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Two thin discs, each of radius $r$ and mass $m$, are held on a rough
horizontal surface with their centres a distance $6r$ apart. A thin light
 elastic band, of natural length $2\pi r$ and modulus $\dfrac{\pi mg}{12}$, 
is wrapped once round
the discs, its straight sections being parallel. The contact between the
elastic band and the discs is smooth. The coefficient of static 
friction between
each disc and the horizontal surface is $\mu$, and each disc
experiences a force due to friction equal to $\mu mg$ when it is
sliding.
The discs are released simultaneously.  If the discs collide, 
they rebound and  a half of their total
kinetic energy is lost in the collision. 
\begin{questionparts}
\item Show that the discs start sliding, but
come to rest before colliding, if and only if \mbox{$\frac23 <\mu <1$}.
\item Show that, if the discs collide at least once, 
their total kinetic energy
just before the first collision is $\frac43 mgr(2-3\mu)$.
\item
Show that if $\frac 4 9 > \mu^2 >\frac{5}{27}$ the discs come to
rest exactly once after the first collision.
\end{questionparts}

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