Physics 181 - Summer 2013 - Experiment #3
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Experiment #3, Measurement of Density and Archimedes' Principle
1 Purpose
1.
To determine the density of a fluid, such as water, by measurement of its mass when confined in a known
volume; e.g., a 25ml pycnometer (glass container).
2.
To determine the density of a cubic-shaped solid, such as aluminum, by measuring its mass and volume.
3.
To become familiar with some basic properties of fluids, such as density, pressure, and buoyant forces,
and to verify Archimedes' principle. The physics of such phenomena is referred to as hydrostatics.
4.
To utilize Archimedes' principle to determine the density of water.
2 Introduction
2.1 Definition of mass density
Many physical properties of solid objects or fluids depend not on their total mass, but on their mass density.
The mass density of a material, designated by the Greek letter ρ (rho), is defined as the amount of mass per unit
volume. Its unit is g/cm3 in cgs, or kg/m3 in SI. For a solid object of mass m, and volume V, we have:
=
m
V
(1)
The density of a homogeneous fluid is similarly defined as the mass (m) of a sample of the fluid divided by the
volume (V) it occupies. Normally, a fluid is confined in a container and therefore its volume is related to the
volume of the container.
Caution: Do not confuse the mass of an object with its weight. The SI units of mass (m) and weight, W, are the
kilogram (kg) and the Newton (N), respectively. The pound (lb) is a unit of weight and not of mass. The weight
and the mass of an object are related by W = mg where g is the acceleration of gravity (980.35 cm/s2).
2.2 Fluid pressure
At any depth h in a fluid, there is a pressure due to the weight of the fluid above this depth. The deeper we
descend, the greater is the pressure due to the overlying liquid. In fact, the pressure increases linearly with the
depth of a fluid if its density remains constant. This can be deduced easily by considering a column of fluid shown
as shaded in Fig. 1, which depicts a container filled with a fluid (f) of density,f. The pressure, P, across any
surface of area A in the fluid is defined in terms of the normal force, F, exerted by the fluid one the side of the
surface: P =F/A (force per unit area), i.e., F = PA.
The volume (V), in Fig 1, of the shaded cylindrical column of height h, and the base area A, is given by V=Ah, so
that the mass of fluid contained in this volume is m = V = f Ah. Therefore, the weight of the column is given by
W = mg = f Ahg.
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Figure 1: Forces on a column of liquid
Figure 2: Pressure P vs. depth h
As the column is stationary, it is in static equilibrium, i.e., the sum of vertical forces is zero. Therefore, its weight,
W, which acts in the downward direction on the base area A, is balanced by the difference between an upward
force F - pushed up from below by the fluid - and the downward force Fo - due to the atmospheric pressure. This
net upward force acting on the column is called the buoyant force B, and its magnitude is equal to the weight (W)
of the column (fluid).
At equilibrium, the pressure P that the fluid exerts upwards on the bottom of this column must equal the
atmospheric pressure Po, plus the weight (W) of the column divided by its area (A), i.e., the weight of the fluid
column is supported by the difference in pressure forces pushing up from the bottom and down from the top of the
fluid column.
P = F/A, Po = Fo/A, W/A = fgh
P = Po + fgh
(2)
(3)
Please note that from equations (2) and (3) we deduce the following: The fluid pressure increases linearly with
depth in a liquid and also linearly with fluid density.
The pressure increases with depth, P - Po, is directly proportional to the depth below the surface of the fluid as
indicated by the linear relationship in Fig. 2. The pressure is propagated uniformly in all directions and across all
cross sections in the fluid. This result is general and applies at any depth since the fluid below that depth does not
contribute to the weight supported by the area A.
Pressure P changes if the atmospheric pressure changes, in fact, it changes by the same amount as the change in
the atmospheric pressure.
2.3 Archimedes' principle and its application
Fig. 3 shows the outline of an irregular object immersed in a fluid and acted on by two forces, one due to its weight
and the other due to the pressure exerted on its surface by the fluid. As shown in section 2.2 above, the pressure in
a fluid increases with depth, causing the force from the fluid pushing up on the bottom of the object to be greater
than that pushing down from the top of the object. The net effect is for the fluid to exert an upward resultant force
on the object.
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Figure 3. Forces on an irregular object immersed in a fluid.
2.3.1 Archimedes' principle
To calculate the net buoyant force on an irregularly shaped object, such as shown in Fig. 3, is difficult. However,
the ancient Greek scientist and mathematician Archimedes (287-212 B.C.) managed to deduce the correct size of
this buoyant force, independent of the shape of the object. The basic ideas in his reasoning are as follows.
