67. (a) We use conservation of mechanical energy to find an expression for ω 2 as a function of the angle θ that the chimney makes with the vertical. The potential energy of the chimney is given by U = Mgh , where M is its mass and h is the altitude of its center of mass above the ground. When the chimney makes the angle with the vertical, h = ( H /2) cos . Initially the potential energy is U i = Mg ( H /2) and the kinetic energy is zero. The kinetic energy is 1 2 2 I when the chimney makes the angle with the vertical, where I is its rotational inertia about its bottom edge. Conservation of energy then leads to MgH Mg H I MgH I / ( /) /) ( 22 1 2 =+ ¡ =− cos 1 2 (c o s ) . 2 θω ω I = MH2 /3 (found using Table 10-2(e) with the parallel axis theorem). Thus 233 ( 9 . 8 0 m / s ) (1 cos ) (1 cos35.0 ) 0.311 rad/s. 55.0 m g H ωθ
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This note was uploaded on 05/17/2011 for the course PHY 2049 taught by Professor Any during the Spring '08 term at University of Florida.