\setcounter{figure}{0}
\setcounter{table}{0}
\hspace*{3.0in}Name:{\hrulefill} \\
\hspace*{3.25in}Date:{\hrulefill}

\begin{Large}
\section{\bf Density}
\end{Large}

\subsection{Introduction}

As we explore the objects in our Solar System, we quickly find out that these objects
come in all kinds of shapes and sizes. The Sun is the largest object in the Solar
System and is so big that more than 1.3 million Earths could fit inside. But the
mass of the Sun is only 333,000 times that of the Earth. If the Sun were made of the
same stuff as the Earth, it should have a mass that is 1.3 million times the mass
of the Earth---obviously, the Sun and the Earth are not composed of the same stuff!
What we have just done is a direct comparison of the {\it densities} of the Sun and 
Earth.  Density is extremely useful for examining what an object is made of, 
especially in astronomy, where nearly all of the objects of interest are very far 
away.\\

In today's lab we will learn about density, both how to measure it, and how to use
it to gain insight into the composition of objects. The average or ``mean''
density is defined as the {\it mass} of the object divided by its {\it volume}.  
We will use grams (g) for mass and cubic centimeters (cm$^3$) for volume.  
The {\it mass} of an object is a measure of how many protons and neutrons (the
``building blocks'' of atoms) the object 
contains. Denser elements, such as gold, possess many more protons and neutrons within 
a cubic centimeter than do less dense materials such as water.\\
\\
\subsection{Mass versus Weight}
Before we go any further, we need to talk about {\it mass} versus {\it weight}.
The {\it weight} of an object is a measure of the {\it force} exerted upon that object 
by the gravitational attraction of a large, nearby body. An object here on the Earth's 
surface with a {\it mass} of 454 grams (grams and kilograms are a measure of the mass 
of an object) has a weight of one pound.  If we do not remove or add any protons or 
neutrons to this object, its mass and density will not change if we move the object 
around.  However, if we move this object to some other location in the Solar System,
where the gravitational attraction is different then what it is at the Earth's surface, 
than the {\it weight} of this object will be different.  For example, if you weigh
150 lbs on Earth, you will only weigh 25 lbs on the Moon, but would weigh 355 lbs on
Jupiter. Thus, weight is not a useful measurement when talking about the bulk
properties of an object---we need to use a quantity that does not depend on {\it where}
an object is located. One such property is {\it mass}. So, even though you often see 
conversions between pounds (unit of weight) and kilograms (unit of mass), those 
conversions are only valid on the Earth's surface (the astronauts floating around 
inside the International Space Station obviously still have mass, even though they are 
``weightless'').

\subsection{Volume}

Now that we have discussed mass, we need to talk about the other quantity
in our equation for density, and that is volume. Volume is pretty easy
to calculate for objects with regular shapes. For example, you probably know how
to calculate the volume of a cube: V = ~$s$~$\times$~$s$ $\times$~$s$~= $s^{\rm 3}$, 
where $s$ is the length of a side of the cube. Let us generalize this to any rectangular 
solid. In Figure  \ref{boxdim-fig}  we show a drawing for a box that has sides labeled 
with ``length,'' ``width,'' and ``height.'' What is its volume? Its volume is V = 
length $\times$ width $\times$ height. If we told you that the length = 10 cm, the 
height = 5 cm, and the depth = 5 cm, what is the box's volume? V = 10 cm $\times$ 5 cm 
$\times$ 5 cm =
250 cubic cm = 250 cm$^{\rm 3}$. Do you now see why volume is measured in cm$^{\rm 3}$?
This where that comes from---everyday objects are ``three dimensional'' in that
they {\it have} volume (cm$^{\rm 3}$,
m$^{\rm 3}$, km$^{\rm 3}$, inches$^{\rm 3}$, miles$^{\rm 3}$). \\

\begin{figure}[htb]
\centerline{{\includegraphics[width=7cm]{boxdim.jpg}}}
\caption{A rectangular solid has sides of length, width, and height.}
\label{boxdim-fig}
\end{figure}

