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

\begin{Large}
\section{Estimating the Earth's Density}
\end{Large}

\subsection{Introduction}
We know, based upon a variety of measurement methods, that the density of the 
Earth is 5.52 grams per cubic centimeter. [This value is equal to 5520 
kilograms per cubic meter. Your initial density estimate in Table \ref{den3} 
should be 
a value similar to this.] This density value clearly indicates that Earth is 
composed of a combination of rocky materials and metallic materials.\\

With this lab exercise, we will obtain some measurements, and use them to 
calculate our own estimate of the Earth's density. Our observations will be 
relatively easy to obtain, but they will involve contacting someone in the 
Boulder, Colorado area (where the University of Colorado is located) to assist 
with our observations. We will then do some calculations to convert our 
measurements into a density estimate.\\

As we have discussed in class, and in previous labs this semester, we can 
calculate the density of an object (say, for instance, a planet, or more 
specifically, the Earth) by knowing that object's mass and volume. It is a 
challenge, using equipment readily available to us, to determine the Earth's 
mass and its volume directly. [There is no mass balance large enough upon which
we can place the Earth, and if we could what would we have available to 
``balance'' the Earth?] But we have through the course of this semester 
discussed physical processes which relate to mass. One such process is the 
gravitational attraction (force) one object exerts upon another. \\

The magnitude of the gravitational force between two objects depends upon both 
the masses of the two objects in question, as well as the distance separating 
the centers of the two objects. Thus, we can use some measure of the Earth's 
gravitational attraction for an object upon its surface to ultimately determine
the Earth's mass. However, there is another piece of information that we 
require, and that is the distance from the Earth's surface to its center: the 
Earth's radius.\\

We will need to determine both the MASS of the Earth and the RADIUS of the 
Earth. Since we will use the magnitude of Earth's gravitational attraction to 
determine Earth's mass, and since this magnitude depends upon the Earth's 
radius, we’ll first determine Earth's circumference (which will lead us to 
the Earth's radius and then to the Earth's volume) and then determine the 
Earth's mass.\\

\subsection{Determining Earth's Radius}

Earlier this semester you read (or should have read!) in your textbook the 
description of Eratosthenes' method, implemented two-thousand plus years ago, 
to determine Earth's circumference. Since the Earth's circumference is related 
to its radius as:
\vspace{0.05in}
\begin{center}
Circumference = 2 $\times$ $\pi$ $\times$ RADIUS (with $\pi$ = ``pi'' = 3.141592…)
\end{center}
\vspace{0.05in}

\begin{flushleft}
and the Earth's volume is a function of its radius:\\
\end{flushleft}

\begin{center}
VOLUME = (4/3) $\times$ $\pi$ $\times$ RADIUS$^{\rm 3}$
\end{center}

\begin{flushleft}
We will implement Eratosthenes' circumference measurement method and end up
with an estimate of the Earth's radius. \\
\end{flushleft}

Now, what measurements did Eratosthenes use to estimate Earth's circumference?
 Eratosthenes, knowing that Earth is spherical in shape, realized that the 
length of an object's shadow would depend upon how far in latitude 
(north-or-south) the object was from being directly beneath the Sun. He 
measured the length of a shadow cast by a vertical post in Egypt at local noon 
on the day of the northern hemisphere summer solstice (June 20 or so). He made 
a measurement at the point directly beneath the Sun (23.5 degrees North, at
the Egyptian city Syene), and 
at a second location further north (Alexandria, Egypt). The two shadow lengths were not identical, 
and it is that difference in shadow length plus the knowledge of how far apart 
the the two posts were from each other (a few hundred kilometers), that 
permitted Eratosthenes to calculate his estimate of Earth's circumference.\\

As we conduct this lab exercise we are not in Egypt, nor is today the seasonal 
date of the northern hemisphere summer solstice (which occurs in June), nor is 
it locally Noon (since our lab times do not overlap with Noon). But, 
nonetheless, we will forge ahead and estimate the Earth's circumference, and 
from this we will estimate the Earth's radius. \\

\begin{flushleft}
{\bf TASKS:}
\end{flushleft}

\begin{itemize}
\item Take a post outside, into the sunlight, and measure the length of the post with the tape measure.

\item Place one end of the post on the ground, and hold the post as vertical as possible.

\item Using the tape measure provided, measure to the nearest 1/2 centimeter the 
length of the shadow cast by the post; this shadow length should be measured 
three times, by three separate individuals; record these shadow lengths in 
Table \ref{den1}.

\item You will be provided with the length of a post and its shadow measured 
simultaneously today in Boulder, Colorado.