Archimedes considered the force exerted on a small portion of the fluid in a container. This small portion is itself
in static equilibrium, otherwise it would rise or sink through the surrounding fluid. Because the small portion is in
static equilibrium, the net force on it must be zero. Hence, the upward buoyant force, B, must equal the downward
gravitational force, W = mg, i.e., the buoyant force on a volume of fluid is equal to the weight of the fluid.
Archimedes further reasoned that if the liquid in this volume were removed and replaced by an object of exactly
the same size and shape as this liquid portion, none of the liquid pressure forces acting on its surface would
change. Because the object is exactly the same shape and volume as the fluid removed, it would fit exactly into the
previous volume without compressing the surrounding fluid. Therefore, he arrived at the conclusion known as
Archimedes' principle, that the net buoyant force B upward on any object immersed in a fluid is equal to the
weight of the fluid displaced.
Using the subscript f for fluid, we have:
B = Wf = mfg = (fVobj)g
(4)
where the weight of the fluid has been expressed in terms of its density, ρf, and its volume, which is equal to the
volume of the object, Vobj.
This principle applies to any object immersed in any gas or fluid, such as a hot-air or helium balloon or a
submarine in water. The balloon (plus any weight it carries) is totally supported by the buoyant force of the
air it displaces.
2.3.2 Application of Archimedes' principle
Using Archimedes' principle we can easily determine whether an object (obj) will sink or float in a given fluid.
This clearly depends on whether the buoyant force acting upward on the object, totally immersed in the fluid, is
greater or less than the weight of the object. Using Archimedes' principle, as expressed by equation (4) to give the
buoyant force in terms of the fluid density f, we can calculate the net downward force as:
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Fnet = W – B
Fnet = mg - Wf
Fnet = g(objVobj) – g(fVobj)
Fnet = g(obj - f )Vobj
(5)
Therefore, if the density of the object, obj, is greater than the density of the fluid f, the net force on the object is
downward and the object sinks, but if the density of the object is less than that of the liquid, the net force is upward
and the object floats. For example, because ice floats in water we conclude that the density of ice is less than the
density of water.
3 Experimental Apparatus and Procedure
3.1 Theory of the balance
The balance shown in Fig. 4 is used in this experiment to measure
unknown masses and forces. An unknown mass, m, placed on the
left-hand pan can be determined by finding a combination of standard
masses, ms, which when placed on the right-hand pan, have a weight Ws=
msg that is equal to the weight of the unknown mass W = mg. In this case,
the forces and torques exerted on both sides of the balance are equal and
there is no net deflection or rotation of the indicator relative to the scale.
The balance is sensitive to the gravitational forces (weight) on the unknown and standard objects and not on their masses. One is able to
calibrate the balance in units of mass (grams) only because the acceleration of gravity is constant. (mg = msg, where g is constant) Note that m =
ms when W = Ws. The balance used for this experiment has mass riders on
two horizontal scales on the front of the balance (as shown in Fig. 4)
which are used as the standard mass.
Figure 4: The trip balance.
3.1.1 Use of the balance to measure the buoyant force
A method for using the balance to measure buoyant forces on objects denser than the fluid is shown in the three
diagrams in Fig. 5. The object is first balanced by placing an equal standard mass on the right-hand pan, as shown
in the center diagram. The object, suspended on a string attached to the pan, is immersed in the liquid, as shown in
the last diagram. The buoyant force is calculated from the amount of mass Δm that must be removed from
the right-hand pan to obtain balance in the presence of the buoyant force acting on the object.
Figure 5: Method to determine the buoyant force using Archimedes' principle.
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The buoyant force is given by:
B = (m)g
(6)
The density of the liquid may then be found by equating this buoyant force to the expression previously derived in
terms of the fluid density and volume of the object:
B = fVobjg
(7)
We solve this for the density of the fluid:
f 
m
Vobj
(8)
We will use this method in the last part of the experiment to determine the density of water, by using an object of
known volume and Archimedes' principle. This expression for the unknown fluid density has the dimensions of
mass per unit volume. g/cm3 = g/ml [1 milliliter of volume equals 1 cubic centimeter (cc) or 1 ml = 1 cm3].
3.2 Measurements
3.2.1 Determination of the density of water
This method to determine the density of water consists of measuring the mass of a
known volume of water using the balance and a 25-ml pycnometer, shown in Fig. 6.
The pycnometer is also known as a volumetric flask, or specific gravity bottle. Its
glass stopper has a capillary opening. It should be filled until the fluid level reaches
the top of the capillary; only then will the volume of the fluid in the pycnometer
equal the number stamped on it. Fill the pycnometer bottle to the top and then,
insert the stopper. Wipe off the excess fluid that overflows from the stopper.
Figure 6: The pycnometer.
Method:
1.
Record the volume of the pycnometer (not exactly 25.00 ml, be careful!)
