Download Lab10_ArchimedesPrinciple

PHY 131/221
Archimedes’ Principle
(revised 4/26/08 J)
Name: ________________________
Date: ______________
Lab Partners: ____________________________________________
Introduction: Why do some objects float and other objects sink when placed in water? The same question
can be asked of floating and sinking objects in air. The key to answering this question is to consider a freebody diagram of an object submerged in a gas or liquid. When the object is completely submerged, but not
resting on a solid surface, the force of gravity pulls downward on the object. If you hold the object in place
and then release it, the object will do one of three things. If the object rises then there must be a force that
opposes gravity and is greater than the weight of the object. If the object remains still then there must be a
force that opposes gravity and equals the weight of the object. Finally, if the object sinks then there must be a
force that opposes gravity and is less than the weight of the object (the force must be opposing gravity since
the object sinks at a rate less than it would under the force of gravity alone).
This force that opposes gravity is called the buoyant force. The buoyant force is essentially the result of the
fact that as you descend into a gas or liquid the pressure increases. As a result, the object experiences a
greater pressure at its bottom than at its top. This difference in pressure gives rise to an upward buoyant
force.
As an interesting aside, the buoyant force may not always be upward. For example, if the pressure in a gas or
liquid increases to the right through the gas or liquid there will be a buoyant force to the left. An example of
this is when you have a helium balloon in your car while you are driving down a road. If you accelerate you
introduce a fictitious force toward the back of the car (you feel this “force” as it pushes you into your seat
like a horizontal force of gravity). The air in your car feels this force as well and the air pressure increases in
the back of the car and decreases in the front (let’s assume your windows are closed). The helium balloon,
not being fixed to your car, feels a force toward the front of the car, opposite this fictitious force. The balloon
thus moves forward as you accelerate forward even though you feel pressed back into your seat.
It can be shown that the buoyant force is equal in magnitude to the weight of volume of gas or liquid that the
object displaces (This is true as long as the pressure differences in the gas or liquid are due to differences in
depth within the gas or liquid and it is gravity that is causing the pressure differences). This is known as
Archimedes’ Principle. This can be understood by considering the differences in pressure on the top and
bottom of the object along with the height of the object. You may have seen this in class, but if not you can
find a discussion of this in just about any physics text.
As an example, suppose you drop a rock, with a volume of 2 cm3, into water. Clearly the rock will sink, but
as it sinks it pushes away, or displaces, a volume of water equal to its volume. The buoyant force on the rock
will be equal to the weight of 2 cm3 of water and since the rock sinks we know that the buoyant force must
be less than the weight of the rock. What does this example imply about the density of water compared to the
density of the rock?
The weight of an object (the rock) is equal to the object’s mass times g, wo = mg, but m = oVo, where o is
the object’s density and Vo is the object’s volume. Thus, wo = o Vo g. Similarly, the weight of the fluid
displaced (the water) equals the density of the fluid times the volume of fluid displaced times g. Thus, wf =
f Vf g. But, the weight of the fluid displaced equals the buoyant force, Fb = f Vf g.
If the object is completely submerged then Vo = Vf and we have:
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Fb  f

wo  o
or
Fb 
f
wo
o
(1)
From this equation we see that if the density of the object is less than the density of the fluid then the
buoyant force will be greater than the object’s weight and the object will accelerate upward. If the object’s
density is greater than the fluid’s density then the object will sink. Finally, if the densities are equal the
object will be in equilibrium and remain in one location, or move at a constant velocity up or down.
If you hang a small metal block by a string and lower it into a beaker of water you can determine the density
of the metal block by measuring the change in tension of the string. When the block is hanging in the air the
tension in the string is equal in magnitude to the weight, wo, of the block (T = wo). When the block is
completely submerged the tension in the string is a measure not of the object’s weight, but of the object’s
apparent weight, w’o. Since the tension and the buoyant force are acting upwards while the object’s actual
weight is acting downward we can write:
w’o + Fb = wo.
(2)
We will be measuring the tension in the string using a scale that is usually used to measure the mass of
objects. The string will pull down on the scale and we will be able to read a mass from the scale. Multiplying
this mass by g gives us the tension in the string which will equal either the real weight, or the apparent
weight of the object.
Substituting equation (1) into equation (2) gives:
wo' 
f
wo  wo
o
Solving for the density of the object gives:
o 
 f w0
wo  wo'
(3)
If we use units of grams for mass and units of cm3 for volume, the f for water is equal to 1g/cm3. Keep in
mind that our discussion above is valid for fully submerged objects, not floating objects.
Part I: Verification of Archimedes’ Principle.
Obtain an overflow can with a short metal tube extending outward a bit below the top of the can. Place the
overflow can on a lab jack set at its lowest position. Place a plastic beaker next to the overflow can so that
the plastic beaker will catch any water that overflows. Fill the overflow can until water starts overflowing
into the plastic beaker, but don’t let too much water overflow. Once water has stopped overflowing carefully
remove the plastic beaker making sure not to disturb the overflow can and weigh the plastic beaker. Record
the plastic beaker’s weight and then place it back beside the overflow can. (Record weights as mass times g
where mass is measured in kilograms and g = 9.8 m/s2.)
Mount a scale on top of a small ring stand. You should find a cylindrical slot on the underside of the scale.
Pick a metal block of any material and hang it with a string (about 6 to 8 inches long) from the metal support
at the bottom of the scale that is directly connected to the scale’s balance pan (the flat horizontal circular
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surface where you usually place objects to measure their mass). With the metal block hanging from the scale
determine the mass of the metal block by balancing the scale.
Carefully move the scale so the metal block is directly over the overflow can and very slowly raise the lab
jack so that the metal block becomes submerged. Water will overflow into the plastic beaker. Once water has
stopped overflowing re-weigh the plastic beaker. Also, you will notice that the scale is no longer balanced.