\item Proceed through the calculations described after Table \ref{den1}, and 
write your answers in the appropriate locations in Table \ref{den1}. 
{\bf (10 points)}
\end{itemize}

\begin{table}[htbp]
\centering
\caption{{\bf Angle Data}}
\begin{tabular}{|c|c|c|c|}
\hline
Location & Post Height & Shadow Length & ~ ~ Angle ~ ~ \\
         & (cm) & (cm) & (Degrees) \\
\hline
Las Cruces Shadow \#1 & &  & \\
\hline
Las Cruces Shadow \#2 &  & & \\
\hline
Las Cruces Shadow \#3 &  & & \\
\hline
Average Las Cruces Angle: &   &   & \\
\hline
Boulder, Colorado    &  & & \\
\hline
\end{tabular}
\label{den1}
\end{table}

\subsection{Angle Determination:}

With a bit of trigonometry we can transform the height and shadow length you
measured into an angle. As shown in Figure \ref{den4} there is a relationship 
between the length (of your shadow in this situation) and the height (of the 
shadow-casting pole in this situation), where:
\vspace{0.08in}
\begin{center}
TANGENT of the ANGLE = far-side length/ near-side length
\end{center}

Since you know the length of the post (the near-side length, which you have 
measured) and the length of the shadow (the far-side length, which you have 
also measured, three separate times), you can determine the shadow angle from 
your measurements, using the ATAN, or TAN$^{\rm -1}$ capability on your 
calculator 
(these functions will give you an angle if you provide the ratio of the height 
to length):

\vspace{0.08in}
\begin{center}
ANGLE = ATAN (shadow length / post length)

or

ANGLE = TAN$^{\rm -1}$(shadow length / post length)
\end{center}

\begin{figure}[htb]
\centerline{{\includegraphics[width=8cm]{EarthDensity/earthdenfig1.jpg}}}
\caption{The geometry of a vertical post sitting in sunlight.}
\label{den4}
\end{figure}


{\bf Calculate the shadow angle for each of your three shadow-length 
measurements, 
and also for the Boulder, Colorado shadow-length measurement.} Write these angle
values in the appropriate locations in Table \ref{den1}. Then calculate the 
average of the three Las Cruces shadow angles, and write the value on the 
``Average Las Cruces Angle'' line.\\

The angles you have determined are: 1) an estimate of the angle (latitude) 
difference between Las Cruces and the latitude at which the Sun appears to be 
directly overhead (which is currently $\sim$ 12 degrees south of the equator 
since 
we are experiencing early northern autumn), $and$  2) the angle (latitude) 
difference between Boulder, Colorado and the latitude at which the Sun appears 
to be directly overhead. The difference (Boulder angle minus Las Cruces angle) 
between these two angles is the angular (latitude) separation between Las 
Cruces and Boulder, Colorado.\\

We will now use this information and our knowledge of the actual distance (in 
kilometers) between Las Cruces' latitude and Boulder's latitude. This distance 
is:  

\begin{center}
{\bf \underline{857 kilometers north-south distance between Las Cruces and Boulder, Colorado}}
\end{center}

\vspace{0.08in}
In the same way that Eratosthenes used his measurements (just like those you 
have made today), we can now determine an estimate of the Earth's circumference.
Using your calculated Boulder Shadow Angle and your Average Las Cruces Shadow
Angle values, calculate the corresponding EARTH CIRCUMFERENCE value, and write
it below:

\begin{center}
{\bf Average Earth Circumference (kilometers) =                        \\
\vspace{0.1in}
857 kilometers $\times$ (360$^{\rm o}$)/(Boulder angle \textemdash Avg LC Angle) =}\\
\vspace{0.1in}
{\bf 857 $\times$ [360$^{\rm o}$/($\underline{\hspace*{0.7in}}$ $-$ $\underline{\hspace*{0.7in}}$)] = $\underline{\hspace*{1.4in}}$ km}  {\bf (2 points)}
\end{center}