Now that we understand how volume is calculated, how do we do it for objects that
have more complicated shapes, like a coke bottle, a car engine, or a human being? You
may have heard the story of Archimedes. Archimedes was asked by the King of Syracuse
(in ancient Greece) to find out if the dentist making a gold crown for one of his
teeth had embezzled some of the gold the king had given him to make this crown (by
adding lead, or another cheaper metal to the crown while keeping some of the gold for 
himself). Archimedes
pondered the problem for a while and hit on the solution while taking a bath.
Archimedes became so excited he ran out into the street naked shouting ``Eureka!''
What Archimedes realized was that you can use water to figure out a solid object's
volume. For example, you could fill a teacup to the brim with water and drop an object 
in the teacup. The amount of water that overflows and collects in the saucer has the 
same volume as that object. All you need to know to figure out the object's 
volume is the conversion from the amount of liquid water to its volume in 
cm$^{\rm 3}$. An example of the process is shown in Fig. \ref{beaker-fig}. \\

In the metric system a gram was defined to be equal to one cubic cm of water, and
one cubic cm of water is identical to 1 ml (where ``ml'' stands for milliliter, i.e., 
one thousandth of a liter). Today we will measure the water
displacement for a variety of objects, and use this conversion directly: 1 ml = 1 
cm$^{\rm 3}$. \\

In this lab you will first determine the densities of ten different natural 
substances, and then we will show you how astronomers use density to give us 
insight into the nature of various objects in our Solar System.

\begin{flushleft}
{\bf Exercise \#1:} Measuring Masses, Volumes and Densities\\
\end{flushleft}

First, we measure the masses of objects using a triple beam balance. At your 
table, your TA has given you a plastic box with a number of compartments 
containing ten different substances, a triple beam balance, several graduated 
cylinders, digital calipers, a ``Eureka'' can (or overflow vessel), a
container of water, and an empty plastic cup. Our first task is to measure 
the masses of all ten of the objects using the triple beam balance. Note: 
these balances are very sensitive, and quite expensive, so treat them with 
care. The first thing you should do is make sure all of the 
weights\footnote[1]{This is the historical name for these sliding masses, as 
the first scales like these were used to measure weight.} are moved to their 
leftmost positions so that their pointers are all on {\it zero}. When this is 
done, and there is no mass on the steel ``pan,'' the lines on the right hand 
part of the scale should line-up with each other {\it \bf exactly}. The scale 
must be balanced before you begin, and the TA, or their helper, has already
done this for you. If the two lines do not line-up, ask your TA for help. \\

To measure the mass of one of the objects, put it on the pan and slide the 
weights over to the right. Note that for this lab, none of our objects require 
movement of the largest weight, just the two smaller weights. You should 
attempt to read the mass of the object to two significant figures---it is 
possible, but quite unlikely, that an object will have a mass of exactly 10.0 
or 20.0 g. If the sliding weight on the ``10 g'' beam falls between units, 
estimate exactly where it is so that you get more precise numbers like 22.15 g 
(all of your masses should be measured to two places beyond the decimal!).\\

\begin{flushleft}
{\bf Task \#1:} Fill in column \#2 (``Mass") of Table~\ref{density-tab} by measuring the
masses of your ten objects. {\bf (10 points)}\\
\end{flushleft}

\begin{figure}[htb]
%\centerline{{\includegraphics[width=7cm]{beakers.pdf}}}
\centerline{{\includegraphics[width=7cm]{beakers.jpg}}}
\caption{The rectangular object displaces 10 ml of water.  Therefore, it has a volume of 10 ml = 10 cm$^3$.} 
\label{beaker-fig}
\end{figure}
\vspace{0.3 in}
\vspace{0.35in}
\begin{table}[htb]
\begin{center}
\caption{The Masses, Volumes, and Densities of the Different Objects.}
\begin{tabular}{|l|c|c|c|c|}\hline
{\bf Object} &{\bf Mass (g)} &Volume of Water&{\bf Volume}&{\bf Density}\\ 
             &&  (ml) & cm$^{\rm 3}$& g/cm$^{\rm 3}$\\
{\bf Column \#1}& {\bf \#2}& {\bf \#3}&{\bf \#4}&{\bf \#5} \\
\hline\hline
Obsidian     &&&&	\\ \hline
Gabbro       &&&&     \\ \hline
Pumice$^{\rm 2}$       &&&&	\\ \hline
Silicon      &&&&	\\ \hline
Magnesium     &&&&\\ \hline
Copper       &&&&       \\ \hline
Iron (Steel) &&&&       \\ \hline
Zinc         &&&&       \\ \hline
Mystery      &&&&       \\ \hline
Aluminum     &&&&       \\ \hline
\end{tabular}
\label{density-tab}
\end{center}
$^{\rm 2}$It is tricky to measure the volume of Pumice, but find a way to 
{\it submerge} the entire stone.
\end{table}

