Since we are sharing our favorite mechanics problems I thought I would submit
one.
A uniform rigid plank of length L and thickness d and mass m is symmetrically
balanced horizontally on a fixed round rigid cylinder of radius R whose
symmetry axis is horizontal in the Earth's gravitational field g as crudely
shown below.
The plank is free to rock (or is it roll?) over the curved upper surface of
the fixed cylinder. The static friction between the plank and the cylinder
is sufficient to prevent any slipping. The system is assumed lossless when
rocking/rolling without slipping.
A. Find the condition(s) that is/are necessary for this balanced state to
be one of stable (rather than unstable) equilibrium.
B. When condition(s) A above obtain find the frequency of small rocking
oscillations of the plank on the cylinder.
C. What is the maximum oscillation amplitude (angle) that the system can
sustain (when condition(s) A hold) without the plank just rolling off of
the cylinder?
D. EXTRA CREDIT If the coefficient of static friction between the surfaces
is mu find the maximum oscillation amplitude (angle) for which no
slipping will occur when rocking. Under what conditions will slipping
occur before unstable rolling, and under what conditions will unstable
rolling occur before slipping?
When answering the above questions notice the role played of the plank's mass
m and the gravitational field strength g in your various answers.
(Hint. This problem's solution has a nicer formulation when using Lagrangian
mechanics than using Newtonian methods involving free body force diagrams and
torques and things.)