%\begin{center}
%{\bf AVERAGE EARTH CIRCUMFERENCE = $\underline{\hspace*{1.4in}}$  kilometers} {\bf (5 points)}
%\end{center}

\begin{flushleft}
The CIRCUMFERENCE value you have just calculated is related to the 
 RADIUS via the equation:\\
\end{flushleft}
\vspace{0.08in}

\begin{center}
EARTH CIRCUMFERENCE = 2 $\times$ $\pi$ $\times$ EARTH RADIUS\\
\vspace{0.13in}
\end{center}

\begin{flushleft}
which can be converted to RADIUS using:\\
\end{flushleft}
\vspace{0.13in}

\begin{center}
EARTH RADIUS = R$_{\rm E}$ = EARTH CIRCUMFERENCE / (2 $\times$ $\pi$)
\vspace{0.08in}
\end{center}
For your calculated CIRCUMFERENCE, calculate that value of the Radius (in 
units of kilometers) in the appropriate location below:
\vspace{0.1in}
\begin{center}
{\bf AVERAGE EARTH RADIUS VALUE = R$_{\rm E}$ = $\underline{\hspace*{1.7in}}$ kilometers} {\bf (3 points)}
\end{center}
\vspace{0.13in}
{\bf Convert this radius (R$_{\rm E}$) from kilometers to meters, and enter
that value in Table \ref{den3}.} {\it (Note we will use the radius in meters 
the rest of this lab.)}
\vspace{0.13in}

You have now obtained one important piece of information (the radius of the 
Earth) needed for determining the density of Earth. We will, in a bit, use this 
radius value to calculate the Earth's volume. Next, we will determine Earth's 
mass, since we need to know both the Earth's volume and its mass in order to be 
able to calculate the Earth's density.\\

\subsection{Determining the Earth's Mass}

The gravitational acceleration (increase of speed with increase of time) that 
a dropped object experiences here at the Earth's surface has a magnitude 
defined by the Equation (thanks to Sir Isaac Newton for working out this 
relationship!) shown below:\\

\begin{center}
Acceleration (meters per second per second) = G $\times$ M$_{\rm E}$/R$_{\rm E}$$^{\rm 2}$
\end{center}
\vspace{0.1in}
Where {\bf M$_{\rm E}$} is the mass of the Earth in {\it kilograms}, 
{\bf R$_{\rm E}$} is the radius of the Earth in units of 
{\it meters}, and the Gravitational Constant, {\bf G} = 6.67 x 10$^{\rm -11}$ meters$^{\rm 3}$/(kg-seconds$^{\rm 2}$).
You have obtained several estimates, and calculated an average value of 
R$_{\rm E}$, above. 
However, you currently have no estimate for M$_{\rm E}$. You can estimate the
Earth's mass from the measured acceleration of an object dropped here at the 
surface of Earth; you will now conduct such an exercise.\\

A falling object, as shown in Figure \ref{den6}, increases its downward speed at the constant rate {\bf ``X''}
(in units of meters per second per second). Thus, as you hold an object in 
your hand, its downward speed is zero meters per second. One second after you 
release the object, its downward speed has increased to {\bf X} meters per 
second. After two seconds of falling, the dropped object has a speed of {\bf 2X}
meter per second, after 3 seconds its downward speed is {\bf 3X} meters per 
second, and so on. So, if we could measure the speed of a falling object at 
some point in time after it is dropped, we could determine the object's 
acceleration rate, and from this determine the Earth's mass (since we know the 
Earth's radius). However, it is difficult to measure the instantaneous speed of
a dropped object.\\

\begin{figure}[htb]
\centerline{{\includegraphics[width=11cm]{EarthDensity/earthdenfig2.jpg}}}
\caption{The distance a dropped object will fall during a time interval
{\it t} is proportional to {\it t$^{\rm 2}$}. A dropped object speeds up
as it falls, so it  travels faster and faster and falls a greater distance
as {\it t} increases.}
\label{den6}
\end{figure}

We can, however, make a different measurement from which we can derive the 
dropped object's acceleration, which will then permit us to calculate the 
Earth's mass. As was pointed out above, before being dropped the object's 
downward speed is zero meters per second. One second after being dropped, the 
object's downward speed is X meters per second. During this one-second 
interval, what was the object's AVERAGE downward speed? Well, if it was zero 
to begin with, and X meters per second after falling for one second, {\bf its 
average fall speed during the one-second interval is:}
\vspace{0.08in}

\begin{flushleft}
{\bf Average Fall speed during first second = (Zero + X) / 2 = X/2 meters per second}, which is just the average of the initial (zero) and final (X) speeds.\\
\vspace{0.08in}
At an average speed of X/2 meters per second during the first second, the 
distance traveled during that one second will be:\\
\vspace{0.08in}

\begin{center}
{\bf (X/2) (meters per second) $\times$ 1 second = (X/2) meters,} \\
\end{center}
\vspace{0.08in}

since:\\
\vspace{0.08in}

\begin{center}
{\bf \small
DISTANCE = AVERAGE SPEED $\times$ TIME = 1/2 $\times$ ACCELERATION x TIME$^{\rm 2}$}\\
\end{center}

\end{flushleft}
So, if we measure the length of time required for a dropped object to fall a 
certain distance, we can calculate the object's acceleration.

