Answer to the Question 03/03
PLANK ON A LOG
The 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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