Now we are going to measure the volumes of these ten objects using the method 
of Archimedes, and the Eureka can. Fill the tin can with water until it
comes out of the spout (you might use one of your graduated cylinders to
collect the overflow). Take your finger and pass it under the bottom of the
spout to remove the drop of water that (usually) clings to the spout. Now,
hold the appropriate sized, empty, dry, graduated cylinder under the spout
and drop in the first object. Read the graduated cylinder to find how much
water was displaced, {\it the volume}, and record this value in the table. 
Repeat the process for all of your objects. Note that the smaller the object, 
the smaller the graduated cylinder you should use. Using a big cylinder with
a small object will lead to errors, as the big cylinders are harder to read to 
high precision. Ask your TA about how to ``read the miniscus'' if you do not 
know what that means.\\
\vspace{0.3 in}

\begin{flushleft}
{\bf Task \#2:} Fill in columns 3 and 4 (again, remember for column \#4, that
 1 ml = 1 cm$^{\rm 3}$). {\bf (10 points)}\\
\end{flushleft}
\vspace{0.1 in}

\begin{flushleft}
{\bf Task \#3:} Fill in the Density column in Table \ref{density-tab}. {\bf (5 points)}
\end{flushleft}
\vspace{0.1 in}

\begin{flushleft}
{\bf Question \# 1:} Think about the process you used to determine the volume.
How accurate do you think it is? Why? How could we improve this technique? {\bf 
(5 points)}
\end{flushleft}
\vspace{2.5 in}

We chose to supply you with several rectangular solids so that we could check 
on how well you measured the volume using the Archimedes method. Now we want 
you to actually measure the volume of the five metal ``cubes'' (do not assume 
they are perfect cubes!) using the digital caliper. You will measure the 
lengths of their sides in mm, but remember to convert to cm (1 cm = 10 mm). 
The digital caliper is easy to operate, but requires two actions: 1) there
is a button that switches between inches and millimeters, we want mm, and 2)
they must be ``zeroed''. To zero the caliper, use the thumbwheel to insure
the jaws are closed, and then hit the ``zero'' button. Open the caliper
slowly to the width necessary to measure the cube, and then close them tight.
Read off the number. It is not a bad idea to zero the caliper before each
object, as repeated motion can cause small errors to creep-in.

\begin{flushleft}
{\bf Task \#4:} Fill in Table \ref{newdensity-tab}. Copy the mass 
measurements from Table \ref{density-tab} for the five metal ``cubes''. 
Calculate the volumes of these ``cubes'' using the caliper. {\bf(5 points)}
\end{flushleft}

\begin{table}[htb]
\begin{center}
\caption{The Masses, Volumes, and Densities of the Metal Cubes.}
\begin{tabular}{|l|c|c|c|c|}\hline
{\bf Object} &{\bf Mass (g)} & ~ ~ ~ $l \times ~w \times ~h$ ~ ~ =&{\bf Volume} cm$^{\rm 3}$&{\bf Density} g/cm$^{\rm 3}$\\ 
\hline
Copper       &&&&       \\ \hline
Iron (Steel) &&&&       \\ \hline
Zinc         &&&&       \\ \hline
Mystery      &&&&       \\ \hline
Aluminum     &&&&       \\ \hline
\end{tabular}
\label{newdensity-tab}
\end{center}
\end{table}