\vspace{0.1in}
\begin{flushleft}
{\bf Tasks:}
\end{flushleft}

\begin{itemize}

\item Using a stopwatch, measure the amount of time required for a dropped object (from the top of the Astronomy Building) to fall 9.0 meters (28.66 feet). 
Different members of your group should take turns making the fall-time 
measurements; write these fall time values for two ``drops'' in the appropriate
location in Table \ref{den2}. {\bf (10 points for a completed table)}

\item {\bf Use the equation: Acceleration = [2.0 x Fall Distance] / [(Time to fall)$^{\rm 2}$]}

and your measured Time to Fall values and the measured distance (9.0 meters) 
of Fall to determine the gravitational acceleration due to the Earth; write 
these acceleration values (in units of meters per second per second) in the 
proper locations in Table \ref{den2}.\\

\item Now, knowing the magnitude of the average acceleration that Earth's 
gravity imposes upon a dropped object, we will now use the ``Gravity'' equation
to get M$_{\rm E}$:

Gravitational acceleration = G $\times$ M$_{\rm E}$/R$_{\rm E}$$^{\rm 2}$ (where
R$_{\rm E}$ must be in meters!)
\end{itemize}



\begin{table}[htbp]
\centering
\caption{{\bf Time of Fall Data}}
\begin{tabular}{|c|c|c|c|}
\hline
 & Time to Fall & Fall Distance & Acceleration \\
\hline
Object Drop \#1 &  & 9 meters & \\
\hline
Object Drop \#2 &  &  9 meters& \\
\hline
 Average  = &  &  & \\
\hline
\end{tabular}
\label{den2}
\end{table}

\begin{flushleft}
By rearranging the Gravity equation to solve for M$_{\rm E}$, we can now make 
an estimate of the Earth's mass:\\
\vspace{0.15in}
{\bf M$_{\rm E}$ = Average Acceleration $\times$ (R$_{\rm E}$)$^{\rm 2}$ / G  = {$\underline{\hspace*{2in}}$} } {\bf (5 points)}

\vspace{0.1in}
%[The factor of 1000 here converts your Average Radius determined in the
%previous section in units of kilometers to a radius in units of meters.] \\

Write the value of M$_{\rm E}$ (in kilograms) in Table \ref{den3} below.
\end{flushleft}
\vspace{0.1in}

\subsection{Determining the Earth's Density}

Now that we have estimates for the mass (M$_{\rm E}$) and radius (R$_{\rm E}$) 
of the Earth, we can easily calculate the density: Density = Mass/Volume. You 
will do this below.\\

\begin{flushleft}
{\bf Tasks:}
\end{flushleft}

\begin{itemize}

\item Calculate the volume (V$_{\rm E}$) of the Earth given your
determination of its radius {\it in meters!}:

\begin{center}
{\bf V$_{\rm E}$ = (4/3) $\times$ $\pi$ $\times$ R$_{\rm E}$$^{\rm 3}$}
\end{center}

and write this value in the appropriate location in Table \ref{den3} below.

\item {\it Divide your value of M$_{\rm E}$ (that you entered in Table 
\ref{den3}) by your estimate of V$_{\rm E}$ that you just calculated
(also written in Table \ref{den3}):} the result will be your estimate of the 
Average Earth Density in units of kilograms per cubic meter. Write this value 
in the appropriate location in Table \ref{den3}.

\item {\it Divide your AVERAGE ESTIMATE OF EARTH'S DENSITY value that you just
calculated by the number 1000.0}; the result will be your estimated Earth 
density value in units of grams per cubic centimeter (the unit in which
most densities are tabulated). Write this value in the appropriate location in 
Table \ref{den3}. 

\begin{table}[htbp]
\centering
\caption{{\bf Data for the Earth}}
\begin{tabular}{|lr|}
\hline
                            &                                 \\
Estimate of Earth's Radius:& $\underline{\hspace*{1.8in}}$ {\bf m} (4 points)\\
                            &                                 \\
Estimate of Earth's Mass:& $\underline{\hspace*{1.8in}}$ {\bf kg}  (4 points)\\
                            &                                 \\
Estimate of Earth's Volume:& $\underline{\hspace*{1.8in}}${\bf m$^{\rm 3}$} (4 points)\\
                            &                                 \\
Estimate of Earth's Density:& $\underline{\hspace*{1.6in}}$ {\bf kg/m$^{\rm 3}$} (4 points)\\
                            &                                 \\
Converted Density of the Earth:& $\underline{\hspace*{1.5in}}$ {\bf gm/cm$^{\rm 3}$} (4 points)\\
                            &                                 \\
\hline
\end{tabular}
\label{den3}
\end{table}
\end{itemize}
\clearpage

