Answer to the Question 03/03
PLANK ON A LOGThe question was:
A thin plank is placed on a log of semicircular cross section. If the plank is slightly tilted it will start oscillating. Find the frequency of the oscillations. Assume that all masses and dimensions are known.
(5/03) The problem has been solved (12/3/03) by Alex Smolyanitskiy (e-mail shurakbh@hotmail.com), (12/3/03) by Chetan Mandayam Nayakar (e-mail mn_chetan@yahoo.com), (1/4/03) by Zoran Hadzibabic (e-mail zoran@mit.edu), and (7/4/03) by Luca Visinelli (e-mail luca.visinelli@libero.it). Alex Smolyanitskiy treated such a problem long before it was published in our QUIZ. Thus, he submitted and extremely detailed solution which even accounts for details that we neglected and can be seen in this postscipt file. A nice solution of the problem also has been submitted (12/5/03) by Matthias Punk (e-mail matze.p@gmx.net) - see his solution in this postscipt file.
The answer: The angular frequency of oscillation will be sqrt{12gR/L2}, where L is the length of ther plank, R is the radius of the log, and g is the acceleration of free fall.
The solution: We first notice that once the plan is deflected by angle {theta} the restoring torque (to the lowest order in {theta} is MgR{theta}, where M and L are the mass and the length of the plank. This should be equated with the moment of interia ML2/12 of the plank around its center multiplied by angular acceleration. This immediately leads to the desired answer. However, the outlined solution disregards the fact that the center of mass of the plank is moving (left-right and up-down), and that the contact point is moving relative to the center of mass. Nevertheless, a detailed analysis of all the approximations, as explained in the solutions of Smolyanitskiy and Punk, shows that for small angle oscillations these "complications" can be neglected.
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