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\Large \bf Laboratory Calculus I: Project \\
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Forces on a Piston
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{\bf The problem:} In an \index{ internal combustion engine} internal combustion engine, each \index{ piston} piston is housed in a \index{cylinder} cylinder and is attached to the \index{ crankshaft} crankshaft 
by a \index{ connecting rod} connecting rod as indicated in the diagram below.  The piston moves back and forth in the cylinder, and, in response the crankshaft
rotates.  When the crankshaft is turning, even  at a constant \index{ angular velocity} angular velocity 
the (linear)  \index{ velocity} velocity and \index{ speed} speed of the piston are constantly changing.
Investigate the effect of the length of the connecting rod on the
velocity and acceleration of the piston. 

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\begin{enumerate}
\item  Suppose  the \index{ crank} crank is 1.5 inches in radius and the connecting rod is 6 inches long and the crank is turning at a constant 1 revolution per second.   Assume the piston is moving on the x-axis and express the position of the point at which the rod connects to the piston (the wrist pin) as a function of t (in seconds).

Hint: Let  r denote the crank radius, l the length of the connecting rod,  $\theta (t)$ the crank angle at time $t$ and  assume $\theta (0)=0$ and that the rotation is counterclockwise.  We can place the center of the crank at the origin. Then if $x(t)$ is the position of the wrist pin at time t,  the law of cosines says that  $$l(t)^2 = r^2 + x(t)^2 -1rx(t)\cos(\theta(t)).$$ Solve for $x(t)$  and choose one of the solutions (e.g. the one for which $x(t) = r+l$ when $t=0$). 

\item Calculate expressions for the  position, velocity, and acceleration of the \index{ wrist pin} wrist pin as a function of time and plot all three on the same graph for t from 0 to 2 seconds. At what times are the velocity and acceleration maximal and minimal? What are the maximum velocity and acceleration?

\item Repeat the  above calculations for  various  lengths of the connecting rod from 2 to 8 and observe the effect of these changes on the velocity and acceleration of the wrist pin.  Note particularly the change in shape of the acceleration curve and formulate a conjecture as to length of the connecting rod for which marks the transition from one shape to the other. Develop additional evidence for your conjecture through additional experiments.
 

\end{enumerate}