\begin{flushleft}
{\bf Question \#2:} Compare the two sets of densities you found for each of the five
metal cubes. How close are they? Assuming the second method was better, which 
substance had the biggest error? Why do you think that happened? 
{\bf (5 points)}\\
\end{flushleft}
\vspace{2.5 in}
\clearpage
~ \\
\vspace{1.5in}
\begin{flushleft}
{\bf Question \#3:} One of the objects in our table was labeled as a ``mystery''
metal. This particular substance is composed of two metals, called an ``alloy.''
You have already measured the density of the two metals that compose this alloy. We now 
want you to figure out which of these two metals are in this alloy. Note that this 
particular alloy is a 50-50 mixture! So its mean density is (Metal A + Metal B)/2.0. 
What are these two metals? Did its color help you decide? {\bf (3 points)}\\
\end{flushleft}
\vspace{1.2 in}
You have just used density to attempt to figure out the composition of an unknown
object. Obviously, we had to tell you additional information to allow you to derive
this answer. Scientists are not so lucky, they have to figure out the compositions of
objects without such hints (though they have additional techniques besides density
to determine what something is made of--you will learn about some of these this
semester). \\

\begin{flushleft}
{\bf Exercise \#2:} Using Density to Understand the Composition of Planets.\\
\end{flushleft}

We now want to show you how density is used in astronomy to figure out the 
compositions of the planets, and other astronomical bodies. As part of Exercise \#1, you 
measured the density of
three rocks: Obsidian, Gabbro, and Pumice. All three of these rocks are the result
of volcanic eruptions. Even though they are volcanic in origin (``igneous rocks''),
both Obsidian and Gabbro have densities similar to most of the rocks on the
Earth's surface. So, what elements are found in Obsidian and Gabbro? Their chemistries
are quite similar. Obsidian is 75\% Silicon dioxide (SiO$_{\rm 2}$), with a little
bit (25\%) of Magnesium (Mg) and Iron (Fe) oxides (MgO, and 
Fe$_{\rm 3}$O$_{\rm 4}$). Gabbro has the same elements, but less Silicon dioxide 
($\sim$~50\%), and more Magnesium and Iron.\\

\begin{flushleft}
{\bf Question \#4:} You measured the densities of (pure) silicon, iron and magnesium
in Exercise \#1. Compare the density of Gabbro and Obsidian to that of pure silicon.
Can you tell that there must be some iron and/or magnesium in these minerals? How?
Which of these two elements {\it must} dominate?
Were your density measurements good enough to demonstrate that Gabbro has less
silicon than Obsidian? 
{\bf (4 points)}\\
\end{flushleft}
\vspace{2.9 in}

Now let's compare the densities of these rocks to two familiar objects: the Earth and 
the Moon. We have listed the mean densities of the Earth and Moon in Table 
\ref{earthden-tab}, along with the density of the Earth's crust. As you can
see, the mean density of the Earth's crust is similar to the value you determined
for Gabbro and/or Obsidian--it better be, as these rocks {\it are from the
Earth's crust}!
\vspace{0.4 in}

\begin{table}[htb]
\begin{center}
\caption{Densities of the Earth and Moon}
\begin{tabular}{|c|c|}\hline
Object  & Density g/cm$ ^{\rm 3}$\\
\hline
Earth & 5.5 \\
\hline
Moon  & 3.3 \\
\hline
Earth's Crust & 3.0 \\
\hline
\end{tabular}
\label{earthden-tab}
\end{center}
\end{table}
\vspace{0.5 in}

\begin{flushleft}
{\bf Question \#5:} Compare the mean densities of the Earth's crust and the Moon.
The leading theory for the formation of the Moon is that a small planet
crashed into the Earth 4.3 billion years ago, and blasted off part of the Earth's
crust. This material went into orbit around the Earth, and condensed to form the
Moon. Do the densities of the Earth's crust and the Moon support this idea? How?
{\bf (4 points)}\\
\end{flushleft}
\vspace{2.5 in}
\clearpage
~ \\
\vspace{1.5in}

\begin{flushleft}
{\bf Question \#6:} If you were asked ``What are the main elements that make-up the
 Moon?'', what would your answer be? Why? {\bf (2 points)}
\end{flushleft}
\vspace{1.4 in}

