\input{labpreamble}
\begin{document}

\subsubsection*{PHSX 221L\hfill Archimedes' Principle\hfill Name:\underline{\hspace*{2in}}}


\noindent
\underline{Object:} To study Archimedes' principle directly as well as 
to use it to determine quantities of substances such as density 
and specific gravity.

\noindent
\underline{Theory:} Archimedes' principle states that {\em an object
placed in a fluid has an upward buoyant force on it that is equal to
the weight of the fluid displaced by the object.} If an object floats
in a fluid it means that it is buoyed up by a force $F_{\rm b}$
greater than or equal to its weight; if it sinks in the fluid if its
weight exceeds that of the fluid it displaces.  The magnitude of the
buoyant force depends only on the weight of the fluid displaced by the
object and does not depend on the weight of the object.

\noindent A consequence of Archimedes' (see page 466 in Serway)
principle is that an object will float in a fluid if its density
$\rho_{\rm o}$ is less than the density of the fluid $\rho_{\rm f}$,
and an object will sink in a fluid if its density is more than the
density of the fluid , and will float in equilibrium at any submerged
depth where it is placed if its density is equal to that of the 
fluid.  Both liquids and gasses are fluids.

\noindent Using the same or similar apparatus one can also measure the
\underline{specific gravity} of an object, which is defined to be the
ratio of its density to the density of water at the same temperature
(usually $4\,^{\rm o}\,$C).  (See example 15.5 in Serway.)  The term
``specific gravity'' might be better expressed as ``relative
density,'' but traditions are hard to change.  Since specific gravity
is the ratio of two densities it is unitless; and since the density of
water $\rho_{\rm w}$ at $4\,^{\rm o}\,$C is 1.000 g/cm$^{3}$ (and
doesn't vary much from that over the temperature range in which water
is a liquid) the specific gravity of a substance is numerically the
same as its density in g/cm$^{3}$.  An object's density ($\rho = m/V$)
can also be measured in other ways (such as in our first lab), but the
purpose here is to use Archimedes' principle.

\noindent An \underline{object more dense than water} will be
completely immersed when lowered into water, so the volume of the
object is the same as the volume of the water displaced by it ($V_{\rm
o} = V_{\rm w}$) and
\begin{equation}\label{heavy1}
	{\rm sp.\ gr.} = { \rho_{\rm o}\over \rho_{\rm w}} = {m_{\rm 
	o}/V_{\rm o}\over m_{\rm w}/V_{\rm w}} = {m_{\rm o}\over m_{\rm w}} 
	= {m_{\rm o}g\over m_{\rm w}g} = {w_{\rm o}\over w_{\rm w}} = 
	{w_{\rm o}\over F_{\rm b}} 
\end{equation}
where $w_{\rm o}$ is the weight of the object and $w_{\rm w}$ is the
weight of the displaced water, and by Archimedes' principle $w_{\rm w}
= F_{\rm b}$.  If attached to a scale, the submerged object will have
an apparent weight (indicated with a primed symbol) of $w_{\rm
o}' = w_{\rm o} - F_{\rm b}$.  (Verify this with a free body diagram.) 
Thus $F_{\rm b} = w_{\rm o} - w_{\rm o}'$ and equation \ref{heavy1} becomes
\begin{equation}\label{heavy2}
	{\rm sp.\ gr.} = {w_{\rm o}\over w_{\rm w}} = {w_{\rm o}\over w_{\rm 
	o} - w_{\rm o}'},
\end{equation}
or, in terms of mass measured on a balance (since $g$ cancels from 
every term),
\begin{equation}\label{heavy3}
		\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
		{\rm sp.\ gr.} = {m_{\rm o}\over m_{\rm o} - m_{\rm o}'}
			\ \ \ \ \ \ \ {\rm\it for\ a\ heavy\ object\ that\ sinks},
\end{equation}
where $m_{\rm o}'$ is the apparent mass of the submerged object.

\pagebreak
\noindent In order to use Archimedes' principle to determine the
specific gravity of an \underline{object that floats} or is less dense
than water, it is necessary to use another object (a sinker) of
sufficient weight and density to submerge the light object completely
so that the volume of the displaced water is equal to the volume of
the object.  Letting the sinker hang from the light object, $w_{1} =
w_{\rm o} + w_{\rm s}'$ is the measured weight of the object and
sinker with only the sinker submerged, and $w_{2} = w_{\rm o}' +
w_{\rm s}'$ is the measured weight when both are submerged.  Then
\begin{equation}\label{light1}
	w_{1}-w_{2} = (w_{\rm o} + w_{\rm s}') - (w_{\rm o}' + w_{\rm s}') = 
	w_{\rm o} - w_{\rm o}'
\end{equation}
or, put in terms of mass since $g$ is constant over the experiment,
\begin{equation}\label{light2}
	m_{1}-m_{2} = m_{\rm o} - m_{\rm o}'.
\end{equation}
Substituting into equation \ref{heavy3} gives the specific gravity 
for the light object:
\begin{equation}\label{light3}
		\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 
		{\rm sp.\ gr.} = {m_{\rm o}\over m_{1} - m_{2}}
		\ \ \ \ \ \ \ {\rm\it for\ a\ light\ object\ that\ floats}.
\end{equation}
	
