surface after the removal of material by erosion. In this portion of the lab we will be
determining the densities of each of the basic rock types in the lithosphere and then using
this information to better understand the forces controlling the surface topography of the
earth.

Equipment needed:
1 piece of limestone
1 piece of granite
1 piece of basalt
1 piece of peridotite
1 graduated beaker or cylinder into which the rocks can fit
1 scale

Directions:
For each sample:
1) Calculate the densities for each of these samples. First weight each sample while it is
dry and record this weight in Table 7.

2) Fill a graduated cylinder about half full of water and record the precise volume of
water (Vo) in Table 7.

3) Gently place the sample in the graduated cylinder, making sure that the water level
does not go over the top of the cylinder (if it does, redo the experiment with less
water in the cylinder initially). Record the new volume in the cylinder (V1) in Table
7.

4) Subtract the original volume of water (Vo) from the new volume with the rock added
(V1). This new volume (VR) is the volume of the rock sample. Calculate the volume
of the rock sample and record it in Table 7.

Volume of rock sample (VR) = V1 - Vo                  (22)

5) Calculate the density of the rock sample using equation 5a (Record all values in Table
7).
Table 7

Volume of         Volume of                  Density of
Weight       water in          water with   Volume of       rock
Rock sample       (grams)      cylinder (Vo)      sample      Rock (VR)      sample
(cm3)             added (V1)     (cm3)        (g/cm3)
(cm3)
Granite

Basalt

Peridotite

Limestone

Questions:
Compare the densities of these three rocks? Will Granite “float” on peridotite?

Will basalt “float” on peridotite?

Will granite “float” on basalt?

Will basalt “float” on granite?

Will peridotite “float” on granite?
Homework
GOL 111: Introductory Oceanography
Isostasy

Use the values of density of the various rock types that you have determined in the lab
and equation (21) to answer the following questions. Feel free to work in groups to
answer these questions, but each person must submit their own copy of the homework. I
will collect the homework next lecture. For simplification, do not consider the thickness
or weight of any solid portion of peridotite in the lithosphere.

Questions:

How deep do the roots of the Appalachian Mountains extend beneath the surface of
the earth? (assume they are made of granite and their elevation above sea-level is
approximately 6000 feet or 2 km)

How deep do the roots of the Himalaya Mountains (assume they are made of limestone
and their elevation above sea-level is approximately 30,000 feet or 10 km)

Given an erosion rate of approximately 5 mm/year, how long will it take to erode the
Himalaya Mountains to sea level if there is no isostatic compensation (no rebound)?

How long will it take to erode the Himalaya Mountains to sea level if we consider
isostatic compensation?

```
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