It is clear from Table \ref{earthden-tab}, that the mean density of the whole Earth is
much higher than the density of its crust. There must be denser material below the
crust, deep inside the Earth.\\

\begin{flushleft}
{\bf Question \#7:} Given that the mean density of the Earth's crust is 3.0 
g/cm$^{\rm 3}$, and the mean density of the whole Earth is 5.5 g/cm$^{\rm 3}$,
what (common) element do you suppose is partially responsible for the higher mean
density of the whole Earth? If we guess, and say that
the Earth is a 50-50 mixture of this element, and the crust material, what density
do you calculate? Does the resulting density compare with that for the whole
Earth? {\bf (4 points)}\\
\end{flushleft}
\vspace{2.5 in}

Now let's return to the rocks in our set of objects. We included Pumice into this
set to show you that nature can sometimes surprise you---have you ever seen a rock
that floats? Would it surprise you to find out that Pumice has almost the same
composition as Gabbro and Obsidian? It is mostly SiO$_{\rm 2}$! So how can this
rock float?! Let's try to answer this.\\

\begin{flushleft}
{\bf Question \#8:} If Pumice has the same basic composition as Gabbro, how might
it have such a low density? [Hint: think about a boat. As you have found out,
cubes of pure metals do not float. But then how does a boat made of iron (steel)
or aluminum actually float? What is found in the boat that fills most of its 
volume?] {\bf (2 points)}\\
\end{flushleft}
\vspace{1.0 in}

\begin{flushleft}
{\bf Question \#9:} Dry air has a density of 0.0012 g/cm$^{\rm 3}$,
let's make an estimate for how much air must be inside Pumice to give it the density 
you measured. Note: this is like the alloy problem you worked on above, but the 
densities of one of the two components in the alloy is essentially zero. 
{\bf (6 points)}\\
\vspace{0.1 in}

You measured the volume of the piece of Pumice along with its mass, and then calculated
its density. We stated that density = mass/volume. But you could re-arrange this 
equation
to read volume = mass/density. {\bf Assume that the density of the material that 
comprises the solid parts of Pumice is the same as that for Gabbro.}\\
\vspace{0.1 in}

a) What would be the volume of a piece of Gabbro that has the same mass as your piece
of Pumice? \\
\end{flushleft}
\begin{center}
Volume(Gabbro) ~ = ~ Mass(Pumice)/Density(Gabbro) ~ = ~ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_   ~ ~ cm$^{\rm 3}$
\\
\end{center}
\begin{flushleft}
b) Now take the value of the volume you just calculated and divide it by the
volume of the Pumice stone that you measured: 
\end{flushleft}
\begin{center}
r = Volume(Gabbro)/Volume(Pumice) ~ = ~ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \%
\\
\end{center}
\vspace{0.1 in}
\begin{flushleft}
This ratio, ``r'', shows you how much of the volume of Pumice is occupied by {\bf rocky
material}. The volume of Pumice occupied by ``air'' is:
\end{flushleft}
\begin{center}
1 - r ~ = ~ \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_ \%
\end{center}
\vspace{0.15 in}
Pumice is formed when lava is explosively ejected from a volcano. Deep in the volcano
the liquid rock is under high pressure and mixed with gas. When this material is
explosively ejected, it is shot into a low pressure environment (air!) and quickly 
expands.  Gas bubbles get trapped inside the rock, and this leads to its unusually low 
density.\\
\clearpage
~ ~ ~\\
%\quad {\bf Material} & \quad {\bf Density (g/cm$^3$)} \\ \hline \hline
%Air (sea level) & 0.0012 \\ \hline
%Helium & 0.0001785 \\ \hline
%Liquid Water	& 1.0		\\ \hline
%Aluminum & 2.7 \\ \hline
%Lead	&  11.4		\\ \hline
%Gold	&  19.3		\\ \hline
\newpage
\hspace*{3.0in}Name:{\hrulefill} \\
\hspace*{3.25in}Date:{\hrulefill}\\