2.
First, balance the empty pycnometer and its stopper and record its mass.
3.
Fill the pycnometer with water until it is full, and insert stopper. Water will overflow the capillary
opening. Wipe off any excess water on the outside of the pycnometer.
4.
Measure and record the mass of the filled flask and stopper.
5.
Empty the pycnometer when finished.
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3.2.2 Measurement of the density of aluminum
Method:
1. Use a caliper to measure the three sides, (La)(Lb)(Lc), of the aluminum block. Record your results to the
nearest tenth of a millimeter. Do not include the string when measuring the length, and be sure that the
caliper is touching the total side of the cube. Just using the tips of the caliper distorts the reading.
2.
Measure and record the aluminum block's mass. A paper clip should be on the right pan so that the mass
of the paper clip can be ignored. The mass of the thread is negligible and ignored.
3.2.3 Verification of Archimedes' principle
Carry out the verification of Archimedes' principle using a device consisting of
a pair of finely machined concentric cylinders, the cross section of which is
shown in Fig. 7. The outer hollow cylinder is closed at one end. The matching
solid cylinder fits inside the outer cylinder. Therefore, the volume of the solid
cylinder 'precisely' equals the inner volume of the hollow cylinder. Determine
whether Archimedes' principle holds by observing the volume of water needed
to re balance the scale when the cylinder is immersed in water. Keep in mind
that the volume in the hollow cylinder equals the volume of the solid cylinder.
Figure 7. Concentric cylinders.
Method:
1.
Do not place the solid cylinder into the hollow one as the string may break when pulled.
2.
Hang the solid cylinder (by use of the paper clip) over the side of the left pan and into an empty glass
beaker.
3.
Place the hollow cylinder on top of the left pan and balance with the sliding standard masses. Record the
value of this combined mass as m1.
4.
Pour water into the glass beaker, making sure that the solid cylinder is totally immersed.
5.
Record your observations as to what happened to the balance as you poured in the water. Which side of
the pan balance is lifted as you poured in the water? Was the solid cylinder forced up by the
buoyant force of water?
6.
Remove any air bubbles which may have formed under the solid cylinder.
7.
Rebalance the scale, and record this value as m2.
8.
Return the sliding masses so that the scale reads m1 from step 3. The solid cylinder should be left
immersed in the water, so the scale at this point will not be balanced!
9.
Add water to the hollow cylinder until the scale is balanced. Be careful not to spill water onto the pan.
(Use the eye dropper when the cylinder is about full.)
10. What happened to the left pan as you added water to the hollow cylinder in step 9? What can you
say about the volume of water you added to the hollow cylinder and the volume of the water (in the
beaker) displaced by the solid cylinder? According to Archimedes's principle, should these volumes
be the same?
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11. Leave the cylinders separated when finished. Dry the hollow cylinder.
3.2.4 Measurement of the volume of the aluminum cube by Archimedes' principle
Method:
1.
Hang the aluminum cube (using string and a hook made from a paper clip) into an empty glass beaker.
2.
Balance and record its mass, m1.
3.
Fill the beaker with water to totally immerse the cube. Record your observation. Which pan goes up?
What causes the unbalance?
4.
Remove any air bubbles which may have formed under the cube.
5.
Rebalance the scale and record the new mass as indicated by the balance, m2.
4 Calculations and Analysis of the Data
4.1 Calculation of the density of water
1.
Compute the mass of water in the pycnometer by calculating the difference in standard mass used to
balance the pycnometer when empty and when full of water.
2.
Compute the density of water (g/ml) by dividing this mass in grams by its volume indicated on the side of
the pycnometer.
3.
Use percent error to compare your result with the accepted value of the density of water at 20oC (page 9).
4.2 Calculation of the volume of the aluminum cube and its density
1.
Calculate the volume of the cube, where Vcube,1 = (L1)(L2)(L3).
2.
From its mass and volume, calculate the density of aluminum. Compare it with the accepted handbook
value given in the Appendix I on page 9. Calculate the percent error between the accepted and
experimental values for the density of the aluminum block.
4.3 Calculations of buoyant force
1.
Using ΔmB, calculate the value of the buoyant force on the solid cylinder when water is added to the glass
beaker in step 4 of 3.2.3. Enter the buoyant force using scientific notation in your lab report.
2.
Note comments in step 5 of 3.2.3, and answer them in this part of your lab report.
3.
Give your response to the questions in step 10 of 3.2.3.
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4.4 Calculation of the volume of the aluminum cube.
As explained in section 2, the density of a liquid is given by:
f=
( m)
V obj
where Vobj is the volume of the aluminum cube, Vcube, - in this case - and where Δm is the amount of mass that must
be removed from the right-hand side of the balance to rebalance the aluminum cube when it is immersed in water.