\noindent Archimedes' principle can also be used to find the specific
gravity of a \underline{liquid}.  A heavy object is weighed first in
air ($w_{\rm o}$) and then when submerged in a liquid ($w_{\rm o}'$). 
Then $w_{\rm o} - (w_{\rm o}')_{\rm liquid}$ is the weight of the
volume of liquid the object displaces, by Archimedes' principle. 
Applying the same procedure for the object in water, $w_{\rm o} -
(w_{\rm o}')_{\rm water}$ is the weight of the volume of water the
object displaces.  Since the two volumes are the same and $g$ is
constant over the experiment, the ratio of the two weights (expressed
in terms of mass) gives the specific gravity of the liquid:
\begin{equation}\label{liquid}
		\ \ \ \ \ \ \ \ \ \ \ \ 
		{\rm sp.\ gr.} = {m_{\rm o} - (m_{\rm o}')_{\rm liquid}\over m_{\rm 
		o} - (m_{\rm o}')_{\rm water}}
		\ \ \ \ \ \ \ {\rm\it for\ a\ liquid}.
\end{equation}


\noindent
\underline{Procedure:} ~In each case below draw a diagram of the
apparatus (see, for example, figures 15.12 and 15.23 in Serway).  Also
take care to avoid air bubbles on submerged objects and make sure they
aren't touching the sides of the beaker.  Measurements repeated three
or four times help reduce errors.

\goodbreak
\vspace{-.1in}
\underline{Direct Proof of Archimedes' principle:}
\vspace{-.25in}
\begin{enumerate}

\item Fill an overflow can with water and place a beaker in position to 
collect overflow.

\item Hang the plummet below the bucket and measure the combined mass 
in air.

\item Move the overflow can below the plummet and lower the set until
the plummet is immersed in the water and measure the mass again.  The
difference between the two weights is the buoyant force on the plummet.

\item Carefully pour the water from the beaker into the bucket; make 
sure no water is left in the beaker.

\item Measure the mass of the bucket (with water) and plummet to 
verify Archimedes principle.

\end{enumerate}

\goodbreak
\vspace{-.1in}
\underline{Specific Gravity of a Heavy Solid Object:}
\vspace{-.25in}
\begin{enumerate}

\item Measure the mass of a heavy object in air $m_{\rm o}$, and the
apparent mass $m_{\rm o}'$ of the same object while it is suspended
by a string and completely immersed in water.

\item Use equation \ref{heavy3} to compute the specific gravity of the
heavy object.

\item If the substance is known, compare your answer to the accepted 
value; if not then speculate on what the substance might be.

\end{enumerate}

\goodbreak
\vspace{-.1in}
\underline{Specific Gravity of a Light Solid Object:}
\vspace{-.25in}
\begin{enumerate}

\item Measure the mass $m_{\rm o}$ of a light object alone in air.  Suspend 
the sinker from the light object and measure the mass of the 
combination with only the sinker immersed in water; this is $m_{1}$.  
Now measure the mass of the combination with both solid objects 
immersed; this is $m_{2}$.

\item Use equation \ref{light3} to compute the specific gravity of the
light object.

\item If the substance is known, compare your answer to the accepted 
value; if not then speculate on what the substance might be.

\end{enumerate}

\goodbreak
\vspace{-.1in}
\underline{Specific Gravity of a Liquid:}
\vspace{-.25in}
\begin{enumerate}

\item Measure the mass of an object in air and then its apparent mass
in the unknown liquid and in water.  (Don't put amber in alcohol as it 
will dissolve.)

\item Use equation \ref{liquid} to compute the specific gravity of the
liquid.

\item Also use a hydrometer to measure the specific gravity of the 
liquid and compare your answers.

\end{enumerate}



\noindent
\underline{Analysis and Questions:}
\vspace{-.25in}
\begin{enumerate}

\item Why does a block of wood apparently lose more than its entire 
weight in air, when completely immersed in water?  How is this 
possible?

\item What becomes of the lost weight of an object when immersed in a 
liquid?  How is the apparent loss of weight explained?

\item Suppose there were a bubble of air on the bottom of the heavy 
object immersed in water; how would this affect the calculations of 
the specific gravity (or density) of the object?

\item What would the apparent weight of an object be if it were 
immersed in a fluid which had the same density as that of the object?

\item Does the air (which is also a fluid) produce an upward buoyant
force that must be taken into account?  For example, if you measure
the mass of a 50 g lead sinker on a pan balance, how much upward
buoyant force is exerted by the air?  Does this affect the accuracy of
the balance reading?

\end{enumerate}


\noindent
\underline{Conclusions:}~~Summarize and evaluate this experiment.

\newpage
\ 

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