\subsection{Take Home Exercise (35 points total)}

For the take-home part of this lab, we are going to explore the densities and compositions
of other objects in the Solar System.\\
\begin{enumerate}
\item Use your textbook, class notes, or other sources to fill in the following
table {\bf (10 points)}:
\vspace{0.1 in}
\begin{table}[htbp]
\begin{center}
\begin{tabular}{|l|c|}\hline
\quad {\bf Object} & \quad {\bf Average Density (g/cm$^3$)} \\ \hline \hline
Sun			&	\\ \hline
Mercury	&		\\ \hline
Venus	&   	\\ \hline
Mars	& 		\\ \hline
Ceres (largest asteroid)  & 2.0 \\ \hline
Jupiter &		\\ \hline
Saturn	& 		\\ \hline
Titan (Saturn's largest moon)  & \\ \hline
Uranus	&		\\ \hline
Neptune	&		\\ \hline
Pluto	&		\\ \hline
Comet Halley (nucleus) &  0.1 \\ \hline
\end{tabular}
\label{denss-tab}
\end{center}
\end{table}
\vspace{0.1 in}
\item Mercury, Venus, Earth, and Mars are classified as Terrestrial planets
(``Terrestrial'' means Earth-like). Do they have similar densities? Do you think
they have similar compositions? Why/Why not? {\bf (3 points)}
\vspace{1.1 in}
\item Jupiter, Saturn, Uranus and Neptune are classified as Jovian planets 
(``Jovian'' means Jupiter-like). Why do you think that is? Compare the densities of the
Jovian planets to that of the Sun. Do you think they are made of similar materials?
Why/why not? {\bf (6 points)}
\clearpage
~
\vspace{0.8 in}
\item Saturn has an unusual density. What would happen if you could put Saturn
into a huge pool/body of water?? (Remember
water has a density of 1 g/cm$^{\rm 3}$, and recall the density and {\it behavior}
 of Pumice.) {\bf (2 points)}
\vspace{1.0 in}

\item The densities of Ceres, Titan and Pluto are very similar. Most astronomers
believe that these three bodies contain large quantities of water ice. If we assume
roughly half of the volume of these bodies is due to water (density = 1 g/cm$^{\rm 3}$)
and half from some other material, what is the approximate mean density of this other
material? Hint: this is identical to the alloy problem you worked-on in lab:
\begin{center}
Density(Ceres) = (1.0 g/cm$^{\rm 3}$ + X g/cm$^{\rm 3}$)/2.0 \\
\end{center}
Just solve for ``X'' (if this hard for you, see the section ``Solving for X'' in
Appendix A at the end of this manual). What material have we been dealing with in this 
lab that has a density with a value {\it similar} to  ``X''? What do you conclude 
about the composition of Ceres, Titan and Pluto? {\bf (8 points)}\\
\vspace{1.2 in}
\item The nucleus of comet Halley has a very low density. We know that comets are mostly
composed of water and other ices, but those other ices still have a higher density 
than that measured for Halley's comet. So, how can we possibly explain this low 
density? [Hint: Look back at Question \#9. Why is Pumice so light, even though it
is a silicate rock?] What does this imply for the nucleus of comet 
Halley?!!] {\bf (6 points)}\\
\end{enumerate}
\clearpage
\subsection{Possible Quiz Questions}

\begin{enumerate}
\item What is the difference between mass and weight?

\item How do you calculate density?

\item What are the physical units on density?

\item How do astronomers use density to study planets?

\item Does the shape of an object affect its density?
\end{enumerate}

\subsection{Extra Credit (ask your TA for permission before attempting, 5 points)}

Look up some information about the element Mercury (chemical symbol ``Hg'').
Note that at room temperature, Mercury is a liquid. You found out above
that, depending on density, some objects will float in water (like pumice).
What is the density of Mercury? So, if you had a beaker full of Mercury,
which of the metals you experimented with in this lab do you think would 
float in Mercury? In Question \# 7, we discussed that the core
of the Earth is much more dense than its crust, and concluded that there
must be a lot
of iron at the center of the Earth. Given what you have just found out
about rather dense materials floating in Mercury, apply this knowledge to 
discuss why the Earth's core is made of molten (=liquid) iron, while the 
crust is made of silicates.\\
\clearpage
~ ~ ~\\