\subsection{In-Lab Questions:}

\begin{enumerate}
\item Is your calculated value of the (Converted) Earth's density GREATER 
THAN, or LESS 
THAN, or EQUAL TO the actual value (see the Introduction) of the Earth's 
density? If your calculated density value is not identical to the known Earth 
density value, calculate the ``percent error'' of your calculated density 
value compared to the actual density value {\bf (2 points)}:\\

\vspace{0.08in}
PERCENT ERROR = 
\vspace{0.08in}
\small{
\begin{displaymath}
\frac{\textrm{ 100\%} \times (\textrm{CALCULATED DENSITY} - \textrm{ ACTUAL DENSITY)}} {\textrm{ ACTUAL DENSITY}}= \underline{\hspace*{0.9in}} \\
\end{displaymath} 
}
\vspace{0.08in}

\item You used the AVERAGE Las Cruces shadow angle in calculating your estimate
of the Earth's density (which you wrote down in Table \ref{den3}). If you had 
used the LARGEST of the three measured Las Cruces shadow angles shown in Table 
\ref{den1}, would the Earth density value that you would calculate with the 
LARGEST Las Cruces shadow angle be larger than or smaller than the Earth 
density value you wrote in Table \ref{den3}? Think before writing your answer! 
Explain your answer. {\bf (5 points)}
\vspace{2.4in}

\item If the Las Cruces to Boulder, Colorado distance was actually 200 km in 
length, but your measured fall times did not change from what you measured, 
would you have calculated a larger or smaller Earth density value? Explain the
reasoning for your answer. {\bf (3 points)}
\vspace{2.0in}
\item If we had conducted this experiment on the Moon rather than here on the
Earth, would your measured values (fall time, angles and angle difference 
between two locations separated north-south by 857 kilometers) be the same as 
here on Earth, or different? Clearly explain your reasoning. [It might help if 
you draw a circle representing Earth and then draw a circle with 1/4$^{\rm th}$
of the radius of the Earth's circle to represent the Moon.] {\bf (5 points)}
\vspace{1.0in}
\end{enumerate}
~ ~ ~\\
\clearpage

\hspace*{3.0in}Name:{\hrulefill} \\
\hspace*{3.25in}Date:{\hrulefill}\\

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

\begin{enumerate}

\item Type a 1.5-2 page Lab Report in which you will address the following 
topics:

a) The estimated density value you arrived at was likely different from the actual Earth density value of 5.52 grams per cubic centimeter; describe 2 or 3 potential errors in your measurements that could possibly play a role in generating your incorrect estimated density value.

b) Describe 2-3 ways in which you could improve the measurement techniques used in lab; keep in mind that NMSU is a state-supported school and thus we do not have infinite resources to purchase expensive sophisticated equipment, so your suggestions should not be too expensive.

c) Describe what you have learned from this lab, what aspects of the lab surprised you, what 
aspects of the lab worked just as you thought they would, etc.
\end{enumerate}

\subsection{Possible Quiz Questions}

\begin{enumerate}

\item What is meant by the ``radius'' of a circle? (Drawing ok)

\item What does the term ``circumference of a circle'' mean?

\item How do you calculate the circumference of a circle if given
the radius? 

\item What is ``pi'' (or $\pi$)? What is the value of pi?

\item What is the volume of a sphere?

\item What does the term ``density'' mean?
\end{enumerate}

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

Astronomers use density to segregate the planets into categories, such
as ``Terrestrial'' and ``Jovian''. Using your book, or another reference,
look up the density of the Sun and Jupiter (or, if you have completed
the previous lab, use the data table you constructed for Take-Home portion of 
that lab). Compare the densities of the Sun and Jupiter. Do you think they
are composed of same elements? Why/why not? What are the two main elements
in the periodic table that dominate the composition of the Sun?
If the material that formed the Sun (and the Sun {\it has} 99.8\% of
the mass of the solar system) was the original ``stuff'' from which all 
of the planets were formed, how did planets like Earth end up with such 
high densities? What do you think might have happened in the distant past 
to the lighter elements? (Hint: think of a helium balloon, or a glass
of water thrown out onto a Las Cruces parking lot in the summer!).
~ ~ ~ \\