Balance the scale to determine the apparent weight of the metal block. This “weight” is really equal to the
actual weight minus the buoyant force. From the change in weight of the block calculate the buoyant force.
Also, from the change in weight of the plastic beaker calculate the weight of the water displaced. Should
these weights be equal? Compare their values by calculating a percent difference.
Part II: Determining the Density of objects more dense than water.
Pick three metal blocks that look like they are made of different materials. Write a short description,
including the color, of each block in the data section. Also, measure the weight of each block and enter this
in Table 1.
Submerge each block in water and measure the apparent weight. For each block calculate the density and use
the calculated density to determine the type of metal the block is made of. You should be able to find
densities of various metals in your textbook.
Part III: Determining the Density of objects less dense than water.
To measure the density of an object that sinks in water we simply let the object sink and then we measured
the object’s apparent weight. We then used equation (3) which was derived from equation (1) assuming that
the volume of the object is equal to the volume of water displaced. These equations will not work for objects
that float since some of the object remains above water and thus the volumes are not equal.
In order to use equation (1) for the buoyant force we must be sure the object is completely submerged. We
can do this by hanging each object from the scale by a string and then hanging a metal block from the object.
The greater density of the metal block will pull the less dense object below the surface of the water.
Pick two objects that will float, but do not get them wet yet. Describe your two objects in the data section.
Hang one of the objects from the scale and measure its weight. Record this in table 2. Hang a metal block
(this block shouldn’t be one that you previously used) from the bottom of the object and record the weight of
the object plus the metal block. Place an empty beaker on a lab jack directly below the hanging masses and
raise the beaker until the metal block is just above the bottom of the beaker. If the object extends above the
top of the beaker even slightly you will need to shorten the length of the string between the object and metal
block otherwise you will not be able to completely submerge the object.
Carefully add water to the beaker being careful not to get the object wet, and not to create bubbles on the
metal block. Once the metal block is submerged stop adding water. Measure the weight of the masses and
record in table 2. Finally, add more water until the object is completely submerged. Measure the weight
again and record. Refer to question 3 for analysis of these measurements.
Proceed to the Questions section.
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Data
Part I: Verification of Archimedes’ Principle:
Initial Weight of plastic beaker = _____________ (kg m/s2)
Weight of hanging metal block = _____________(kg m/s2)
Weight of hanging metal block when fully submerged = ______________(kg m/s2)
Final weight of plastic beaker = _______________( kg m/s2)
Weight of water displaced = _____________ (kg m/s2)
Measurement of Buoyant force from change in metal block’s weight = _____________(kg m/s2)
Percent error = ______________
Part II: Density of three metal blocks
Block 1: Description ________________________________________________________________
______________________________________________________________
Block 2: Description ________________________________________________________________
______________________________________________________________
Block 3: Description ________________________________________________________________
______________________________________________________________
Block
Weight (kg m/s2)
Weight while
submerged (kg m/s2)
Density (kg/m3)
Type of Metal
1
2
3
Table 1
Part III: Density of floating objects.
Description of object 1: ____________________________________________________________
Description of object 2: ____________________________________________________________
Object
Weight
(kg m/s2)
Weight of object
and metal block
(wa) (kg m/s2)
Weight of object and
metal block with only
block submerged
(wb) (kg m/s2)
Weight with both
submerged
(wc) (kg m/s2)
Density of
Object
(kg/m3)
1
2
Table 2
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Questions
1.
Using the results of part I are you satisfied that the buoyant force equals the weight of the water
displaced? Explain why or why not.
2.
In part I your instructor may have asked you to use a graduated cylinder instead of the overflow can.
Explain the measurements that you made and how those measurements lead you to conclude that
Archimedes’ Principle is correct.
3.
In part II you determined the type of metal that made up each block. How certain are you that each
block is made of the metal you think it is made of? For each block calculate the percent error between
your measured density and the density of the metal that you think the block is made of.
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4.
In part III you measured the weight of a low density object and then hung a metal object from the low
density object and measured their combined weight three times. a) When neither was submerged, b)
when only the metal object was submerged, and c) when both were submerged. In total, you measured
four different “weights.”
Draw a free body diagram for metal object/low-density object system for each of the three cases. Treat
the metal object and the low-density object as one combined object so that you can ignore the tension
in the string between them.
Label all forces. Label the tension in the string between the scale and the object as wa, wb, and wc in
cases a), b), and c) respectively.
In case a) there is no buoyant force. In case b) there is a buoyant force on the metal object. In case c)
there is also a buoyant force on the low-density object. In each case the buoyant force is given by
equation (1), but can be measured by considering the change in apparent weight as the metal object and
low-density object are submerged. Using equations that you determine from the three cases along with
only the four measured weights described above, calculate the density of the low-density object. Show
all your work below and enter the calculated densities in Table 2.
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5.
Suppose bubbles formed on the surface of the objects that your were submerging. How would these
bubbles affect the measurement of the density of the objects? Would the bubbles make the measured
densities too large, or too small? Explain.
6.
Suppose there is a small pond near your home. A measuring stick is mounted at the end of a pier in
order to measure the height of the water. Suppose the height of the water is 100 cm. You are carrying a
large boulder as you step into a small boat tied to the pier. As you step into the boat the water’s height
rises to 105 cm. Without making waves you paddle out to the middle of the pond. While floating there
you decide to slowly drop the boulder into the water. Initially there are some ripples on the surface of
the water, but after a few minutes the water is calm. What is the height of the water now? (a)
Unchanged. (b) greater than 105 cm, (c) less than 105 cm but greater than 100 cm, or (d) less than 100
cm. Explain
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