Problem source

A smooth cylinder with circular cross-section of radius $a$ is held
with its axis horizontal. A~light elastic band of unstretched 
length $2\pi a$ and modulus of elasticity
$\lambda$ is wrapped round the circumference of the cylinder, so that it forms a circle 
in a plane perpendicular to the axis of the cylinder. A particle of mass $m$ is then
attached to the  rubber band at its lowest point and released from rest.
\begin{questionparts}
\item Given that the particle falls to a distance $2a$ below the
below the axis of the cylinder, but no further, show that
\[
\lambda = \frac{9\pi m g}{(3\sqrt3-\pi)^2} \;.
\]
\item Given instead that the particle reaches its maximum speed at a
distance $2a$ below the axis of the cylinder, find a similar expression for 
$\lambda$\,.
\end{questionparts}