\input{labpreamble}
\begin{document}


\subsubsection*{PHSX 221L\hfill Torques and Equilibrium\hfill Name:\underline{\hspace*{2in}}}

% \underline{Title:} Torques and Equilibrium

\underline{Object:} To study center of gravity, equilibrium, and the 
effects of different placement of forces.

\underline{Introduction:} A \underline{torque} results from a force 
{\bf F} acting at a 
distance {\bf r} from a specific point; torques produce or tend to produce 
rotational acceleration about the point.  Torque is a vector quantity 
given by 
\begin{equation}\label{torque}
	\boldsymbol{\tau} = \boldsymbol{r}\times\boldsymbol{F};
\end{equation}
its magnitude is given by
\begin{equation}
\tau = rF\sin\theta
\end{equation}
where $\theta$ is the angle between ${\bf r}$ and ${\bf 
F}$, and its direction is given by the right hand rule.  The angle $\theta$ 
will be 90 degrees for the torques in this experiment, so the math is 
simplified.

A body is said to be in \underline{equilibrium} if there is no net force 
and no 
net torque acting on the body.  Stated mathematically, a body is in 
equilibrium if
\begin{equation}
	\ \ \ \ \ \ \ \ \ \Sigma\boldsymbol{F} = 0\ \ \ \ \ \ \ \ \ \ \ {\rm (translational\ equalibrium)}
\end{equation}
and
\begin{equation}\label{equalib}
\ \ \ \ \ \ \Sigma\boldsymbol{\tau} = 0\ \ \ \ \ \ \ \ \ \ \ {\rm (rotational\ equalibrium)}.
\end{equation}

The \underline{center of gravity} of a body is defined as the point at 
which a single upward force can balance the gravitational attraction 
on all parts of the body for any orientation of the body.  An 
equivalent definition is the point through which the sum of all the 
torques due to the gravitational forces on all the little elements 
(parts) of mass of the body is zero.  Effectively, the entire mass of 
the body can be assumed to act at the center of gravity.  If the 
gravitational field is uniform throughout the body then the center of 
gravity coincides with the center of mass.  A body that is uniform and 
symmetric has its center of gravity at the center of symmetry.


\underline{Procedure:}  Strive for $< 1\%$ error in this experiment.

\underline{Center of Gravity of Meter Stick}
\vspace{-.2in} 
\begin{enumerate}
	\item Use a laboratory balance to find the mass of a meter stick.
	\item Suspend the meter stick in a knife-edge clamp and carefully 
	slide it back and forth until it balances.  Record the corresponding 
	marking on the meter stick as the center of gravity.  
	\item Estimate the uncertainty 
	of this measurement by moving the stick slightly back and forth in 
	its support to see at what point it deviates from horizontal.
	\item Now suspend your meter stick at the 30.0~cm mark.  Put a clamp 
	at the 5.0~cm mark and hang enough mass on a hanger there to achieve 
	equilibrium.  Record the total amount of mass (including clamp and 
	hanger) at the 5.0~cm mark on your accurate and clear diagram showing 
	precise locations and amounts of the various masses.
    \item Compute the mass of the meter stick by assuming the 
    counterclockwise torques equal the clockwise torques when the 
    stick is in equilibrium (remember, the torque due to the weight 
    of the meter stick is calculated as if all of the meter stick's 
    mass were concentrated at its center of gravity).
	\item Compute the percent difference between the mass of the meter 
	stick as derived from the equilibrium equations and its mass from the 
	laboratory balance.
	\item Fold an eighth-sheet of paper in half a couple of times and place it on the meter 
	stick at the 99~cm mark and note the results.
\end{enumerate}

\underline{Equilibrium}
\begin{enumerate}
\vspace{-.2in} 	
	\item Suspend the meter stick at the 35.0~cm mark.  Using the 
	clamps and hangers, suspend a mass of about 230.0~g (including clamp and 
	hanger) at the 5.0~cm mark, and a mass of about 100.0~g (including clamp and 
	hanger) on the other side of the suspension point.  Move 
	this second mass until equilibrium is achieved, and then record the 
	total masses, including the knife-edge clamps, and their positions.

	\item Using equation \ref{torque} compute the total counterclockwise 
	torque and the total clockwise torque.
	
	\item Compute the percent difference between the counterclockwise 
	 and the clockwise torques.
	
	\item Suspend the meter stick at its center of gravity found in the first 
	part of the procedure.  Hang a mass of about 200.0~g (including clamp and 
	hanger) at the 20.0~cm mark and a mass of about 100.0~g (including clamp and 
	hanger) at the 55.0~cm mark.  Find a third mass that will put the 
	meter stick in equilibrium when it is on the same side of the support 
	as the 100~g mass.  Again make an accurate and clear diagram showing 
	precise locations and amounts of the various masses.

	\item Using equation \ref{torque} compute the total counterclockwise 
	torque and the total clockwise torque.
	
	\item Compute the percent difference between the counterclockwise 
	 and the clockwise torques.
\end{enumerate}

\underline{Value of Unknown Mass}
\vspace{-.2in} 
\begin{enumerate}
	
	\item Suspend the meter stick at its center of gravity found in 
	the first part of the procedure.  Hang the unknown mass at the 
	10.0~cm mark.  Then hang a known mass on the other side of the 
	suspension point and slide it back and forth until equilibrium is 
	achieved.  Again make an accurate and clear diagram showing 
	precise locations and amounts of the various masses.
	
	\item Use the principles of equilibrium (equation \ref{equalib}) to
	determine the value of the unknown mass.
	
	\item Weigh (mass) the unknown mass on the beam balance; note the 
	similarity in operation between your meter stick ``balance'' and 
	the laboratory balance.
	
	\item Compute the percent error between the two determinations of 
	the mass.
	
	\item Repeat this procedure for your own unknown mass, such as a set 
	of keys or a calculator, if your instructor says there is enough time.
	
\end{enumerate}


\underline{Discussion Questions:} 
\vspace{-.2in}
\begin{enumerate}

	\item If you were careful (both in taking data and analyzing it)
	during this lab, your percent differences will be very
	small---less than in most prior experiments, although you used the
	same equipment, such as meter sticks, balances, etc.  Why are the
	percent differences so low in this experiment?
	
	\item What evidence supports the idea that the weight of an extended 
	object acts at the center of gravity?  
	
	\item In a couple of procedures you had to 
	include the torque due to the weight of the meter stick itself; in others you 
	didn't; why?
	
	\item In this experiment you used mass units on force quantities.  
	Usually this would be considered incorrect and would lead to 
	incorrect results when used in computations.  Why were you able to 
	get away with this in this experiment?

	\item Could you use a modified ``meter stick balance'' to measure the very 
	small masses (thousandths of a gram)?  How?

	\item In step 2.1 of the procedure you balanced torques about the 35 
	cm mark.  Recalculate your torques about the 0.0 cm mark.  Find all 
	the CW and then all the CCW torques and see if the totals equal.  
	(Don't forget to add the torques exerted at the suspension point: 
	the weight of the meter stick and all the masses on it contributes a 
	CW torque and the normal force exerted upward by the suspension 
	device contributes a CCW torque.)  Recalculating torques about a new 
	axis is an easy way to check your equilibrium calculations.


\end{enumerate}

\underline{Conclusions:} 

\newpage\ 

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