A rectangular box leans on a frictionless wall with one corner and rests on a frictionless floor with another corner. It starts sliding down. When will the box become detached from the wall? (Assume that all the dimensions are given.)
(1/06) This problem has been solved correctly (30/8/05) by Ioannis Florakis, from University of Athens, Greece (e-mail iflorakis@yahoo.gr), (6/9/05) by Chetan Mandayam Nayakar, a student at India Insttitute of Technology, Madras, India (e-mail mn_chetan@yahoo.com), (10/10/05) by Ruben Mazzoleni (e-mail ruben.mazzoleni@tiscali.it), and (16/10/05) by Yaniv Hefetz from the Faculty of Applied Mathematics and Theoretical Physics at the University of Cambridge, England (e-mail yaniv-he@zahav.net.il).
The solution: If the box touches the wall, then position of the center of mass can be uniquely related to the angle θ of incline of the box. Consequentlly, the horizontal velocity can be related to the rate of change of θ. Since the law of energy conservation allows determination of the (translational+rotational) kinetic energy as a function of θ, we can express the horizontal velocity vx of the center-of-mass of the box in terms of the angle. We further note, that as long as the box touches the wall, the wall acts on the box accelerating its center of mass. However, at the point when the box disconnects from the wall, the horizontal component of acceleration vanishes, i.e. ax=dvx/dt= (dvx/dθ)(dθ/dt)=0, leading to condition dvx/dθ=0. This condition determines the point of separtation of the box from the wall. This condition leads to complicated equation for θ which can only be solved numerically. Ruben Mazzoleni provided a nice detailed solution of this problems that can be found in the following PDF file. (Similar detailed solution was provided by Yaniv Hefetz.) The solution of Ioannis Florakis (which can be found in the following PDF file) demonstrates an approximate analytical solution of the problem when the box is very thin and the problem resembles the classical case of "ladder leaning on a wall."
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