1.
Use m1 and m2 to calculate Δm for water. In addition, answer part 3 of 3.2.4.
2.
Use the handbook value of the density of water at 20oC, given in the Appendix I on page 9, and the
corresponding Δm in water, to obtain the volume of the aluminum cube. Compare this value using
percent error to the volume calculated in step 1 of 4.2 above. Use 4.2 as the accepted value.
5 Questions
1.
In a short table, compare your experimental results with the accepted densities. This is a simple repeat of
4.1.3 and 4.2.2. Calculate and list the percent error between corresponding accepted and experimental
values of the densities.
2.
Discuss qualitatively the errors (random and/or systematic) in your various measurements of densities.
3. Calculate the buoyant force on the aluminum cube exerted by water using your data from 4.4.1. Use cgs
units (centimeter-grams-second). Your answer should be in scientific notation.
4.
500 ml of glycerin is poured into a liter container. The aluminum cube used in this experiment is dropped
into the same container. Would the cube sink? Show your calculations. See Appendix I on page 9. Use
Vcube,1.
5. Determine the volume of a hydrogen balloon just large enough to almost lift a 75 kg person in a 50 kg
basket (includes mass of balloon) off the ground. Here, assume that Fnet = 0.00 N when the balloon almost
starts to lift, and that the weight of the person, basket and inflated balloon, is equal to the weight of the air
displaced. A balloon (plus any weight it carries) is totally supported by the buoyant force of the air it
displaces. See Appendix I on page 9.
6 Conclusion
This section should have a clear statement of the results of the experiment and the extent to which the results are in
agreement with the theory being tested. Use percent error to compare your experimental result to the accepted or
theoretical value. If percent error was done earlier, you are still required to do it in this section.
To make this comparison meaningful, you should include the impact of the experimental error on your results.
This includes errors in plotting and reading linear graphs when determining their slope and intercept. Possible
systematic and procedural errors should be included. Please include a statement of what you have learned, a
critique of the experiment, and any suggestions you have which you think could improve the experiment or the lab
handout.
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The Measurement of Density and Archimedes' Principle Appendix I
Densities of some common materials at T = 20oC. (The densities of the gases listed are given for 20oC and 1
atmospheric pressure.)
Substance
Density (g/cm3)
Air
1.293 x10-3
Aluminum
2.70
Glycerin
1.26
Hydrogen
0.0899 x10-3
Water
0.99823
(g/ml = grams/cm3 = g/cc)
Data Sheet
3.2.1 Determination of the density of Water
Measurements:
1. Volume of the pycnometer
________ ml
2. Mass of the empty pycnometer and its stopper ________ g
3. Mass of the full pycnometer and its stopper
________ g
4. Temperature of the water
________ oC
 T = ________ oC
5. Error in reading thermometer
6. Error in reading scale balance
 m = ________ g
Computations:
1. Compute the mass of the water in the pycnometer
2. Compute the density of the water
3. Handbook value of the density of the water (page 9)
________ g
________ g/ml
________ g/ml
4. Percent error ________ %
3.2.2 Measurement of the density of aluminum
Measurements:
1. Length of each side of the aluminum cube:
L1 = _________ cm, L2 = _________ cm, L3 = __________ cm
2. Mass of the aluminum cube:
__________ g
 L = _______ cm
3. Error in reading caliper
Computations:
1. Volume of the aluminum cube, Vcube,1 ________ cc
(Note: 1cc = 1cm3 = 1ml)
2. Density of the aluminum cube
________ g/cc
3. Handbook value of the density of aluminum ________ g/cc
4. Percent error ________ %
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3.2.3 Verification of the Archimedes' principle
Measurements:
Step 3: Combined mass of solid and hollow cylinders, m1 ________ g
Step 7: Recorded mass when scale is rebalanced after solid
________ g
cylinder is immersed in water, m2
Change in mass due to immersion in water,
________ g
ΔmB = m1 - m2
Calculations:
1. Compute the value of the buoyant force on the solid cylinder when immersed in water in step 4 of
3.2.3.
g =980.35 cm/s2
B = (ΔmB)g
B = ____________ dyne Use scientific notation. (1 dyne = 1 g-cm/s2)
3.2.4 Second determination of the volume of the cube:
Measurements:
1. Mass of the aluminum cube in air ________ g
2. Mass of the aluminum cube immersed in water ________ g
Computations:
1. Calculate Δm for water
Δmwater _______ g
2. Use the handbook value of the density of water at 20oC and the corresponding Δm in water to obtain
the volume of the aluminum cube. Compare this value to the volume calculated in 4.2.1.
Vcube,2 = Δmwater / ρf = ________ cc
Percent error in volumes ________ %