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{\bf ADVANCED DYNAMICS --- PHY 4241/5227}\\\vspace {2mm} 
{\bf HOME AND CLASS WORK -- SET 3}\\%
\vspace {3mm} (January 16, 2009) \\
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\item{(12)} Use Lagrange multipliers for the following problems.
\begin{enumerate}
\item Determine the constraint force for the motion on a circle (no 
      other forces acting) as function of the velocity and radius.
\item An inclined plane is given by the constraint $f=y-y_0+ax=0$.
      Gravity acts parallel to the $y$ direction. Find the constraint
      forces in $x$ and $y$ directions. Compare the equations for 
      $\ddot{x}$ and $\ddot{y}$ with those found from Newton's laws.
\end{enumerate}
Due January 26 before class (5 points). Read M\&T chapter 7.5.

\item{(13)} A cylinder is released at rest (and suitably positioned)
at the top of a frictionles inclined plane. What happens? Multiple
choice: (a) The cylinder roles down the plane. (b) The cylinder slides
down the plane. 
Due January 21 in class (1~point).
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\item{(14)} The ladder of the figure is kept in equilibrium by
constraint forces $F_y=\lambda_y \hat{y}$ and $F_x=\lambda_x\hat{x}$.
Use Lagrange multipliers to show the relationship 
$$ mg=\lambda_y-(y_0/x_0)\lambda_x\ .$$
Discuss the special cases $\lambda_y=0$ and $\lambda_x=0$.
Hint: Approximate the ladder by two masses $m+m$ at the endpoints, 
which are rigidly connected. Then, use three constraint functions, 
$f_x$, $f_y$ and $f$.  Due January 30 before class (10~points).
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% \caption{Ladder for assignment~14.} \label{fig_lader}
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