\section{M-10   Young's Modulus of Elasticity and Hooke's Law}

\noindent{ OBJECTIVE}:
\begin{enumerate}
\item[] \latex{\vspace*{-.05in}}To study the elastic 
properties of piano wire under tension 
(Hooke's law and Young's modulus).
\end{enumerate}


\noindent { APPARATUS}:
\begin{enumerate}
\item[] \latex{\vspace*{-.05in}}Frame 
for holding the steel wire, optical lever, 
telescope \& support 
stand for measuring elongation (alternatively, dial gauges), 1 kg slotted 
masses, micrometer, tape measure.
\end{enumerate}

\noindent {INTRODUCTION}:\begin{enumerate}
\item[] \latex{\vspace*{-.05in}}Young's modulus 
$M_Y$ is the ratio of longitudinal stress 
to the resultant longitudinal strain:
$$M_Y = \frac{\mbox{stress}}{\mbox{strain}} = \frac{\Delta F/A}{\Delta L/L}$$
where
\begin{eqnarray*}
\Delta F & = & \mbox{longitudinal force in newtons}\\
A & = & \mbox{area in square meters}\\
\Delta L & = & \mbox{elongation}\\
L & = & \mbox{length of wire undergoing the elongation 
(not the total length!)} 
\end{eqnarray*}
\noindent Note that 
$$\Delta F = \left\{ \frac{M_YA}{L} \right\} \Delta L = k(\Delta L) $$
is Hooke's law where $k$ is a constant if the elastic limit is not exceeded.
\end{enumerate}


\noindent {\ SUGGESTIONS}: 
\begin{enumerate}
\item \latex{\vspace*{-.08in}}First read about the optical lever (Appendix C) and about parallax 
and focusing a telescope (Appendix D).  Also be sure that the frame holding 
the wire, and the stand holding the telescope are on solid bases.  If you 
use a table, avoid leaning on it.  With the telescope check the table sag 
resulting from leaning on it (least sag when units are near the table legs).
\item Adjust the height of the platform holding the optical lever so that 
the long arm of the lever is approximately horizontal.
\end{enumerate}


\noindent {PROCEDURE}:
\begin{enumerate}
\item \latex{\vspace*{-.08in}}Put a load of three kilograms on the wire to straighten it.
\item Measure the successive deflections as you increase the load one 
kilogram at a time up to a total of 10 kg (i.e., 7 kg additional).
\item Repeat (2) but reducing the load 1 kg at a time.
\item Convert differences in scale readings to elongations (see Appendix C).
\item Plot total elongation as abscissa against load in Newtons as 
ordinate.  Find the average slope (i.e. $\Delta F/\Delta L = k)$.
\item Measure $A$ and $L$; compute Young's modulus.  The value for piano wire 
is $\simeq~20~\times~10^{10}~N/m^2$.
\end{enumerate}

\input{toolbar/m11_tb1}

\noindent {QUESTIONS}:
\begin{enumerate}
\item \latex{\vspace*{-.05in}}Piano wire has a tensile strength (breaking stress) of 19 to 
23 $\times$ 10$^8$ N/m$^2$.  (The elastic limit may be about 0.7 the breaking 
stress).  Calculate the maximum load your wire could stand.  At what load 
would you pass the elastic limit?  

\noindent Note that the wire fails at a stress 
which is $~$100 x less than Young's modulus.  Can you understand why the two 
values are not inconsistent?  (Hint: If the stress equaled $M_Y$, what 
would be the strain?).
\item Discuss the sources of error in this experiment and estimate the 
reliability of your result.  Is the accepted value for piano wire steel 
within the limits you have estimated?
\item How could you detect slipping of the wire in the chuck during the 
experiment?
\item How could you modify the experiment so as to detect and correct for any 
sagging of the support frame?
\item Poisson's ratio, $\left(\frac{-\Delta r/r}{\Delta L/L}\right)$, 
for steel is about 0.3. 

\noindent Could you notice the decrease in diameter of the wire in this 
experiment by use of a micrometer caliper?  If so, try it.
\end{enumerate}

\begin{enumerate}
\item[] \latex{\vspace*{-.05in}}{\bf Suggested additional experiment}:  Determine the dependence 
of elongation on stretching force for a rubber 
band over a wide range of elongations (up to 3 or more times the unstretched 
length).  Does it obey Hooke's law?  Can you suggest reasons for its 
behavior? 
\end{enumerate}

